diff b34550f..b1f9bc4 simplicial_volume_scalar_bound_v2.txt
diff --git a/simplicial_volume_scalar_bound_v2.txt b/simplicial_volume_scalar_bound_v2.txt
index 30a2670..f55e98f 100644
--- a/simplicial_volume_scalar_bound_v2.txt
+++ b/simplicial_volume_scalar_bound_v2.txt
@@ -6,10 +6,10 @@
-\usetikzlibrary{arrows.meta,calc,positioning}
+usetikzlibrary{arrows.meta,calc,positioning}
-\hypersetup{
+hypersetup{
pdftitle={An integral scalar curvature bound for real simplicial volume},
}
@@ -44,16 +44,16 @@
-For every integer $n\ge3$ there is a constant $a_n>0$, depending only on
+For every integer $nge3$ there is a constant $a_n>0$, depending only on
$n$, such that every closed connected oriented smooth $n$-manifold $M$
and every smooth Riemannian metric $g$ on $M$ satisfy
-\[
- \int_M(\Scal_g^-)^{n/2}\,dV_g\ \ge\ a_n\,\|M\|.
-\]
-Here $\Scal_g^-=\max\{0,-\Scal_g\}$ and $\|M\|$ is the real simplicial
+[
+ int_M(Scal_g^-)^{n/2},dV_g ge a_n,|M|.
+]
+Here $Scal_g^-=max{0,-Scal_g}$ and $|M|$ is the real simplicial
volume of $M$. The left side is a local analytic quantity: it is
dimensionless, so it does not change under a constant rescaling
-$g\mapsto\lambda^2g$, and it measures both how negative the scalar
+$gmapstolambda^2g$, and it measures both how negative the scalar
curvature becomes and over how much volume. The right side is a global
topological invariant, independent of the metric. The theorem says that
negative scalar curvature is a resource: without enough of it, measured
@@ -62,94 +62,94 @@ large simplicial volume.
The proof has two inputs. Qualitatively, a closed manifold admitting a
metric of nonnegative scalar curvature has zero real simplicial volume;
-for spin universal covers this vanishing theorem is the result of [cite], and it is used here once. Quantitatively, a prescribed-curvature graph flow deforms the metric and is iterated to build an entropy metric with a large energy coefficient, and it converts the bounded-cohomology detector of $\|M\|$ into measured comparison degrees. Those degrees are
+for spin universal covers this vanishing theorem is the result of [cite], and it is used here once. Quantitatively, a prescribed-curvature graph flow deforms the metric and is iterated to build an entropy metric with a large energy coefficient, and it converts the bounded-cohomology detector of $|M|$ into measured comparison degrees. Those degrees are
bounded by a weighted integral that, after the deformations, is
-comparable to $\Vol_g(M)$. All constants depend only on $n$.
+comparable to $Vol_g(M)$. All constants depend only on $n$.
-\par\noindent{\footnotesize\ttfamily src-ma2026.pdf}\par
+parnoindent{footnotesizettfamily src-ma2026.pdf}par
The real simplicial volume of a closed connected oriented
-$n$-manifold $M$ is the $\ell^1$-seminorm of its real fundamental class:
-\[
- \|M\|=\inf\Bigl\{\sum_j|a_j|:\ \sum_j a_j\sigma_j
- is a finite real singular cycle representing [M]_{\R}\Bigr\}.
-\]
-It is the cheapest $\ell^1$ cost of writing the fundamental class as a
+$n$-manifold $M$ is the $ell^1$-seminorm of its real fundamental class:
+[
+ |M|=infBigl{sum_j|a_j|: sum_j a_jsigma_j
+ is a finite real singular cycle representing [M]_{R}Bigr}.
+]
+It is the cheapest $ell^1$ cost of writing the fundamental class as a
sum of singular simplices with real coefficients. It depends only on the
topology of $M$ and not on a metric, it is a homotopy invariant, and it
vanishes for many manifolds.
-In 1986 Gromov proposed \cite[Conjecture~3.A, Remark~(B),
+In 1986 Gromov proposed cite[Conjecture~3.A, Remark~(B),
Equation~(8$'$)]{Gromov1986} that the integral of the $n/2$-power of the
-negative part of scalar curvature should bound $\|M\|$ above with a
+negative part of scalar curvature should bound $|M|$ above with a
coefficient depending only on $n$. Writing
-\(\Scal_g^-:=\max\{0,-\Scal_g\}\), the quantity
-\[
- \int_M(\Scal_g^-)^{n/2}\,dV_g
-\]
-is invariant under a constant rescaling $g\mapsto\lambda^2g$, so it is
+(Scal_g^-:=max{0,-Scal_g}), the quantity
+[
+ int_M(Scal_g^-)^{n/2},dV_g
+]
+is invariant under a constant rescaling $gmapstolambda^2g$, so it is
dimensionless; and it records both how negative the scalar curvature
becomes and over how much volume it does so.
-\par\noindent{\footnotesizehttps://doi.org/10.1007/BFb0075649}\par
+parnoindent{footnotesizehttps://doi.org/10.1007/BFb0075649}par
[Main theorem]
-For every integer $n\ge3$ there is a constant $a_n>0$, depending only on
+For every integer $nge3$ there is a constant $a_n>0$, depending only on
$n$, such that every closed connected oriented smooth $n$-manifold $M$
and every smooth Riemannian metric $g$ on $M$ satisfy
- \int_M(\Scal_g^-)^{n/2}\,dV_g\ \ge\ a_n\,\|M\|.
+ int_M(Scal_g^-)^{n/2},dV_g ge a_n,|M|.
The constant is uniform in both $M$ and $g$. As a representative case,
-if $\Scal_g\ge-n(n-1)$, which is the natural normalization in dimension
-$n$, then $\Scal_g^-\le n(n-1)$ and the theorem reads
-\[
- \Vol_g(M)\ \ge\ a_n\,[n(n-1)]^{-n/2}\|M\|.
-\]
+if $Scal_gge-n(n-1)$, which is the natural normalization in dimension
+$n$, then $Scal_g^-le n(n-1)$ and the theorem reads
+[
+ Vol_g(M) ge a_n,[n(n-1)]^{-n/2}|M|.
+]
The integral form is strictly stronger than this. It permits scalar
curvature to be arbitrarily negative, provided the region where it is
negative has small volume: a tiny ball of huge negative curvature
-contributes almost nothing to $\int_M(\Scal_g^-)^{n/2}dV_g$.
+contributes almost nothing to $int_M(Scal_g^-)^{n/2}dV_g$.
Theorem~thm:main is closely related to the pointwise-uniform
assertion
- \Scal_g\ge-\kappa^2\quad⇒\quad
- \|M\|\le C_n\kappa^n\Vol_g(M),\qquad \kappa\ge0 .
+ Scal_gge-kappa^2quad⇒quad
+ |M|le C_nkappa^nVol_g(M),qquad kappage0 .
-For a fixed dimension $n\ge3$ the integral statement for all metrics on
+For a fixed dimension $nge3$ the integral statement for all metrics on
all such manifolds and (eq:pointwise-form) for all metrics on all
such manifolds are equivalent by a Yamabe rescaling; see
-\cite[Lemma~2.1]{MinZhengZhu2026}.
+cite[Lemma~2.1]{MinZhengZhu2026}.
-\par\noindent{\footnotesize\ttfamily src-min2026.pdf}\par
+parnoindent{footnotesizettfamily src-min2026.pdf}par
Several related statements were known before the general case.
-\item Braun and Sauer [cite] prove a macroscopic
+item Braun and Sauer [cite] prove a macroscopic
counterpart when the volumes of unit balls in the universal cover are
uniformly bounded above; their constant depends also on that bound.
-\item Ma, Wang, Xie, Yu and Zhu [cite] prove that
- $\|M\|=0$ when the universal cover is spin and $\Scal_g\ge0$. This is
+item Ma, Wang, Xie, Yu and Zhu [cite] prove that
+ $|M|=0$ when the universal cover is spin and $Scal_gge0$. This is
the qualitative vanishing result used below.
-\item Min, Zheng and Zhu [cite] prove the integral
+item Min, Zheng and Zhu [cite] prove the integral
estimate with coefficient $27/2$ for every metric on every closed
- K\"ahler surface, and for an infinite family of non-K\"ahler
+ K"ahler surface, and for an infinite family of non-K"ahler
symplectic four-manifolds.
-\par\noindent{\footnotesize\ttfamily src-braunsauer2021.pdf}\par
+parnoindent{footnotesizettfamily src-braunsauer2021.pdf}par
The one external input is the qualitative vanishing
theorem of
@@ -168,35 +168,35 @@ and the last stage composes them.
-\item Conformal reduction to $\Scal_g=-1$
+item Conformal reduction to $Scal_g=-1$
Proposition~prop:yamabe-reduction.
In every conformal class the Yamabe minimizer has constant scalar
- curvature. If $\|M\|>0$, the vanishing theorem forces that constant to
+ curvature. If $|M|>0$, the vanishing theorem forces that constant to
be negative, and a constant rescaling normalizes it to $-1$.
- H\"older's inequality then bounds the volume of the normalized metric
- by $\int_M(\Scal_g^-)^{n/2}dV_g$ of the original metric. So
+ H"older's inequality then bounds the volume of the normalized metric
+ by $int_M(Scal_g^-)^{n/2}dV_g$ of the original metric. So
Theorem~thm:main follows from the normalized estimate
- \|M\|\le C_n\Vol_g(M)\qquadwhenever \Scal_g\equiv-1,
+ |M|le C_nVol_g(M)qquadwhenever Scal_gequiv-1,
which is Theorem~thm:scalar-volume.
-\item A detector for $\|M\|$.
+item A detector for $|M|$.
Proposition~prop:detector.
- Duality between the $\ell^1$-seminorm on homology and the supremum
+ Duality between the $ell^1$-seminorm on homology and the supremum
seminorm on bounded cohomology, together with the Mapping Theorem,
produces a bounded group cocycle $c$ of norm at most one whose value
- $I$ on the fundamental class is at least $\|M\|/2$. The cocycle
- becomes a closed $n$-form $\eta$ on the simplex of probability vectors,
+ $I$ on the fundamental class is at least $|M|/2$. The cocycle
+ becomes a closed $n$-form $eta$ on the simplex of probability vectors,
and an equivariant probability map $F$ from the universal cover pulls
- $\eta$ back with total integral $I$. So (eq:normalized-target)
- follows once one bounds $I$ by $C_n\Vol_g(M)$.
+ $eta$ back with total integral $I$. So (eq:normalized-target)
+ follows once one bounds $I$ by $C_nVol_g(M)$.
-\item The graph engine.
+item The graph engine.
Replace a metric $h$ by the metric induced on the
- graph of a function $f$, and shift the weight $\phi$ so that the
- weighted volume $e^\phi dV_h$ does not change. The weighted graph
+ graph of a function $f$, and shift the weight $phi$ so that the
+ weighted volume $e^phi dV_h$ does not change. The weighted graph
identity, Lemma~lem:graph-identity, writes the change of weighted
scalar curvature as a square of a prescribed quantity $D$ plus
nonnegative errors. Since $D$ can be prescribed, a solvability theorem
@@ -205,9 +205,9 @@ and the last stage composes them.
Lemma~lem:graph-localization) permits making that square
systematically positive. Sweeping the coordinates of $F$ in order
yields Proposition~prop:sweep: one pass improves the energy of $F$
- at a cost bounded by an explicit nonnegative reserve $\Pi_p$.
+ at a cost bounded by an explicit nonnegative reserve $Pi_p$.
-\item Iteration to an entropy metric.
+item Iteration to an entropy metric.
A conformal bootstrap raises the energy
coefficient from a very small value to a fixed dimensional one
(Lemma~lem:entropy-bootstrap). Phases of sweeps then raise it to
@@ -215,24 +215,24 @@ and the last stage composes them.
expansion controlled (Lemma~lem:entropy-accumulation) and the
residual gradient absolutely small (Lemma~lem:entropy-phase-exit).
The output is an entropy datum with
- \[
- \wscal(h,\phi)\ge-C+2W-n\log E+E\,e_h(F),\qquad
- e_h(F)^{1/2}\le E^{-1/100},
- \]
- where $W=\log(dV_h/\rho)\ge0$ and $\rho\le dV_g$
+ [
+ wscal(h,phi)ge-C+2W-nlog E+E,e_h(F),qquad
+ e_h(F)^{1/2}le E^{-1/100},
+ ]
+ where $W=log(dV_h/rho)ge0$ and $rhole dV_g$
(Theorem~thm:entropy).
-\item Degree and transfer.
+item Degree and transfer.
The measured degree
estimate, Theorem~thm:degree, bounds the signed degree of a map
from a measured family of $d$-manifolds to a $d$-cube by an integral of
- $\exp(-\wscal/2)$ times a penalty for each coordinate gradient. The
+ $exp(-wscal/2)$ times a penalty for each coordinate gradient. The
transfer builds such a map whose degree is the
detector pairing $I$ and whose tangential and normal lengths are
- $b_*\sqrt E$ and $b_*√(qE)$. The factor $E^{n/2}$ produced by the
+ $b_*sqrt E$ and $b_*√(qE)$. The factor $E^{n/2}$ produced by the
degree bound cancels the $E^{-n/2}$ from the product volume, the
remaining $e^{-W}$ cancels the volume expansion, and what survives is
- an integral of $\rho$, hence $\Vol_g(M)$
+ an integral of $rho$, hence $Vol_g(M)$
(Theorem~thm:scalar-volume).
@@ -241,7 +241,7 @@ the tangent directions of the comparison family contribute the volume
$L_T^m$, the normal directions contribute the reciprocal
$L_T^{-m}L_N^{-n}$ of the degree denominator, and the entropy inequality
contributes $E^{n/2}e^{-W}$; the three factors multiply to
-$b_*^{-n}q^{-n/2}\rho$, in which no power of $E$ appears.
+$b_*^{-n}q^{-n/2}rho$, in which no power of $E$ appears.
The three technical engines are the graph engine, the entropy
@@ -270,15 +270,15 @@ of constant scalar curvature whose sign is the Yamabe invariant; the
vanishing theorem forces that sign to be negative, and rescaling
normalizes the constant to $-1$. Bounded cohomology supplies a bounded
cocycle of norm at most one, pairing with the fundamental class in an
-amount $I\ge\|M\|/2$, and a closed form on a simplex of probability
+amount $Ige|M|/2$, and a closed form on a simplex of probability
vectors that an equivariant probability map pulls back with the same
integral $I$. Graph deformations supply the identity
-\[
- \wscal(h',\phi')-\wscal(h,\phi)
- =D^2-2U(D)+|\II|^2+|d\log L|^2
-\]
+[
+ wscal(h',phi')-wscal(h,phi)
+ =D^2-2U(D)+|II|^2+|dlog L|^2
+]
together with the fact that $D$ can be prescribed by solving the
-quasilinear equation $D=p(x,z,\nabla z)$ with $∂_zp<0$; this is
+quasilinear equation $D=p(x,z,nabla z)$ with $∂_zp<0$; this is
what makes the change of weighted scalar curvature positive on demand.
Geometric measure theory supplies the measured degree and its bound by a
weighted integral with gradient penalties, obtained by slicing, weighted
@@ -295,8 +295,8 @@ Use the following result, stated here in the precise form needed.
-Let \(M\) be a closed connected oriented smooth manifold of positive
-dimension. If \(M\) admits a smooth Riemannian metric with nonnegative
+Let (M) be a closed connected oriented smooth manifold of positive
+dimension. If (M) admits a smooth Riemannian metric with nonnegative
scalar curvature, then its real simplicial volume is zero.
@@ -305,213 +305,213 @@ identically. Theorem~thm:vanishing-input is the only external result
invoked below without proof.
The Yamabe reformulation of the scalar curvature inequality is
-standard; \cite[Lemma 2.1]{MinZhengZhu2026} records the equivalence of
+standard; cite[Lemma 2.1]{MinZhengZhu2026} records the equivalence of
its uniform pointwise and integral forms. Include the short
classwise reduction that identifies the use of the vanishing input.
-Fix \(n\ge3\). Suppose that a constant \(C_n<∞\) has the following
-property: every closed connected oriented \(n\)-manifold with a smooth
-metric \(h\) of constant scalar curvature \(-1\) satisfies
+Fix (nge3). Suppose that a constant (C_n<∞) has the following
+property: every closed connected oriented (n)-manifold with a smooth
+metric (h) of constant scalar curvature (-1) satisfies
- \|M\|\le C_n\Vol_h(M).
+ |M|le C_nVol_h(M).
-Then every smooth Riemannian metric \(g\) on every such manifold
+Then every smooth Riemannian metric (g) on every such manifold
satisfies
- \|M\|\le C_n\int_M(\Scal_g^-)^{n/2}\,dV_g .
+ |M|le C_nint_M(Scal_g^-)^{n/2},dV_g .
Proof.
-The assertion is immediate when \(\|M\|=0\), so assume that
-\(\|M\|>0\). Write \(Y(M,[g])\) for the Yamabe constant of the
-conformal class of \(g\):
-\[
+The assertion is immediate when (|M|=0), so assume that
+(|M|>0). Write (Y(M,[g])) for the Yamabe constant of the
+conformal class of (g):
+[
Y(M,[g])=
-\inf_{u\in C^∞(M),\,u>0}
-\frac{\displaystyle\int_M
- ((4(n-1))/(n-2)|du|_g^2+\Scal_g u^2)dV_g}
-{\displaystyle(\int_Mu^{2n/(n-2)}\,dV_g)^{(n-2)/n}}.
-\]
+inf_{uin C^∞(M),,u>0}
+frac{displaystyleint_M
+ ((4(n-1))/(n-2)|du|_g^2+Scal_g u^2)dV_g}
+{displaystyle(int_Mu^{2n/(n-2)},dV_g)^{(n-2)/n}}.
+]
The Yamabe theorem, developed through the work of Yamabe, Trudinger,
Aubin, and Schoen [cite],
gives a conformal metric attaining this value and having constant
-scalar curvature; see also \cite[Section 11]{LeeParker1987}.
+scalar curvature; see also cite[Section 11]{LeeParker1987}.
Its scalar curvature is negative by
Theorem~thm:vanishing-input. After a constant rescaling, it becomes a
-metric \(h\) with \(\Scal_h=-1\). Evaluating the Yamabe quotient at the
+metric (h) with (Scal_h=-1). Evaluating the Yamabe quotient at the
constant function for this minimizing metric gives
-\[
- -Y(M,[g])=\Vol_h(M)^{2/n}.
-\]
-For each positive \(u\), H\"older's inequality gives
-\[
-\int_M\Scal_g u^2\,dV_g
-\ge
--(\int_M(\Scal_g^-)^{n/2}\,dV_g)^{2/n}
- (\int_Mu^{2n/(n-2)}\,dV_g)^{(n-2)/n}.
-\]
+[
+ -Y(M,[g])=Vol_h(M)^{2/n}.
+]
+For each positive (u), H"older's inequality gives
+[
+int_MScal_g u^2,dV_g
+ge
+-(int_M(Scal_g^-)^{n/2},dV_g)^{2/n}
+ (int_Mu^{2n/(n-2)},dV_g)^{(n-2)/n}.
+]
The gradient term in the Yamabe quotient is nonnegative. Taking the
infimum therefore yields
-\[
+[
-Y(M,[g])
-\le (\int_M(\Scal_g^-)^{n/2}\,dV_g)^{2/n}.
-\]
-Consequently \(\Vol_h(M)\le\int_M(\Scal_g^-)^{n/2}\,dV_g\), and
+le (int_M(Scal_g^-)^{n/2},dV_g)^{2/n}.
+]
+Consequently (Vol_h(M)leint_M(Scal_g^-)^{n/2},dV_g), and
(eq:scalar-minus-one-target) implies
(eq:yamabe-transfer).
-\par\noindent{\footnotesize J. Math. Pures Appl. (9) 55 (1976), 269--296}\par
-\par\noindent{\footnotesizehttps://doi.org/10.4310/jdg/1214439291}\par
-\par\noindent{\footnotesize\ttfamily src-trudinger1968.pdf}\par
-\par\noindent{\footnotesize\ttfamily src-yamabe1960.pdf}\par
-\par\noindent{\footnotesize\ttfamily src-leeparker1987.pdf}\par
+parnoindent{footnotesize J. Math. Pures Appl. (9) 55 (1976), 269--296}par
+parnoindent{footnotesizehttps://doi.org/10.4310/jdg/1214439291}par
+parnoindent{footnotesizettfamily src-trudinger1968.pdf}par
+parnoindent{footnotesizettfamily src-yamabe1960.pdf}par
+parnoindent{footnotesizettfamily src-leeparker1987.pdf}par
-For the rest of the proof one fixes \(M^n\) and \(g\) with
-\(\Scal_g=-1\), and prove (eq:scalar-minus-one-target). Continue to assume \(\|M\|>0\).
+For the rest of the proof one fixes (M^n) and (g) with
+(Scal_g=-1), and prove (eq:scalar-minus-one-target). Continue to assume (|M|>0).
-Let \(\Gamma=\pi_1(M)\). This group is countable because \(M\) has a
+Let (Gamma=pi_1(M)). This group is countable because (M) has a
finite triangulation. Use two standard facts about bounded
cohomology with trivial real coefficients. First, duality between the
-\(\ell^1\)-seminorm on homology and the supremum seminorm on bounded
+(ell^1)-seminorm on homology and the supremum seminorm on bounded
cohomology gives
-\|M\|=
-\sup{<\beta,[M]_{\R}>:
- \beta\in H_b^n(M;\R),\ \|\beta\|_∞\le1}.
+|M|=
+sup{<beta,[M]_{R}>:
+ betain H_b^n(M;R), |beta|_∞le1}.
-See \cite[Section 1.1]{Gromov1982}.
-Second, the isometric Mapping Theorem identifies \(H_b^k(M;\R)\) with
-\(H_b^k(\Gamma;\R)\) for every \(k\ge2\), through the classifying map.
+See cite[Section 1.1]{Gromov1982}.
+Second, the isometric Mapping Theorem identifies (H_b^k(M;R)) with
+(H_b^k(Gamma;R)) for every (kge2), through the classifying map.
The version used here applies to connected countable CW spaces and
preserves the seminorms. Use the Mapping Theorem of Gromov
-\cite[Section 3.1]{Gromov1982} and its canonical group formulation in
-\cite[Theorems 8.3 and 8.6]{IvanovNotes}.
+cite[Section 3.1]{Gromov1982} and its canonical group formulation in
+cite[Theorems 8.3 and 8.6]{IvanovNotes}.
-\par\noindent{\footnotesize\ttfamily src-gromov1982.pdf}\par
-\par\noindent{\footnotesize\ttfamily src-ivanov2017.pdf}\par
+parnoindent{footnotesizettfamily src-gromov1982.pdf}par
+parnoindent{footnotesizettfamily src-ivanov2017.pdf}par
-Use the homogeneous model for group cochains. Thus an \(n\)-cochain
-is a bounded function on \(\Gamma^{n+1}\), invariant under simultaneous
+Use the homogeneous model for group cochains. Thus an (n)-cochain
+is a bounded function on (Gamma^{n+1}), invariant under simultaneous
left translation, and its coboundary is the alternating sum obtained by
omitting one argument. Alternation is a cochain projection of norm at
most one and is equivariantly cochain homotopic to the identity through
maps bounded in each degree. It therefore preserves bounded cohomology
-classes \cite[Section~4.10, Proposition~4.26]{Frigerio2016}.
+classes cite[Section~4.10, Proposition~4.26]{Frigerio2016}.
By (eq:bounded-duality), the Mapping Theorem, and
approximation of the seminorm by cocycle representatives, there is a
homogeneous alternating cocycle
-\[
- c:\Gamma^{n+1}→\R,\qquad \|c\|_∞\le1,
-\]
+[
+ c:Gamma^{n+1}→R,qquad |c|_∞le1,
+]
whose corresponding bounded singular cohomology class has value
- I:=<[c],[M]_{\R}>\ge\frac12\|M\|.
+ I:=<[c],[M]_{R}>gefrac12|M|.
For completeness, one can choose a class in
-(eq:bounded-duality) with value at least \(3\|M\|/4\), choose a
-group representative of norm at most \(1+ε\), alternate it,
-and divide by \(1+ε\), with \(0<ε\le1/2\). Replacing
+(eq:bounded-duality) with value at least (3|M|/4), choose a
+group representative of norm at most (1+ε), alternate it,
+and divide by (1+ε), with (0<εle1/2). Replacing
the cocycle by its negative first, if necessary, fixes the sign.
-\par\noindent{\footnotesize\ttfamily src-frigerio2016.pdf}\par
+parnoindent{footnotesizettfamily src-frigerio2016.pdf}par
Allow a finite number of copies of every group label. More
-precisely, let \(\mathcal L=\Gamma\times\{1,...,s\}\), with the free
-left \(\Gamma\)-action on the first factor, and evaluate \(c\) on
-\(\mathcal L^{n+1}\) by projection to \(\Gamma^{n+1}\). The same
-notation will be used when further finite copies of \(\mathcal L\) are
-introduced. Extend \(c\) multilinearly to finitely supported vectors
-on \(\mathcal L\).
+precisely, let (mathcal L=calL=Gammatimes{1,...,s}), with the free
+left (Gamma)-action on the first factor, and evaluate (c) on
+(mathcal L^calL^{n+1}) by projection to (Gamma^{n+1}). The same
+notation will be used when further finite copies of (mathcal LcalL) are
+introduced. Extend (c) multilinearly to finitely supported vectors
+on (mathcal LcalL).
Let
-\[
-\Delta_{\mathcal L}=
-{p=(p_\ell)_{\ell\in\mathcal L}:
- p_\ell\ge0,\ \sum_\ell p_\ell=1,\ p has finite support}.
-\]
-On each finite-dimensional face define the \(n\)-form
+[
+Delta_{mathcal L}calL=
+{p=(p_ell)_{ellinmathcal L}calL:
+ p_ellge0, sum_ell p_ell=1, p has finite support}.
+]
+On each finite-dimensional face define the (n)-form
-\eta_p(v_1,...,v_n)=n!\,c(p,v_1,...,v_n).
+eta_p(v_1,...,v_n)=n!,c(p,v_1,...,v_n).
These definitions agree on common faces.
-Call a map \(F=(F_\ell)\) with nonnegative coordinates and
-\(\sum_\ell F_\ell^2=1\) a root probability map. For the maps
+Call a map (F=(F_ell)) with nonnegative coordinates and
+(sum_ell F_ell^2=1) a root probability map. For the maps
used here, whose coordinate supports are locally finite, the squared
-coordinates \(p_\ell=F_\ell^2\) give the associated map to
-\(\Delta_{\mathcal L}\).
+coordinates (p_ell=F_ell^2) give the associated map to
+(Delta_{mathcal L}calL).
-The form \(\eta\) is closed on every finite-dimensional face of
-\(\Delta_{\mathcal L}\). There is a smooth \(\Gamma\)-equivariant map
-\[
- F:\widetilde M→\ell^2(\mathcal L),\qquad
- F_\ell\ge0,\qquad \sum_\ell F_\ell^2=1,
-\]
+The form (eta) is closed on every finite-dimensional face of
+(Delta_{mathcal L}calL). There is a smooth (Gamma)-equivariant map
+[
+ F:widetilde M→ell^2(mathcal L),calL),qquad
+ F_ellge0,qquad sum_ell F_ell^2=1,
+]
with finitely many label orbits and locally finite compact supports for
-its coordinates. If \(p_\ell=F_\ell^2\), then the descended form
+its coordinates. If (p_ell=F_ell^2), then the descended form
satisfies
- \int_M p^*\eta=I.
+ int_M p^*eta=I.
Proof.
-Write \(ε(v)=\sum_\ell v_\ell\) for the mass of a finitely
+Write (ε(v)=sum_ell v_ell) for the mass of a finitely
supported vector. Multilinear extension of the cocycle equation gives
-\[
-0=\sum_{j=0}^{n+1}(-1)^jε(w_j)
- c(w_0,...,\widehat w_j,...,w_{n+1}).
-\]
-Choose \(w_0\) of mass one and let \(w_1,...,w_{n+1}\) have mass
-zero. It follows that \(c(w_1,...,w_{n+1})=0\). All tangent vectors
+[
+0=sum_{j=0}^{n+1}(-1)^jε(w_j)
+ c(w_0,...,widehatwidehat w_j,...,w_w_j,...,w_{n+1}).
+]
+Choose (w_0) of mass one and let (w_1,...,w_{n+1}) have mass
+zero. It follows that (c(w_1,...,w_{n+1})=0). All tangent vectors
to a probability simplex have mass zero. Differentiating
-(eq:probability-form) now shows \(d\eta=0\).
+(eq:probability-form) now shows (deta=0).
-To construct \(F\), choose finitely many evenly covered open sets in
-\(M\) and smooth nonnegative bumps subordinate to them whose nonzero
-sets cover \(M\). Lift each bump to one chosen sheet and take all its
-deck translates, denoting the resulting functions by \(b_\ell\).
-Their supports are compact and locally finite on \(\widetilde M\).
+To construct (F), choose finitely many evenly covered open sets in
+(M) and smooth nonnegative bumps subordinate to them whose nonzero
+sets cover (M). Lift each bump to one chosen sheet and take all its
+deck translates, denoting the resulting functions by (b_ell).
+Their supports are compact and locally finite on (widetilde M).
Set
-\[
- F_\ell=\frac{b_\ell}{(\sum_j b_j^2)^{1/2}}.
-\]
+[
+ F_ell=frac{b_ell}{(sum_j b_j^2)^{1/2}}.
+]
The denominator is positive, and the stated properties follow.
-Choose a finite oriented triangulation of \(M\). On its lift, assign
-one label to every vertex equivariantly, and let \(p^{aff}\)
+Choose a finite oriented triangulation of (M). On its lift, assign
+one label to every vertex equivariantly, and let (p^{aff})
be the probability map affine in barycentric coordinates on each
simplex. The homotopy
-\((1-t)p+t p^{aff}\) is equivariant and has locally finite
-support. Applying Stokes' theorem to each simplex times \([0,1]\), and
+((1-t)p+t p^{aff}) is equivariant and has locally finite
+support. Applying Stokes' theorem to each simplex times ([0,1]), and
cancelling the common faces, shows that its pullback integral is
constant. On a simplex with ordered vertex labels
-\(\ell_0,...,\ell_n\), write
-\[
-p^{aff}=e_{\ell_0}
-+\sum_{j=1}^nt_j(e_{\ell_j}-e_{\ell_0}).
-\]
+(ell_0,...,ell_n), write
+[
+p^{aff}=e_{ell_0}
++sum_{j=1}^nt_j(e_{ell_j}-e_{ell_0}).
+]
Alternation makes the value of
-\(c(p^{aff},dp^{aff},...,dp^{aff})\)
+(c(p^{aff},dp^{aff},...,dp^{aff}))
on the coordinate frame constant and equal to
-\(c(\ell_0,...,\ell_n)\). The coordinate simplex has volume
-\(1/n!\), so the factor \(n!\) in (eq:probability-form) makes its
+(c(ell_0,...,ell_n)). The coordinate simplex has volume
+(1/n!), so the factor (n!) in (eq:probability-form) makes its
integral exactly this cocycle value. Summing over the oriented
fundamental triangulation gives (eq:detector-integral). The
identification with the chosen group class is the usual cochain
@@ -525,63 +525,63 @@ construction.
The deformations below use auxiliary probability choices. Record their integration convention so that the detector value remains
unambiguous.
-Let \((Ω,\mu)\) be a standard Borel probability space with a
-measure-preserving \(\Gamma\)-action. A measured equivariant family
-over \(M\) consists of \(\Gamma\)-equivariant Borel data on
-\(\widetilde M\timesΩ\), smooth in the \(\widetilde M\) variable.
+Let ((Ω,mu)) be a standard Borel probability space with a
+measure-preserving (Gamma)-action. A measured equivariant family
+over (M) consists of (Gamma)-equivariant Borel data on
+(widetilde MtimesΩ), smooth in the (widetilde M) variable.
Geometrically one may regard the diagonal quotient
-\[
- \mathcal M=(\widetilde M\timesΩ)/\Gamma
-\]
+[
+ mathcal M=(calM=(widetilde MtimesΩ)/Gamma
+]
as a space whose leaves are ordinary smooth manifolds. If
-\(\Gamma_\omega\) is the stabilizer of \(\omega\), the corresponding
-leaf is \(\widetilde M/\Gamma_\omega\); the deck action is free and
-proper in the \(\widetilde M\) factor.
+(Gamma_omega) is the stabilizer of (omega), the corresponding
+leaf is (widetilde M/Gamma_omega); the deck action is free and
+proper in the (widetilde M) factor.
Related equivariant measured bundles are central to the macroscopic
-volume argument of Braun and Sauer \cite[Sections 1.2 and 4]{BraunSauer2021}.
+volume argument of Braun and Sauer cite[Sections 1.2 and 4]{BraunSauer2021}.
Here only Borel transverse data are needed, and the following direct
integration convention.
Choose a finite evenly covered
-atlas \(U_a\) of \(M\), chosen lifts \(\widetilde U_a\), and a smooth
-partition of unity \(\chi_a\) subordinate to the atlas. For an
-integrable equivariant leafwise \(n\)-form \(\alpha\), set
+atlas (U_a) of (M), chosen lifts (widetilde U_a), and a smooth
+partition of unity (chi_a) subordinate to the atlas. For an
+integrable equivariant leafwise (n)-form (alpha), set
-\int_{\mathcal M}\alpha
+int_{mathcal M}calMalpha
=
-\sum_a\int_Ω\int_{\widetilde U_a}
- (\chi_a\circ\pi)\,\alpha_\omega\,d\mu(\omega).
+sum_aint_Ωint_{widetilde U_a}
+ (chi_acircpi),alpha_omega,dmu(omega).
-Equivariance and invariance of \(\mu\) show that this value is
+Equivariance and invariance of (mu) show that this value is
independent of the lifts and partition. The same convention applies
-to nonnegative densities. Write \(dV_h\) for the resulting measured
-Riemannian density when \(h=h_\omega\) is a leafwise metric.
+to nonnegative densities. Write (dV_h) for the resulting measured
+Riemannian density when (h=h_omega) is a leafwise metric.
The data used in the first deformation will have bounds in a fixed
-finite lifted atlas that are uniform over \(\omega\). These bounds
+finite lifted atlas that are uniform over (omega). These bounds
may depend on the instance: the original manifold and metric and the
finite sequence of constructions already made. For the forms used
here, these bounds ensure integrability over the compact base and
justify differentiation and Fubini. Leafwise Stokes' theorem follows from
(eq:measured-integral): apply ordinary Stokes' theorem in each
lifted chart to the partitioned form; the terms containing
-\(d\chi_a\) cancel on overlaps after the measure-preserving changes of
+(dchi_a) cancel on overlaps after the measure-preserving changes of
transverse coordinate.
In particular, suppose that two equivariant probability maps
-\(p^0,p^1\) have uniformly locally finite coordinate supports and
+(p^0,p^1) have uniformly locally finite coordinate supports and
uniform leafwise derivative bounds. Their affine homotopy has the
same properties. Stokes' theorem on the measured product with
-\([0,1]\) gives
-\[
-\int_{\mathcal M}(p^0)^*\eta
+([0,1]) gives
+[
+int_{mathcal M}calM(p^0)^*eta
=
-\int_{\mathcal M}(p^1)^*\eta.
-\]
+int_{mathcal M}calM(p^1)^*eta.
+]
Thus every probability map constructed below continues to detect the
-same number \(I\). Enlarging \(Ω\) by an independent probability
+same number (I). Enlarging (Ω) by an independent probability
space also preserves the integral by Fubini. Discard invariant
null sets whenever needed. More general measured spaces of leaves
will enter the degree estimate in the degree estimate below; their chart and
@@ -596,214 +596,214 @@ curvature to control this change. A localized version of the construction will
estimate for one sweep through the coordinates of a root probability
map.
-For a metric \(h\) and a smooth weight \(\phi\), write
-\[
- \wscal(h,\phi)=\Scal_h-2\Delta_h\phi-|d\phi|_h^2,
- \qquad
- \diver_\phi V=e^{-\phi}\diver(e^\phi V),
- \qquad
- \Delta_\phi=\diver_\phi\nabla .
-\]
-Thus the weighted operators use \(e^\phi dV_h\). The scalar quantity
-is Perelman's \(R^m\), with his \(f=-\phi\)
-\cite[Section~1.3]{Perelman2002}. The estimates below concern its
+For a metric (h) and a smooth weight (phi), write
+[
+ wscal(h,phi)=Scal_h-2Delta_hphi-|dphi|_h^2,
+ qquad
+ diver_phi V=e^{-phi}diver(e^phi V),
+ qquad
+ Delta_phi=diver_phinabla .
+]
+Thus the weighted operators use (e^phi dV_h). The scalar quantity
+is Perelman's (R^m), with his (f=-phi)
+cite[Section~1.3]{Perelman2002}. The estimates below concern its
behavior under particular graph deformations. For antecedents of
the graph and vertical Jacobi identities, see
[cite]; one includes the
calculation with the conventions needed here.
-\par\noindent{\footnotesize\ttfamily src-perelman2002.pdf}\par
+parnoindent{footnotesizettfamily src-perelman2002.pdf}par
[Weighted graph identity]
-Let \((X,h)\) be a smooth Riemannian manifold, let \(\phi,f\) be
-smooth real functions, and let \(N>0\). Set
-\[
- L=(1+N^2|df|_h^2)^{1/2},\qquad v=L^{-1},\qquad
- U=(N\nabla_h f)/(L),\qquad
- h'=h+N^2df ⊗ df,\qquad \phi'=\phi-\log L.
-\]
-View \(h'\) as the induced metric on the graph
-\(Σ=\{(x,Nf(x)):x\in X\}\) in
-\((X\times\R,h+dz^2)\), and extend \(\phi\) constantly in the
-\(z\)-direction. Orient the graph by
-\(\nu=(-U,v)\), put
-\(\II(V,W)=<\bar\nabla_V\nu,W>\), and let
-\[
- H=\tr_Σ\II,\qquad
- D=H+∂_\nu\phi=-\diver_\phi U .
-\]
-Here \(D\), and its derivative \(U(D)\), are regarded as functions on
-\(X\) through the graph projection. Then
-
- \wscal(h',\phi')-\wscal(h,\phi)
- =D^2-2U(D)+|\II|_{h'}^2+|d\log L|_{h'}^2.
+Let ((X,h)) be a smooth Riemannian manifold, let (phi,f) be
+smooth real functions, and let (N>0). Set
+[
+ L=(1+N^2|df|_h^2)^{1/2},qquad v=L^{-1},qquad
+ U=(Nnabla_h f)/(L),qquad
+ h'=h+N^2df ⊗ df,qquad phi'=phi-log L.
+]
+View (h') as the induced metric on the graph
+(Σ={(x,Nf(x)):xin X}) in
+((XtimesR,h+dz^2)), and extend (phi) constantly in the
+(z)-direction. Orient the graph by
+(nu=(-U,v)), put
+(II(V,W)=<barnabla_Vnu,W>), and let
+[
+ H=tr_ΣII,qquad
+ D=H+∂_nuphi=-diver_phi U .
+]
+Here (D), and its derivative (U(D)), are regarded as functions on
+(X) through the graph projection. Then
+
+ wscal(h',phi')-wscal(h,phi)
+ =D^2-2U(D)+|II|_{h'}^2+|dlog L|_{h'}^2.
Moreover,
- e^{\phi'}dV_{h'}=e^\phi dV_h,\qquad
- N^2|df|_{h'}^2=1-v^2=|U|_h^2,\qquad
- |d\log L|_{h'}\le L|U|_h|\II|_{h'} .
+ e^{phi'}dV_{h'}=e^phi dV_h,qquad
+ N^2|df|_{h'}^2=1-v^2=|U|_h^2,qquad
+ |dlog L|_{h'}le L|U|_h|II|_{h'} .
Proof.
-Write \(\bar h=h+dz^2\), and suppress pullback by the graph
+Write (barbar h=h+dz^2h=h+dz^2), and suppress pullback by the graph
projection. The formula for the graph mean curvature gives
-\(H=-\diver U\), while \(∂_\nu\phi=-U(\phi)\); hence the stated
-formula for \(D\) holds. The rank-one determinant formula gives
-\(dV_{h'}=L\,dV_h\), proving the first assertion in
+(H=-diver U), while (∂_nuphi=-U(phi)); hence the stated
+formula for (D) holds. The rank-one determinant formula gives
+(dV_{h'}=L,dV_h), proving the first assertion in
(eq:graph-basic-bounds). The inverse rank-one formula gives its
second assertion.
Let
-\[
- P=|\II|^2+
- (\Ric_{\bar h}-\bar\nabla^2\phi)(\nu,\nu).
-\]
+[
+ P=|II|^2+
+ (Ric_{barbar h}h-barnabla^2phi)(nu,nu).
+]
The Gauss equation and the decompositions of the Laplacian and
-gradient of \(\phi\) give
+gradient of (phi) give
- \Scal_{h'}&=\Scal_{\bar h}-2\Ric_{\bar h}(\nu,\nu)
- +H^2-|\II|^2,\\
- \Delta_{\bar h}\phi
- &=\Delta_Σ(\phi|_Σ)
- +H∂_\nu\phi+\bar\nabla^2\phi(\nu,\nu),\\
- |d(\phi|_Σ)|^2&=|d\phi|_{\bar h}^2-(∂_\nu\phi)^2.
+ Scal_{h'}&=Scal_{barbar h}h-2Ric_{barbar h}h(nu,nu)
+ +H^2-|II|^2,
+ Delta_{barbar h}hphi
+ &=Delta_Σ(phi|_Σ)
+ +H∂_nuphi+barnabla^2phi(nu,nu),
+ |d(phi|_Σ)|^2&=|dphi|_{barbar h}h^2-(∂_nuphi)^2.
-Since \(\wscal(\bar h,\phi)=\wscal(h,\phi)\), their combination is
+Since (wscal(barbar h,h,phi)=wscal(h,phi)), their combination is
- \wscal(h',\phi|_Σ)
- =\wscal(h,\phi)+D^2-2P+|\II|^2.
+ wscal(h',phi|_Σ)
+ =wscal(h,phi)+D^2-2P+|II|^2.
-Adding \(\log v\) to the restricted weight changes the left side by
-\[
- -2\Delta_{Σ,\phi}\log v-|d\log v|^2
- =-\frac{2\Delta_{Σ,\phi}v}{v}+|d\log v|^2 .
-\]
+Adding (log v) to the restricted weight changes the left side by
+[
+ -2Delta_{Σ,phi}log v-|dlog v|^2
+ =-frac{2Delta_{Σ,phi}v}{v}+|dlog v|^2 .
+]
The tangential part of vertical translation is
-\[
- T=∂_z-v\nu=(vU,1-v^2).
-\]
-For a hypersurface variation with normal speed \(v\), the variation
-of \(H\) is
-\(-\Delta_Σ v-(|\II|^2+\Ric_{\bar h}(\nu,\nu))v\).
-The additional variation of \(∂_\nu\phi\) is
-\[
- v\bar\nabla^2\phi(\nu,\nu)
- -< d(\phi|_Σ),dv>_Σ .
-\]
-Including the tangential velocity \(T\), and using that vertical
-translation preserves the metric, weight, and \(D\), therefore gives
-
- (\Delta_{Σ,\phi}+P)v=T(D).
+[
+ T=∂_z-vnu=(vU,1-v^2).
+]
+For a hypersurface variation with normal speed (v), the variation
+of (H) is
+(-Delta_Σ v-(|II|^2+Ric_{barbar h}h(nu,nu))v).
+The additional variation of (∂_nuphi) is
+[
+ vbarnabla^2phi(nu,nu)
+ -< d(phi|_Σ),dv>_Σ .
+]
+Including the tangential velocity (T), and using that vertical
+translation preserves the metric, weight, and (D), therefore gives
+
+ (Delta_{Σ,phi}+P)v=T(D).
-The horizontal part of \(T\) is \(vU\), so \(T(D)=vU(D)\).
+The horizontal part of (T) is (vU), so (T(D)=vU(D)).
Substitution in (eq:graph-gauss-weighted) proves
(eq:graph-identity). Finally, differentiating
-\(v=<\nu,∂_z>\) tangentially gives
-\(dv(V)=\II(V,T)\). Since \(|T|^2=1-v^2=|U|^2\), it follows that
-\(|d\log v|\le v^{-1}|U||\II|\), which is the last bound in
+(v=<nu,∂_z>) tangentially gives
+(dv(V)=II(V,T)). Since (|T|^2=1-v^2=|U|^2), it follows that
+(|dlog v|le v^{-1}|U||II|), which is the last bound in
(eq:graph-basic-bounds).
A one-coordinate calculation.:
The square and differentiated terms in the graph identity suggest
-an ideal target for \(ND\), the scaled weighted mean curvature.
-Let \(F\) be one smooth input coordinate, restrict
-to a region where \(f>0\), and fix \(ε,a>0\). Also let
-\(h_{in}\le h\) be a comparison metric. In the later sweep,
-the incoming energy is measured in \(h_{in}\), while \(h\)
+an ideal target for (ND), the scaled weighted mean curvature.
+Let (F) be one smooth input coordinate, restrict
+to a region where (f>0), and fix (ε,a>0). Also let
+(h_{in}le h) be a comparison metric. In the later sweep,
+the incoming energy is measured in (h_{in}), while (h)
already includes the earlier graphs.
-Put \(A=|U|_h\). Where \(A>0\), write \(U=Ae\) with \(|e|_h=1\).
+Put (A=|U|_h). Where (A>0), write (U=Ae) with (|e|_h=1).
The longitudinal unit tangent to the graph is
-\(\tau=(ve,A)\). Define
-\[
- x=-\II(\tau,\tau),\qquad X_{∥}=Nx,\qquad t=A^2/f,\qquad
- B=1-|e|_{h_{in}}^2\in[0,1].
-\]
-At \(A=0\), choose any \(h\)-unit vector \(e\), put \(\tau=(e,0)\),
-and use the same definitions. With \(\tau^\flat\) denoting the dual
-covector in the graph metric \(h'\), set
-\[
- \II^{rem}=\II+x\,\tau^\flat ⊗ \tau^\flat .
-\]
-The component of \(\II^{rem}\) in the
-\(\tau^\flat ⊗ \tau^\flat\) direction vanishes, so
+(tau=(ve,A)). Define
+[
+ x=-II(tau,tau),qquad X_{∥}=Nx,qquad t=A^2/f,qquad
+ B=1-|e|_{h_{in}}^2in[0,1].
+]
+At (A=0), choose any (h)-unit vector (e), put (tau=(e,0)),
+and use the same definitions. With (tau^flat) denoting the dual
+covector in the graph metric (h'), set
+[
+ II^{rem}=II+x,tau^flat ⊗ tau^flat .
+]
+The component of (II^{rem}) in the
+(tau^flat ⊗ tau^flat) direction vanishes, so
- NU(f)=LA^2,\qquad
- |\II|_{h'}^2=x^2+|\II^{rem}|_{h'}^2,\qquad
- \tr_{h'}\II^{rem}=H+x,\\
- x=e(A)\quadwhere A>0.
+ NU(f)=LA^2,qquad
+ |II|_{h'}^2=x^2+|II^{rem}|_{h'}^2,qquad
+ tr_{h'}II^{rem}=H+x,
+ x=e(A)quadwhere A>0.
-For the last identity, use \(\nu=(-Ae,v)\) and \(A^2+v^2=1\) to
+For the last identity, use (nu=(-Ae,v)) and (A^2+v^2=1) to
compute
-\[
- \II(\tau,\tau)=-v^2e(A)+Av\,e(v)=-e(A).
-\]
+[
+ II(tau,tau)=-v^2e(A)+Av,e(v)=-e(A).
+]
The first identity in (eq:graph-longitudinal) follows directly
-from the definitions of \(U,L,A\).
+from the definitions of (U,L,A).
-Define the ideal target for \(ND\) by
+Define the ideal target for (ND) by
Z=ε a(F^2/f-f)+A^2/(2f).
-Since \(NU(A^2)=2A^2X_{∥}\), with both sides zero at \(A=0\),
+Since (NU(A^2)=2A^2X_{∥}), with both sides zero at (A=0),
differentiation gives
-2NU(Z)
={}&-4ε a(F/f)NAe(F)
- +2ε a(F^2/f^2+1)LA^2 \notag\\
+ +2ε a(F^2/f^2+1)LA^2 notag
&-2X_{∥}t+Lt^2.
-The first term couples the prescribed \(F^2/f\) to the incoming
-energy. The \(A^2/(2f)\) term completes the longitudinal square:
+The first term couples the prescribed (F^2/f) to the incoming
+energy. The (A^2/(2f)) term completes the longitudinal square:
X_{∥}^2-2X_{∥}t+Lt^2
- =(X_{∥}-t)^2+(L-1)t^2,\\
+ =(X_{∥}-t)^2+(L-1)t^2,
&ε N^2|dF|_{h_{in}}^2
- -4ε a(F/f)NAe(F)\\
- &\quad=ε|N\,dF
- -2a(F/f)A\,h_{in}(e,\cdot)|_{h_{in}}^2
- -4ε a^2(F/f)^2A^2(1-B)\\
- &\quad\ge-4ε a^2(F/f)^2A^2(1-B).
+ -4ε a(F/f)NAe(F)
+ &quad=ε|N,dF
+ -2a(F/f)A,h_{in}(e,cdot)|_{h_{in}}^2
+ -4ε a^2(F/f)^2A^2(1-B)
+ &quadge-4ε a^2(F/f)^2A^2(1-B).
Consequently
- &Z^2-2NU(Z)+N^2|\II|_{h'}^2+N^2|d\log L|_{h'}^2
- +ε N^2|dF|_{h_{in}}^2 \notag\\
- \ge{}&Z^2+(X_{∥}-t)^2+N^2|\II^{rem}|_{h'}^2
- +N^2|d\log L|_{h'}^2 \notag\\
+ &Z^2-2NU(Z)+N^2|II|_{h'}^2+N^2|dlog L|_{h'}^2
+ +ε N^2|dF|_{h_{in}}^2 notag
+ ge{}&Z^2+(X_{∥}-t)^2+N^2|II^{rem}|_{h'}^2
+ +N^2|dlog L|_{h'}^2 notag
&+ε A^2[(2a-4a^2)F^2/f^2+2a]
- +4ε a^2(F^2/f^2)A^2B \notag\\
+ +4ε a^2(F^2/f^2)A^2B notag
&+(L-1)[2ε a(F^2/f^2+1)A^2+t^2].
-At \(A=0\), \(t=0\) and
-\((X_{∥}-t)^2+N^2|\II^{rem}|^2=N^2|\II|^2\); every other
+At (A=0), (t=0) and
+((X_{∥}-t)^2+N^2|II^{rem}|^2=N^2|II|^2); every other
direction-dependent term vanishes. Thus the inequality is independent
of the arbitrary direction at such points.
-This calculation is algebraic and assumes no equation for \(D\).
-The complete equation below realizes \(ND=Z-\beta f\) up to a positive
+This calculation is algebraic and assumes no equation for (D).
+The complete equation below realizes (ND=Z-beta f) up to a positive
floor, and localization introduces controlled cutoff errors. A sweep
will also have to account for normalization of the output map and the
interaction of the ordered graph metrics.
@@ -817,7 +817,7 @@ boundary, and with uniform bounds on the metric, its inverse, and
all their coordinate derivatives. These bounds, as well as the
chart size, may depend on the instance. A function is uniformly
smooth if all its coordinate derivatives, including its value, are
-uniformly bounded on these charts. For a weight \(\phi\), only
+uniformly bounded on these charts. For a weight (phi), only
positive-order derivatives are required to be uniformly bounded.
The uniform Hölder space below means the Banach space
obtained by taking the supremum of the corresponding Hölder norms in
@@ -834,34 +834,34 @@ the degree estimate.
[Prescribed graphs on a complete manifold]
-Let \((X,h)\) be a complete smooth Riemannian manifold without
+Let ((X,h)) be a complete smooth Riemannian manifold without
boundary, with bounded geometry as above, and suppose the
-positive-order derivatives of \(\phi\) are uniformly bounded. For
-a graph of height \(z\), set
-\[
- L_z=(1+|dz|_h^2)^{1/2},\quad v_z=L_z^{-1},\quad
- U_z=(\nabla_hz)/(L_z),\quad
- D_z=-\diver_\phi U_z .
-\]
-Consider either of the following equations \(D_z=p(x,z,v_z)\).
-
-
-\renewcommand{\labelenumi}{(\roman{enumi)}}
-\item Let \(N,ε,a>0\), \(0<\delta\le1\), and \(H_*\ge0\).
- Suppose \(F_i,\beta\) are uniformly smooth,
- \(|F_i|\le1\), and \(0\le\beta\le H_*\). On the region \(z>0\),
+positive-order derivatives of (phi) are uniformly bounded. For
+a graph of height (z), set
+[
+ L_z=(1+|dz|_h^2)^{1/2},quad v_z=L_z^{-1},quad
+ U_z=(nabla_hz)/(L_z),quad
+ D_z=-diver_phi U_z .
+]
+Consider either of the following equations (D_z=p(x,z,v_z)).
+
+
+renewcommand{labelenumi}{(roman{enumi)}}
+item Let (N,ε,a>0), (0<deltale1), and (H_*ge0).
+ Suppose (F_i,beta) are uniformly smooth,
+ (|F_i|le1), and (0lebetale H_*). On the region (z>0),
prescribe
p(x,z,v)=
- (ε a(F_i(x)^2+\delta^2)+(1-v^2)/2)/(z)
- -(ε a+\beta(x))(z)/(N^2).
+ (ε a(F_i(x)^2+delta^2)+(1-v^2)/2)/(z)
+ -(ε a+beta(x))(z)/(N^2).
-\item Let \(e,w>0\) be constants. Suppose \(B,f_0\) are uniformly
- smooth and \(B\ge B_{\min}>0\). On the region
- \(|(f_0(x)-ez)/w|<\pi/2\), prescribe
+item Let (e,w>0) be constants. Suppose (B,f_0) are uniformly
+ smooth and (Bge B_{min}>0). On the region
+ (|(f_0(x)-ez)/w|<pi/2), prescribe
- p(x,z,v)=B(x)\tan((f_0(x)-ez)/w).
+ p(x,z,v)=B(x)tan((f_0(x)-ez)/w).
@@ -869,47 +869,47 @@ Consider either of the following equations \(D_z=p(x,z,v_z)\).
In either case there is a smooth solution of bounded height with
uniform bounds on all its derivatives, and it lies uniformly inside
the stated open prescription region. Its angle satisfies
-\(v_z\ge v_0>0\). In Case (i) it satisfies
+(v_zge v_0>0). In Case (i) it satisfies
- c_{\min}:=\delta√((ε a)/(ε a+H_*))
- \ \le\ f:=z/N\ \le 2.
+ c_{min}:=delta√((ε a)/(ε a+H_*))
+ le f:=z/N le 2.
-In Case (ii) there is \(\eta>0\) such that
-\[
- |(f_0-ez)/(w)|\le(\pi)/(2)-\eta;
- \qquad
- |ez-f_0|<(\pi w)/(2),\quad
- |z|\le(\|f_0\|_∞+\pi w/2)/(e).
-\]
-The constants \(v_0,\eta\) and the derivative bounds may depend on
+In Case (ii) there is (eta>0) such that
+[
+ |(f_0-ez)/(w)|le(pi)/(2)-eta;
+ qquad
+ |ez-f_0|<(pi w)/(2),quad
+ |z|le(|f_0|_∞+pi w/2)/(e).
+]
+The constants (v_0,eta) and the derivative bounds may depend on
all the indicated parameters and common bounds of the data.
The solution is unique among bounded-height solutions whose
-\(C^{2,\alpha}\) norms in all buffered charts are uniformly bounded
-for some \(0<\alpha<1\), and whose values have a uniform margin inside
+(C^{2,alpha}) norms in all buffered charts are uniformly bounded
+for some (0<alpha<1), and whose values have a uniform margin inside
the prescription region. Each such solution may have its own
exponent, bound, and margin.
-The graph metric \(h_z=h+dz ⊗ dz\) is again complete and has
-bounded geometry, and \(\phi_z=\phi-\log L_z\) has uniform
+The graph metric (h_z=h+dz ⊗ dz) is again complete and has
+bounded geometry, and (phi_z=phi-log L_z) has uniform
positive-order derivatives. In particular,
-\[
- e^{\phi_z}dV_{h_z}=e^\phi dV_h,\qquad
+[
+ e^{phi_z}dV_{h_z}=e^phi dV_h,qquad
|dz|_{h_z}^2=1-v_z^2<1.
-\]
-For the family assertion, let \(\mathcal T\) be a standard Borel
+]
+For the family assertion, let (mathcal TcalT) be a standard Borel
parameter space indexing complete problems of a fixed dimension.
Assume a common chart radius and buffers, common bounds for the
metric, its inverse, and all their chart derivatives, and common
-bounds for the positive-order derivatives of \(\phi\). Require
+bounds for the positive-order derivatives of (phi). Require
common bounds, including order zero, for all chart derivatives of
-\(F_i,\beta\) in Case (i) or \(B,f_0\) in Case
-(ii). In Case (i), let \(N,ε,a,\delta\)
+(F_i,beta) in Case (i) or (B,f_0) in Case
+(ii). In Case (i), let (N,ε,a,delta)
range in fixed compact subsets of their permitted positive ranges,
-and require a common finite upper bound on \(H_*\). In Case
-(ii), let \(e,w\) range in fixed compact subsets of
-\((0,∞)\), and require \(B\ge B_{\min}\) for one common
-\(B_{\min}>0\). Suppose also that the data and chart overlaps are
+and require a common finite upper bound on (H_*). In Case
+(ii), let (e,w) range in fixed compact subsets of
+((0,∞)), and require (Bge B_{min}) for one common
+(B_{min}>0). Suppose also that the data and chart overlaps are
Borel and that the charts accessible along each leaf admit a
countable Borel enumeration. Then the preceding estimates for the
solutions and their graph metrics and weights hold with constants common to the family,
@@ -924,49 +924,49 @@ Give existence and the estimates together, since the estimates
also give uniqueness and measurable dependence.
Height barriers and comparison.
-In Case (i), replace \(F_i,\beta\) by \(tF_i,t\beta\),
-\(0\le t\le1\). At \(t=0\) the constant \(f=\delta\) is a
-solution. For every \(t\), the constant graphs \(f=c_{\min}\) and
-\(f=2\) are a lower and an upper barrier, respectively. Indeed their
-weighted mean curvature is zero, and at angle \(v=1\) the right side
+In Case (i), replace (F_i,beta) by (tF_i,tbeta),
+(0le tle1). At (t=0) the constant (f=delta) is a
+solution. For every (t), the constant graphs (f=c_{min}) and
+(f=2) are a lower and an upper barrier, respectively. Indeed their
+weighted mean curvature is zero, and at angle (v=1) the right side
of (eq:graph-prescription-positive) is nonnegative at
-\(f=c_{\min}\) and strictly negative at \(f=2\). In Case
-(ii), replace \(f_0\) by \(tf_0\), starting with \(z=0\).
-For \(\tau=\pi/2-\eta\), the graphs
-\[
- z_-=(tf_0-w\tau)/(e),\qquad
- z_+=(tf_0+w\tau)/(e)
-\]
-have arguments \(+\tau\) and \(-\tau\), respectively. Their
+(f=c_{min}) and strictly negative at (f=2). In Case
+(ii), replace (f_0) by (tf_0), starting with (z=0).
+For (tau=pi/2-eta), the graphs
+[
+ z_-=(tf_0-wtau)/(e),qquad
+ z_+=(tf_0+wtau)/(e)
+]
+have arguments (+tau) and (-tau), respectively. Their
weighted mean curvatures have a common bound independent of
-\(\eta\), because their derivatives do not involve \(\tau\).
-Choose \(\eta>0\) so small that \(B_{\min}\tan\tau\) exceeds this
-bound. These are then lower and upper barriers for every \(t\).
+(eta), because their derivatives do not involve (tau).
+Choose (eta>0) so small that (B_{min}tantau) exceeds this
+bound. These are then lower and upper barriers for every (t).
They give the asserted height and pole margins.
-Both comparisons use strict decrease of \(p\) in its height
+Both comparisons use strict decrease of (p) in its height
argument, with the other arguments held fixed:
- p_z&=-(ε a(F_i^2+\delta^2)+(1-v^2)/2)/(z^2)
- -(ε a+\beta)/(N^2)
- \le-(ε a)/(N^2)
- &&in Case (i), \notag\\
- p_z&=-(eB)/(w)\sec^2((f_0-ez)/w)
- \le-\frac{eB_{\min}}w
+ p_z&=-(ε a(F_i^2+delta^2)+(1-v^2)/2)/(z^2)
+ -(ε a+beta)/(N^2)
+ le-(ε a)/(N^2)
+ &&in Case (i), notag
+ p_z&=-(eB)/(w)sec^2((f_0-ez)/w)
+ le-frac{eB_{min}}w
&&in Case (ii).
At an interior positive maximum of the difference between a
solution and an upper barrier, their gradients, hence their angles,
-agree. The negative elliptic principal part of \(D_z\) gives a
+agree. The negative elliptic principal part of (D_z) gives a
nonnegative difference of mean curvatures there, whereas strict
height decrease gives a negative difference of prescriptions.
This is impossible. The lower comparison and uniqueness follow in
the same way. On a noncompact manifold the maximum can be replaced
by a supremum: take uniform charts centered at an extremizing
-sequence, subtract the value of \(\phi\) at each center, and pass to
+sequence, subtract the value of (phi) at each center, and pass to
a local limit. For any individual solution in the stated uniqueness
-class, its uniform \(C^{2,\alpha}\) bound and the coefficient bounds
+class, its uniform (C^{2,alpha}) bound and the coefficient bounds
give such a limit, which attains the extremum at the center of an
interior chart. This
justifies the same comparison without compactness.
@@ -974,33 +974,33 @@ justifies the same comparison without compactness.
The angle bound.
For a solution, (eq:graph-translation) and the chain rule give
- (\Delta_{Σ,\phi}+P)v_z
+ (Delta_{Σ,phi}+P)v_z
=v_z< p_x,U_z>
+p_z(1-v_z^2)+p_vT(v_z).
-Here \(p_x\) differentiates with \(z,v\) fixed, and use was made of \(T(z)=1-v_z^2\). The ambient curvature and weight bounds imply
-\(P\ge-C\), since the remaining term \(|\II|^2\) is nonnegative.
-At an infimum of \(v_z\), taken by the same recentering procedure,
-\(\Delta_{Σ,\phi}v_z\ge0\) and \(T(v_z)=0\). Consequently, if
-\(v_z\le1/2\) there,
+Here (p_x) differentiates with (z,v) fixed, and use was made of (T(z)=1-v_z^2). The ambient curvature and weight bounds imply
+(Pge-C), since the remaining term (|II|^2) is nonnegative.
+At an infimum of (v_z), taken by the same recentering procedure,
+(Delta_{Σ,phi}v_zge0) and (T(v_z)=0). Consequently, if
+(v_zle1/2) there,
- (1-v_z^2)|p_z|\le v_z(C+|p_x|).
+ (1-v_z^2)|p_z|le v_z(C+|p_x|).
-The height margins just proved bound \(p_x\) uniformly, and
-(eq:graph-height-monotonicity) bounds \(|p_z|\) below by a
-positive constant. Since \(1-v_z^2\ge3/4\) in this case,
+The height margins just proved bound (p_x) uniformly, and
+(eq:graph-height-monotonicity) bounds (|p_z|) below by a
+positive constant. Since (1-v_z^2ge3/4) in this case,
(eq:graph-angle-infimum) gives a uniform positive lower bound
-for \(v_z\). In using recentered limits at this stage, a solution
-is first in an individual uniform \(C^{2,\alpha}\) class. Ordinary
+for (v_z). In using recentered limits at this stage, a solution
+is first in an individual uniform (C^{2,alpha}) class. Ordinary
local bootstrapping gives its individual higher regularity before
the common estimates are deduced, so the differentiations and
limits in this argument are justified.
Higher estimates.
In a uniform chart let
-\(\omega=e^\phi√(\det h)\), and divide \(\omega\) by its value
+(omega=e^phi√(det h)), and divide (omega) by its value
at the chart center. The resulting density has a common upper
bound and a positive lower bound, and all its derivatives are
bounded. The equation is of the form
@@ -1008,44 +1008,44 @@ bounded. The equation is of the form
-∂_j A^j(x,∂ z)=F(x,z,∂ z).
-For example \(A^j=\omega h^{jk}z_k/L_z\) after the normalization.
+For example (A^j=omega h^{jk}z_k/L_z) after the normalization.
On the established height and gradient ranges the flux and the
right side are smooth with uniformly bounded derivatives, and
-\(∂ A^j/∂ z_k\) is uniformly elliptic. Differentiating
+(∂ A^j/∂ z_k) is uniformly elliptic. Differentiating
(eq:graph-divergence-equation) gives linear divergence
-equations for \(∂_kz\). Their leading coefficients are
-bounded and uniformly elliptic. Gradient dependence of \(F\)
+equations for (∂_kz). Their leading coefficients are
+bounded and uniformly elliptic. Gradient dependence of (F)
produces a bounded first-order coefficient; the remaining terms
are a bounded function plus the divergence of a bounded vector
-field. The term \(F_z∂_kz\) is included in the bounded
+field. The term (F_z∂_kz) is included in the bounded
source, since the gradient is already bounded.
The interior De Giorgi--Nash estimate for these equations, with
-bounded lower-order coefficients and source data in \(L^{s/2}\)
-and \(L^s\) for \(s>\dim X\), gives a common Hölder bound for
-\(∂ z\); see \cite[Theorem~8.24]{GilbargTrudinger}.
+bounded lower-order coefficients and source data in (L^{s/2})
+and (L^s) for (s>dim X), gives a common Hölder bound for
+(∂ z); see cite[Theorem~8.24]{GilbargTrudinger}.
The coefficients in the nondivergence form of the original
equation are now Hölder continuous. Interior Schauder estimates
-give a uniform \(C^{2,\alpha}\) bound for some \(\alpha>0\), after
+give a uniform (C^{2,alpha}) bound for some (alpha>0), after
shrinking within the chart buffers. Differentiating and applying
Schauder estimates again yields uniform bounds for every
derivative. All these estimates are local with common buffers, so
-their constants are uniform throughout \(X\).
+their constants are uniform throughout (X).
Continuation.
-Work in uniform \(C^{2,\alpha}\) spaces on the open set of functions
-with a uniform height margin. The equation \(D_z-p(x,z,v_z)=0\)
-defines a smooth map to uniform \(C^{0,\alpha}\). At a solution
+Work in uniform (C^{2,alpha}) spaces on the open set of functions
+with a uniform height margin. The equation (D_z-p(x,z,v_z)=0)
+defines a smooth map to uniform (C^{0,alpha}). At a solution
its linearization has negative uniformly elliptic principal part,
uniform Hölder coefficients, and zeroth-order coefficient
-\(-p_z\) bounded below by a positive constant. This linear
+(-p_z) bounded below by a positive constant. This linear
operator is invertible on the uniform spaces. To see
surjectivity, exhaust a noncompact component by smooth relatively
compact domains and solve the linear Dirichlet problem with zero
boundary data on each domain. The maximum principle bounds the
supremum of the solutions by the supremum of the right side
divided by the positive zeroth-order lower bound. Interior
-Schauder estimates then give the same \(C^{2,\alpha}\) bound in
+Schauder estimates then give the same (C^{2,alpha}) bound in
every buffered chart once it lies in the exhaustion. A diagonal
local limit solves the equation on the component with the uniform
bound. On a compact component the corresponding closed-manifold
@@ -1064,11 +1064,11 @@ specified above, so existence follows. Comparison gives the
claimed uniqueness.
The graph estimates now follow from
-(eq:graph-basic-bounds). In particular \(h_z\ge h\), and
+(eq:graph-basic-bounds). In particular (h_zge h), and
the gradient bound also gives a uniform upper comparison in the
-given charts. Thus \(h_z\) is complete and has uniform buffered
+given charts. Thus (h_z) is complete and has uniform buffered
charts with bounds on all derivatives. The corresponding
-derivative bounds on \(\log L_z\) give those for \(\phi_z\).
+derivative bounds on (log L_z) give those for (phi_z).
This also permits successive applications of the lemma.
Borel dependence.
@@ -1084,7 +1084,7 @@ The chart data and changes of coordinates are Borel by
hypothesis. The resulting Borel relation between data and local
representatives has a singleton fiber over each set of data,
by existence and uniqueness. The Borel injection theorem
-\cite[Corollary~15.2]{Kechris1995},
+cite[Corollary~15.2]{Kechris1995},
applied to its projection to the data, makes the inverse
projection Borel. This proves measurable dependence. The
coding refers to a full prescribed problem on the same complete
@@ -1095,8 +1095,8 @@ uniqueness proves equivariance.
-\par\noindent{\footnotesizehttps://doi.org/10.1007/978-3-642-61798-0}\par
-\par\noindent{\footnotesizehttps://doi.org/10.1007/978-1-4612-4190-4}\par
+parnoindent{footnotesizehttps://doi.org/10.1007/978-3-642-61798-0}par
+parnoindent{footnotesizehttps://doi.org/10.1007/978-1-4612-4190-4}par
The positive solution in Lemma~lem:graph-prescription is
@@ -1109,127 +1109,127 @@ procedure that enter the sweep.
[Localization of the positive graph]
-Let \(X,h,\phi\) satisfy the hypotheses of
-Lemma~lem:graph-prescription, with \(X\) connected, and let
-\(h_{ref}\le h\) be a complete second metric. Require the
-coefficients of \(h_{ref}\), its inverse, and all their
+Let (X,h,phi) satisfy the hypotheses of
+Lemma~lem:graph-prescription, with (X) connected, and let
+(h_{ref}le h) be a complete second metric. Require the
+coefficients of (h_{ref}), its inverse, and all their
coordinate derivatives to have uniform bounds in the same buffered
-charts used for \(h\).
-Suppose \(F_i\) is uniformly smooth, \(|F_i|\le1\), with compact
-support \(K\), and let \(\alpha\ge0\) be a uniformly smooth
+charts used for (h).
+Suppose (F_i) is uniformly smooth, (|F_i|le1), with compact
+support (K), and let (alphage0) be a uniformly smooth
function. Fix
-\[
- ε>0,\qquad N^2=lε,\qquad
- a=\tfrac12+\theta,\qquad 0<l\le1,\qquad l\le\theta\le\tfrac1{16}.
-\]
-There is a radius \(R_{loc}\), depending only on the
+[
+ ε>0,qquad N^2=lε,qquad
+ a=tfrac12+theta,qquad 0<lle1,qquad llethetaletfrac1{16}.
+]
+There is a radius (R_{loc}), depending only on the
dimension and a fixed sufficiently small numerical constant
-\(c_0>0\), with the following property. For every sufficiently
-large \(H_*\), one can choose smooth functions
-\(\beta_s\ge0\) and \(0\le\chi\le1\) such that, with
-\(\beta=\beta_s+\alpha\),
-
- 0\le\beta\le H_*,
- \qquad
- N|d\beta_s|_h
- \le c_0(\thetaε^{3/2}+\beta_s^{3/2}).
+(c_0>0), with the following property. For every sufficiently
+large (H_*), one can choose smooth functions
+(beta_sge0) and (0lechile1) such that, with
+(beta=beta_s+alpha),
+
+ 0lebetale H_*,
+ qquad
+ N|dbeta_s|_h
+ le c_0(thetaε^{3/2}+beta_s^{3/2}).
-The function \(\beta_s\) vanishes on a neighborhood of \(K\).
-The function \(\chi\) is one there, is supported in
-\(\{\dist_{h_{ref}}(x,K)<R_{loc}\}\), and changes
-from one to zero only within a region where \(F_i=0\) and
-\(\beta=H_*\). This region and the support of \(d\chi\) have fixed
-positive buffers in \(h_{ref}\), independent of \(H_*\).
-The derivatives of \(\chi\) are bounded independently of \(H_*\),
-\(\chi\) is flat on its zero set, and \(|d\chi|_h^2\le C\chi\).
+The function (beta_s) vanishes on a neighborhood of (K).
+The function (chi) is one there, is supported in
+({dist_{h_{ref}}(x,K)<R_{loc}}), and changes
+from one to zero only within a region where (F_i=0) and
+(beta=H_*). This region and the support of (dchi) have fixed
+positive buffers in (h_{ref}), independent of (H_*).
+The derivatives of (chi) are bounded independently of (H_*),
+(chi) is flat on its zero set, and (|dchi|_h^2le Cchi).
-Set \(\delta=H_*^{-2}\), and let \(\widetilde f=z/N\) be the
+Set (delta=H_*^{-2}), and let (widetilde f=z/N) be the
complete solution of (eq:graph-prescription-positive) for
-this \(\beta,\delta\). Define \(f=\chi\widetilde f\) and form the
-graph of height \(Nf\). Denote its quantities by \(L,U,D,\II\),
-put \(A=|U|_h\), and set, where \(f>0\),
+this (beta,delta). Define (f=chiwidetilde f) and form the
+graph of height (Nf). Denote its quantities by (L,U,D,II),
+put (A=|U|_h), and set, where (f>0),
ND^0=ε a(F_i^2/f-f)+A^2/(2f).
-The expressions \(A^2/f\) and \(ND^0\) are extended by zero
-where \(f=0\). The bump \(f\) is smooth,
-\(0\le f\le2\), is positive on \(K\), and is flat on its zero set.
-On the region \(\chi=1\) it satisfies
+The expressions (A^2/f) and (ND^0) are extended by zero
+where (f=0). The bump (f) is smooth,
+(0le fle2), is positive on (K), and is flat on its zero set.
+On the region (chi=1) it satisfies
- ND=ND^0-\beta f+ε a\delta^2/f .
+ ND=ND^0-beta f+ε adelta^2/f .
On the region where the cutoff changes, the following quantities
-tend uniformly to zero as \(H_*\to∞\), with the preceding
+tend uniformly to zero as (H_*to∞), with the preceding
data and parameters fixed:
- N^2|\wscal(h+N^2df ⊗ df,\phi-\log L)
- -\wscal(h,\phi)|,\quad
- A^2/f,\quad |ND^0|,\quad \beta f,\\
- |ND|,\quad N|\II|,\quad N|d\log L|_{h+N^2df ⊗ df}.
+ N^2|wscal(h+N^2df ⊗ df,phi-log L)
+ -wscal(h,phi)|,quad
+ A^2/f,quad |ND^0|,quad beta f,
+ |ND|,quad N|II|,quad N|dlog L|_{h+N^2df ⊗ df}.
-In particular, on all of \(X\) the function
-\(err:=ND-(ND^0-\beta f)\) tends uniformly to zero.
+In particular, on all of (X) the function
+(err:=ND-(ND^0-beta f)) tends uniformly to zero.
All data retain the corresponding uniform bounds for each fixed
-choice of \(H_*\), with the positive-order convention for weights.
+choice of (H_*), with the positive-order convention for weights.
For the family assertion, let a standard Borel parameter space index
such problems of a fixed dimension. Require common sizes and buffers
for these simultaneous charts, and common bounds there for
-\(h,h^{-1},h_{ref},h_{ref}^{-1}\) and all their
+(h,h^{-1},h_{ref},h_{ref}^{-1}) and all their
coordinate derivatives. Require common bounds for all derivatives,
-including values, of \(F_i,\alpha\), and for the positive-order
-derivatives of \(\phi\). Let \((ε,l,\theta)\) range in a fixed
+including values, of (F_i,alpha), and for the positive-order
+derivatives of (phi). Let ((ε,l,theta)) range in a fixed
compact subset of the permitted parameter set. Require the data,
-including \(h_{ref}\), to
+including (h_{ref}), to
be Borel in the charts and preserved by the equivariant identifications
under consideration. Require also the Borel chart overlaps and
countable Borel enumerations of accessible charts in
Lemma~lem:graph-prescription. Then the choices are Borel and
commute with those identifications; their bounds for each fixed
-\(H_*\), and the limits above as \(H_*\to∞\), are uniform over
-the family. When \(K\) is empty one may take \(f=0\) and omit the
+(H_*), and the limits above as (H_*to∞), are uniform over
+the family. When (K) is empty one may take (f=0) and omit the
operation.
Proof.
First construct the outer coefficient while keeping its
transition width bounded. Starting at a fixed positive distance
-from \(K\), a scalar profile \(b\) can increase from zero to
-\(H_*\) at speed at most
-\[
- (c_0)/(C N)(\thetaε^{3/2}+b^{3/2}),
-\]
-where the fixed factor \(C\) leaves room for the estimates below.
+from (K), a scalar profile (b) can increase from zero to
+(H_*) at speed at most
+[
+ (c_0)/(C N)(thetaε^{3/2}+b^{3/2}),
+]
+where the fixed factor (C) leaves room for the estimates below.
Choose the speed comparable to this expression during the
increase, with smooth flattening at either end. The width needed
is at most
- C N\int_0^∞\frac{dt}
- {\thetaε^{3/2}+t^{3/2}}
- =C\frac{N}{\theta^{1/3}\sqrtε}
- =C\frac{\sqrt l}{\theta^{1/3}}\le C .
+ C Nint_0^∞frac{dt}
+ {thetaε^{3/2}+t^{3/2}}
+ =Cfrac{N}{theta^{1/3}sqrtε}
+ =Cfrac{sqrt l}{theta^{1/3}}le C .
For example one can use the primitive of the reciprocal of a smooth
positive function comparable to
-\(\thetaε^{3/2}+t^{3/2}\) as the new profile coordinate,
+(thetaε^{3/2}+t^{3/2}) as the new profile coordinate,
and compose with fixed smooth flattenings. The last inequality
-uses \(l\le\theta\le1/16\).
+uses (llethetale1/16).
One may use this profile in a smooth approximation of
-\(\dist_{h_{ref}}(x,K)\) with gradient at most \(2\)
+(dist_{h_{ref}}(x,K)) with gradient at most (2)
and a uniformly small error. Here is a construction with the
needed uniformity. Shrinking the controlled charts gives normal
charts at a fixed positive radius, by the uniform geodesic
equation estimates. Average the distance function in such a
-chart centered at \(x\) against a fixed smooth radial kernel in
-\(T_xX\), at a sufficiently small radius. Written as an integral
-against a kernel in \((x,y)\), this is a smooth function of \(x\).
+chart centered at (x) against a fixed smooth radial kernel in
+(T_xX), at a sufficiently small radius. Written as an integral
+against a kernel in ((x,y)), this is a smooth function of (x).
The error is at most the averaging radius. To bound the first
derivative, move the center and parallel transport the averaging
vectors. At a sufficiently small radius the displacement of
@@ -1239,279 +1239,279 @@ bound follows. Higher derivatives have uniform bounds from the
smooth local kernels: in differentiating the normalized kernel
one may subtract the distance at the center, leaving a uniformly
bounded oscillation on its support. These derivative bounds hold in
-the common buffered charts, since the \(h_{ref}\) exponential
+the common buffered charts, since the (h_{ref}) exponential
maps and kernels have uniformly bounded derivatives there. The inverse metric
-inequality \(h^{-1}\le h_{ref}^{-1}\) then bounds the
-gradient of the composed profile in \(h\).
+inequality (h^{-1}le h_{ref}^{-1}) then bounds the
+gradient of the composed profile in (h).
-Choose this raw profile \(b\) to vanish near \(K\), to equal
-\(H_*\) past the width in (eq:graph-cutoff-width), and to
-leave fixed outer buffers. To accommodate \(\alpha\), take a
+Choose this raw profile (b) to vanish near (K), to equal
+(H_*) past the width in (eq:graph-cutoff-width), and to
+leave fixed outer buffers. To accommodate (alpha), take a
smooth nondecreasing saturation
-\(\sigma_{H_*}:[0,∞)\to[0,H_*]\) that equals the identity
-up to \(H_*/3\), equals \(H_*\) from \(H_*\) on, and has a
+(sigma_{H_*}:[0,∞)to[0,H_*]) that equals the identity
+up to (H_*/3), equals (H_*) from (H_*) on, and has a
derivative bounded by a numerical constant. Put
-\[
- \beta=\sigma_{H_*}(b+\alpha),\qquad
- \beta_s=\sigma_{H_*}(b+\alpha)-\alpha .
-\]
-Take \(H_*\) much larger than the value and derivative bounds
-of the already fixed \(\alpha\). Then \(\beta_s\ge0\), and it
-equals \(b\) near \(K\). Wherever the saturation differs from
-the identity, both \(\beta_s\) and the scale of \(b\) are
-comparable to \(H_*\). The formula
-\[
- d\beta_s=\sigma_{H_*}'(b+\alpha)\,db
- +(\sigma_{H_*}'(b+\alpha)-1)d\alpha
-\]
+[
+ beta=sigma_{H_*}(b+alpha),qquad
+ beta_s=sigma_{H_*}(b+alpha)-alpha .
+]
+Take (H_*) much larger than the value and derivative bounds
+of the already fixed (alpha). Then (beta_sge0), and it
+equals (b) near (K). Wherever the saturation differs from
+the identity, both (beta_s) and the scale of (b) are
+comparable to (H_*). The formula
+[
+ dbeta_s=sigma_{H_*}'(b+alpha),db
+ +(sigma_{H_*}'(b+alpha)-1)dalpha
+]
therefore proves (eq:graph-spatial-cutoff): choose the
initial fixed speed factor large enough to handle the first
-term, and then choose \(H_*\) large enough for the fixed
-\(d\alpha\) term to be a small fraction of
-\(H_*^{3/2}/N\). Where saturation is the identity this is just
-the raw profile bound. Where \(b=H_*\), the total \(\beta\)
-equals \(H_*\). A cutoff supported beyond fixed additional
+term, and then choose (H_*) large enough for the fixed
+(dalpha) term to be a small fraction of
+(H_*^{3/2}/N). Where saturation is the identity this is just
+the raw profile bound. Where (b=H_*), the total (beta)
+equals (H_*). A cutoff supported beyond fixed additional
buffers can consequently be chosen once at a radius
-\(R_{loc}\) as in the statement. Take it to be the
+(R_{loc}) as in the statement. Take it to be the
square of a smooth nonnegative cutoff that is flat at its outer
-end. This gives \(|d\chi|^2\le C\chi\) and the other asserted
+end. This gives (|dchi|^2le Cchi) and the other asserted
cutoff properties. If some outer regions are empty, the
corresponding assertions are vacuous.
Next prove that the complete solution can be cut on this
plateau. All constants in the rest of the proof may use the
-fixed data and parameters, but not \(H_*\). Write
-\[
+fixed data and parameters, but not (H_*). Write
+[
c_H=H_*^{-2}√((ε a)/(ε a+H_*))
- \asymp H_*^{-5/2}.
-\]
+ asymp H_*^{-5/2}.
+]
By (eq:graph-height-positive),
-\(c_H\le\widetilde f\le2\). Cover the cutoff region by inner chart
+(c_Hlewidetilde fle2). Cover the cutoff region by inner chart
boxes, each with a fixed succession of larger buffered boxes whose
-closures lie in the plateau \(F_i=0,\ \beta=H_*\). Their sizes may
-use the current \(h\)-chart bounds inside the fixed
-\(h_{ref}\) buffers; all these bounds were fixed before
-\(H_*\), so the boxes and their buffers have sizes independent of
-\(H_*\). Our target is that, for every integer \(j\ge0\) and every
-\(M>0\), there is \(C_{j,M}\) such that, for all sufficiently large
-\(H_*\),
-
- \|\widetilde f-c_H\|_{C^j}
- \le C_{j,M}H_*^{-M},
+closures lie in the plateau (F_i=0, beta=H_*). Their sizes may
+use the current (h)-chart bounds inside the fixed
+(h_{ref}) buffers; all these bounds were fixed before
+(H_*), so the boxes and their buffers have sizes independent of
+(H_*). Our target is that, for every integer (jge0) and every
+(M>0), there is (C_{j,M}) such that, for all sufficiently large
+(H_*),
+
+ |widetilde f-c_H|_{C^j}
+ le C_{j,M}H_*^{-M},
on the inner boxes, with norms in these fixed coordinates.
The constants use only the indicated data and parameters fixed
-before \(H_*\). The family assertion of
+before (H_*). The family assertion of
Lemma~lem:graph-prescription is applied separately for each
-fixed \(H_*\); the estimates that follow establish this
-superpolynomial decay uniformly as \(H_*\) grows.
+fixed (H_*); the estimates that follow establish this
+superpolynomial decay uniformly as (H_*) grows.
There is first a global angle bound
- v\ge\frac{c_1}{1+√(H_*)}
+ vgefrac{c_1}{1+√(H_*)}
for the uncut graph. In fact, at a recentered infimum with
-\(v\le1/2\), the formulas for \(p_x,p_z\) and
+(vle1/2), the formulas for (p_x,p_z) and
(eq:graph-spatial-cutoff) give
-\[
- |p_z|\ge c_2(1+\beta+\widetilde f^{-2}),\qquad
- |p_x|\le C(\widetilde f^{-1}
- +\widetilde f(1+\beta^{3/2})).
-\]
-For the first bound use \(1-v^2\ge3/4\) in \(p_z\); for the
-second use the fixed bounds on \(F_i,\alpha\) and their
-derivatives. Since \(\widetilde f\le2\) and \(\beta\le H_*\),
+[
+ |p_z|ge c_2(1+beta+widetilde f^{-2}),qquad
+ |p_x|le C(widetilde f^{-1}
+ +widetilde f(1+beta^{3/2})).
+]
+For the first bound use (1-v^2ge3/4) in (p_z); for the
+second use the fixed bounds on (F_i,alpha) and their
+derivatives. Since (widetilde fle2) and (betale H_*),
the ratio of the right side of the second estimate, plus a
constant, to the right side of the first is at most
-\(C(1+√(H_*))\). Equation
+(C(1+√(H_*))). Equation
(eq:graph-angle-infimum) proves
(eq:graph-global-plateau-angle). This recentering is done
-for each fixed \(H_*\), using the individual regularity supplied
+for each fixed (H_*), using the individual regularity supplied
by Lemma~lem:graph-prescription.
On each of the outer plateau boxes the equation is
- ND=(ε a\delta^2)/(\widetilde f)
- +(1-v^2)/(2\widetilde f)
- -(ε a+H_*)\widetilde f,
- \qquad
- (ε a+H_*)c_H^2=ε a\delta^2 .
+ ND=(ε adelta^2)/(widetilde f)
+ +(1-v^2)/(2widetilde f)
+ -(ε a+H_*)widetilde f,
+ qquad
+ (ε a+H_*)c_H^2=ε adelta^2 .
There is an exponential value estimate on an inner box:
- 0\le\widetilde f-c_H\le C\exp(-c_3√(H_*)).
+ 0lewidetilde f-c_Hle Cexp(-c_3√(H_*)).
-To prove it, let \(d_{face}\) be ordinary coordinate
+To prove it, let (d_{face}) be ordinary coordinate
distance to a face of the outer box and put
-\[
- w_1=\sum_{faces}3
- \exp(-\gamma√(H_*)\,d_{face}).
-\]
-On the boundary, \(c_H+w_1\ge3\ge\widetilde f\).
+[
+ w_1=sum_{faces}3
+ exp(-gamma√(H_*),d_{face}).
+]
+On the boundary, (c_H+w_1ge3gewidetilde f).
Its first and second coordinate derivatives are bounded by
-\(C\gamma√(H_*)w_1\) and \(C\gamma^2H_*w_1\),
+(Cgamma√(H_*)w_1) and (Cgamma^2H_*w_1),
respectively. Direct differentiation of the weighted graph
divergence therefore bounds its mean curvature from below by
-\[
- -C(\gamma^2H_*+\gamma√(H_*))w_1 .
-\]
+[
+ -C(gamma^2H_*+gamma√(H_*))w_1 .
+]
For this graph, the height terms in its prescription satisfy
-\[
- (ε a\delta^2)/(N(c_H+w_1))
- -\frac{(ε a+H_*)(c_H+w_1)}N
+[
+ (ε adelta^2)/(N(c_H+w_1))
+ -frac{(ε a+H_*)(c_H+w_1)}N
=-(ε a+H_*)/(N)w_1
(1+(c_H)/(c_H+w_1)).
-\]
+]
The remaining angle term is at most
-\(C\gamma^2H_*w_1\), because
-\((1-v^2)/(c_H+w_1)\le
- N^2|dw_1|^2/(c_H+w_1)\le C\gamma^2H_*w_1\).
-Choosing \(\gamma>0\) sufficiently small and then \(H_*\)
-sufficiently large makes \(c_H+w_1\) an upper barrier.
-Comparison with this barrier and the lower constant \(c_H\)
+(Cgamma^2H_*w_1), because
+((1-v^2)/(c_H+w_1)le
+ N^2|dw_1|^2/(c_H+w_1)le Cgamma^2H_*w_1).
+Choosing (gamma>0) sufficiently small and then (H_*)
+sufficiently large makes (c_H+w_1) an upper barrier.
+Comparison with this barrier and the lower constant (c_H)
proves (eq:graph-plateau-value) on any inner box a fixed
distance from the faces.
On a smaller box the angle is bounded below independently of
-\(H_*\). To see this, choose a smooth chart cutoff
-\(0\le\xi\le1\), supported in a region where
+(H_*). To see this, choose a smooth chart cutoff
+(0lexile1), supported in a region where
(eq:graph-plateau-value) holds and equal to one on the
-smaller box. Minimize \(v/\xi\) over \(\{\xi>0\}\).
+smaller box. Minimize (v/xi) over ({xi>0}).
The global lower bound (eq:graph-global-plateau-angle)
-makes a minimum with value below \(1/2\), if one exists,
+makes a minimum with value below (1/2), if one exists,
interior, and at that minimum
-\[
- v<\xi/2,\qquad \xi\ge c/√(H_*),\qquad
- \nabla_Σ v=(v/\xi)\nabla_Σ\xi,\qquad
- \Delta_{Σ,\phi}v\ge(v/\xi)\Delta_{Σ,\phi}\xi.
-\]
+[
+ v<xi/2,qquad xige c/√(H_*),qquad
+ nabla_Σ v=(v/xi)nabla_Σxi,qquad
+ Delta_{Σ,phi}vge(v/xi)Delta_{Σ,phi}xi.
+]
The formula for the restriction of an ambient function to the
graph gives
-\(\Delta_{Σ,\phi}\xi\ge-C(1+|\II|)\). The positive
-\(|\II|^2\) term in \(P\), and completion of its square against
-\(C|\II|/\xi\), show that the left side of
-(eq:graph-angle-equation) is at least \(-CH_*v\).
-On this plateau \(p_x=0\). Since \(v<1/2\), the \(p_z\) term
-on its right side is at most \(-c/\widetilde f^2\).
-Also \(p_v=-v/(N\widetilde f)\) and
-\(|T(v)|\le Cv/\xi\), so that right side is at most
-\[
- -(c)/(\widetilde f^2)
- +\frac{C√(H_*)\,v}{\widetilde f}.
-\]
-Combining the two bounds and multiplying by \(\widetilde f^2\)
+(Delta_{Σ,phi}xige-C(1+|II|)). The positive
+(|II|^2) term in (P), and completion of its square against
+(C|II|/xi), show that the left side of
+(eq:graph-angle-equation) is at least (-CH_*v).
+On this plateau (p_x=0). Since (v<1/2), the (p_z) term
+on its right side is at most (-c/widetilde f^2).
+Also (p_v=-v/(Nwidetilde f)) and
+(|T(v)|le Cv/xi), so that right side is at most
+[
+ -(c)/(widetilde f^2)
+ +frac{C√(H_*),v}{widetilde f}.
+]
+Combining the two bounds and multiplying by (widetilde f^2)
would give
-\[
- c\le CH_*\widetilde f^2+
- C√(H_*)\,\widetilde f .
-\]
+[
+ cle CH_*widetilde f^2+
+ C√(H_*),widetilde f .
+]
By (eq:graph-plateau-value),
-\(\widetilde f\le2c_H\) for large \(H_*\), and the last right
-side tends to zero. This contradiction proves \(v\ge1/2\) on
+(widetilde fle2c_H) for large (H_*), and the last right
+side tends to zero. This contradiction proves (vge1/2) on
the smaller box.
It remains to control derivatives on the cutoff region without
using any growth rate for the global elliptic constants.
-In a still smaller box, rescale coordinates \(x=x_0+c_Hy\) and
-write \(\widetilde f=c_Hg\). The weighted graph curvature in
-the rescaled chart is \(c_HD\), and its right side becomes
+In a still smaller box, rescale coordinates (x=x_0+c_Hy) and
+write (widetilde f=c_Hg). The weighted graph curvature in
+the rescaled chart is (c_HD), and its right side becomes
c_Hp(x,Nc_Hg,v)
- &=(ε a\delta^2+(1-v^2)/2)/(Ng)
- -(c_H^2(ε a+H_*)g)/(N)\\
- &=(ε a\delta^2+(1-v^2)/2)/(Ng)
- -(ε a\delta^2g)/(N).
+ &=(ε adelta^2+(1-v^2)/2)/(Ng)
+ -(c_H^2(ε a+H_*)g)/(N)
+ &=(ε adelta^2+(1-v^2)/2)/(Ng)
+ -(ε adelta^2g)/(N).
-Here \(1\le g\le2\) for large \(H_*\), and the angle bound gives
+Here (1le gle2) for large (H_*), and the angle bound gives
a uniform gradient bound. The rescaled metric and normalized
weight have uniformly bounded derivatives; the displayed
right side and all its derivatives are uniformly bounded on
these ranges. The local higher estimates in the proof of
Lemma~lem:graph-prescription, applied on unit boxes in
these coordinates, consequently give uniform bounds for every
-derivative of \(g\). Scaling back gives bounds of the form
-\(C_jc_H^{1-j}\) for the \(j\)-th derivatives of
-\(\widetilde f\). Interior interpolation between these polynomial
+derivative of (g). Scaling back gives bounds of the form
+(C_jc_H^{1-j}) for the (j)-th derivatives of
+(widetilde f). Interior interpolation between these polynomial
bounds and the exponential bound (eq:graph-plateau-value), on
further fixed buffered boxes, proves (eq:graph-plateau-jets).
Indeed the usual finite-difference interpolation of a
-\(j\)-th derivative between the supremum norm and a higher
+(j)-th derivative between the supremum norm and a higher
derivative norm gives an exponential factor times a power of
-\(c_H^{-1}\), which is smaller than every power of \(H_*^{-1}\).
+(c_H^{-1}), which is smaller than every power of (H_*^{-1}).
Now cut off. On the transition region,
-\(\widetilde f\asymp c_H\), its positive-order derivatives are
-smaller than every power of \(H_*^{-1}\), and the derivatives
-of \(\chi\) are fixed. On \(\{\chi>0\}\),
-\[
- (|d(\chi\widetilde f)|^2)/(\chi\widetilde f)
- \le
- 2\widetilde f\,(|d\chi|^2)/(\chi)
- +2\chi\,(|d\widetilde f|^2)/(\widetilde f)
+(widetilde fasymp c_H), its positive-order derivatives are
+smaller than every power of (H_*^{-1}), and the derivatives
+of (chi) are fixed. On ({chi>0}),
+[
+ (|d(chiwidetilde f)|^2)/(chiwidetilde f)
+ le
+ 2widetilde f,(|dchi|^2)/(chi)
+ +2chi,(|dwidetilde f|^2)/(widetilde f)
=O(c_H).
-\]
-It follows that \(A^2/f\le N^2|df|^2/f\) tends to zero.
-All fixed derivatives of \(f\) are \(O(c_H)\) there.
-The graph formulas therefore give \(ND,N|\II|\), and
-\(N|d\log L|\) tending to zero. They also control
-\(\nabla D\), so (eq:graph-identity) gives the first
-limit in (eq:graph-cutoff-small). Finally \(F_i=0\) in
-this region, \(ND^0=-ε af+A^2/(2f)\), and
-\(\beta f\le H_*\,O(c_H)\to0\). This proves every limit in
+]
+It follows that (A^2/fle N^2|df|^2/f) tends to zero.
+All fixed derivatives of (f) are (O(c_H)) there.
+The graph formulas therefore give (ND,N|II|), and
+(N|dlog L|) tending to zero. They also control
+(nabla D), so (eq:graph-identity) gives the first
+limit in (eq:graph-cutoff-small). Finally (F_i=0) in
+this region, (ND^0=-ε af+A^2/(2f)), and
+(beta fle H_*,O(c_H)to0). This proves every limit in
(eq:graph-cutoff-small). To verify the extensions at the zero
-set, write \(\chi=\zeta^2\), where \(\zeta\) is smooth and flat there.
-On \(\{\zeta>0\}\),
-\[
+set, write (chi=zeta^2), where (zeta) is smooth and flat there.
+On ({zeta>0}),
+[
(|df|_h^2)/(f)
- =4\widetilde f\,|d\zeta|_h^2
- +4\zeta< d\zeta,d\widetilde f>_h
- +\zeta^2(|d\widetilde f|_h^2)/(\widetilde f).
-\]
-The right side extends smoothly and flatly across \(\{\zeta=0\}\),
-since \(\widetilde f>0\). The same is therefore true of
-\[
+ =4widetilde f,|dzeta|_h^2
+ +4zeta< dzeta,dwidetilde f>_h
+ +zeta^2(|dwidetilde f|_h^2)/(widetilde f).
+]
+The right side extends smoothly and flatly across ({zeta=0}),
+since (widetilde f>0). The same is therefore true of
+[
(A^2)/(f)
- =(N^2)/(1+N^2|df|_h^2)\,(|df|_h^2)/(f).
-\]
-Moreover, \(F_i\) vanishes on a neighborhood of the zero set of
-\(f\). Thus \(ND^0\) extends smoothly by zero there, and all jets
-of \(f=\zeta^2\widetilde f\) vanish on its zero set.
+ =(N^2)/(1+N^2|df|_h^2),(|df|_h^2)/(f).
+]
+Moreover, (F_i) vanishes on a neighborhood of the zero set of
+(f). Thus (ND^0) extends smoothly by zero there, and all jets
+of (f=zeta^2widetilde f) vanish on its zero set.
On the uncut region the last term
-\(\varpi=ε a\delta^2/f\) in
+(varpi=ε adelta^2/f) in
(eq:graph-core-prescription) has
- \|\varpi\|_∞
- \le\delta√(ε a(ε a+H_*))→0,
- \qquad
- -2NU(\varpi)
- =(2ε a\delta^2 L A^2)/(f^2)\ge0 .
+ |varpi|_∞
+ ledelta√(ε a(ε a+H_*))→0,
+ qquad
+ -2NU(varpi)
+ =(2ε adelta^2 L A^2)/(f^2)ge0 .
-Since \(ND^0-\beta f\ge-(ε a+\beta)f\), its contribution
+Since (ND^0-beta fge-(ε a+beta)f), its contribution
inside the curvature square satisfies
- (ND^0-\beta f+\varpi)^2
- \ge (ND^0-\beta f)^2
- -2ε a\delta^2(ε a+H_*).
+ (ND^0-beta f+varpi)^2
+ ge (ND^0-beta f)^2
+ -2ε adelta^2(ε a+H_*).
The last loss tends to zero. These estimates on the core and
(eq:graph-cutoff-small) on the transition prove the
-claimed bound for \(err\). The curvature estimates
+claimed bound for (err). The curvature estimates
use the derivative sign in (eq:graph-floor-derivative)
and the full transition estimates, as well as that bound on
-the value of \(err\).
+the value of (err).
-Every construction has uniform bounds for fixed \(H_*\).
-For Borel data, the distance to the open nonzero set of \(F_i\)
+Every construction has uniform bounds for fixed (H_*).
+For Borel data, the distance to the open nonzero set of (F_i)
is Borel: using the enumerated charts it is the infimum of
lengths of a countable dense collection of chart paths ending
-where \(F_i\ne0\). The normalized smoothing kernels and all
+where (F_ine0). The normalized smoothing kernels and all
subsequent scalar operations are Borel and equivariant.
Lemma~lem:graph-prescription supplies these same
properties for the complete solution. This proves the final
@@ -1524,219 +1524,219 @@ probability map. The order of the operations matters: a graph
enlarges the metric available to all subsequent coordinates. Use finitely many colors so that graphs in the same color have
disjoint supports and can be performed simultaneously.
For a root probability map, its energy is
-\[
- e_h(F)=|dF|_{h,\HS}^2=\sum_i|dF_i|_h^2.
-\]
+[
+ e_h(F)=|dF|_{h,HS}^2=sum_i|dF_i|_h^2.
+]
Fix a smooth nonnegative nondecreasing function
-\(\Psi:[0,∞)\to[0,∞)\) such that
+(Psi:[0,∞)to[0,∞)) such that
- \Psi(s)=0\quad(s\le8),\qquad
- \Psi(s)\le Cs^2,\qquad |\Psi'(s)|^2\le C\Psi(s),\qquad
- \Psi(s)\ge cs^2\quad(s\ge R_\Psi),
+ Psi(s)=0quad(sle8),qquad
+ Psi(s)le Cs^2,qquad |Psi'(s)|^2le CPsi(s),qquad
+ Psi(s)ge cs^2quad(sge R_Psi),
-where \(R_\Psi\) and the constants are numerical. Such a
+where (R_Psi) and the constants are numerical. Such a
function can be obtained as the square of a smooth
-nondecreasing function that is zero up to \(8\), has bounded
-derivative, and equals \(s-8\) from a fixed larger value on.
-Use either \(\psi=0\), or \(\psi=\kappa\Psi\) for a small
-constant \(\kappa>0\). In the latter case,
-\[
- \psi\le C\kappa s^2,\qquad
- |\psi'|^2\le C\kappa\psi,\qquad
- \psi(s)\ge c\kappa s^2\quad(s\ge R_\Psi).
-\]
+nondecreasing function that is zero up to (8), has bounded
+derivative, and equals (s-8) from a fixed larger value on.
+Use either (psi=0), or (psi=kappaPsi) for a small
+constant (kappa>0). In the latter case,
+[
+ psile Ckappa s^2,qquad
+ |psi'|^2le Ckappapsi,qquad
+ psi(s)ge ckappa s^2quad(sge R_Psi).
+]
Notation for an ordered sweep.:
Introduce the pointwise quantities for any candidate sequence of
graphs before asserting that the sequence can be constructed. The
pass parameters are
- ε>0,\qquad N^2=lε,\qquad
- a=\tfrac12+\theta,\qquad \lambda=\tfrac12-2\theta,\qquad
- 0<l\le1,\quad l\le\theta\le\tfrac1{16},
+ ε>0,qquad N^2=lε,qquad
+ a=tfrac12+theta,qquad lambda=tfrac12-2theta,qquad
+ 0<lle1,quad llethetaletfrac1{16},
-with \(N>0\). On one leaf, start with \((h_0,\phi_0,F)\), where
-\(F_i\ge0\) and \(\sum_iF_i^2=1\). Suppose nonnegative smooth
-bumps \(f_i\) form a locally finite family ordered in finitely many
+with (N>0). On one leaf, start with ((h_0,phi_0,F)), where
+(F_ige0) and (sum_iF_i^2=1). Suppose nonnegative smooth
+bumps (f_i) form a locally finite family ordered in finitely many
colors, with disjoint supports within each color, and that
-\(F_i\ne0\) implies \(f_i>0\). Attach also nonnegative functions
-\(\beta_i\). At this stage these are candidate data, with no graph
+(F_ine0) implies (f_i>0). Attach also nonnegative functions
+(beta_i). At this stage these are candidate data, with no graph
equation imposed.
-Apply in color order the graph of height \(N\) times the sum of the
-bumps in that color. On the support of \(f_i\), write
-\(h_{i-1},\phi_{i-1}\) for the pair before its color and
-\(h_i,\phi_i\) for the pair afterward; write \(j<i\) for an earlier
-color. Let \(L_i,U_i,D_i,\II_i,H_i\) be the individual graph
-quantities for \(f_i\), and let \(h_+,\phi_+\) be the final pair.
-The subscript \(p\) denotes the current pass in \(W_p\) and in the
-reserve \(\Pi_p\) below.
+Apply in color order the graph of height (N) times the sum of the
+bumps in that color. On the support of (f_i), write
+(h_{i-1},phi_{i-1}) for the pair before its color and
+(h_i,phi_i) for the pair afterward; write (j<i) for an earlier
+color. Let (L_i,U_i,D_i,II_i,H_i) be the individual graph
+quantities for (f_i), and let (h_+,phi_+) be the final pair.
+The subscript (p) denotes the current pass in (W_p) and in the
+reserve (Pi_p) below.
The graph identities give the output and its normalization:
- h_+=h_0+N^2\sum_i df_i ⊗ df_i,\qquad
- W_p:=\sum_i\log L_i=\log(dV_{h_+}/dV_{h_0})\ge0,\\
- \phi_+=\phi_0-W_p,\qquad
- e^{\phi_+}dV_{h_+}=e^{\phi_0}dV_{h_0},\qquad
- r:=\sum_i f_i^2>0,\qquad Y_i=f_i/\sqrt r .
+ h_+=h_0+N^2sum_i df_i ⊗ df_i,qquad
+ W_p:=sum_ilog L_i=log(dV_{h_+}/dV_{h_0})ge0,
+ phi_+=phi_0-W_p,qquad
+ e^{phi_+}dV_{h_+}=e^{phi_0}dV_{h_0},qquad
+ r:=sum_i f_i^2>0,qquad Y_i=f_i/sqrt r .
The sums are pointwise finite, with at most one nonzero bump per
-color. The implication \(F_i\ne0⇒ f_i>0\) ensures
-\(r>0\). In all the quantities below, indices are restricted to
-\(f_i>0\).
+color. The implication (F_ine0⇒ f_i>0) ensures
+(r>0). In all the quantities below, indices are restricted to
+(f_i>0).
For each such index, use the longitudinal notation of the
-one-coordinate calculation, with \(h=h_{i-1}\),
-\(h'=h_i\), and \(h_{in}=h_0\):
-\[
- A_i=|U_i|_{h_{i-1}},\qquad U_i=A_i e_i,\qquad
- x_i=-\II_i(\tau_i,\tau_i),\qquad X_i=Nx_i,\qquad
+one-coordinate calculation, with (h=h_{i-1}),
+(h'=h_i), and (h_{in}=h_0):
+[
+ A_i=|U_i|_{h_{i-1}},qquad U_i=A_i e_i,qquad
+ x_i=-II_i(tau_i,tau_i),qquad X_i=Nx_i,qquad
t_i=A_i^2/f_i.
-\]
-Here \(\tau_i=(L_i^{-1}e_i,A_i)\) and \(x_i=e_i(A_i)\) when
-\(A_i>0\). At \(A_i=0\) use any unit direction as in
-(eq:graph-longitudinal). In the graph metric \(h_i\), define
+]
+Here (tau_i=(L_i^{-1}e_i,A_i)) and (x_i=e_i(A_i)) when
+(A_i>0). At (A_i=0) use any unit direction as in
+(eq:graph-longitudinal). In the graph metric (h_i), define
- \II_i^{rem}
- :=\II_i+x_i\,\tau_i^\flat ⊗ \tau_i^\flat,\\
- |\II_i|^2=x_i^2+|\II_i^{rem}|^2,\qquad
- \tr\II_i^{rem}=H_i+x_i .
+ II_i^{rem}
+ :=II_i+x_i,tau_i^flat ⊗ tau_i^flat,
+ |II_i|^2=x_i^2+|II_i^{rem}|^2,qquad
+ trII_i^{rem}=H_i+x_i .
-In particular, at \(A_i=0\) the combination
-\((X_i-t_i)^2+N^2|\II_i^{rem}|^2=N^2|\II_i|^2\)
+In particular, at (A_i=0) the combination
+((X_i-t_i)^2+N^2|II_i^{rem}|^2=N^2|II_i|^2)
is independent of the chosen direction. The losses from comparing
the current graph with the input and final metrics are
-\[
- B_i=1-|e_i|_{h_0}^2,\qquad
+[
+ B_i=1-|e_i|_{h_0}^2,qquad
C_i=
- 1-N^2|df_i|_{h_+}^2/A_i^2,&A_i>0,\\
+ 1-N^2|df_i|_{h_+}^2/A_i^2,&A_i>0,
0,&A_i=0.
-\]
+]
Metric monotonicity and the inverse graph formula give
-\(0\le B_i,C_i\le1\).
+(0le B_i,C_ile1).
-The normalization gives probability weights \(\nu_i=f_i^2/r\).
-Write \(\E_\nu b_i=\sum_i\nu_i b_i\). The mean and centered
+The normalization gives probability weights (nu_i=f_i^2/r).
+Write (E_nu b_i=sum_inu_i b_i). The mean and centered
quantities used in the estimate are
- m=\sum_iA_i^2,\qquad
- u_i=(A_i^2)/(ε\nu_i),\qquad
- \bar u=\E_\nu u_i=\frac mε,\\
- \sigma_i=(F_i^2)/(\nu_i),\qquad \E_\nu\sigma_i=1,\\
- M_0=a(1-r)-\bar u/2,\qquad
- V_i=a(\sigma_i-1)+\lambda(u_i-\bar u),\qquad
- \Var_\nu u=\E_\nu(u_i-\bar u)^2 .
+ m=sum_iA_i^2,qquad
+ u_i=(A_i^2)/(εnu_i),qquad
+ barbar u=u=E_nu u_i=frac mε,
+ sigma_i=(F_i^2)/(nu_i),qquad E_nusigma_i=1,
+ M_0=a(1-r)-barbar u/2,u/2,qquad
+ V_i=a(sigma_i-1)+lambda(u_i-barbar u),u),qquad
+ Var_nu u=E_nu(u_i-barbar u)^2u)^2 .
-The identity for \(\E_\nu\sigma_i\) uses
-\(F_i\ne0⇒ f_i>0\); the individual ratios \(\sigma_i\)
+The identity for (E_nusigma_i) uses
+(F_ine0⇒ f_i>0); the individual ratios (sigma_i)
need not be at most one.
Finally, define two nonnegative quantities:
- T_0={}&\sum_i A_i^2[
- C_i+4a^2\sigma_iB_i+(L_i-1)\{2a(\sigma_i+r)+u_i\}],
- \\
- \mathcal X={}&\sum_i[
- (X_i-t_i)^2+N^2|\II_i^{rem}|^2
- +N^2|d\log L_i|_{h_i}^2+\beta_i^2f_i^2].
+ T_0={}&sum_i A_i^2[
+ C_i+4a^2sigma_iB_i+(L_i-1){2a(sigma_i+r)+u_i}],
+
+ mathcal X=calX={}&sum_i[
+ (X_i-t_i)^2+N^2|II_i^{rem}|^2
+ +N^2|dlog L_i|_{h_i}^2+beta_i^2f_i^2].
-The quantity \(T_0\) records the gains from the metric comparisons
+The quantity (T_0) records the gains from the metric comparisons
and from each graph's stretching. Its full coefficient will yield
-the sharp quadratic term \(m^2/n\) in the entropy iteration.
-The quantity \(\mathcal X\) retains the derivative squares and the
+the sharp quadratic term (m^2/n) in the entropy iteration.
+The quantity (mathcal XcalX) retains the derivative squares and the
applied coefficients. Together with the mean and variance squares
below, it controls graph curvature; those moment squares also control
the normalization.
[One sweep]
-There is a numerical \(c>0\) with the following property.
-For a sweep without boost, set \(\psi=0\). For a boosted sweep,
-given \(ε_*>0\), there are constants
-\(\kappa>0\), \(c_*>0\), and \(C_*<∞\), depending only on
-\(ε_*\), for which one may use \(\psi=\kappa\Psi\) whenever
-\(ε\geε_*\).
-
-Let \(h_0,\phi_0,F\) be a smooth equivariant Borel family on
-\(\widetilde M\timesΩ\), with bounds uniform in the lifted
-charts for the instance. Suppose \(h_0\) is complete, its
-geometry and the positive-order derivatives of \(\phi_0\) have
+There is a numerical (c>0) with the following property.
+For a sweep without boost, set (psi=0). For a boosted sweep,
+given (ε_*>0), there are constants
+(kappa>0), (c_*>0), and (C_*<∞), depending only on
+(ε_*), for which one may use (psi=kappaPsi) whenever
+(εgeε_*).
+
+Let (h_0,phi_0,F) be a smooth equivariant Borel family on
+(widetilde MtimesΩ), with bounds uniform in the lifted
+charts for the instance. Suppose (h_0) is complete, its
+geometry and the positive-order derivatives of (phi_0) have
the uniform bounds above, and
-\[
- F=(F_i)_{i\in\mathcal L},\qquad F_i\ge0,\qquad
- \sum_iF_i^2=1
-\]
+[
+ F=(F_i)_{iinmathcal L}calL,qquad F_ige0,qquad
+ sum_iF_i^2=1
+]
has finitely many free label orbits and uniformly locally finite
compact coordinate supports. For each orbit representative,
suppose all its transverse supports lie in a fixed compact
-subset of \(\widetilde M\). Let \(h_{ref}\le h_0\)
+subset of (widetilde M). Let (h_{ref}le h_0)
be a complete equivariant Borel leafwise metric whose coefficients,
inverse, and all coordinate derivatives have uniform bounds in the
-same lifted buffered charts as \(h_0\). Suppose a Borel
+same lifted buffered charts as (h_0). Suppose a Borel
equivariant coloring
-\(\chi:\mathcal L\timesΩ\to\{1,...,k\}\), \(k<∞\),
-has been chosen such that, for each retained \(\omega\) and distinct
-labels \(i,j\) with \(\chi(i,\omega)=\chi(j,\omega)\), the
-\(R_{loc}\)-neighborhoods in \(h_{ref}(\omega)\)
-of \(\supp F_i(\cdot,\omega)\) and \(\supp F_j(\cdot,\omega)\)
+(chi:mathcal LcalLtimesΩto{1,...,k}), (k<∞),
+has been chosen such that, for each retained (omega) and distinct
+labels (i,j) with (chi(i,omega)=chi(j,omega)), the
+(R_{loc})-neighborhoods in (h_{ref}(omega))
+of (supp F_i(cdot,omega)) and (supp F_j(cdot,omega))
are disjoint. Fix the pass parameters in
-(eq:sweep-parameters), with \(ε\geε_*\) in
+(eq:sweep-parameters), with (εgeε_*) in
the boosted case.
-For any \(\delta_0,\eta_0>0\), there is a candidate sweep as above
-with smooth nonnegative bumps \(f_i\le2\), supported in the
-indicated enlargements, positive on the supports of \(F_i\), and
+For any (delta_0,eta_0>0), there is a candidate sweep as above
+with smooth nonnegative bumps (f_ile2), supported in the
+indicated enlargements, positive on the supports of (F_i), and
flat on their zero sets. Its output is given by
-(eq:sweep-output). The lower bound for \(r\) is uniform
+(eq:sweep-output). The lower bound for (r) is uniform
for the instance. The bumps, coefficients, all intermediate and
final graph metrics and weights, their smooth graph quantities, and
-\(Y\) are equivariant and Borel and retain the corresponding uniform
+(Y) are equivariant and Borel and retain the corresponding uniform
chart bounds, with the positive-order convention for weights.
-The map \(Y\) is again a
+The map (Y) is again a
root probability map with the same label orbits.
-For each \(i\), the coefficient furnished by
+For each (i), the coefficient furnished by
Lemma~lem:graph-localization has the form
- \beta_i=\beta_{s,i}+ε\psi(R_i),\qquad
- R_i=\sum_{j<i}f_j^2 ,
+ beta_i=beta_{s,i}+εpsi(R_i),qquad
+ R_i=sum_{j<i}f_j^2 ,
-Here \(\beta_{s,i}\) vanishes near \(\supp F_i\), and the coefficients
-satisfy (eq:graph-spatial-cutoff) in \(h_{i-1}\).
+Here (beta_{s,i}) vanishes near (supp F_i), and the coefficients
+satisfy (eq:graph-spatial-cutoff) in (h_{i-1}).
Writing
-\[
+[
ND_i^0=ε a(F_i^2/f_i-f_i)+A_i^2/(2f_i)
-\]
-on \(\{f_i>0\}\), extended by zero off the bump, one has
+]
+on ({f_i>0}), extended by zero off the bump, one has
- ND_i=ND_i^0-\beta_i f_i+err_i,\qquad
- \sum_i|err_i|^2\le\eta_0 .
+ ND_i=ND_i^0-beta_i f_i+err_i,qquad
+ sum_i|err_i|^2leeta_0 .
Define the nonnegative reserve
- \Pi_p:={}&
- T_0+cε(M_0^2+\E_\nu V_i^2+\theta\Var_\nu u)
- \\
- &+c\frac rε\mathcal X
- +c_*ε r^6\indic_{\{r\ge C_*\}},
+ Pi_p:={}&
+ T_0+cε(M_0^2+E_nu V_i^2+thetaVar_nu u)
+
+ &+cfrac rεmathcal XcalX
+ +c_*ε r^6indic_{{rge C_*}},
@@ -1744,24 +1744,24 @@ where the final term is omitted in a sweep without boost. Then,
pointwise,
Q:=(rN^2)/(ε)[
- &\wscal(h_+,\phi_+)-\wscal(h_0,\phi_0)
+ &wscal(h_+,phi_+)-wscal(h_0,phi_0)
+ε e_{h_0}(F)-ε e_{h_+}(Y)]
- \notag\\
- &\ge\Pi_p-4\theta^2m-\frac rε\delta_0 .
+ notag
+ &gePi_p-4theta^2m-frac rεdelta_0 .
-The numerical constant \(c\) is independent of the instance, the
+The numerical constant (c) is independent of the instance, the
number of colors, and the pass parameters. The errors
-\(\delta_0,\eta_0\) can be prescribed independently before the pass.
+(delta_0,eta_0) can be prescribed independently before the pass.
Proof.
Process the colors in their prescribed order. At a label in
-the current color, the sum \(R_i\) of earlier bumps and all its
+the current color, the sum (R_i) of earlier bumps and all its
derivatives have common bounds for that color, so the function
-\(\alpha_i=ε\psi(R_i)\) is fixed and uniformly smooth.
+(alpha_i=εpsi(R_i)) is fixed and uniformly smooth.
Apply Lemma~lem:graph-localization with this
-\(\alpha_i\), using a common sufficiently large \(H_*\) for
+(alpha_i), using a common sufficiently large (H_*) for
the color. Uniqueness of the complete solutions and the
equivariant cutoff construction make all choices equivariant.
Their Borel dependence follows from the same lemma. Bumps in
@@ -1772,49 +1772,49 @@ bounds. The reference metric is unchanged during the pass, so the
simultaneous chart control required by
Lemma~lem:graph-localization persists for the next color.
-At any point some \(F_i\) is nonzero, and its cutoff is one
+At any point some (F_i) is nonzero, and its cutoff is one
there. The lower barrier for that color gives
-\(f_i\ge c_{\min}>0\). Taking the minimum over the finitely
+(f_ige c_{min}>0). Taking the minimum over the finitely
many colors gives the stated instance-uniform lower bound for
-\(r\); also \(r\le4k\). Thus \(Y\) is smooth with uniform
-bounds and \(\sum_iY_i^2=1\). The graph determinant and
+(r); also (rle4k). Thus (Y) is smooth with uniform
+bounds and (sum_iY_i^2=1). The graph determinant and
density identities prove (eq:sweep-output).
By making the error in each color small enough that its square
-is at most \(\eta_0/k\), the last conclusion of
+is at most (eta_0/k), the last conclusion of
Lemma~lem:graph-localization gives
(eq:sweep-prescription-error).
Prove the curvature estimate in the units before
-multiplication by \(r/ε\). Write
-\[
- G=N^2[\wscal(h_+,\phi_+)-\wscal(h_0,\phi_0)
+multiplication by (r/ε). Write
+[
+ G=N^2[wscal(h_+,phi_+)-wscal(h_0,phi_0)
+ε e_{h_0}(F)-ε e_{h_+}(Y)],
- \qquad Z_i=ND_i^0,
-\]
-so \(Q=(r/ε)G\). For each \(i\), the one-coordinate calculation
-applies with \(h=h_{i-1}\), \(h'=h_i\), and
-\(h_{in}=h_0\). First account for the difference between
-the ideal target \(Z_i\) for \(ND_i\) and the realized prescription.
+ qquad Z_i=ND_i^0,
+]
+so (Q=(r/ε)G). For each (i), the one-coordinate calculation
+applies with (h=h_{i-1}), (h'=h_i), and
+(h_{in}=h_0). First account for the difference between
+the ideal target (Z_i) for (ND_i) and the realized prescription.
For the moment omit the floor in
-(eq:graph-core-prescription). Replacing \(Z_i\) by
-\(Z_i-\beta_i f_i\) in the first two terms of
-(eq:graph-identity), after multiplying by \(N^2\),
+(eq:graph-core-prescription). Replacing (Z_i) by
+(Z_i-beta_i f_i) in the first two terms of
+(eq:graph-identity), after multiplying by (N^2),
adds exactly
- &(Z_i-\beta_i f_i)^2-2NU_i(Z_i-\beta_i f_i)
- -Z_i^2+2NU_i(Z_i)\\
- &\hspace{1cm}=\beta_i^2f_i^2
- +2ε a\beta_i(f_i^2-F_i^2)
- +(2L_i-1)\beta_iA_i^2
- +2NA_if_i e_i(\beta_i).
+ &(Z_i-beta_i f_i)^2-2NU_i(Z_i-beta_i f_i)
+ -Z_i^2+2NU_i(Z_i)
+ &hspace{1cm}=beta_i^2f_i^2
+ +2ε abeta_i(f_i^2-F_i^2)
+ +(2L_i-1)beta_iA_i^2
+ +2NA_if_i e_i(beta_i).
-Since \(L_i\ge1\), this is at least
+Since (L_ige1), this is at least
- \mathcal B_i:=
- \beta_i^2f_i^2+2ε a\beta_i f_i^2+\beta_i A_i^2
- -2ε a\beta_iF_i^2
- -2NA_if_i|d\beta_i|_{h_{i-1}}.
+ mathcal B_i:=calB_i:=
+ beta_i^2f_i^2+2ε abeta_i f_i^2+beta_i A_i^2
+ -2ε abeta_iF_i^2
+ -2NA_if_i|dbeta_i|_{h_{i-1}}.
@@ -1822,28 +1822,28 @@ On the uncut region the actual floor has nonnegative
differentiated contribution by
(eq:graph-floor-derivative), and its square loses at
most the quantity in (eq:graph-floor-square).
-On a cutoff transition \(F_i=0\) and \(d\beta_i=0\).
+On a cutoff transition (F_i=0) and (dbeta_i=0).
There (eq:graph-cutoff-small) makes the actual scalar
change small. It also makes every term on the right of the
single-step expansion just obtained small:
-\(\beta_i A_i^2=(\beta_i f_i)t_i\),
-the longitudinal terms are controlled by \(N|\II_i|\) and
-\(t_i\), and the other terms are controlled by
-\(Z_i,\beta_i f_i,A_i\). The first term in
+(beta_i A_i^2=(beta_i f_i)t_i),
+the longitudinal terms are controlled by (N|II_i|) and
+(t_i), and the other terms are controlled by
+(Z_i,beta_i f_i,A_i). The first term in
(eq:graph-ideal-derivative) is zero. Thus this
expanded lower bound still holds on a cutoff transition with
an arbitrarily small additive error in these units. It is
also valid off the bumps, where all its terms vanish.
Choose the floor and transition errors for each color so their
-sum is at most \(\delta_0\) at every point. Telescoping
+sum is at most (delta_0) at every point. Telescoping
(eq:graph-identity) over the colors gives
- G\ge \sum_i[&
- Z_i^2-2NU_i(Z_i)+N^2|\II_i|^2
- +N^2|d\log L_i|_{h_i}^2
- +ε N^2|dF_i|_{h_0}^2+\mathcal B_i]
- \notag\\
- &-ε N^2e_{h_+}(Y)-\delta_0 .
+ Gge sum_i[&
+ Z_i^2-2NU_i(Z_i)+N^2|II_i|^2
+ +N^2|dlog L_i|_{h_i}^2
+ +ε N^2|dF_i|_{h_0}^2+mathcal B_i]calB_i]
+ notag
+ &-ε N^2e_{h_+}(Y)-delta_0 .
This argument uses the full differentiated floor and cutoff
@@ -1853,218 +1853,218 @@ estimates, rather than differentiating the value error in
The input energy is already included in
(eq:graph-ideal-bound). For the output, normalization is an
orthogonal projection in the target:
-\[
- dY=r^{-1/2}(I-Y ⊗ Y)\,df .
-\]
+[
+ dY=r^{-1/2}(I-Y ⊗ Y),df .
+]
The rank-one inverse identity and metric monotonicity give
-\[
- N^2|df_i|_{h_i}^2=A_i^2,\qquad
- N^2|df_i|_{h_+}^2=A_i^2(1-C_i)\le A_i^2 .
-\]
+[
+ N^2|df_i|_{h_i}^2=A_i^2,qquad
+ N^2|df_i|_{h_+}^2=A_i^2(1-C_i)le A_i^2 .
+]
Consequently
N^2e_{h_+}(Y)
- \le\frac1r\sum_i A_i^2(1-C_i).
+ lefrac1rsum_i A_i^2(1-C_i).
Put
-\[
- K_i=(X_i-t_i)^2+N^2|\II_i^{rem}|^2
- +N^2|d\log L_i|_{h_i}^2,\qquad
- g_i=a(\sigma_i-r)+u_i/2 .
-\]
+[
+ K_i=(X_i-t_i)^2+N^2|II_i^{rem}|^2
+ +N^2|dlog L_i|_{h_i}^2,qquad
+ g_i=a(sigma_i-r)+u_i/2 .
+]
The definitions imply
-\[
- Z_i=(ε f_i)/(r)g_i,\quad
- \frac rε\sum_iZ_i^2=ε\E_\nu g_i^2,\quad
- \frac rε(L_i-1)t_i^2=(L_i-1)A_i^2u_i .
-\]
+[
+ Z_i=(ε f_i)/(r)g_i,quad
+ frac rεsum_iZ_i^2=εE_nu g_i^2,quad
+ frac rε(L_i-1)t_i^2=(L_i-1)A_i^2u_i .
+]
Use these identities, (eq:graph-ideal-bound), and
(eq:sweep-output-energy) in (eq:sweep-raw).
-After multiplication by \(r/ε\), the result is
+After multiplication by (r/ε), the result is
- Q\ge{}&T_0+\frac rε\sum_iK_i
- +ε\E_\nu g_i^2
- +\sum_i A_i^2[(2a-4a^2)\sigma_i+2ar-1]
- +\frac rε(\sum_i\mathcal B_i-\delta_0).
+ Qge{}&T_0+frac rεsum_iK_i
+ +εE_nu g_i^2
+ +sum_i A_i^2[(2a-4a^2)sigma_i+2ar-1]
+ +frac rε(sum_imathcal B_i-calB_i-delta_0).
-Indeed \(rF_i^2/f_i^2=\sigma_i\). The terms with \(B_i\) and
-\(L_i-1\) in (eq:graph-ideal-bound) become the corresponding
-terms of \(T_0\), while the output contributes
-\(-A_i^2+A_i^2C_i\). The remaining height terms give the displayed
+Indeed (rF_i^2/f_i^2=sigma_i). The terms with (B_i) and
+(L_i-1) in (eq:graph-ideal-bound) become the corresponding
+terms of (T_0), while the output contributes
+(-A_i^2+A_i^2C_i). The remaining height terms give the displayed
quadratic expression.
The remaining quadratic expression has an exact mean and
-variance decomposition. Use \(\E_\nu\sigma_i=1\) and
-\(\E_\nu u_i=\bar u\) from (eq:sweep-statistics), and write
-\(w_i=u_i-\bar u\). Dividing the quadratic expression in
-(eq:sweep-expanded) by \(ε\), its contribution
+variance decomposition. Use (E_nusigma_i=1) and
+(E_nu u_i=barbar uu) from (eq:sweep-statistics), and write
+(w_i=u_i-barbar uu). Dividing the quadratic expression in
+(eq:sweep-expanded) by (ε), its contribution
from the means is
- [a(1-r)+\bar u/2]^2+\bar u(2a-4a^2+2ar-1)
- &= [a(1-r)-\bar u/2]^2-(2a-1)^2\bar u\\
- &=M_0^2-4\theta^2\bar u .
+ [a(1-r)+barbar u/2]^2+u/2]^2+barbar u(2a-4a^2+2ar-1)u(2a-4a^2+2ar-1)
+ &= [a(1-r)-barbar u/2]^2-(2a-1)^2u/2]^2-(2a-1)^2barbar uu
+ &=M_0^2-4theta^2barbar uu .
The coefficient of
-\(\E_\nu[w_i(\sigma_i-1)]\) in the centered expression is
-\[
- a+(2a-4a^2)=3a-4a^2=2a\lambda .
-\]
+(E_nu[w_i(sigma_i-1)]) in the centered expression is
+[
+ a+(2a-4a^2)=3a-4a^2=2alambda .
+]
Completing its square therefore gives the identity
- ε\E_\nu g_i^2
- &+\sum_i A_i^2[(2a-4a^2)\sigma_i+2ar-1] \notag\\
+ εE_nu g_i^2
+ &+sum_i A_i^2[(2a-4a^2)sigma_i+2ar-1] notag
&=
- ε(M_0^2+\E_\nu V_i^2+
- (\tfrac14-\lambda^2)\Var_\nu u)
- -4\theta^2m .
+ ε(M_0^2+E_nu V_i^2+
+ (tfrac14-lambda^2)Var_nu u)
+ -4theta^2m .
Here
-\(\tfrac14-\lambda^2=2\theta-4\theta^2\ge7\theta/4\).
+(tfrac14-lambda^2=2theta-4theta^2ge7theta/4).
This supplies a reserve for all three squares in the
proposition. Finish by paying the terms in
-\(\mathcal B_i\) from fixed fractions of this reserve and of
-the nonnegative \(\beta_i\) terms.
-
-Write \(s_i=\beta_{s,i}\) and \(b_i=ε\psi(R_i)\), so
-\(\beta_i=s_i+b_i\) with both summands nonnegative.
-In particular \(\beta_i^2\ge s_i^2+b_i^2\).
-The negative \(F_i^2\) term in
-(eq:sweep-beta-correction) has no \(s_i\) part, because
-\(s_i\) vanishes near the support of \(F_i\).
+(mathcal B_icalB_i) from fixed fractions of this reserve and of
+the nonnegative (beta_i) terms.
+
+Write (s_i=beta_{s,i}) and (b_i=εpsi(R_i)), so
+(beta_i=s_i+b_i) with both summands nonnegative.
+In particular (beta_i^2ge s_i^2+b_i^2).
+The negative (F_i^2) term in
+(eq:sweep-beta-correction) has no (s_i) part, because
+(s_i) vanishes near the support of (F_i).
The spatial derivative term satisfies
2NA_if_i|ds_i|
- &\le2c_0 A_if_i(\thetaε^{3/2}+s_i^{3/2}) \notag\\
- &\le c_0(s_i^2f_i^2+s_iA_i^2)
- +Cc_0\theta(ε^2f_i^2+A_i^4/f_i^2).
+ &le2c_0 A_if_i(thetaε^{3/2}+s_i^{3/2}) notag
+ &le c_0(s_i^2f_i^2+s_iA_i^2)
+ +Cc_0theta(ε^2f_i^2+A_i^4/f_i^2).
-For the first pair of terms use \(2xy\le x^2+y^2\).
-For the other pair, divide by \(ε^2f_i^2\) and use
-\(t\le1+t^4\) with \(t=A_i/(\sqrtε f_i)\).
-This derivative cost occurs only in rows with \(F_i=0\), hence
-\(\sigma_i=0\). In those rows,
-\[
- V_i=-a+\lambda w_i,\qquad
- ar+\bar u/2=a-M_0=\lambda w_i-V_i-M_0 .
-\]
-Since \(a\ge1/2\) and \(|\lambda|\le1/2\), these equations give
-\[
- r+u_i\le2(|M_0|+|V_i|+|w_i|),\qquad
- r^2+u_i^2\le12(M_0^2+V_i^2+w_i^2).
-\]
-After multiplication by \(r/ε\), the last cost in
+For the first pair of terms use (2xyle x^2+y^2).
+For the other pair, divide by (ε^2f_i^2) and use
+(tle1+t^4) with (t=A_i/(sqrtε f_i)).
+This derivative cost occurs only in rows with (F_i=0), hence
+(sigma_i=0). In those rows,
+[
+ V_i=-a+lambda w_i,qquad
+ ar+barbar u/2=a-M_0=u/2=a-M_0=lambda w_i-V_i-M_0 .
+]
+Since (age1/2) and (|lambda|le1/2), these equations give
+[
+ r+u_ile2(|M_0|+|V_i|+|w_i|),qquad
+ r^2+u_i^2le12(M_0^2+V_i^2+w_i^2).
+]
+After multiplication by (r/ε), the last cost in
(eq:sweep-spatial-cost), summed over these rows, is
-\[
- Cc_0ε\theta
- \sum_{\{\sigma_i=0\}}\nu_i(r^2+u_i^2).
-\]
+[
+ Cc_0εtheta
+ sum_{{sigma_i=0}}nu_i(r^2+u_i^2).
+]
The preceding bound shows that it is a small fraction of the
-three squares in (eq:sweep-mean-identity) when \(c_0\)
+three squares in (eq:sweep-mean-identity) when (c_0)
is a sufficiently small numerical constant. The first cost
in (eq:sweep-spatial-cost) is paid by a small fraction
-of \(\beta_i^2f_i^2+\beta_iA_i^2\).
-This proves the required absorption when \(\psi=0\), leaving
+of (beta_i^2f_i^2+beta_iA_i^2).
+This proves the required absorption when (psi=0), leaving
a fixed fraction of every asserted square.
-Suppose now that \(\psi=\kappa\Psi\) and
-\(ε\geε_*\). Metric addition over earlier
+Suppose now that (psi=kappaPsi) and
+(εgeε_*). Metric addition over earlier
colors implies, as quadratic forms,
-\[
- h_{i-1}\ge N^2\sum_{j<i}df_j ⊗ df_j .
-\]
-For a unit vector in \(h_{i-1}\), the vector
-\((Ndf_j)_{j<i}\) consequently has Euclidean norm at most one.
+[
+ h_{i-1}ge N^2sum_{j<i}df_j ⊗ df_j .
+]
+For a unit vector in (h_{i-1}), the vector
+((Ndf_j)_{j<i}) consequently has Euclidean norm at most one.
Cauchy--Schwarz applied to
-\(dR_i=2\sum_{j<i}f_jdf_j\) proves
+(dR_i=2sum_{j<i}f_jdf_j) proves
- |dR_i|_{h_{i-1}}\le2√(R_i)/N .
+ |dR_i|_{h_{i-1}}le2√(R_i)/N .
-For any fixed small \(\zeta>0\), the derivative bound in
+For any fixed small (zeta>0), the derivative bound in
(eq:sweep-boost-profile) and Young's inequality give
-\[
+[
2NA_if_i|db_i|
- \le4ε A_if_i√(R_i)\,|\psi'(R_i)|
- \le\zetaε\psi(R_i)A_i^2
- +C_\zetaε\kappa f_i^2R_i .
-\]
-The first term is a small fraction of \(\beta_iA_i^2\).
+ le4ε A_if_i√(R_i),|psi'(R_i)|
+ lezetaεpsi(R_i)A_i^2
+ +C_zetaεkappa f_i^2R_i .
+]
+The first term is a small fraction of (beta_iA_i^2).
For the other costs, use
-\[
- \sum_i f_i^2R_i
- =\tfrac12(r^2-\sum_i f_i^4)\le r^2/2,
- \qquad
- \sum_i\psi(R_i)F_i^2\le C\kappa r^2\sum_iF_i^2=C\kappa r^2.
-\]
-The profile and its derivative vanish up to \(8\). Thus the
-sum of all remaining boost costs in the units of \(G\) is at
+[
+ sum_i f_i^2R_i
+ =tfrac12(r^2-sum_i f_i^4)le r^2/2,
+ qquad
+ sum_ipsi(R_i)F_i^2le Ckappa r^2sum_iF_i^2=Ckappa r^2.
+]
+The profile and its derivative vanish up to (8). Thus the
+sum of all remaining boost costs in the units of (G) is at
most
- C(ε+ε^2)\kappa r^2\indic_{\{r\ge8\}}.
+ C(ε+ε^2)kappa r^2indic_{{rge8}}.
-Two positive terms cover this cost. First, for \(r\ge8\),
-\[
- -M_0=a(r-1)+\bar u/2\ge7r/16,
- \qquad
- (ε^2)/(r)M_0^2\ge cε^2r .
-\]
+Two positive terms cover this cost. First, for (rge8),
+[
+ -M_0=a(r-1)+barbar u/2u/2ge7r/16,
+ qquad
+ (ε^2)/(r)M_0^2ge cε^2r .
+]
The latter is the mean square in
-(eq:sweep-mean-identity) expressed in the units of \(G\).
-Second, the increments \(f_i^2\) in \(R_i\) are at most \(4\).
-For \(r\) beyond a fixed threshold, the rows with
-\(R_i\ge r/2\) carry total increment at least
-\(r/2-4\ge r/4\). In fact the first prefix crossing \(r/2\)
-overshoots it by at most \(4\). Once \(r/2\ge R_\Psi\), the
+(eq:sweep-mean-identity) expressed in the units of (G).
+Second, the increments (f_i^2) in (R_i) are at most (4).
+For (r) beyond a fixed threshold, the rows with
+(R_ige r/2) carry total increment at least
+(r/2-4ge r/4). In fact the first prefix crossing (r/2)
+overshoots it by at most (4). Once (r/2ge R_Psi), the
lower profile bound on each of these rows gives
- \sum_i f_i^2 b_i^2
- =ε^2\sum_i f_i^2\psi(R_i)^2
- \ge cε^2\kappa^2 r^5
- \qquad(r\ge C_*).
+ sum_i f_i^2 b_i^2
+ =ε^2sum_i f_i^2psi(R_i)^2
+ ge cε^2kappa^2 r^5
+ qquad(rge C_*).
To make the parameter choice explicit, let
-\(K_*=C(1+ε_*^{-1})\). The cost
+(K_*=C(1+ε_*^{-1})). The cost
(eq:sweep-boost-cost) is at most
-\(K_*ε^2\kappa r^2\) on \(r\ge8\).
-Choose \(\kappa\) so small that
-\(\kappa^{-1/2}\ge C_*\) and \(K_*\sqrt\kappa\) is smaller
+(K_*ε^2kappa r^2) on (rge8).
+Choose (kappa) so small that
+(kappa^{-1/2}ge C_*) and (K_*sqrtkappa) is smaller
than the fixed fractions of the preceding two reserves
available for this purpose. If
-\(8\le r\le\kappa^{-1/2}\), this cost is at most
-\(K_*\sqrt\kappa\,ε^2r\). If
-\(r\ge\kappa^{-1/2}\), it is at most
-\(K_*\sqrt\kappa\,ε^2\kappa^2r^5\).
+(8le rlekappa^{-1/2}), this cost is at most
+(K_*sqrtkappa,ε^2r). If
+(rgekappa^{-1/2}), it is at most
+(K_*sqrtkappa,ε^2kappa^2r^5).
It is therefore covered in both ranges. Fixed fractions of
the mean square and of (eq:sweep-boost-square) remain.
One can simultaneously retain a fixed fraction of
-\(\sum_i\beta_i^2f_i^2\) for \(\mathcal X\): reserve, for
+(sum_ibeta_i^2f_i^2) for (mathcal XcalX): reserve, for
example, one half of this sum at the outset and use
-\(\beta_i^2\ge s_i^2+b_i^2\) on the other half for the
-spatial and boost estimates. Take \(c_0,\zeta,\kappa\) small
+(beta_i^2ge s_i^2+b_i^2) on the other half for the
+spatial and boost estimates. Take (c_0,zeta,kappa) small
enough for these fixed allocations.
Insert the absorptions into (eq:sweep-expanded) and
(eq:sweep-mean-identity). A fixed fraction of
-\(\sum_iK_i+\sum_i\beta_i^2f_i^2=\mathcal X\) and of the
+(sum_iK_i+sum_ibeta_i^2f_i^2=mathcal XcalX) and of the
three mean and variance squares survives. The remaining
part of (eq:sweep-boost-square), after multiplication
-by \(r/ε\), is
-\(c_*ε r^6\indic_{\{r\ge C_*\}}\), with
-\(c_*\) a positive multiple of \(\kappa^2\).
-The terms \(T_0\) and \(-4\theta^2m\) have not been spent.
-Choose the constant \(c\) in (eq:sweep-positive-reserve)
-below the fixed numerical fractions that survive for \(\mathcal X\)
+by (r/ε), is
+(c_*ε r^6indic_{{rge C_*}}), with
+(c_*) a positive multiple of (kappa^2).
+The terms (T_0) and (-4theta^2m) have not been spent.
+Choose the constant (c) in (eq:sweep-positive-reserve)
+below the fixed numerical fractions that survive for (mathcal XcalX)
and the mean and variance squares. These fractions are independent
-of \(ε_*\), because \(\kappa\) was chosen to fit the fixed
+of (ε_*), because (kappa) was chosen to fit the fixed
allocations.
This proves (eq:sweep-inequality).
@@ -2072,117 +2072,117 @@ Finally, the choices of errors impose no estimate on the
growth of analytic constants from one color to the next.
For a fixed color, the reference buffers are fixed for the
pass, and the smaller analytic boxes and all previous
-geometry bounds are fixed before its \(H_*\) is chosen.
+geometry bounds are fixed before its (H_*) is chosen.
There are finitely many colors, so the raw scalar error can
-be made at most \(\delta_0/k\) per color and the squared
-value error at most \(\eta_0/k\) per color. Later choices
-of \(H_*\) may be arbitrarily larger. The summed errors
+be made at most (delta_0/k) per color and the squared
+value error at most (eta_0/k) per color. Later choices
+of (H_*) may be arbitrarily larger. The summed errors
are therefore precisely those asserted in
(eq:sweep-prescription-error) and
(eq:sweep-inequality).
This section turns a root probability map for a metric of scalar curvature
-\(-1\) into a map with a large energy coefficient in a weighted scalar
+(-1) into a map with a large energy coefficient in a weighted scalar
inequality. The accompanying volume term is the logarithm of the actual
volume expansion from a density no larger than the original volume density.
Also make the gradient absolutely small and control all label
overlaps by one graph. The last two properties allow the energy coefficient
to be chosen larger than the combinatorial quantities used later.
-Write \(\exp^{[0]}(t)=t\) and
-\(\exp^{[j+1]}(t)=\exp(\exp^{[j]}(t))\). Call a function
-\(\mathcal B:[1,∞)\to[0,∞)\) elementary if, for some fixed
-\(H\in\N\cup\{0\}\) and \(A,B>0\),
-\[
- \mathcal B(q)\leq \exp^{[H]}(B(1+q)^A)\qquad(q\geq1).
-\]
+Write (exp^{[0]}(t)=t) and
+(exp^{[j+1]}(t)=exp(exp^{[j]}(t))). Call a function
+(mathcal B:[1,∞)calB:[1,∞)to[0,∞)) elementary if, for some fixed
+(HinNcup{0}) and (A,B>0),
+[
+ mathcal B(q)calB(q)leq exp^{[H]}(B(1+q)^A)qquad(qgeq1).
+]
When the function is prescribed in a fixed dimension, its parameters
-\(H,A,B\) are allowed to depend on that dimension.
+(H,A,B) are allowed to depend on that dimension.
[Entropy metrics]
-Let \((M^n,g)\) be a closed connected Riemannian manifold, where \(n\geq3\)
-and \(\Scal_g=-1\), and put \(\Gamma=\pi_1(M)\). Let
-\((Ω_0,\mu_0)\) be a standard Borel probability space with a
-measure-preserving \(\Gamma\)-action; the one-point space is allowed.
-Suppose \(\mathcal L_0\) is a countable set with finitely many free
-\(\Gamma\)-orbits and
-\[
- F^{in}:\widetilde M\timesΩ_0→
- \ell^2(\mathcal L_0),\qquad
- F^{in}_\ell\geq0,\qquad
- \sum_{\ell\in\mathcal L_0}(F^{in}_\ell)^2=1
-\]
+Let ((M^n,g)) be a closed connected Riemannian manifold, where (ngeq3)
+and (Scal_g=-1), and put (Gamma=pi_1(M)). Let
+((Ω_0,mu_0)) be a standard Borel probability space with a
+measure-preserving (Gamma)-action; the one-point space is allowed.
+Suppose (mathcal L_0calL_0) is a countable set with finitely many free
+(Gamma)-orbits and
+[
+ F^{in}:widetilde MtimesΩ_0→
+ ell^2(mathcal L_0),calL_0),qquad
+ F^{in}_ellgeq0,qquad
+ sum_{ellinmathcal L_0}calL_0(F^{in}_ell)^2=1
+]
is equivariant, Borel in the transverse variable, and smooth along
-\(\widetilde M\). Assume the following uniformity for this input. In
-charts lifted from a fixed finite atlas on \(M\), with fixed inner buffers,
+(widetilde M). Assume the following uniformity for this input. In
+charts lifted from a fixed finite atlas on (M), with fixed inner buffers,
the supports are locally finite with a uniform bound on their number,
and all leafwise derivatives of the coordinate functions have uniform
-bounds. For each orbit representative \(\ell\) there is a compact set
-\(K_\ell\subset\widetilde M\) such that
-\(\supp F^{in}_{\gamma\ell}(\,\cdot\,,\omega)
-\subset\gamma K_\ell\) for all \(\gamma\in\Gamma\) and almost every
-\(\omega\). The bounds and compact sets in these assumptions may depend
+bounds. For each orbit representative (ell) there is a compact set
+(K_ellsubsetwidetilde M) such that
+(supp F^{in}_{gammaell}(,cdot,,omega)
+subsetgamma K_ell) for all (gammainGamma) and almost every
+(omega). The bounds and compact sets in these assumptions may depend
on the input.
-For every prescribed elementary function \(\mathcal B\) and every finite
-number \(E_{\min}\), there are a standard Borel probability
-\(\Gamma\)-space \((Ω,\mu)\) extending \((Ω_0,\mu_0)\), a
-countable set \(\mathcal L\) with finitely many free \(\Gamma\)-orbits, and
-equivariant leafwise data on \(\widetilde M\timesΩ\): a metric \(h\),
-a function \(\phi\), a positive density \(\rho\), and a root probability
-map \(F\) with labels \(\mathcal L\). There are also \(E>E_{\min}\), an
-integer \(q\geq n+1\), and a constant \(C=C(n,\mathcal B)<∞\) such
+For every prescribed elementary function (mathcal BcalB) and every finite
+number (E_{min}), there are a standard Borel probability
+(Gamma)-space ((Ω,mu)) extending ((Ω_0,mu_0)), a
+countable set (mathcal LcalL) with finitely many free (Gamma)-orbits, and
+equivariant leafwise data on (widetilde MtimesΩ): a metric (h),
+a function (phi), a positive density (rho), and a root probability
+map (F) with labels (mathcal LcalL). There are also (E>E_{min}), an
+integer (qgeq n+1), and a constant (C=C(n,mathcal B)<∞calB)<∞) such
that, on an invariant conull set of transverse parameters,
- &\rho\leq dV_g,\qquad
- W:=\log(dV_h)/(\rho)\geq0,\\
- &\wscal(h,\phi)\geq
- -C+2(W-\frac n2\log E)+E e_h(F),
- \qquad e_h(F)^{1/2}\leq E^{-1/100},\\
- &E>\mathcal B(q).
+ &rholeq dV_g,qquad
+ W:=log(dV_h)/(rho)geq0,
+ &wscal(h,phi)geq
+ -C+2(W-frac n2log E)+E e_h(F),
+ qquad e_h(F)^{1/2}leq E^{-1/100},
+ &E>mathcal B(q).calB(q).
Here the extension means that there is an equivariant measure-preserving
-map \(Ω\toΩ_0\). All output objects are Borel transversely and
+map (ΩtoΩ_0). All output objects are Borel transversely and
smooth along leaves. In the same lifted charts, the coefficients of
-\(h\), their derivatives of every order, and the coefficients of \(h^{-1}\)
-have uniform bounds; the function \(\phi\), the density ratio
-\(\rho/dV_g\), its reciprocal, and their derivatives of every order have
-uniform bounds. The coordinates of \(F\) have the same derivative and
+(h), their derivatives of every order, and the coefficients of (h^{-1})
+have uniform bounds; the function (phi), the density ratio
+(rho/dV_g), its reciprocal, and their derivatives of every order have
+uniform bounds. The coordinates of (F) have the same derivative and
uniform local finiteness properties as the input. There are equivariant
compact sets containing their supports, uniformly in the transverse
parameter. All these output bounds may depend on the input and on the
chosen finite construction.
-There is, in addition, a fixed \(\Gamma\)-invariant graph \(\mathcal G\)
-on \(\mathcal L\), independent of the transverse parameter, of finite
-maximum degree \(\Delta\). It contains an edge between every pair of
+There is, in addition, a fixed (Gamma)-invariant graph (mathcal GcalG)
+on (mathcal LcalL), independent of the transverse parameter, of finite
+maximum degree (Delta). It contains an edge between every pair of
distinct labels that are simultaneously nonzero at any point of any leaf.
Take
-\[
- q\geq \max\{n+1,\#(\Gamma\backslash\mathcal L),\Delta+1\}
-\]
+[
+ qgeq max{n+1,#(Gammabackslashmathcal L),calL),Delta+1}
+]
and a Borel equivariant coloring
-\(\chi:\mathcal L\timesΩ\to\{1,...,q\}\) such that labels of
-graph distance one or two have different colors. The set \(\mathcal L\)
-can be taken to consist of finitely many copies of \(\mathcal L_0\).
-The constant \(C\) is independent of the manifold, metric, input family,
-all the uniform bounds just specified, and \(E_{\min}\).
+(chi:mathcal LcalLtimesΩto{1,...,q}) such that labels of
+graph distance one or two have different colors. The set (mathcal LcalL)
+can be taken to consist of finitely many copies of (mathcal L_0calL_0).
+The constant (C) is independent of the manifold, metric, input family,
+all the uniform bounds just specified, and (E_{min}).
Prove the theorem using Proposition~prop:sweep. The bootstrap
-will fix the density \(\rho\), which then remains fixed through all later
+will fix the density (rho), which then remains fixed through all later
sweeps and conformal changes. For
-\(W=\log(dV_h/\rho)\), the balance propagated after the bootstrap is
-\[
- \wscal(h,\phi)-ε e_h(F)-2W+n\logε\geq-C+P,
- \qquad W\geq0,\quad P\geq0.
-\]
+(W=log(dV_h/rho)), the balance propagated after the bootstrap is
+[
+ wscal(h,phi)-ε e_h(F)-2W+nlogεgeq-C+P,
+ qquad Wgeq0,quad Pgeq0.
+]
First raise an initially small energy coefficient to a dimensional
positive value. Subsequent sweeps have a summable scalar loss and
-retain the nonnegative surplus \(P\). At the end of a group of sweeps,
+retain the nonnegative surplus (P). At the end of a group of sweeps,
that surplus pays for a conformal change which makes the gradient
absolutely small. The small gradient then controls how fast the fixed
graph of label overlaps can grow in the next group of sweeps.
@@ -2191,24 +2191,24 @@ also occur in [cite];
the proof below establishes the forcing and entropy estimates needed here.
-First supply the colors required for the sweeps. Let \(\mathcal H\)
+First supply the colors required for the sweeps. Let (mathcal HcalH)
be any fixed invariant graph of maximum degree
-\(\Delta_{\mathcal H}<∞\) on a countable label set \(\mathcal L\).
+(Delta_{mathcal H}calH<∞) on a countable label set (mathcal LcalL).
Adjoin to the transverse space independent uniform priorities in
-\([0,1]^{\mathcal L}\), with the \(\Gamma\)-action permuting coordinates.
+([0,1]^{mathcal L}calL), with the (Gamma)-action permuting coordinates.
This is a standard Borel probability extension. Direct every edge
from the larger priority to the smaller, after discarding the invariant
null set on which two priorities coincide. This set is null because the
label set is countable and the priorities have continuous laws. For a
fixed vertex the probability of a strictly descending path of length
-\(k\) is at most \(\Delta_{\mathcal H}^k/(k+1)!\): there are at most
-\(\Delta_{\mathcal H}^k\) paths, and the priorities of the \(k+1\)
+(k) is at most (Delta_{mathcal H}calH^k/(k+1)!): there are at most
+(Delta_{mathcal H}calH^k) paths, and the priorities of the (k+1)
distinct vertices in a given descending path have the required order
-with probability \(1/(k+1)!\). The probability of arbitrarily long
+with probability (1/(k+1)!). The probability of arbitrarily long
such paths is therefore zero. Since the label set is countable, on
an invariant conull set every vertex has finite descending depth.
Greedily assign to a vertex the least color in
-\(\{1,...,\Delta_{\mathcal H}+1\}\) unused by its lower-priority
+({1,...,Delta_{mathcal H}calH+1}) unused by its lower-priority
neighbors, recursively in that finite depth. This defines a proper
coloring. It is Borel, since each vertex's color is determined in
some finite descending neighborhood, and it is equivariant, since
@@ -2217,12 +2217,12 @@ to the graph square separates vertices of distance at most two.
For the initial map and throughout the bootstrap, graphs of the
required finite degree exist with input-dependent bounds. To see
-this, choose a compact set in \(\widetilde M\) whose translates cover
+this, choose a compact set in (widetilde M) whose translates cover
the universal cover. Each label support lies in one of finitely many
fixed compact sets translated by its group label. A support
enlargement by a bounded distance in any one of the current uniformly
controlled metrics is contained in an input-dependent compact
-enlargement in the lifted metric \(g\). Properness of the deck action
+enlargement in the lifted metric (g). Properness of the deck action
implies that only finitely many translates of any of these compact
sets can meet another. Joining all such possible pairs, over all
transverse parameters, gives one invariant finite-degree graph for
@@ -2235,510 +2235,510 @@ will control the degree relative to the eventual energy coefficient.
Here and below all the notation of a single sweep is that of
Proposition~prop:sweep. In particular, its input is
-\((h_0,\phi_0,F)\), its output is \((h_+,\phi_+,Y)\), and
-\[
+((h_0,phi_0,F)), its output is ((h_+,phi_+,Y)), and
+[
- N^2=lε,\qquad a=\tfrac12+\theta,\qquad
- \lambda=\tfrac12-2\theta,\\
- r=\sum_i f_i^2>0,\qquad Y=f/\sqrt r,\qquad
- W_p=\sum_i\log L_i,\qquad m=\sum_i A_i^2.
+ N^2=lε,qquad a=tfrac12+theta,qquad
+ lambda=tfrac12-2theta,
+ r=sum_i f_i^2>0,qquad Y=f/sqrt r,qquad
+ W_p=sum_ilog L_i,qquad m=sum_i A_i^2.
-\]
-The weights \(\nu_i=f_i^2/r\) sum to one. Recall the probability
+]
+The weights (nu_i=f_i^2/r) sum to one. Recall the probability
statistics and their centered combinations from
(eq:sweep-statistics):
-\[
+[
- u_i=(A_i^2)/(ε\nu_i),\qquad
- \sigma_i=(F_i^2)/(\nu_i),\qquad
- \bar u=\frac mε,\\
- M_0=a(1-r)-(\bar u)/(2),\qquad
- V_i=a(\sigma_i-1)+\lambda(u_i-\bar u),\qquad
+ u_i=(A_i^2)/(εnu_i),qquad
+ sigma_i=(F_i^2)/(nu_i),qquad
+ barbar u=u=frac mε,
+ M_0=a(1-r)-(barbar u)/(2),u)/(2),qquad
+ V_i=a(sigma_i-1)+lambda(u_i-barbar u),u),qquad
t_i=(A_i^2)/(f_i).
-\]
+]
Indices of vanishing bumps are omitted. In particular
-\(\E_\nu u_i=\bar u\) and \(\E_\nu\sigma_i=1\); the ratios \(\sigma_i\)
-need not be at most one. Write \(\E\) and \(\Var\) for
+(E_nu u_i=barbar uu) and (E_nusigma_i=1); the ratios (sigma_i)
+need not be at most one. Write (E) and (Var) for
this finite weighted mean and variance in the pointwise calculations.
The graph operations give
- \frac{dV_{h_+}}{dV_{h_0}}=e^{W_p},\qquad
- \phi_+=\phi_0-W_p,\qquad
- N^2 e_{h_+}(Y)\leq \frac mr.
+ frac{dV_{h_+}}{dV_{h_0}}=e^{W_p},qquad
+ phi_+=phi_0-W_p,qquad
+ N^2 e_{h_+}(Y)leq frac mr.
The added graph tensors are nonnegative. Thus, as quadratic forms,
-\(h_0\leq h_{i-1}\leq h_i\leq h_+\); the inverse forms reverse these
-inequalities. In particular, for every covector \(\alpha\),
-\[
- |\alpha|_{h_+}\leq|\alpha|_{h_i}
- \leq|\alpha|_{h_{i-1}}\leq|\alpha|_{h_0}.
-\]
-The positive reserve \(\Pi_p\) is defined in
+(h_0leq h_{i-1}leq h_ileq h_+); the inverse forms reverse these
+inequalities. In particular, for every covector (alpha),
+[
+ |alpha|_{h_+}leq|alpha|_{h_i}
+ leq|alpha|_{h_{i-1}}leq|alpha|_{h_0}.
+]
+The positive reserve (Pi_p) is defined in
(eq:sweep-positive-reserve), with coefficient exactly one on
-\(T_0\). In a sweep without boost its final term is omitted; in a
-boosted sweep its constants \(c_*,C_*\) may depend on the fixed lower
-threshold for \(ε\). The sweep inequality reads
+(T_0). In a sweep without boost its final term is omitted; in a
+boosted sweep its constants (c_*,C_*) may depend on the fixed lower
+threshold for (ε). The sweep inequality reads
Q&:=(rN^2)/(ε)
- [\wscal(h_+,\phi_+)-\wscal(h_0,\phi_0)
- +ε e_{h_0}(F)-ε e_{h_+}(Y)]\\
- &\geq \Pi_p-4\theta^2m-\frac rε\delta_0 .
+ [wscal(h_+,phi_+)-wscal(h_0,phi_0)
+ +ε e_{h_0}(F)-ε e_{h_+}(Y)]
+ &geq Pi_p-4theta^2m-frac rεdelta_0 .
-The positive errors \(\delta_0,\eta_0\) can be prescribed independently
+The positive errors (delta_0,eta_0) can be prescribed independently
before each sweep. The construction realizes those errors by choosing
the analytic parameters for each color after the geometry of the
preceding colors has been fixed.
[Conformal bootstrap]
-There are constants \(ε_*>0\) and \(C_{boot}<4\),
-depending only on \(n\), with the following property. Starting with
+There are constants (ε_*>0) and (C_{boot}<4),
+depending only on (n), with the following property. Starting with
the input of Theorem~thm:entropy, finitely many sweeps without
boost, followed individually by conformal changes, produce a metric
-\(h_*\), a root probability map \(F_*\), and
-\(ε_0\in[ε_*,2ε_*]\) such that
+(h_*), a root probability map (F_*), and
+(ε_0in[ε_*,2ε_*]) such that
- \rho:=dV_{h_*}\leq dV_g,\qquad
- \Scal_{h_*}\geq-C_{boot}+ε_0e_{h_*}(F_*).
+ rho:=dV_{h_*}leq dV_g,qquad
+ Scal_{h_*}geq-C_{boot}+ε_0e_{h_*}(F_*).
The metric and map have the uniformity and support properties in
Theorem~thm:entropy. The number of sweeps may depend on the input.
Proof.
-Use sweeps with \(\phi_0=0\), no boost, and \(l=\theta\), where \(l>0\)
+Use sweeps with (phi_0=0), no boost, and (l=theta), where (l>0)
will be a sufficiently small fixed dimensional number. Then
-\(\phi_+=-W_p\). After a sweep set
-\[
+(phi_+=-W_p). After a sweep set
+[
k=e^{-2W_p/(n-1)}h_+ .
-\]
-The scalar conformal formula and the definition of \(\wscal\) give
+]
+The scalar conformal formula and the definition of (wscal) give
- \Scal_k=e^{2W_p/(n-1)}
- (\wscal(h_+,\phi_+)+\frac{|dW_p|_{h_+}^2}{n-1}).
+ Scal_k=e^{2W_p/(n-1)}
+ (wscal(h_+,phi_+)+frac{|dW_p|_{h_+}^2}{n-1}).
The same reset contracts the volume density:
dV_k=e^{-nW_p/(n-1)}dV_{h_+}
- =e^{-W_p/(n-1)}dV_{h_0}\leq dV_{h_0}.
+ =e^{-W_p/(n-1)}dV_{h_0}leq dV_{h_0}.
-Claim that there are dimensional \(C_b\) and \(ε_b>0\) such
-that, whenever \(2\leq C_0\leq4\), \(0<ε\leqε_b\), and
-\(\Scal_{h_0}\geq-C_0+ε e_{h_0}(F)\), one can choose the errors
+Claim that there are dimensional (C_b) and (ε_b>0) such
+that, whenever (2leq C_0leq4), (0<εleqε_b), and
+(Scal_{h_0}geq-C_0+ε e_{h_0}(F)), one can choose the errors
in the sweep so that
- \Scal_k\geq-(C_0+C_b lε)+(1+l)ε e_k(Y).
+ Scal_kgeq-(C_0+C_b lε)+(1+l)ε e_k(Y).
First derive the budget needed for this step. The reset gives
-\(e_k(Y)=e^{2W_p/(n-1)}e_{h_+}(Y)\). Substituting the definition of
-\(Q\) and the input scalar inequality into
+(e_k(Y)=e^{2W_p/(n-1)}e_{h_+}(Y)). Substituting the definition of
+(Q) and the input scalar inequality into
(eq:entropy-bootstrap-reset), and dropping its nonnegative
gradient term, shows that it suffices to have
-\[
- (ε Q)/(rN^2)\geq
+[
+ (ε Q)/(rN^2)geq
lε e_{h_+}(Y)+C_0(1-e^{-2W_p/(n-1)})
-C_b lε e^{-2W_p/(n-1)}.
-\]
-Multiply by \(rN^2\), and use
-\(N^2 e_{h_+}(Y)\leq m/r\) and \(N^2=lε\). A sufficient budget
+]
+Multiply by (rN^2), and use
+(N^2 e_{h_+}(Y)leq m/r) and (N^2=lε). A sufficient budget
for (eq:entropy-bootstrap-step) is therefore
- ε Q\geq
+ ε Qgeq
lε m+C_0lε r(1-e^{-2W_p/(n-1)})
-C_b l^2ε^2r e^{-2W_p/(n-1)}.
-To pay this budget when \(ε\) is small, one needs control of both
-\(m\) and the deviation of \(r\) from one. The sweep inequality implies
+To pay this budget when (ε) is small, one needs control of both
+(m) and the deviation of (r) from one. The sweep inequality implies
- ε Q\geq
+ ε Qgeq
c_n(ε^2(1-r)^2+m^2)
- -C_nε l^2m-r\delta'_0 ,
-
-where \(\delta'_0>0\) can be chosen arbitrarily small. Here is a proof.
-Fix a sufficiently small dimensional \(m_b>0\). If \(m\geq m_b\) and
-\(ε_b\) is small enough, then \(\bar u=m/ε\geq4a\). Since
-\[
- -M_0=ar+(\bar u)/(2)-a,
-\]
-the square \(ε^2M_0^2\) controls
-\(ε^2(1-r)^2+m^2\). This is one of the positive terms in
-\(ε\Pi_p\).
-
-Suppose next that \(m<m_b\). The prescription term \(D_i^0\) of
+ -C_nε l^2m-rdelta'_0 ,
+
+where (delta'_0>0) can be chosen arbitrarily small. Here is a proof.
+Fix a sufficiently small dimensional (m_b>0). If (mgeq m_b) and
+(ε_b) is small enough, then (barbar u=m/εu=m/εgeq4a). Since
+[
+ -M_0=ar+(barbar u)/(2)-a,u)/(2)-a,
+]
+the square (ε^2M_0^2) controls
+(ε^2(1-r)^2+m^2). This is one of the positive terms in
+(εPi_p).
+
+Suppose next that (m<m_b). The prescription term (D_i^0) of
Proposition~prop:sweep satisfies
-\[
+[
ND_i^0+t_i=(ε f_i)/(r)
- [V_i+M_0+(\lambda+\tfrac12)\bar u+
- (\tfrac32-\lambda)u_i].
-\]
-Both displayed coefficients of \(\bar u\) and \(u_i\) are positive and
-bounded below. Moving \(V_i+M_0\) to the other side, taking positive
+ [V_i+M_0+(lambda+tfrac12)barbar u+u+
+ (tfrac32-lambda)u_i].
+]
+Both displayed coefficients of (barbar uu) and (u_i) are positive and
+bounded below. Moving (V_i+M_0) to the other side, taking positive
parts, and summing squares therefore yields
- \sum_i t_i^2\leq
- (Cε^2)/(r)(\E V_i^2+M_0^2)
- +C\sum_i(ND_i^0+t_i)_+^2 .
+ sum_i t_i^2leq
+ (Cε^2)/(r)(E V_i^2+M_0^2)
+ +Csum_i(ND_i^0+t_i)_+^2 .
The prescription error relation in Proposition~prop:sweep is
-\(ND_i=ND_i^0-\beta_i f_i+err_i\), with
-\(\sum_i|err_i|^2\) arbitrarily small. Moreover, the
-longitudinal component of \(\II_i\) is \(-X_i/N\), so
-\[
+(ND_i=ND_i^0-beta_i f_i+err_i), with
+(sum_i|err_i|^2) arbitrarily small. Moreover, the
+longitudinal component of (II_i) is (-X_i/N), so
+[
ND_i+t_i=-(X_i-t_i)
- +N(\tr\II_i^{rem}-U_i(\phi_{i-1})).
-\]
-The terms \((X_i-t_i)^2\), \(N^2|\II_i^{rem}|^2\), and
-\(\beta_i^2f_i^2\) occur in \(\mathcal X\).
-For the remaining term, choose \(m_b<1/4\). Then
-\(L_i\leq C\) and \(|d\log L_i|_{h_i}\leq C A_i|\II_i|\).
+ +N(trII_i^{rem}-U_i(phi_{i-1})).
+]
+The terms ((X_i-t_i)^2), (N^2|II_i^{rem}|^2), and
+(beta_i^2f_i^2) occur in (mathcal XcalX).
+For the remaining term, choose (m_b<1/4). Then
+(L_ileq C) and (|dlog L_i|_{h_i}leq C A_i|II_i|).
Because the metrics increase during the sweep and
-\(\phi_{i-1}=-\sum_{j<i}\log L_j\), Cauchy--Schwarz gives
-\[
+(phi_{i-1}=-sum_{j<i}log L_j), Cauchy--Schwarz gives
+[
- \sum_iN^2U_i(\phi_{i-1})^2
- &\leq C m^2\sum_jN^2|\II_j|^2\\
- &\leq C' m^2(\mathcal X+\sum_jt_j^2).
+ sum_iN^2U_i(phi_{i-1})^2
+ &leq C m^2sum_jN^2|II_j|^2
+ &leq C' m^2(mathcal X+calX+sum_jt_j^2).
-\]
+]
For example, the first line follows by bounding
-\(|d\phi_{i-1}|_{h_{i-1}}\) by
-\(C\sum_{j<i}A_j|\II_j|\), squaring with
-\(\sum_jA_j^2=m\), and then summing the factor \(A_i^2\).
-Choosing \(m_b\) small absorbs the last sum of \(t_j^2\) in
+(|dphi_{i-1}|_{h_{i-1}}) by
+(Csum_{j<i}A_j|II_j|), squaring with
+(sum_jA_j^2=m), and then summing the factor (A_i^2).
+Choosing (m_b) small absorbs the last sum of (t_j^2) in
(eq:entropy-bootstrap-t). Since
-\[
- m^2=(\sum_i f_i t_i)^2\leq r\sum_i t_i^2,
- \qquad
- ε^2(1-r)^2\leq C(ε^2M_0^2+m^2),
-\]
-fractions of the \(M_0^2,\E V_i^2,\mathcal X\) terms in
-\(ε\Pi_p\) control both required squares. The arbitrarily
-small errors in the prescription contribute \(r\delta'_0\).
-The term \(-4\theta^2m\) in (eq:entropy-sweep-form) is
-\(-4l^2m\). This proves (eq:entropy-bootstrap-coercivity).
+[
+ m^2=(sum_i f_i t_i)^2leq rsum_i t_i^2,
+ qquad
+ ε^2(1-r)^2leq C(ε^2M_0^2+m^2),
+]
+fractions of the (M_0^2,E V_i^2,mathcal XcalX) terms in
+(εPi_p) control both required squares. The arbitrarily
+small errors in the prescription contribute (rdelta'_0).
+The term (-4theta^2m) in (eq:entropy-sweep-form) is
+(-4l^2m). This proves (eq:entropy-bootstrap-coercivity).
Now compare the right side of
(eq:entropy-bootstrap-coercivity) with the required budget.
-When \(m\geq m_b\), the first two terms on the right of
+When (mgeq m_b), the first two terms on the right of
(eq:entropy-bootstrap-budget) are at most
-\(lε(m+4r)\). The squares
-\(m^2+ε^2(1-r)^2\) dominate this expression with a fixed
-fraction to spare after \(l\) is chosen small and then \(ε_b\)
-is decreased if needed. One may check this separately for \(r\leq2\),
-where \(m\geq m_b\) controls the constant part, and for \(r>2\), where
-Young's inequality controls \(ε r\) by
-\(ε^2(r-1)^2\) and a sufficiently small fixed part of \(m^2\).
-The term \(C_nε l^2m\) is covered at the same time.
-
-When \(m<m_b\), the identity
-\(L_i=(1-A_i^2)^{-1/2}\) gives \(W_p\leq C m\).
-For \(r\notin(1/2,2)\), Young's inequality applied to
-\(lε m(1+r)\) again gives the required domination by the two
-squares, with \(l\) small. For \(1/2<r<2\), the positive terms on the
+(lε(m+4r)). The squares
+(m^2+ε^2(1-r)^2) dominate this expression with a fixed
+fraction to spare after (l) is chosen small and then (ε_b)
+is decreased if needed. One may check this separately for (rleq2),
+where (mgeq m_b) controls the constant part, and for (r>2), where
+Young's inequality controls (ε r) by
+(ε^2(r-1)^2) and a sufficiently small fixed part of (m^2).
+The term (C_nε l^2m) is covered at the same time.
+
+When (m<m_b), the identity
+(L_i=(1-A_i^2)^{-1/2}) gives (W_pleq C m).
+For (rnotin(1/2,2)), Young's inequality applied to
+(lε m(1+r)) again gives the required domination by the two
+squares, with (l) small. For (1/2<r<2), the positive terms on the
right of (eq:entropy-bootstrap-budget), together with the term
-\(C_nε l^2m\) from the coercivity estimate, are at most
-\(C_nlε m\). Here
-\[
- C_nlε m\leq (c_n)/(2)m^2+C'_n l^2ε^2.
-\]
-On this region \(r e^{-2W_p/(n-1)}\) has a dimensional positive lower
-bound. Therefore choose \(C_b\) dimensionally to cover the
-last term. These comparisons can be made strict by increasing \(C_b\)
-and decreasing \(l,ε_b\). They leave a positive multiple of
-\(l^2ε^2r\) in the middle region, and positive square margins
-outside it; dividing the latter margins by \(r\) gives a positive
-lower bound for fixed \(l,ε\). Consequently choose
-\(\delta'_0\) small enough for the sweep. This proves
+(C_nε l^2m) from the coercivity estimate, are at most
+(C_nlε m). Here
+[
+ C_nlε mleq (c_n)/(2)m^2+C'_n l^2ε^2.
+]
+On this region (r e^{-2W_p/(n-1)}) has a dimensional positive lower
+bound. Therefore choose (C_b) dimensionally to cover the
+last term. These comparisons can be made strict by increasing (C_b)
+and decreasing (l,ε_b). They leave a positive multiple of
+(l^2ε^2r) in the middle region, and positive square margins
+outside it; dividing the latter margins by (r) gives a positive
+lower bound for fixed (l,ε). Consequently choose
+(delta'_0) small enough for the sweep. This proves
(eq:entropy-bootstrap-step).
-The input map has a finite uniform energy bound in \(g\). Thus for
-some positive, possibly input-dependent \(ε_{in}\)
+The input map has a finite uniform energy bound in (g). Thus for
+some positive, possibly input-dependent (ε_{in})
small enough,
-\(\Scal_g=-1\geq-2+ε_{in}e_g(F^{in})\).
-Choose a dimensional \(ε_*>0\) below \(ε_b\), small
-enough that \(2+2C_bε_*<4\), and decrease
-\(ε_{in}\) further so that
-\(0<ε_{in}<ε_*\). Iterate
+(Scal_g=-1geq-2+ε_{in}e_g(F^{in})).
+Choose a dimensional (ε_*>0) below (ε_b), small
+enough that (2+2C_bε_*<4), and decrease
+(ε_{in}) further so that
+(0<ε_{in}<ε_*). Iterate
(eq:entropy-bootstrap-step) until the coefficient first reaches
-\(ε_*\). The output coefficient is at most
-\((1+l)ε_*\leq2ε_*\), and
-\[
- \sum_{bootstrap sweeps}lε
- =ε_0-ε_{in}\leq2ε_*.
-\]
-This bounds the total scalar cost by \(2C_bε_*\), independently
+(ε_*). The output coefficient is at most
+((1+l)ε_*leq2ε_*), and
+[
+ sum_{bootstrap sweeps}lε
+ =ε_0-ε_{in}leq2ε_*.
+]
+This bounds the total scalar cost by (2C_bε_*), independently
of the number of sweeps. By (eq:entropy-bootstrap-density),
every reset contracts the volume density. The final density
-\(\rho=dV_{h_*}\) therefore satisfies
+(rho=dV_{h_*}) therefore satisfies
(eq:entropy-bootstrap-output). The sweep construction and
finitely many smooth conformal changes preserve all the stated
input-uniform properties.
-Fix \(ε_*\) from Lemma~lem:entropy-bootstrap. Now use
+Fix (ε_*) from Lemma~lem:entropy-bootstrap. Now use
the boost in Proposition~prop:sweep, choosing its constant
-\(\kappa>0\) sufficiently small depending on this fixed threshold.
+(kappa>0) sufficiently small depending on this fixed threshold.
All constants arising from the boost are consequently dimensional.
The increment estimate has a coarse region, where a fixed fraction of
the reserve remains after paying the budget, and a region near
-\(m=0,r=1\), where the exact quadratic coefficient will matter.
+(m=0,r=1), where the exact quadratic coefficient will matter.
[Entropy increment]
-There are dimensional constants \(m_0>0\), \(D_*\geq1\), \(l_0>0\),
-\(\gamma_n>0\), and \(C_n<∞\) with the following property.
-For \(ε\geqε_*\), take a boosted sweep with
-\[
- 0<l\leq l_0,\qquad \theta=D_*l\leq\tfrac1{16},
- \qquad N^2=lε.
-\]
+There are dimensional constants (m_0>0), (D_*geq1), (l_0>0),
+(gamma_n>0), and (C_n<∞) with the following property.
+For (εgeqε_*), take a boosted sweep with
+[
+ 0<lleq l_0,qquad theta=D_*lleqtfrac1{16},
+ qquad N^2=lε.
+]
The errors can be chosen so that
- \wscal(h_+,\phi_+)-(1+l)ε e_{h_+}(Y)
- \geq{}&\wscal(h_0,\phi_0)-ε e_{h_0}(F)\\
- &+2W_p-n\log(1+l)-C_nl^2+P_p ,
+ wscal(h_+,phi_+)-(1+l)ε e_{h_+}(Y)
+ geq{}&wscal(h_0,phi_0)-ε e_{h_0}(F)
+ &+2W_p-nlog(1+l)-C_nl^2+P_p ,
where the nonnegative, pointwise surplus is
- P_p=\gamma_n(ε)/(rN^2)\Pi_p\,
- \indic_{\{m\geq m_0\}\,\cup\,\{|1-r|\geq1/4\}}.
+ P_p=gamma_n(ε)/(rN^2)Pi_p,
+ indic_{{mgeq m_0},cup,{|1-r|geq1/4}}.
-In particular, on the indicated region \(P_p\) retains a fixed fraction
+In particular, on the indicated region (P_p) retains a fixed fraction
of every positive term in the sweep inequality.
Proof.
The output energy bound in (eq:entropy-sweep-density) shows that,
-in the units of \(Q\), it suffices to prove
+in the units of (Q), it suffices to prove
- Q\geq lm+2lrW_p-nrl\log(1+l)-C_nr l^3
+ Qgeq lm+2lrW_p-nrllog(1+l)-C_nr l^3
+(rN^2)/(ε)P_p .
-The term \(lm\) pays for the additional \(lε e_{h_+}(Y)\).
-Choose \(\delta_0\leqε l^3\) in
+The term (lm) pays for the additional (lε e_{h_+}(Y)).
+Choose (delta_0leqε l^3) in
(eq:entropy-sweep-form). It remains to compare the other terms
there with the budget in (eq:entropy-budget).
Use a trace identity for the amount of the new graph energy that
remains in the final metric. Set
-\[
- m_f:=\sum_i A_i^2(1-C_i)=N^2\sum_i|df_i|_{h_+}^2.
-\]
+[
+ m_f:=sum_i A_i^2(1-C_i)=N^2sum_i|df_i|_{h_+}^2.
+]
As matrices,
-\(\sum_iN^2df_i ⊗ df_i=h_+-h_0\). Hence
+(sum_iN^2df_i ⊗ df_i=h_+-h_0). Hence
- m_f=n-\tr(h_+^{-1}h_0)
- \leq n(1-e^{-2W_p/n}).
+ m_f=n-tr(h_+^{-1}h_0)
+ leq n(1-e^{-2W_p/n}).
The inequality is arithmetic--geometric mean applied to the positive
-eigenvalues of \(h_+^{-1}h_0\), whose product is \(e^{-2W_p}\).
+eigenvalues of (h_+^{-1}h_0), whose product is (e^{-2W_p}).
There also holds
L_i=(1-A_i^2)^{-1/2}.
-First choose a small dimensional \(m_0>0\), in particular
-\(m_0\llε_*\) and \(m_0<1/4\). Claim that
+First choose a small dimensional (m_0>0), in particular
+(m_0llε_*) and (m_0<1/4). Claim that
- T_0+ε M_0^2\geq c_n(m+rW_p)
- \quadif m\geq m_0
- or if m<m_0,\ |1-r|\geq\tfrac14.
+ T_0+ε M_0^2geq c_n(m+rW_p)
+ quadif mgeq m_0
+ or if m<m_0, |1-r|geqtfrac14.
-For \(m\geq2n\), the \(C_i\) terms of \(T_0\) give
-\(T_0\geq m-m_f\geq m-n\geq m/2\).
-For \(m_0\leq m\leq2n\), first suppose that every \(A_i^2\) is at most
-a small number \(\eta=\eta(n,m_0)>0\). Then
-\[
- 2W_p=\sum_i-\log(1-A_i^2)\leq(1+C\eta)m.
-\]
+For (mgeq2n), the (C_i) terms of (T_0) give
+(T_0geq m-m_fgeq m-ngeq m/2).
+For (m_0leq mleq2n), first suppose that every (A_i^2) is at most
+a small number (eta=eta(n,m_0)>0). Then
+[
+ 2W_p=sum_i-log(1-A_i^2)leq(1+Ceta)m.
+]
By (eq:entropy-trace),
-\[
- m-m_f\geq m-n(1-e^{-(1+C\eta)m/n})\geq c_n m.
-\]
-For the last inequality, the same expression with \(\eta=0\) divided
-by \(m\) has a positive minimum on \([m_0,2n]\), and then \(\eta\)
-is chosen small. If instead some \(A_i^2>\eta\), the self term of
-that index in \(T_0\) is at least \(c_n(r+u_i)\). It gives a
-dimensional positive bound if \(r\) is bounded below. If \(r\) is
-small, \(M_0=a(1-r)-\bar u/2\) is bounded away from zero unless \(\bar u\)
-lies in a fixed compact subinterval of \((0,∞)\). In the former
-case \(ε M_0^2\geq c_nε_*\). In the latter,
-\(ε=m/\bar u\leq C_n\) and
-\(u_i=A_i^2/(ε\nu_i)\geq c_n\), since \(\nu_i\leq1\).
-Each bound controls \(m\leq2n\). This proves the \(m\) part for
-\(m\geq m_0\).
-
-For the volume part, indices with \(A_i\geq1/4\) satisfy
-\[
- A_i^2(L_i-1)2ar\geq c\,r(L_i-1)\geq c\,r\log L_i.
-\]
-For all other indices, \(\log L_i\leq C A_i^2\), so their contribution
-to \(rW_p\) is at most \(Crm\). If \(r\leq2\), the bound for \(m\)
-already covers this. If \(r>2\), then
-\(|M_0|\geq c(r+\bar u)\), and
-\(ε M_0^2\geq cε r\bar u=c\,rm\).
-This proves (eq:entropy-coarse-coercivity) when \(m\geq m_0\).
-When \(m<m_0\), one has \(\bar u\leq m_0/ε_*\) small and
-\[
- W_p=\frac m2+O(\sum_iA_i^4)\leq C m.
-\]
-If \(|1-r|\geq1/4\), choosing \(m_0\) still smaller gives
-\(|M_0|\geq c|1-r|\). The quantity
-\(ε(1-r)^2\) controls \(m+rm\): for bounded \(r\) use
-\(ε\geqε_*\) and \(m\leq m_0\), and for \(r\geq2\)
-use \((1-r)^2\geq r^2/4\). This proves the claim in the remaining
+[
+ m-m_fgeq m-n(1-e^{-(1+Ceta)m/n})geq c_n m.
+]
+For the last inequality, the same expression with (eta=0) divided
+by (m) has a positive minimum on ([m_0,2n]), and then (eta)
+is chosen small. If instead some (A_i^2>eta), the self term of
+that index in (T_0) is at least (c_n(r+u_i)). It gives a
+dimensional positive bound if (r) is bounded below. If (r) is
+small, (M_0=a(1-r)-barbar u/2u/2) is bounded away from zero unless (barbar uu)
+lies in a fixed compact subinterval of ((0,∞)). In the former
+case (ε M_0^2geq c_nε_*). In the latter,
+(ε=m/barbar uuleq C_n) and
+(u_i=A_i^2/(εnu_i)geq c_n), since (nu_ileq1).
+Each bound controls (mleq2n). This proves the (m) part for
+(mgeq m_0).
+
+For the volume part, indices with (A_igeq1/4) satisfy
+[
+ A_i^2(L_i-1)2argeq c,r(L_i-1)geq c,rlog L_i.
+]
+For all other indices, (log L_ileq C A_i^2), so their contribution
+to (rW_p) is at most (Crm). If (rleq2), the bound for (m)
+already covers this. If (r>2), then
+(|M_0|geq c(r+barbar u)u)), and
+(ε M_0^2geq cε rbarbar u=cu=c,rm).
+This proves (eq:entropy-coarse-coercivity) when (mgeq m_0).
+When (m<m_0), one has (barbar uuleq m_0/ε_*) small and
+[
+ W_p=frac m2+O(sum_iA_i^4)leq C m.
+]
+If (|1-r|geq1/4), choosing (m_0) still smaller gives
+(|M_0|geq c|1-r|). The quantity
+(ε(1-r)^2) controls (m+rm): for bounded (r) use
+(εgeqε_*) and (mleq m_0), and for (rgeq2)
+use ((1-r)^2geq r^2/4). This proves the claim in the remaining
case.
On the region of (eq:entropy-coarse-coercivity), the positive
-reserve \(\Pi_p\) consequently controls \(m+rW_p\). The positive terms
-\(lm+2lrW_p\) in (eq:entropy-budget) use at most \(C_nl\Pi_p\),
-and \(4\theta^2m\) uses at most \(C_nD_*^2l^2\Pi_p\). For \(l\)
-sufficiently small after \(D_*\), a fixed fraction of \(\Pi_p\)
-remains. Choose \(\gamma_n\) below that fraction. The error term
-\(r\delta_0/ε\) is covered by \(C_nr l^3\). This proves
+reserve (Pi_p) consequently controls (m+rW_p). The positive terms
+(lm+2lrW_p) in (eq:entropy-budget) use at most (C_nlPi_p),
+and (4theta^2m) uses at most (C_nD_*^2l^2Pi_p). For (l)
+sufficiently small after (D_*), a fixed fraction of (Pi_p)
+remains. Choose (gamma_n) below that fraction. The error term
+(rdelta_0/ε) is covered by (C_nr l^3). This proves
(eq:entropy-budget) on the region where the surplus is retained.
It remains to prove the budget when
- m<m_0,\qquad |1-r|<\tfrac14.
+ m<m_0,qquad |1-r|<tfrac14.
-Here \(T_0\geq c_nm^2\). Indeed, expanding the exponential in
+Here (T_0geq c_nm^2). Indeed, expanding the exponential in
(eq:entropy-trace) and using
-\(W_p=m/2+O(\sum_iA_i^4)\) gives
-\[
- \sum_i A_i^2 C_i=m-m_f
- \geq c_n m^2-C_n\sum_iA_i^4
-\]
-after decreasing \(m_0\). The self terms
-\(\sum_i A_i^2(L_i-1)2ar\) are at least
-\(c_n\sum_iA_i^4\), by \(r>3/4\) and \(L_i-1\geq A_i^2/2\).
+(W_p=m/2+O(sum_iA_i^4)) gives
+[
+ sum_i A_i^2 C_i=m-m_f
+ geq c_n m^2-C_nsum_iA_i^4
+]
+after decreasing (m_0). The self terms
+(sum_i A_i^2(L_i-1)2ar) are at least
+(c_nsum_iA_i^4), by (r>3/4) and (L_i-1geq A_i^2/2).
Adding a sufficiently small fraction of the first estimate to the
-second proves \(T_0\geq c_nm^2\).
-Since \(W_p\leq Cm\) on this region, this controls the budget whenever
-\(m\geq Kl\), for a fixed sufficiently large dimensional \(K\).
-In making this conclusion, first fix \(K\); after \(D_*\) is chosen
-below, take \(l\) small enough that \(4D_*^2l^2m\) is also a small
-fraction of \(m^2\).
-
-What remains is the critical regime \(m\leq Kl\), still under
+second proves (T_0geq c_nm^2).
+Since (W_pleq Cm) on this region, this controls the budget whenever
+(mgeq Kl), for a fixed sufficiently large dimensional (K).
+In making this conclusion, first fix (K); after (D_*) is chosen
+below, take (l) small enough that (4D_*^2l^2m) is also a small
+fraction of (m^2).
+
+What remains is the critical regime (mleq Kl), still under
(eq:entropy-small-region). Here
-\(W_p=m/2+O(m^2)\) and \(3/4<r<5/4\), so the budget expands as
-\[
- lm+2lrW_p-nrl\log(1+l)
+(W_p=m/2+O(m^2)) and (3/4<r<5/4), so the budget expands as
+[
+ lm+2lrW_p-nrllog(1+l)
=2lm-nl^2+O_n(l^3)+O_n(l^2)|r-1|.
-\]
+]
The target quadratic term is covered exactly by
-\[
+[
(m^2)/(n)-2lm+nl^2
- =(\frac m{\sqrt n}-\sqrt n\,l)^2\geq0.
-\]
-It remains to recover \(m^2/n\) from \(T_0\), while paying the
-displayed \(r-1\) error and the errors in that recovery. This is where
-the coefficient one on \(T_0\) in \(\Pi_p\) is needed.
-
-In particular \(e^{W_p}=1+O(m)\). Metric addition gives
-\(h_0\leq h_j\leq h_+\) as quadratic forms.
-Every eigenvalue of \(h_0^{-1}h_+\) is at least one, and their product
-is \(e^{2W_p}\). Each eigenvalue is therefore at most that product,
+ =(frac m{sqrt n}-sqrt n,l)^2geq0.
+]
+It remains to recover (m^2/n) from (T_0), while paying the
+displayed (r-1) error and the errors in that recovery. This is where
+the coefficient one on (T_0) in (Pi_p) is needed.
+
+In particular (e^{W_p}=1+O(m)). Metric addition gives
+(h_0leq h_jleq h_+) as quadratic forms.
+Every eigenvalue of (h_0^{-1}h_+) is at least one, and their product
+is (e^{2W_p}). Each eigenvalue is therefore at most that product,
which proves the comparison
-\[
- h_0\leq h_j\leq h_+\leq e^{2W_p}h_0=(1+O(m))h_0.
-\]
-Normalize the directions \(e_i\) in the \(h_0\) inner product and write
-the resulting unit vectors as \(\bar e_i\). Put
-\(\xi_{ij}=(\bar e_i\cdot\bar e_j)^2\). Before approximating the metric
+[
+ h_0leq h_jleq h_+leq e^{2W_p}h_0=(1+O(m))h_0.
+]
+Normalize the directions (e_i) in the (h_0) inner product and write
+the resulting unit vectors as (barbar e_ie_i). Put
+(xi_{ij}=(barbar e_ie_icdotbarbar e_j)^2e_j)^2). Before approximating the metric
losses, their exact earlier- and later-color telescopes are
-\[
- B_i=\sum_{j<i}(Ndf_j(e_i))^2,\qquad
- C_i=\frac1{A_i^2}\sum_{j>i}A_j^2(Ndf_i(e_j))^2
- \quad(A_i>0).
-\]
+[
+ B_i=sum_{j<i}(Ndf_j(e_i))^2,qquad
+ C_i=frac1{A_i^2}sum_{j>i}A_j^2(Ndf_i(e_j))^2
+ quad(A_i>0).
+]
The first identity follows from
-\(h_{i-1}=h_0+N^2\sum_{j<i}df_j ⊗ df_j\) and
-\(|e_i|_{h_{i-1}}=1\). The second follows by telescoping the rank-one
+(h_{i-1}=h_0+N^2sum_{j<i}df_j ⊗ df_j) and
+(|e_i|_{h_{i-1}}=1). The second follows by telescoping the rank-one
inverse formula
-\[
- h_j^{-1}=h_{j-1}^{-1}-A_j^2 e_j e_j^{\mathsf T}
-\]
-from \(h_i\) to \(h_+\), and using
-\(N^2|df_i|_{h_i}^2=A_i^2\). Now
-\(Ndf_j=A_jL_j e_j^\flat\), with the dual taken in \(h_{j-1}\).
-The common metric comparison and \(L_j=1+O(m)\) show that each summand
-in the exact formulas differs from its \(h_0\) expression by
-\(O(m)A_j^2\). Their sums therefore give, for \(A_i\ne0\),
+[
+ h_j^{-1}=h_{j-1}^{-1}-A_j^2 e_j e_j^{mathsf T}sfT
+]
+from (h_i) to (h_+), and using
+(N^2|df_i|_{h_i}^2=A_i^2). Now
+(Ndf_j=A_jL_j e_j^flat), with the dual taken in (h_{j-1}).
+The common metric comparison and (L_j=1+O(m)) show that each summand
+in the exact formulas differs from its (h_0) expression by
+(O(m)A_j^2). Their sums therefore give, for (A_ine0),
- B_i=\sum_{j<i}A_j^2\xi_{ij}+O(m^2),\qquad
- C_i=\sum_{j>i}A_j^2\xi_{ij}+O(m^2).
+ B_i=sum_{j<i}A_j^2xi_{ij}+O(m^2),qquad
+ C_i=sum_{j>i}A_j^2xi_{ij}+O(m^2).
-Inserting (eq:entropy-BC-expansion) into \(T_0\) yields
+Inserting (eq:entropy-BC-expansion) into (T_0) yields
- T_0\geq
- \tr[(\sum_i A_i^2\bar e_i\bar e_i^{\mathsf T})^2]
+ T_0geq
+ tr[(sum_i A_i^2barbar e_ie_ibarbar e_i^e_i^{mathsf T}sfT)^2]
-C_n(m^3+m^2|r-1|
- +m\sum_iA_i^2(1-\sigma_i)_+).
+ +msum_iA_i^2(1-sigma_i)_+).
-To see the leading coefficient, the \(B_i\) and \(C_i\) terms each
+To see the leading coefficient, the (B_i) and (C_i) terms each
give one copy of
-\(\sum_{i<j}A_i^2A_j^2\xi_{ij}\). In the \(B_i\) terms use
-\(4a^2\geq1\), \(B_i\geq0\); replacing \(\sigma_i\) by one costs at most
-\(Cm\sum_iA_i^2(1-\sigma_i)_+\). The self terms use
-\(L_i-1\geq A_i^2/2\) and \(2a\geq1\). They give
-\(\sum_iA_i^4\), up to the displayed \(r-1\) and \(\sigma_i-1\) costs.
-The errors in (eq:entropy-BC-expansion) sum to \(O(m^3)\).
+(sum_{i<j}A_i^2A_j^2xi_{ij}). In the (B_i) terms use
+(4a^2geq1), (B_igeq0); replacing (sigma_i) by one costs at most
+(Cmsum_iA_i^2(1-sigma_i)_+). The self terms use
+(L_i-1geq A_i^2/2) and (2ageq1). They give
+(sum_iA_i^4), up to the displayed (r-1) and (sigma_i-1) costs.
+The errors in (eq:entropy-BC-expansion) sum to (O(m^3)).
This proves (eq:entropy-sharp-T). The trace in that estimate
-is at least \(m^2/n\), since the matrix is positive semidefinite
-with trace \(m\).
+is at least (m^2/n), since the matrix is positive semidefinite
+with trace (m).
The last error in (eq:entropy-sharp-T) is paid for by the
centered squares. From
-\(a(\sigma_i-1)=V_i-\lambda(u_i-\bar u)\), Cauchy--Schwarz and Young's
-inequality give, for every fixed \(c'>0\),
+(a(sigma_i-1)=V_i-lambda(u_i-barbar u)u)), Cauchy--Schwarz and Young's
+inequality give, for every fixed (c'>0),
- m\sum_iA_i^2(1-\sigma_i)_+
- &\leq C mε\,\E[u_i(|V_i|+|u_i-\bar u|)]\\
- &\leq c'ε(\E V_i^2+\theta\Var u)
- +C_{c'}(m^2)/(\theta)ε(\bar u^2+\Var u).
+ msum_iA_i^2(1-sigma_i)_+
+ &leq C mε,E[u_i(|V_i|+|u_i-barbar u|)]u|)]
+ &leq c'ε(E V_i^2+thetaVar u)
+ +C_{c'}(m^2)/(theta)ε(barbar u^2+u^2+Var u).
-Choose \(c'\) to use a small fraction of the squares in \(\Pi_p\).
-The coefficient of \(ε\Var u\) in the last term, relative to
-\(\thetaε\Var u\), is at most \(C_{c'}K^2/D_*^2\), because
-\(m\leq Kl\) and \(\theta=D_*l\). Choose \(D_*\) sufficiently large
-after \(K\) so that this term is absorbed as well. The remaining term
+Choose (c') to use a small fraction of the squares in (Pi_p).
+The coefficient of (εVar u) in the last term, relative to
+(thetaεVar u), is at most (C_{c'}K^2/D_*^2), because
+(mleq Kl) and (theta=D_*l). Choose (D_*) sufficiently large
+after (K) so that this term is absorbed as well. The remaining term
is
-\[
- C_{c'}(m^2)/(\theta)ε \bar u^2
- =C_{c'}(m^4)/(\thetaε)=O_n(l^3).
-\]
-Here \(ε\geqε_*\), and \(K,D_*,ε_*\) are already
+[
+ C_{c'}(m^2)/(theta)ε barbar u^2u^2
+ =C_{c'}(m^4)/(thetaε)=O_n(l^3).
+]
+Here (εgeqε_*), and (K,D_*,ε_*) are already
fixed dimensionally. Also
-\[
- |r-1|\leq C(|M_0|+\bar u).
-\]
-Consequently a term \(O_n(l^2)|r-1|\) is bounded by a small fraction
-of \(ε M_0^2\) plus \(O_n(l^3)\): use Young's inequality on
-\(l^2|M_0|\), and use \(\bar u\leq Kl/ε_*\) on the other part.
-The errors \(m^3\), \(m^2|r-1|\), and \(4D_*^2l^2m\) in the present
-region are therefore covered with a remaining error \(O_n(l^3)\).
+[
+ |r-1|leq C(|M_0|+barbar u).u).
+]
+Consequently a term (O_n(l^2)|r-1|) is bounded by a small fraction
+of (ε M_0^2) plus (O_n(l^3)): use Young's inequality on
+(l^2|M_0|), and use (barbar uuleq Kl/ε_*) on the other part.
+The errors (m^3), (m^2|r-1|), and (4D_*^2l^2m) in the present
+region are therefore covered with a remaining error (O_n(l^3)).
The just established mean-square estimate also covers the
-\(O_n(l^2)|r-1|\) term in the critical budget. The recovered
-\(m^2/n\) then covers \(2lm-nl^2\) by the displayed square.
-Since \(r\) is bounded above and below, the remaining \(O_n(l^3)\)
-terms are of the form \(C_nr l^3\). This proves
+(O_n(l^2)|r-1|) term in the critical budget. The recovered
+(m^2/n) then covers (2lm-nl^2) by the displayed square.
+Since (r) is bounded above and below, the remaining (O_n(l^3))
+terms are of the form (C_nr l^3). This proves
(eq:entropy-budget) in (eq:entropy-small-region).
-The choices just made have a definite order: first \(m_0\) after
-\(ε_*\), then \(K\), then \(D_*\), and finally \(l_0>0\)
-small enough for all the comparisons and for \(D_*l_0\leq1/16\).
+The choices just made have a definite order: first (m_0) after
+(ε_*), then (K), then (D_*), and finally (l_0>0)
+small enough for all the comparisons and for (D_*l_0leq1/16).
The errors in each sweep are chosen afterward. Thus every constant
in (eq:entropy-increment) is dimensional.
@@ -2746,650 +2746,650 @@ in (eq:entropy-increment) is dimensional.
[Bounds supplied by the surplus]
For every sweep of Lemma~lem:entropy-increment,
- W_p\leq C_nε^{1/4}(ε+P_p)^{3/4}.
+ W_pleq C_nε^{1/4}(ε+P_p)^{3/4}.
-There is also a dimensional \(ε_{exit}\) such that,
-if \(ε\geqε_{exit}\) and \(r\leq1/4\), then
+There is also a dimensional (ε_{exit}) such that,
+if (εgeqε_{exit}) and (rleq1/4), then
- P_p&\geq c_n(ε)/(lr)
- (1+(m^2)/(ε^2)),\\
- \sum_i(D_i^2+|\II_i|^2+|d\log L_i|_{h_i}^2)
- &\leq C_n(P_p)/(\theta).
+ P_p&geq c_n(ε)/(lr)
+ (1+(m^2)/(ε^2)),
+ sum_i(D_i^2+|II_i|^2+|dlog L_i|_{h_i}^2)
+ &leq C_n(P_p)/(theta).
The errors in the prescription may simultaneously be chosen to have
-\(\sum_i|err_i|^2\leq1\).
+(sum_i|err_i|^2leq1).
Proof.
-If \(\bar u=m/ε\) is larger than a fixed dimensional number, then
-\(m\geq m_0\), so the surplus retains the mean square and boost.
-Since \(|M_0|\geq c(r+\bar u)\) in this range and \(l\leq1\),
-\[
- P_p\geq c_nε\,
- \frac{(r+\bar u)^2+c_*r^6\indic_{\{r\geq C_*\}}}{r}
- \geq c'_nε \bar u^{5/3}.
-\]
-For the last inequality, if \(r\leq \bar u^{1/3}\) use \(\bar u^2/r\);
-if \(r\geq \bar u^{1/3}\) use \(r^5\), taking the threshold for \(\bar u\)
-large enough that \(\bar u^{1/3}\geq C_*\). It follows in all cases that
-
- m\leq C_nε(1+P_p/ε)^{3/5}.
-
-If any \(A_i>1/2\), then \(m>1/4>m_0\), and the self terms in \(T_0\)
+If (barbar u=m/εu=m/ε) is larger than a fixed dimensional number, then
+(mgeq m_0), so the surplus retains the mean square and boost.
+Since (|M_0|geq c(r+barbar u)u)) in this range and (lleq1),
+[
+ P_pgeq c_nε,
+ frac{(r+barbar u)^2+c_*r^6u)^2+c_*r^6indic_{{rgeq C_*}}}{r}
+ geq c'_nε barbar u^u^{5/3}.
+]
+For the last inequality, if (rleq barbar u^u^{1/3}) use (barbar u^2/ru^2/r);
+if (rgeq barbar u^u^{1/3}) use (r^5), taking the threshold for (barbar uu)
+large enough that (barbar u^u^{1/3}geq C_*). It follows in all cases that
+
+ mleq C_nε(1+P_p/ε)^{3/5}.
+
+If any (A_i>1/2), then (m>1/4>m_0), and the self terms in (T_0)
give
-\[
- P_p\geq c_n\sum_{A_i>1/2}(L_i-1).
-\]
-Indeed their contribution to \(T_0\) is at least
-\(c r\sum_{A_i>1/2}(L_i-1)\), and
-\(ε/(rN^2)=1/(lr)\geq1/r\).
-For \(A_i\leq1/2\), \(\log L_i\leq C A_i^2\).
-There are at most \(4m\) indices with \(A_i>1/2\). Concavity of
-\(\log(1+s)\) on those indices, together with the last estimate,
+[
+ P_pgeq c_nsum_{A_i>1/2}(L_i-1).
+]
+Indeed their contribution to (T_0) is at least
+(c rsum_{A_i>1/2}(L_i-1)), and
+(ε/(rN^2)=1/(lr)geq1/r).
+For (A_ileq1/2), (log L_ileq C A_i^2).
+There are at most (4m) indices with (A_i>1/2). Concavity of
+(log(1+s)) on those indices, together with the last estimate,
therefore gives
- W_p\leq C_n m(1+\log(1+P_p/m))
+ W_pleq C_n m(1+log(1+P_p/m))
-when \(m>0\); when \(m=0\), \(W_p=0\).
+when (m>0); when (m=0), (W_p=0).
To justify the bound when the number of those indices is smaller
-than \(4m\), use that \(s\log(1+b/s)\) is increasing in \(s>0\).
-Put \(x=m/ε\), \(y=P_p/ε\). The elementary bound
-\(1+\log(1+t)\leq C(1+t)^{1/4}\) for \(t\geq0\), applied with
-\(t=y/x\), makes the right side of
-(eq:entropy-log-surplus), divided by \(ε\), at most
-\[
+than (4m), use that (slog(1+b/s)) is increasing in (s>0).
+Put (x=m/ε), (y=P_p/ε). The elementary bound
+(1+log(1+t)leq C(1+t)^{1/4}) for (tgeq0), applied with
+(t=y/x), makes the right side of
+(eq:entropy-log-surplus), divided by (ε), at most
+[
C_n x^{3/4}(x+y)^{1/4}
- \leq C_n(1+y)^{7/10}
- \leq C_n(1+y)^{3/4},
-\]
+ leq C_n(1+y)^{7/10}
+ leq C_n(1+y)^{3/4},
+]
where (eq:entropy-m-surplus) was used in the middle inequality.
This proves (eq:entropy-pass-volume).
-Now suppose \(r\leq1/4\). The surplus retains all positive terms
-because \(|1-r|\geq3/4\). Outside a fixed compact subinterval of
-\(\bar u\in(0,∞)\), the expression \(M_0=a(1-r)-\bar u/2\) satisfies
-\(M_0^2\geq c(1+\bar u^2)\). On that compact interval, for \(ε\)
+Now suppose (rleq1/4). The surplus retains all positive terms
+because (|1-r|geq3/4). Outside a fixed compact subinterval of
+(barbar uuin(0,∞)), the expression (M_0=a(1-r)-barbar u/2u/2) satisfies
+(M_0^2geq c(1+barbar u^2)u^2)). On that compact interval, for (ε)
sufficiently large,
-\[
- T_0\geq m-m_f\geqε \bar u-n
- \geq c_nε(1+\bar u^2)
-\]
+[
+ T_0geq m-m_fgeqε barbar u-nu-n
+ geq c_nε(1+barbar u^2)u^2)
+]
by (eq:entropy-trace). Multiplying the appropriate retained
-term by \(\gamma_n/(lr)\) proves the first inequality in
+term by (gamma_n/(lr)) proves the first inequality in
(eq:entropy-last-surplus).
For the second, the retained derivative terms imply
- \mathcal X\leq C_n N^2P_p,\qquad
+ mathcal XcalXleq C_n N^2P_p,qquad
(ε^2)/(rN^2)
- (\E V_i^2+\theta\Var u)\leq C_nP_p.
+ (E V_i^2+thetaVar u)leq C_nP_p.
Also
-\[
- \sum_i t_i^2=(ε^2)/(r)(\bar u^2+\Var u).
-\]
+[
+ sum_i t_i^2=(ε^2)/(r)(barbar u^2+u^2+Var u).
+]
The first inequality of (eq:entropy-last-surplus) controls
-the \(\bar u^2\) part of this sum divided by \(N^2\), and the variance
+the (barbar u^2u^2) part of this sum divided by (N^2), and the variance
term in (eq:entropy-X-retained) controls the other part with
-a factor \(1/\theta\). Thus
-\(\sum_i t_i^2/N^2\leq C_nP_p/\theta\). Since \(\mathcal X\) contains
-\((X_i-t_i)^2+N^2|\II_i^{rem}|^2+N^2|d\log L_i|^2\), this proves the
-claimed bounds for \(\II_i\) and \(d\log L_i\).
+a factor (1/theta). Thus
+(sum_i t_i^2/N^2leq C_nP_p/theta). Since (mathcal XcalX) contains
+((X_i-t_i)^2+N^2|II_i^{rem}|^2+N^2|dlog L_i|^2), this proves the
+claimed bounds for (II_i) and (dlog L_i).
-For \(D_i\), use
-\[
+For (D_i), use
+[
ND_i^0=(ε f_i)/(r)
- (a(\sigma_i-r)+(u_i)/(2)),\qquad
- \E \sigma_i^2\leq C(1+\E V_i^2+\Var u).
-\]
-Here the second inequality follows by solving the definition of \(V_i\)
-for \(\sigma_i-1\). Since \(r\leq1/4\),
-\[
- \sum_i(D_i^0)^2\leq
+ (a(sigma_i-r)+(u_i)/(2)),qquad
+ E sigma_i^2leq C(1+E V_i^2+Var u).
+]
+Here the second inequality follows by solving the definition of (V_i)
+for (sigma_i-1). Since (rleq1/4),
+[
+ sum_i(D_i^0)^2leq
(Cε^2)/(rN^2)
- (1+\bar u^2+\E V_i^2+\Var u)
- \leq C_nP_p/\theta.
-\]
-The relation \(ND_i=ND_i^0-\beta_i f_i+err_i\) finishes the
-estimate: \(\sum_i\beta_i^2f_i^2\leq\mathcal X\), and one chooses \(\sum_i|err_i|^2\leq1\). Its contribution \(1/N^2\) is
-bounded by \(C_nP_p\) using the first inequality of
-(eq:entropy-last-surplus), \(ε\geq1\), and \(r\leq1/4\).
+ (1+barbar u^2+u^2+E V_i^2+Var u)
+ leq C_nP_p/theta.
+]
+The relation (ND_i=ND_i^0-beta_i f_i+err_i) finishes the
+estimate: (sum_ibeta_i^2f_i^2leqmathcal XcalX), and one chooses (sum_i|err_i|^2leq1). Its contribution (1/N^2) is
+bounded by (C_nP_p) using the first inequality of
+(eq:entropy-last-surplus), (εgeq1), and (rleq1/4).
-The step size has two jobs: the scalar losses \(O_n(l^2)\) must be
-summable, and the number of support enlargements before reaching \(E\)
-must grow at most cubically in \(\log E\). Since one sweep changes
-\(\log(1+ε)\) by a quantity comparable to \(l\), one takes
- l(ε)=(l_*)/((1+\log(1+ε))^2),
+The step size has two jobs: the scalar losses (O_n(l^2)) must be
+summable, and the number of support enlargements before reaching (E)
+must grow at most cubically in (log E). Since one sweep changes
+(log(1+ε)) by a quantity comparable to (l), one takes
+ l(ε)=(l_*)/((1+log(1+ε))^2),
-where the dimensional \(l_*>0\) is chosen below \(l_0\) from
+where the dimensional (l_*>0) is chosen below (l_0) from
Lemma~lem:entropy-increment. Thus all the parameter choices in
-that lemma, including \(\theta=D_*l\), apply.
+that lemma, including (theta=D_*l), apply.
The fixed exponents in the next two lemmas leave room, respectively,
for the number of sweeps, the conformal scalar cost, the outgoing
volume estimate, and the final gradient target:
-\[
- 0.26=\tfrac14+0.01,\qquad
- 0.26+0.04<(1+0.32)/(4),\qquad
- 0.26+\tfrac34(0.04)<0.32,\qquad
- 0.02>\tfrac1{100}.
-\]
+[
+ 0.26=tfrac14+0.01,qquad
+ 0.26+0.04<(1+0.32)/(4),qquad
+ 0.26+tfrac34(0.04)<0.32,qquad
+ 0.02>tfrac1{100}.
+]
The estimates below show how each margin is used.
[Accumulation of entropy passes]
-Suppose at parameter \(ε_{in}\geqε_*\) that
-\[
- \wscal(h,\phi)\geq-C+2W-n\logε_{in}
+Suppose at parameter (ε_{in}geqε_*) that
+[
+ wscal(h,phi)geq-C+2W-nlogε_{in}
+ε_{in}e_h(F)+P_{in},
- \qquad W=\log(dV_h/\rho)\geq0,\quad P_{in}\geq0.
-\]
+ qquad W=log(dV_h/rho)geq0,quad P_{in}geq0.
+]
Successive sweeps of Lemma~lem:entropy-increment, with
-\(ε_{new}=(1+l(ε))ε\), preserve this
-form of inequality: at the final parameter \(E\) the surplus is
-\(P=P_{in}+\sum_pP_p\), the volume logarithm is
-\(W_{new}=W+\sum_pW_p\), and the increase of the scalar
-constant is \(C_n\sum_p l(ε_p)^2\). Along the entire increasing
+(ε_{new}=(1+l(ε))ε), preserve this
+form of inequality: at the final parameter (E) the surplus is
+(P=P_{in}+sum_pP_p), the volume logarithm is
+(W_{new}=W+sum_pW_p), and the increase of the scalar
+constant is (C_nsum_p l(ε_p)^2). Along the entire increasing
sequence of parameters,
- \sum_p l(ε_p)^2\leq C_n,\qquad
- \#\{p:ε_p<E\}\leq C_n(1+\log^3(1+E)).
+ sum_p l(ε_p)^2leq C_n,qquad
+ #{p:ε_p<E}leq C_n(1+log^3(1+E)).
If the initial volume logarithm is zero, then, for all sufficiently
-large dimensional \(E\),
+large dimensional (E),
- W_{new}\leq E^{0.26}(E+P)^{0.75}.
+ W_{new}leq E^{0.26}(E+P)^{0.75}.
-The same assertion holds if the initial data have parameter \(E_0\),
-surplus \(P_0\), and
-\[
- W\leq E_0^{0.32}(E_0+P_0)^{0.75},
-\]
-provided \(E\geq E_0\) is sufficiently large dimensionally and
-\(E_0^{0.32}\leq\tfrac12E^{0.26}\).
+The same assertion holds if the initial data have parameter (E_0),
+surplus (P_0), and
+[
+ Wleq E_0^{0.32}(E_0+P_0)^{0.75},
+]
+provided (Egeq E_0) is sufficiently large dimensionally and
+(E_0^{0.32}leqtfrac12E^{0.26}).
Proof.
The accumulation assertion follows by adding
(eq:entropy-increment). The energy coefficients telescope
multiplicatively, and
-\(\sum_p\log(1+l(ε_p))=\log(E/ε_{in})\).
+(sum_plog(1+l(ε_p))=log(E/ε_{in})).
The density identity in (eq:entropy-sweep-density) gives the
-sum of the \(W_p\).
+sum of the (W_p).
-Put \(t_p=\log(1+ε_p)\). Since
-\(ε_p\geqε_*>0\) and \(l_*\) is fixed small,
-\[
+Put (t_p=log(1+ε_p)). Since
+(ε_pgeqε_*>0) and (l_*) is fixed small,
+[
t_{p+1}-t_p
- =\log(1+l(ε_p)(ε_p)/(1+ε_p))
- \asymp_n l(ε_p)
+ =log(1+l(ε_p)(ε_p)/(1+ε_p))
+ asymp_n l(ε_p)
=(l_*)/((1+t_p)^2).
-\]
-Comparison with the integrals of \((1+t)^2\) and \((1+t)^{-2}\)
+]
+Comparison with the integrals of ((1+t)^2) and ((1+t)^{-2})
respectively proves the number bound and
-\[
- \sum_p l(ε_p)^2
- \leq C_n l_*\int_{t_0}^{∞}(dt)/((1+t)^2)\leq C_n.
-\]
-The comparison is valid on each interval \([t_p,t_{p+1}]\), since
+[
+ sum_p l(ε_p)^2
+ leq C_n l_*int_{t_0}^{∞}(dt)/((1+t)^2)leq C_n.
+]
+The comparison is valid on each interval ([t_p,t_{p+1}]), since
its endpoints differ by a uniformly bounded small number.
-Let \(k\) be the number of sweeps in the group under consideration.
+Let (k) be the number of sweeps in the group under consideration.
By Lemma~lem:entropy-surplus, since
-\(ε_p\leq E\) and \(P_p\leq P\),
-\[
- \sum_pW_p\leq C_n k E^{1/4}(E+P)^{3/4}.
-\]
-For all sufficiently large \(E\),
-\(C_n k\leq \tfrac12 E^{0.01}\) by
+(ε_pleq E) and (P_pleq P),
+[
+ sum_pW_pleq C_n k E^{1/4}(E+P)^{3/4}.
+]
+For all sufficiently large (E),
+(C_n kleq tfrac12 E^{0.01}) by
(eq:entropy-summability). This proves
(eq:entropy-phase-volume) for zero initial volume logarithm.
-In the other case \(P\geq P_0\) and \(E\geq E_0\), so the initial
-term is at most \(E_0^{0.32}(E+P)^{0.75}\). The stated condition on
-\(E_0\) pays for this with the other half of
-\(E^{0.26}(E+P)^{0.75}\).
+In the other case (Pgeq P_0) and (Egeq E_0), so the initial
+term is at most (E_0^{0.32}(E+P)^{0.75}). The stated condition on
+(E_0) pays for this with the other half of
+(E^{0.26}(E+P)^{0.75}).
Call a finite group of these sweeps a phase. There is no
conformal change inside a phase. Each sweep adds the nonnegative
-tensor in (eq:sweep-output); hence, if \(h^0\) is its starting
-metric and \(h_+\) its final metric, the successive metrics and their
+tensor in (eq:sweep-output); hence, if (h^0) is its starting
+metric and (h_+) its final metric, the successive metrics and their
inverses satisfy
-\[
- h^0\leq h^{(1)}\leq...\leq h_+,\qquad
- (h_+)^{-1}\leq...\leq(h^{(1)})^{-1}\leq(h^0)^{-1}.
-\]
+[
+ h^0leq h^{(1)}leq...leq h_+,qquad
+ (h_+)^{-1}leq...leq(h^{(1)})^{-1}leq(h^0)^{-1}.
+]
In particular, the starting metric is an admissible reference metric
for every sweep in the phase, and
-\(e_{h_+}(Y^0)\leq e_{h^0}(Y^0)\) for its starting map \(Y^0\).
+(e_{h_+}(Y^0)leq e_{h^0}(Y^0)) for its starting map (Y^0).
Now finish a phase with an absolute gradient bound. Normalizing
-the last bumps alone divides by \(\sqrt r\), which can make the
-gradient large where \(r\) is small. First keep a small copy of
+the last bumps alone divides by (sqrt r), which can make the
+gradient large where (r) is small. First keep a small copy of
the starting map in the normalization; its gradient is controlled by
the preceding metric comparison. A conformal enlargement confined
-to the small-\(r\) region then reduces the remaining gradient there,
+to the small-(r) region then reduces the remaining gradient there,
and the last sweep's surplus pays its cost.
[Exit from a phase]
-Consider a phase starting with root map \(Y^0\) and metric \(h^0\), and
-let \(K_0\) be a finite uniform bound for \(e_{h^0}(Y^0)^{1/2}\).
-Suppose its last sweep has input parameter \(ε\), output
-parameter \(E=(1+l(ε))ε\), and output
-\((h_+,\phi_+,Y)\). Suppose, with \(C\geq0\) and \(P\geq P_p\) including the
+Consider a phase starting with root map (Y^0) and metric (h^0), and
+let (K_0) be a finite uniform bound for (e_{h^0}(Y^0)^{1/2}).
+Suppose its last sweep has input parameter (ε), output
+parameter (E=(1+l(ε))ε), and output
+((h_+,phi_+,Y)). Suppose, with (Cgeq0) and (Pgeq P_p) including the
surplus from that last sweep, that
- \wscal(h_+,\phi_+)&\geq-C+2W-n\log E+E e_{h_+}(Y)+P,\\
- W&=\log(dV_{h_+}/\rho)\geq0,\qquad
- W\leq E^{0.26}(E+P)^{0.75}.
+ wscal(h_+,phi_+)&geq-C+2W-nlog E+E e_{h_+}(Y)+P,
+ W&=log(dV_{h_+}/rho)geq0,qquad
+ Wleq E^{0.26}(E+P)^{0.75}.
-If \(E\) is sufficiently large in terms of \(n,K_0\), there are a
-metric \(\widehat h\), a weight \(\widehat\phi\), and a root map \(Z\)
+If (E) is sufficiently large in terms of (n,K_0), there are a
+metric (widehatwidehat hh), a weight (widehatphi), and a root map (Z)
on two copies of the labels such that
- \wscal(\widehat h,\widehat\phi)
- &\geq-(C+C_n)+2\widehat W-n\log E
- +E e_{\widehat h}(Z)+P',\\
- \widehat W&=\log(dV_{\widehat h}/\rho)\geq0,\qquad
- \widehat W\leq E^{0.32}(E+P')^{0.75},\\
- e_{\widehat h}(Z)^{1/2}&\leq E^{-1/100}.
+ wscal(widehatwidehat h,h,widehatphi)
+ &geq-(C+C_n)+2widehatwidehat W-nW-nlog E
+ +E e_{widehatwidehat h}h(Z)+P',
+ widehatwidehat W&=W&=log(dV_{widehatwidehat h}h/rho)geq0,qquad
+ widehatwidehat WWleq E^{0.32}(E+P')^{0.75},
+ e_{widehatwidehat h}h(Z)^{1/2}&leq E^{-1/100}.
-Here \(P'\geq0\), and the added scalar cost \(C_n\) is independent
-of \(K_0\) and of all other instance constants. The changes preserve
+Here (P'geq0), and the added scalar cost (C_n) is independent
+of (K_0) and of all other instance constants. The changes preserve
the uniform smoothness and support properties of
Theorem~thm:entropy.
Proof.
-In this proof all derivatives, unless otherwise indicated, use \(h_+\).
-Write \(r=\sum_i f_i^2\) for the last sweep, set \(s=\sqrt r\), and put
-\[
- c_E=\frac1{\sqrt E(1+K_0)},\qquad
- Z=\frac{(f,c_EY^0)}{√(r+c_E^2)}.
-\]
+In this proof all derivatives, unless otherwise indicated, use (h_+).
+Write (r=sum_i f_i^2) for the last sweep, set (s=sqrt r), and put
+[
+ c_E=frac1{sqrt E(1+K_0)},qquad
+ Z=frac{(f,c_EY^0)}{√(r+c_E^2)}.
+]
The two vectors in the numerator use disjoint copies of the labels.
The map is a smooth root probability map because its denominator is
-at least \(c_E>0\). Writing the copies as \((i,0)\) and \((i,1)\),
-one calls \(i\) their common parent label at the start of the phase;
+at least (c_E>0). Writing the copies as ((i,0)) and ((i,1)),
+one calls (i) their common parent label at the start of the phase;
the sweeps within the phase do not change the label set. The first
copy is supported on the last bumps and the second on the starting
map. This parent assignment will be used to compare support overlaps
between phases. By (eq:entropy-trace),
-\[
- N^2\sum_i|df_i|_{h_+}^2=m_f\leq n,\qquad
- |ds|_{h_+}^2\leq n/N^2.
-\]
-Monotonicity of the phase metrics also gives \(e_{h_+}(Y^0)\leq K_0^2\).
+[
+ N^2sum_i|df_i|_{h_+}^2=m_fleq n,qquad
+ |ds|_{h_+}^2leq n/N^2.
+]
+Monotonicity of the phase metrics also gives (e_{h_+}(Y^0)leq K_0^2).
-Since \(Y\) and \(Y^0\) are unit vectors in orthogonal spaces, their
+Since (Y) and (Y^0) are unit vectors in orthogonal spaces, their
derivatives are perpendicular to their respective vectors. Differentiating
-\(Z=(sY,c_EY^0)/√(s^2+c_E^2)\) thus gives the exact identity
-\[
- e(Z)=\frac r{r+c_E^2}e(Y)+(c_E^2)/(r+c_E^2)e(Y^0)
+(Z=(sY,c_EY^0)/√(s^2+c_E^2)) thus gives the exact identity
+[
+ e(Z)=frac r{r+c_E^2}e(Y)+(c_E^2)/(r+c_E^2)e(Y^0)
+(c_E^2)/((r+c_E^2)^2)|ds|^2.
-\]
-For \(r\geq1/4\), it follows that
-\[
- E(e(Z)-e(Y))\leq
+]
+For (rgeq1/4), it follows that
+[
+ E(e(Z)-e(Y))leq
(Ec_E^2K_0^2)/(r+c_E^2)
+(Ec_E^2n)/(N^2(r+c_E^2)^2)
- \leq 4+(16n)/(N^2)\leq C_n
-\]
-when \(E\) is large, since \(N^2=lε\) tends to infinity under
-(eq:entropy-step-schedule). For \(r\leq1/4\), the norm of the
-derivative of normalization is at most \(1/√(r+c_E^2)\), so
+ leq 4+(16n)/(N^2)leq C_n
+]
+when (E) is large, since (N^2=lε) tends to infinity under
+(eq:entropy-step-schedule). For (rleq1/4), the norm of the
+derivative of normalization is at most (1/√(r+c_E^2)), so
- e(Z)\leq(n/N^2+c_E^2K_0^2)/(r+c_E^2).
+ e(Z)leq(n/N^2+c_E^2K_0^2)/(r+c_E^2).
The first inequality of (eq:entropy-last-surplus) now implies
-\[
- E e(Z)\leq (C_n)/(l(r+c_E^2))
- \leq (C_n)/(ε)P_p.
-\]
-Thus replacing \(Y\) by \(Z\) in
+[
+ E e(Z)leq (C_n)/(l(r+c_E^2))
+ leq (C_n)/(ε)P_p.
+]
+Thus replacing (Y) by (Z) in
(eq:entropy-before-exit) costs a dimensional constant on
-\(\{r\geq1/4\}\) and an arbitrarily small fraction of \(P_p\) on
-\(\{r\leq1/4\}\), once \(E\) is dimensionally large. In particular,
-no factor \(K_0\) occurs in this scalar cost.
+({rgeq1/4}) and an arbitrarily small fraction of (P_p) on
+({rleq1/4}), once (E) is dimensionally large. In particular,
+no factor (K_0) occurs in this scalar cost.
-Choose a smooth function \(u=u(r)\geq0\), supported in
-\(\{r\leq E^{-0.32}\}\), with
+Choose a smooth function (u=u(r)geq0), supported in
+({rleq E^{-0.32}}), with
- e^u\asymp
- \min(E^{0.02},\max(1,E^{-0.16}/\sqrt r)),
- \qquad |(du)/(dr)|\leq\frac C r.
+ e^uasymp
+ min(E^{0.02},max(1,E^{-0.16}/sqrt r)),
+ qquad |(du)/(dr)|leqfrac C r.
For example, smooth, with bounded-width transitions on the logarithmic
scale, the function obtained by truncating
-\(-\tfrac12\log r-0.16\log E\) to the interval
-\([0,0.02\log E]\). The smoothing may vanish whenever the untruncated
+(-tfrac12log r-0.16log E) to the interval
+([0,0.02log E]). The smoothing may vanish whenever the untruncated
function is nonpositive, and changes its value by a bounded amount;
this gives both the support assertion and the comparison in
(eq:entropy-u-choice). Define
-\[
- \widehat h=e^{2u}h_+,\qquad
- \widehat\phi=\phi_+-(n-1)u,\qquad
- \widehat W=W+nu.
-\]
-In particular \(dV_{\widehat h}=e^{nu}dV_{h_+}\), so this is the
-claimed volume logarithm. Differentiating \(r\) gives
-\[
- |dr|\leq2\sqrt r(\sum_i|df_i|^2)^{1/2}
- \leq\frac{2√(nr)}N,\qquad
- |du|\leq(C_n)/(N\sqrt r).
-\]
-The coefficient \(n-1\) in the new weight cancels the Laplacian of
-\(u\) in the weighted conformal formula. With
-\(b=\nabla_{h_+}\phi_+\), that formula is
-
- \wscal(\widehat h,\widehat\phi)
- =e^{-2u}[\wscal(h_+,\phi_+)+2b(u)-(n-1)|du|^2].
-
-
-With the fixed density \(\rho\), a graph sweep changes \(W\) by
-\(+W_p\) and \(\phi\) by \(-W_p\), so it preserves \(\phi+W\).
+[
+ widehatwidehat h=e^h=e^{2u}h_+,qquad
+ widehatphi=phi_+-(n-1)u,qquad
+ widehatwidehat W=W+nu.W=W+nu.
+]
+In particular (dV_{widehatwidehat h}h=e^{nu}dV_{h_+}), so this is the
+claimed volume logarithm. Differentiating (r) gives
+[
+ |dr|leq2sqrt r(sum_i|df_i|^2)^{1/2}
+ leqfrac{2√(nr)}N,qquad
+ |du|leq(C_n)/(Nsqrt r).
+]
+The coefficient (n-1) in the new weight cancels the Laplacian of
+(u) in the weighted conformal formula. With
+(b=nabla_{h_+}phi_+), that formula is
+
+ wscal(widehatwidehat h,h,widehatphi)
+ =e^{-2u}[wscal(h_+,phi_+)+2b(u)-(n-1)|du|^2].
+
+
+With the fixed density (rho), a graph sweep changes (W) by
+(+W_p) and (phi) by (-W_p), so it preserves (phi+W).
The present exit instead satisfies
-\[
- \widehat\phi+\widehat W=\phi_++W+u.
-\]
-Thus after an exit \(W\) need not equal \(-\phi\), and its volume
+[
+ widehatphi+widehatwidehat W=W=phi_++W+u.
+]
+Thus after an exit (W) need not equal (-phi), and its volume
bound supplies no bound for the gradient of the weight. Need a
-bound on \(b(u)\) which uses no bound on the size of \(b\).
-It is enough to estimate this on the support of \(u\), which lies in
-\(\{r\leq1/4\}\) for large \(E\). At such a point view all the graphs
+bound on (b(u)) which uses no bound on the size of (b).
+It is enough to estimate this on the support of (u), which lies in
+({rleq1/4}) for large (E). At such a point view all the graphs
of the last sweep in the common
-Euclidean extension of \((T_x\widetilde M,h_0)\) by their graph
+Euclidean extension of ((T_xwidetilde M,h_0)) by their graph
directions, one for each active color at that point. Write
-\(\mathbf n_i\) for the successive unit normals. They are
+(mathbf n_ibfn_i) for the successive unit normals. They are
orthonormal: a later
normal lies in the preceding graph plus its new vertical direction
and is perpendicular to every earlier normal. In this space, each
gradient is embedded through its graph and padded by zero in the
-unused later directions. Writing \(b_i\) for the embedded gradient
-of \(\phi_i\), orthogonal projection to the new graph gives the exact
+unused later directions. Writing (b_i) for the embedded gradient
+of (phi_i), orthogonal projection to the new graph gives the exact
recurrence
-\[
- b_i=b_{i-1}-< b_{i-1},\mathbf n_i>\mathbf n_i
- -\nabla_{h_i}\log L_i,
-\]
+[
+ b_i=b_{i-1}-< b_{i-1},mathbf n_i>bfn_i>mathbf n_ibfn_i
+ -nabla_{h_i}log L_i,
+]
where the last gradient is embedded in the same space. The normal
coefficient is
-\(< b_{i-1},\mathbf n_i>=-U_i(\phi_{i-1})\), and
+(< b_{i-1},mathbf n_i>=-U_i(bfn_i>=-U_i(phi_{i-1})), and
its magnitude satisfies
-\[
- |U_i(\phi_{i-1})|\leq |D_i|+\sqrt n\,|\II_i|,
-\]
-because \(D_i=\tr\II_i-U_i(\phi_{i-1})\).
+[
+ |U_i(phi_{i-1})|leq |D_i|+sqrt n,|II_i|,
+]
+because (D_i=trII_i-U_i(phi_{i-1})).
By orthonormality and (eq:entropy-last-surplus), the norm of
the sum of these normal corrections is at most
-\(C_n√(P_p/\theta)\).
+(C_n√(P_p/theta)).
For the subtracted tangent gradients, split the indices according as
-\(A_i\leq1/2\) or \(A_i>1/2\). In the first set
-\(|d\log L_i|\leq C A_i|\II_i|\); in the second there are at most
-\(4m\) indices. Cauchy--Schwarz and
+(A_ileq1/2) or (A_i>1/2). In the first set
+(|dlog L_i|leq C A_i|II_i|); in the second there are at most
+(4m) indices. Cauchy--Schwarz and
(eq:entropy-last-surplus) consequently bound the sum of their
-norms by \(C_n√(mP_p/\theta)\). The final tangent embedding sends
-a vector \(v\) to \((v,(Ndf_i(v))_i)\). Summing the recurrence, the
+norms by (C_n√(mP_p/theta)). The final tangent embedding sends
+a vector (v) to ((v,(Ndf_i(v))_i)). Summing the recurrence, the
initial gradient has no vertical component in this common extension,
while the vertical component of the final gradient is
-\((Ndf_i(b))_i\).
+((Ndf_i(b))_i).
The normal and tangent bounds therefore prove
-\[
- \sum_i(Ndf_i(b))^2\leq C_n(1+m)P_p/\theta.
-\]
+[
+ sum_i(Ndf_i(b))^2leq C_n(1+m)P_p/theta.
+]
It follows that
-\[
- |b(u)|\leq\frac C r|b(r)|
- \leq C_n(((1+m)P_p)/(\theta N^2r))^{1/2}
- \leq \frac{C_nP_p}{√(\theta E)}.
-\]
-For the last inequality use \(E\asympε\), \(N^2=lε\),
+[
+ |b(u)|leqfrac C r|b(r)|
+ leq C_n(((1+m)P_p)/(theta N^2r))^{1/2}
+ leq frac{C_nP_p}{√(theta E)}.
+]
+For the last inequality use (Easympε), (N^2=lε),
and
-\[
- P_p\geq(c_n)/(lr)
+[
+ P_pgeq(c_n)/(lr)
(ε+(m^2)/(ε))
- \geq (c'_n(1+m))/(lr)
-\]
-for \(ε\geq1\). The same first surplus bound gives
+ geq (c'_n(1+m))/(lr)
+]
+for (εgeq1). The same first surplus bound gives
- |du|^2\leq(C_n)/(N^2r)\leq(C_nP_p)/(E^2).
+ |du|^2leq(C_n)/(N^2r)leq(C_nP_p)/(E^2).
-All these estimates are only needed where \(u\ne0\), a subset of
-\(\{r\leq1/4\}\) for large \(E\). Since
-\(\theta E\asymp_n E/(1+\log E)^2\) tends to infinity, the two
+All these estimates are only needed where (une0), a subset of
+({rleq1/4}) for large (E). Since
+(theta Easymp_n E/(1+log E)^2) tends to infinity, the two
derivative terms in (eq:entropy-conformal-exit) cost an
-arbitrarily small fraction of \(P_p\).
+arbitrarily small fraction of (P_p).
There is enough surplus on this region also for the change in the
-entropy term. From \(r\leq E^{-0.32}\) and
+entropy term. From (rleq E^{-0.32}) and
(eq:entropy-last-surplus),
-\[
- P\geq P_p\geq c_n E^{1.32}.
-\]
-Using (eq:entropy-before-exit) and \(e^{2u}\leq C E^{0.04}\),
+[
+ Pgeq P_pgeq c_n E^{1.32}.
+]
+Using (eq:entropy-before-exit) and (e^{2u}leq C E^{0.04}),
one obtains there
- \frac{e^{2u}W}{P}\leq
- C_nE^{0.04+0.26}P^{-0.25}\leq C_nE^{-0.03},
- \qquad
- \frac{e^{2u}nu}{P}\leq C_nE^{-1.28}\log E.
+ frac{e^{2u}W}{P}leq
+ C_nE^{0.04+0.26}P^{-0.25}leq C_nE^{-0.03},
+ qquad
+ frac{e^{2u}nu}{P}leq C_nE^{-1.28}log E.
-In the first bound use was made of \(E+P\leq C_nP\), valid for large \(E\)
+In the first bound use was made of (E+Pleq C_nP), valid for large (E)
on this region. Both quantities tend to zero with dimensional
constants.
-Now assemble these estimates. After replacing \(Y\) by \(Z\), write
-the retained surplus as \(\alpha_n P\), where \(\alpha_n>0\) is fixed
-dimensionally. Put \(B_u=2|b(u)|+(n-1)|du|^2\). Substituting in
+Now assemble these estimates. After replacing (Y) by (Z), write
+the retained surplus as (alpha_n P), where (alpha_n>0) is fixed
+dimensionally. Put (B_u=2|b(u)|+(n-1)|du|^2). Substituting in
(eq:entropy-conformal-exit) and moving the new entropy term
to the left gives
- &\wscal(\widehat h,\widehat\phi)
- -E e_{\widehat h}(Z)-2(W+nu)+n\log E\\
- &\quad\geq -(C+C_n)
- +e^{-2u}[\alpha_n P-B_u
+ &wscal(widehatwidehat h,h,widehatphi)
+ -E e_{widehatwidehat h}h(Z)-2(W+nu)+nlog E
+ &quadgeq -(C+C_n)
+ +e^{-2u}[alpha_n P-B_u
-2(e^{2u}-1)W-2nu e^{2u}]
- +(1-e^{-2u})n\log E .
+ +(1-e^{-2u})nlog E .
-Use was made of \(e^{-2u}(C+C_n)\leq C+C_n\). The last term is nonnegative
-for \(E\geq1\). Equations (eq:entropy-exit-derivatives) and
+Use was made of (e^{-2u}(C+C_n)leq C+C_n). The last term is nonnegative
+for (Egeq1). Equations (eq:entropy-exit-derivatives) and
(eq:entropy-exit-volume-cost) leave at least a fixed fraction
-of \(P\) in the bracket on the support of \(u\); off that support the
+of (P) in the bracket on the support of (u); off that support the
volume and derivative costs vanish. Thus set
-\[
- P'=\alpha'_n e^{-2u}P
-\]
-for a fixed \(\alpha'_n>0\), proving the scalar inequality in
+[
+ P'=alpha'_n e^{-2u}P
+]
+for a fixed (alpha'_n>0), proving the scalar inequality in
(eq:entropy-after-exit). The only scalar constant charged was
the dimensional replacement cost.
The volume estimate at exit follows as well. Since
-\(P\leq C_ne^{2u}P'\leq C_nE^{0.04}P'\) and \(nu\leq C_n\log E\),
-\[
- \widehat W\leq
- C_n E^{0.26+0.04(0.75)}(E+P')^{0.75}+C_n\log E
- \leq E^{0.32}(E+P')^{0.75}
-\]
-for all sufficiently large dimensional \(E\).
+(Pleq C_ne^{2u}P'leq C_nE^{0.04}P') and (nuleq C_nlog E),
+[
+ widehatwidehat WWleq
+ C_n E^{0.26+0.04(0.75)}(E+P')^{0.75}+C_nlog E
+ leq E^{0.32}(E+P')^{0.75}
+]
+for all sufficiently large dimensional (E).
Finally, (eq:entropy-Z-energy) and the conformal scaling of
energy give
-\[
- e_{\widehat h}(Z)^{1/2}
- \leq e^{-u}\frac{\sqrt n/N+c_EK_0}{√(r+c_E^2)}.
-\]
+[
+ e_{widehatwidehat h}h(Z)^{1/2}
+ leq e^{-u}frac{sqrt n/N+c_EK_0}{√(r+c_E^2)}.
+]
The choice (eq:entropy-u-choice) implies
-\[
- \frac{e^{-u}}{√(r+c_E^2)}
- \leq C(\frac{E^{-0.02}}{c_E}+E^{0.16}).
-\]
-Using \(\sqrt E/N\leq C_n(1+\log E)\), \(c_EK_0\leq E^{-1/2}\),
-and \(N\leq\sqrt E\), one concludes that
-
- e_{\widehat h}(Z)^{1/2}
- \leq C_n(
- \frac{(1+K_0)(1+\log E)}{E^{0.02}}+\frac{E^{0.16}}{N}).
-
-The first summand is \(o(E^{-0.01})\) for fixed \(K_0\); the second is
-\(O_n((1+\log E)E^{-0.34})\). Taking \(E\) large enough in terms of
-\(n,K_0\) proves the final inequality in
-(eq:entropy-after-exit). The functions \(u\) and the normalizing
+[
+ frac{e^{-u}}{√(r+c_E^2)}
+ leq C(frac{E^{-0.02}}{c_E}+E^{0.16}).
+]
+Using (sqrt E/Nleq C_n(1+log E)), (c_EK_0leq E^{-1/2}),
+and (Nleqsqrt E), one concludes that
+
+ e_{widehatwidehat h}h(Z)^{1/2}
+ leq C_n(
+ frac{(1+K_0)(1+log E)}{E^{0.02}}+frac{E^{0.16}}{N}).
+
+The first summand is (o(E^{-0.01})) for fixed (K_0); the second is
+(O_n((1+log E)E^{-0.34})). Taking (E) large enough in terms of
+(n,K_0) proves the final inequality in
+(eq:entropy-after-exit). The functions (u) and the normalizing
denominator are smooth with finite bounds in each fixed construction.
-The supports of \(Z\) are those of the last bumps and of \(Y^0\).
+The supports of (Z) are those of the last bumps and of (Y^0).
Thus the asserted uniformity and compact support properties persist.
Proof. [Proof of Theorem~thm:entropy]
-Fix parameters \(H,A,B\) for the prescribed elementary bound and set
-\(J=H+2\). This number of phases is fixed before any phase target
+Fix parameters (H,A,B) for the prescribed elementary bound and set
+(J=H+2). This number of phases is fixed before any phase target
is chosen. The first target will be chosen after the bootstrap data
and their uniform constants have been fixed; the estimates below
hold for every sufficiently large choice of that target.
Apply Lemma~lem:entropy-bootstrap. At its output
-\(W=\log(dV_{h_*}/\rho)=0\), \(\phi=0\), and
-\(ε_0\in[ε_*,2ε_*]\). Increasing a dimensional
+(W=log(dV_{h_*}/rho)=0), (phi=0), and
+(ε_0in[ε_*,2ε_*]). Increasing a dimensional
constant once, (eq:entropy-bootstrap-output) has the form
-\[
- \wscal(h_*,0)\geq-C_n+2W-n\logε_0
+[
+ wscal(h_*,0)geq-C_n+2W-nlogε_0
+ε_0 e_{h_*}(F_*),
-\]
-because \(ε_0\) ranges in a fixed dimensional interval.
+]
+because (ε_0) ranges in a fixed dimensional interval.
Use the schedule (eq:entropy-step-schedule) from now on.
The bound (eq:entropy-summability) applies to the entire
increasing sequence of energy coefficients, even when it is
partitioned into phases.
Choose a large first target and run the first phase until the parameter
-first reaches it; call the actual output parameter \(E_1\). The
-overshoot is at most a factor \(1+l_*\). In every sweep of this phase
-use \(h_*\) as the reference metric. The possible support enlargement
+first reaches it; call the actual output parameter (E_1). The
+overshoot is at most a factor (1+l_*). In every sweep of this phase
+use (h_*) as the reference metric. The possible support enlargement
per sweep, including its separation buffers, has a dimensional radius
in this reference metric by Proposition~prop:sweep. The total
-radius is therefore at most \(C_n(1+\log^3(1+E_1))\).
+radius is therefore at most (C_n(1+log^3(1+E_1))).
Record a bound for a single fixed graph sufficient for every
coloring in this phase and for all cooccurrences at its exit. All
-data before the choice of the first target, including \(h_*\) and its
-uniform comparison with \(g\), have already been fixed. The lifted
-metric \(h_*\) is at least a positive input-dependent multiple of \(g\).
-For a fixed finite generating set of \(\Gamma\), cocompactness implies
+data before the choice of the first target, including (h_*) and its
+uniform comparison with (g), have already been fixed. The lifted
+metric (h_*) is at least a positive input-dependent multiple of (g).
+For a fixed finite generating set of (Gamma), cocompactness implies
that word length is bounded above by an affine input-dependent
-function of lifted \(g\)-distance between translates of fixed compact
+function of lifted (g)-distance between translates of fixed compact
fundamental sets. For instance, subdividing a curve into bounded
length pieces and following the finitely many translates meeting
successive covering sets gives this bound. The number of group
-elements of word length at most \(L\) is at most exponential in \(L\).
+elements of word length at most (L) is at most exponential in (L).
It follows that the graph joining every pair of labels whose possible
enlarged supports in the first phase meet, or come within a required
buffer, has degree at most
- \Delta_1+1\leq
- \exp(C_{inst}(1+\log E_1)^3).
+ Delta_1+1leq
+ exp(C_{inst}(1+log E_1)^3).
-Here \(C_{inst}\) is finite and fixed before the
+Here (C_{inst}) is finite and fixed before the
first target is chosen. It includes the initial number of label
orbits and the compact support bounds. The two copies of labels
used at exit and their possible cooccurrences change this bound only
-by a fixed factor, which is absorbed in \(C_{inst}\).
+by a fixed factor, which is absorbed in (C_{inst}).
-Take \(E_1\) sufficiently large for
+Take (E_1) sufficiently large for
Lemma~lem:entropy-accumulation and
-Lemma~lem:entropy-phase-exit; in the latter the initial \(K_0\)
-is the finite input-dependent gradient bound of \(F_*\).
+Lemma~lem:entropy-phase-exit; in the latter the initial (K_0)
+is the finite input-dependent gradient bound of (F_*).
Now there holds the scalar inequality, the volume bound with exponent
-\(0.32\), and the absolute gradient bound at \(E_1\).
+(0.32), and the absolute gradient bound at (E_1).
Denote the exit metric, map, and fixed cooccurrence graph by
-\(h_1,F_1,\mathcal G_1\).
+(h_1,F_1,mathcal G_1calG_1).
Next isolate how a small gradient controls the graph in a later
-phase. Suppose a phase starts with metric \(h_j\), root map \(F_j\)
-with \(e_{h_j}(F_j)^{1/2}\leq K\), and a fixed graph \(\mathcal G_j\)
-containing all its possible cooccurrences. Let \(R\) bound the
+phase. Suppose a phase starts with metric (h_j), root map (F_j)
+with (e_{h_j}(F_j)^{1/2}leq K), and a fixed graph (mathcal G_jcalG_j)
+containing all its possible cooccurrences. Let (R) bound the
following pairwise distance: whenever two possible enlarged supports
meet or come close enough to require separation by a buffer, there
-are points \(x,y\) in the old nonzero supports of their respective
-parent labels with \(\dist_{h_j}(x,y)\leq R\). If a closure is
+are points (x,y) in the old nonzero supports of their respective
+parent labels with (dist_{h_j}(x,y)leq R). If a closure is
involved, allow an arbitrarily small additional distance. Suppose
-\[
- KR<\sqrt2.
-\]
+[
+ KR<sqrt2.
+]
Choosing the additional distance small enough, the gradient bound gives
-\[
- \|F_j(x)-F_j(y)\|_{\ell^2}
- \leq K\dist_{h_j}(x,y)<\sqrt2.
-\]
+[
+ |F_j(x)-F_j(y)|_{ell^2}
+ leq Kdist_{h_j}(x,y)<sqrt2.
+]
Two nonnegative unit vectors with disjoint supports have distance
-exactly \(\sqrt2\). Thus \(F_j(x)\) and \(F_j(y)\) share a nonzero
-label, say \(k\). The first parent and \(k\) cooccur at \(x\), and
-the second parent and \(k\) cooccur at \(y\). The two parents are
-therefore at graph distance at most two in \(\mathcal G_j\). This
+exactly (sqrt2). Thus (F_j(x)) and (F_j(y)) share a nonzero
+label, say (k). The first parent and (k) cooccur at (x), and
+the second parent and (k) cooccur at (y). The two parents are
+therefore at graph distance at most two in (mathcal G_jcalG_j). This
argument applies separately at each transverse parameter; because
-\(\mathcal G_j\) already contains all possible cooccurrences, the
+(mathcal G_jcalG_j) already contains all possible cooccurrences, the
conclusion holds in that one fixed graph.
-After an exit at \(E_j\), one chooses a target that both meets this
+After an exit at (E_j), one chooses a target that both meets this
overlap condition and grows by one exponential level:
- T_{j+1}=\exp(E_j^{1/1000}),
+ T_{j+1}=exp(E_j^{1/1000}),
-and call the first parameter reaching it \(E_{j+1}\). Thus
-\[
- T_{j+1}\leq E_{j+1}\leq(1+l_*)T_{j+1}.
-\]
-The choice \(1/1000\) leaves the strict inequality
-\(3/1000<1/100\): the cubic logarithmic cost of support enlargement
+and call the first parameter reaching it (E_{j+1}). Thus
+[
+ T_{j+1}leq E_{j+1}leq(1+l_*)T_{j+1}.
+]
+The choice (1/1000) leaves the strict inequality
+(3/1000<1/100): the cubic logarithmic cost of support enlargement
will be smaller than the gain from the preceding gradient bound.
-For large \(E_j\) this target satisfies
-\(E_j^{0.32}\leq\tfrac12 E_{j+1}^{0.26}\). The outgoing bound with
-exponent \(0.32\) from phase \(j\) and
+For large (E_j) this target satisfies
+(E_j^{0.32}leqtfrac12 E_{j+1}^{0.26}). The outgoing bound with
+exponent (0.32) from phase (j) and
Lemma~lem:entropy-accumulation therefore give the bound with
-exponent \(0.26\) before the next exit. The next initial gradient
-bound is \(K_0\leq E_j^{-1/100}\leq1\), so all further thresholds
+exponent (0.26) before the next exit. The next initial gradient
+bound is (K_0leq E_j^{-1/100}leq1), so all further thresholds
for Lemma~lem:entropy-phase-exit are dimensional. Increasing
the first target once ensures these conclusions in every one of a
prescribed finite number of phases.
-Use \(h_j\), the starting metric, as the reference metric in phase
-\(j+1\). Let \(R_{j+1}\) bound the pairwise old-support distance in
+Use (h_j), the starting metric, as the reference metric in phase
+(j+1). Let (R_{j+1}) bound the pairwise old-support distance in
the preceding argument, including both support enlargement radii and
all separation buffers. The number of sweeps gives
-\[
- R_{j+1}\leq C_n(1+\log^3 E_{j+1})
- \leq C_n E_j^{3/1000}.
-\]
+[
+ R_{j+1}leq C_n(1+log^3 E_{j+1})
+ leq C_n E_j^{3/1000}.
+]
Consequently
- R_{j+1}E_j^{-1/100}\leq C_n E_j^{-7/1000}→0.
+ R_{j+1}E_j^{-1/100}leq C_n E_j^{-7/1000}→0.
-For large \(E_j\), the overlap condition therefore holds with
-\(K=E_j^{-1/100}\). For every relevant pair the preceding estimate
+For large (E_j), the overlap condition therefore holds with
+(K=E_j^{-1/100}). For every relevant pair the preceding estimate
becomes
-\[
- \|F_j(x)-F_j(y)\|_{\ell^2}
- \leq E_j^{-1/100}\dist_{h_j}(x,y)<\sqrt2.
-\]
+[
+ |F_j(x)-F_j(y)|_{ell^2}
+ leq E_j^{-1/100}dist_{h_j}(x,y)<sqrt2.
+]
This uses only the starting metric of the phase, without a comparison
-constant between that metric and \(g\).
+constant between that metric and (g).
-It follows that every separation graph needed in phase \(j+1\) is
-contained in the square \(\mathcal G_j^2\), after identifying labels
+It follows that every separation graph needed in phase (j+1) is
+contained in the square (mathcal G_j^2calG_j^2), after identifying labels
with their parents at the start of the phase. At exit there are two
copies of each parent. Joining two such copies whenever their parents
are equal or at distance at most two gives a fixed graph
-\(\mathcal G_{j+1}\) containing all new cooccurrences. If
-\(\Delta_j\) is the degree of \(\mathcal G_j\), this construction gives
+(mathcal G_calG_{j+1}) containing all new cooccurrences. If
+(Delta_j) is the degree of (mathcal G_jcalG_j), this construction gives
- \Delta_{j+1}+1\leq2(\Delta_j+1)^2.
+ Delta_{j+1}+1leq2(Delta_j+1)^2.
It also doubles the number of label orbits. The preliminary coloring
construction applies to these graphs before each sweep.
@@ -3398,73 +3398,73 @@ construction remain finite for the input, even though they need
not satisfy a useful quantitative bound. They do not enter
(eq:entropy-graph-recurrence).
-After \(J\) phases, apply the coloring argument to
-\(\mathcal G_J^2\), whose maximum degree is at most \(\Delta_J^2\).
-Take \(q\) to be the maximum of its degree plus one, the number of
-label orbits, and \(n+1\). Equations (eq:entropy-first-graph) and
+After (J) phases, apply the coloring argument to
+(mathcal G_J^2calG_J^2), whose maximum degree is at most (Delta_J^2).
+Take (q) to be the maximum of its degree plus one, the number of
+label orbits, and (n+1). Equations (eq:entropy-first-graph) and
(eq:entropy-graph-recurrence) imply
- \log q\leq C_{inst,J,n}(1+\log E_1)^3.
+ log qleq C_{inst,J,n}(1+log E_1)^3.
-For fixed \(J,n\) the coefficient here is fixed before \(E_1\) is
-chosen. The graph \(\mathcal G_J\) and this coloring have exactly
+For fixed (J,n) the coefficient here is fixed before (E_1) is
+chosen. The graph (mathcal G_JcalG_J) and this coloring have exactly
the properties asserted in the theorem.
It remains to make the first target large enough for the already fixed
-\(J=H+2\). Let
-\(\mathcal B(q)\leq\exp^{[H]}(B(1+q)^A)\) be the prescribed
+(J=H+2). Let
+(mathcal B(q)calB(q)leqexp^{[H]}(B(1+q)^A)) be the prescribed
elementary bound. Equation (eq:entropy-q-growth) implies
-\[
- \mathcal B(q)\leq
- \exp^{[H+1]}(C'_{inst,J,n,A,B}
- (1+\log E_1)^3)
-\]
-for large \(E_1\). On the other hand, the recurrence
-(eq:entropy-phase-targets) gives, for each fixed \(J\geq2\),
+[
+ mathcal B(q)calB(q)leq
+ exp^{[H+1]}(C'_{inst,J,n,A,B}
+ (1+log E_1)^3)
+]
+for large (E_1). On the other hand, the recurrence
+(eq:entropy-phase-targets) gives, for each fixed (Jgeq2),
- E_J\geq\exp^{[J-1]}(c_J E_1^{1/1000})
+ E_Jgeqexp^{[J-1]}(c_J E_1^{1/1000})
-for some \(c_J>0\) and all sufficiently large \(E_1\).
+for some (c_J>0) and all sufficiently large (E_1).
One proves this by induction: raising a fixed-height exponential
-tower with argument \(c t\) to the power \(1/1000\) still dominates
-a tower of the same height with argument \(c't\), for some \(c'>0\)
-and all large \(t\); the next application of \(\exp\) adds a level.
+tower with argument (c t) to the power (1/1000) still dominates
+a tower of the same height with argument (c't), for some (c'>0)
+and all large (t); the next application of (exp) adds a level.
For height one this follows from
-\((e^{ct})^{1/1000}=e^{ct/1000}\); for higher heights, reducing the
+((e^{ct})^{1/1000}=e^{ct/1000}); for higher heights, reducing the
bottom argument by any fixed factor absorbs the fixed outer
multiplicative factor. This proves the induction.
-With the already chosen \(J=H+2\), the tower heights on the
+With the already chosen (J=H+2), the tower heights on the
two sides are now the same, and
-\[
+[
c_J E_1^{1/1000}>
- C'_{inst,J,n,A,B}(1+\log E_1)^3
-\]
-for all sufficiently large \(E_1\). Thus \(E_J>\mathcal B(q)\).
+ C'_{inst,J,n,A,B}(1+log E_1)^3
+]
+for all sufficiently large (E_1). Thus (E_J>mathcal B(q)calB(q)).
Choose the first target still larger to have
-\(E_J>E_{\min}\), to satisfy the initial gradient threshold involving
-\(K_0\), and to meet all dimensional thresholds used above.
+(E_J>E_{min}), to satisfy the initial gradient threshold involving
+(K_0), and to meet all dimensional thresholds used above.
These choices affect only the size of the final parameter and the
finite uniform bounds of the output objects.
Finally one accounts for the scalar constant. The bootstrap charges
-at most \(2C_bε_*\), independently of its input-dependent
+at most (2C_bε_*), independently of its input-dependent
number of sweeps. The initial logarithmic normalization has a
-dimensional cost because \(ε_0\in[ε_*,2ε_*]\).
+dimensional cost because (ε_0in[ε_*,2ε_*]).
All entropy sweep losses together are bounded by
-\(C_n\sum l^2\leq C_n\). Each exit charges at most \(C_n\), and their
-number \(J=H+2\) depends only on the prescribed elementary function
-and \(n\). The input gradient bound \(K_0\), the initial small
-\(ε_{in}\), the compact support radii, and all metric
+(C_nsum l^2leq C_n). Each exit charges at most (C_n), and their
+number (J=H+2) depends only on the prescribed elementary function
+and (n). The input gradient bound (K_0), the initial small
+(ε_{in}), the compact support radii, and all metric
and elliptic bounds occur only in the choices of the first target,
graphs, and arbitrarily small analytic errors. None contributes to
-the scalar constant. Discard the final nonnegative surplus \(P'\);
+the scalar constant. Discard the final nonnegative surplus (P');
the remaining inequality is (eq:entropy-contract) with
-\(C=C(n,\mathcal B)\).
+(C=C(n,mathcal B)calB)).
-The density \(\rho\) has been unchanged since the bootstrap, so
-\(\rho\leq dV_g\). The volume logarithm starts at zero and only
-increases by \(W_p\geq0\) and \(nu\geq0\). Finitely many sweeps and
+The density (rho) has been unchanged since the bootstrap, so
+(rholeq dV_g). The volume logarithm starts at zero and only
+increases by (W_pgeq0) and (nugeq0). Finitely many sweeps and
exits preserve the asserted smoothness, uniformity, compact supports,
and finite number of free label orbits. The transverse spaces were
enlarged only by the probability products used for the colorings,
@@ -3484,13 +3484,13 @@ makes this statement, and the integrations in its proof, well defined.
Use the following model of a measured space of leaf manifolds. The
underlying space $X$ is standard Borel and has a countable collection of
flow boxes
-\[
- \kappa_a:T_a\times Q_a→ X,
-\]
+[
+ kappa_a:T_atimes Q_a→ X,
+]
whose images cover $X$ and which are Borel isomorphisms onto their images.
Here $Q_a$ is an open box
-in $\mathbb R^d$, $T_a$ is standard Borel, and $T_a$ carries a
-$\sigma$-finite measure $\mu_a$. The plaques $\kappa_a(\{t\}\times Q_a)$
+in $mathbb R^d$,bbR^d$, $T_a$ is standard Borel, and $T_a$ carries a
+$sigma$-finite measure $mu_a$. The plaques $kappa_a({t}times Q_a)$
fit together as oriented smooth $d$-manifolds, each Hausdorff and second
countable, and each plaque is open in its leaf. The overlap of two boxes is a countable union of pieces on
which the change of transverse coordinate is a Borel bijection between
@@ -3501,7 +3501,7 @@ functions and tensor fields below are Borel in the boxes and smooth on
plaques. This convention also applies to an open subset along the leaves,
whose leaves are the connected components of that subset.
-The measures $d\mu_a(t)\,dV_h$ in the boxes define a single measured
+The measures $dmu_a(t),dV_h$ in the boxes define a single measured
Riemannian volume, denoted simply by $dV_h$. The same convention, with
leafwise area in place of volume, defines $dA_h$ on regular submanifolds.
One may partition among countably many boxes to define either integral;
@@ -3513,26 +3513,26 @@ transversal. Discard such saturations when an assertion holds on
almost all plaques.
For $t>0$ write $Q_t^d=(-t,t)^d$; in dimension zero this denotes one point.
-Write $|G|_∞=\max_i|G_i|$, with the empty maximum interpreted as
+Write $|G|_∞=max_i|G_i|$, with the empty maximum interpreted as
zero.
[Controlled measured band]
Let $T>0$. A controlled measured $d$-band over $Q_T^d$ consists of a
measured space of oriented $d$-manifolds as above, a leafwise metric $h$,
-a function $\phi$, and a map $G=(G_1,...,G_d):X\to Q_T^d$.
-For every $0<t<T$ the part $\{|G|_∞\le t\}$ has a countable flow-box
+a function $phi$, and a map $G=(G_1,...,G_d):Xto Q_T^d$.
+For every $0<t<T$ the part ${|G|_∞le t}$ has a countable flow-box
cover with the following properties.
-\item The boxes have one common positive size and fixed positive buffers:
+item The boxes have one common positive size and fixed positive buffers:
their smaller concentric boxes cover the indicated part. They can be
-chosen inside $\{|G|_∞<t'\}$ for some $t<t'<T$. After rescaling
+chosen inside ${|G|_∞<t'}$ for some $t<t'<T$. After rescaling
coordinates one may, and does, call them unit boxes with buffers.
-\item In these boxes the coefficients of $h$ and $h^{-1}$ and all their
+item In these boxes the coefficients of $h$ and $h^{-1}$ and all their
leafwise coordinate derivatives are bounded by constants common to the
cover. The same holds for all derivatives of $G$ and for all
-positive-order derivatives of $\phi$. No bound on the value of $\phi$ is
+positive-order derivatives of $phi$. No bound on the value of $phi$ is
required.
-\item The sum of the measures of the transversals of this cover is finite.
+item The sum of the measures of the transversals of this cover is finite.
There is a common bound on the number of plaques of the cover whose
buffered boxes meet any one plaque of the cover.
@@ -3541,19 +3541,19 @@ given band. The leaves are not assumed complete, and $G$ is not assumed
proper.
-For $d\ge1$, the stated control implies the version of Stokes' theorem used here. To
+For $dge1$, the stated control implies the version of Stokes' theorem used here. To
see this, choose a smooth cutoff in each buffered box which is one on its
inner box. Dividing the cutoffs by their sum gives a leafwise smooth,
Borel partition of unity on a slightly smaller inner band. The overlap
bound makes the sum locally finite and bounds all its derivatives. The
metric and inverse-metric bounds give uniform bounds for these derivatives
in geometric norms. Consequently, for a leafwise smooth $(d-1)$-form
-$\eta$ supported over a compact subcube, with $\eta$ and $d\eta$ uniformly
+$eta$ supported over a compact subcube, with $eta$ and $deta$ uniformly
bounded there, chartwise Stokes and Fubini give
- \int_X d\eta=0.
+ int_X deta=0.
-Indeed, each cutoff multiple of $\eta$ is compactly supported in its
+Indeed, each cutoff multiple of $eta$ is compactly supported in its
plaque, and the sum of the absolute integrals is bounded by a constant
times the sum of the transverse measures. The terms differentiating the
partition cancel because its sum is one. The same argument gives measured
@@ -3563,58 +3563,58 @@ cutoff multiple of a test form and then sum. This includes test forms
whose pullbacks have noncompact support in a leaf. Always verify
the indicated local bounds before using current Stokes.
-If $\omega$ is a smooth, compactly supported $d$-form on $Q_T^d$ with
-$\int_{Q_T^d}\omega=1$, define
+If $omega$ is a smooth, compactly supported $d$-form on $Q_T^d$ with
+$int_{Q_T^d}omega=1$, define
- \mathcal D_X(G)=\int_X G^*\omega.
+ mathcal D_X(G)=calD_X(G)=int_X G^*omega.
The integral is absolute: its support lies in an inner band, the
derivatives of $G$ there are bounded, and that band has finite measured
-volume. For $d\ge1$, a compactly supported top form on a cube of integral
+volume. For $dge1$, a compactly supported top form on a cube of integral
zero has a compactly supported primitive. Applying (eq:degree-stokes) to the
pullback of this primitive shows that (eq:degree-definition) is
-independent of $\omega$. This is the measured degree, with the
+independent of $omega$. This is the measured degree, with the
usual signed orientation convention. For $d=0$ it means the measured
-signed count of the oriented points. Abbreviate it to $\mathcal D$
+signed count of the oriented points. Abbreviate it to $mathcal D$calD$
when the band and map are understood.
[Measured degree estimate]
There are absolute constants $C_0,C_1>0$ with the following property.
-Let $q\ge2$, let $d$ be an integer with $0\le d\le q$, and let $\tau>0$.
-For $0\le r\le d$ set
+Let $qge2$, let $d$ be an integer with $0le dle q$, and let $tau>0$.
+For $0le rle d$ set
- T_r=2\tau+(r\tau)/(q),\qquad
- b_r=\tau(1+\frac1{2q}+\frac r{4q^2}).
+ T_r=2tau+(rtau)/(q),qquad
+ b_r=tau(1+frac1{2q}+frac r{4q^2}).
-Suppose $(X,h,\phi,G)$ is a controlled measured $d$-band over $Q_{T_d}^d$
-in the sense of Definition~def:degree-band. Let $B\ge1$ and
+Suppose $(X,h,phi,G)$ is a controlled measured $d$-band over $Q_{T_d}^d$
+in the sense of Definition~def:degree-band. Let $Bge1$ and
$L_1,...,L_d>0$. Suppose throughout $X$ that
- L_i|dG_i|_h\le B-\frac dq\qquad(1\le i\le d),
+ L_i|dG_i|_hle B-frac dqqquad(1le ile d),
and suppose that
- \frac{C_0(1+B)^{2/3}}{L_i}<(\tau)/(10q)
- \qquad(1\le i\le d).
+ frac{C_0(1+B)^{2/3}}{L_i}<(tau)/(10q)
+ qquad(1le ile d).
Define $H_i(G)=2(L_i|dG_i|_h-1)_+$. Then
- |\mathcal D_X(G)|(2\tau)^d
- \le \frac{\exp(C_1d/q)}{\prod_{i=1}^dL_i}
- \int_{\{|G|_∞<b_d\}}
- \exp(\frac{-\wscal(h,\phi)+\sum_{i=1}^dH_i(G)}2)dV_h.
+ |mathcal D_X(G)|(2calD_X(G)|(2tau)^d
+ le frac{exp(C_1d/q)}{prod_{i=1}^dL_i}
+ int_{{|G|_∞<b_d}}
+ exp(frac{-wscal(h,phi)+sum_{i=1}^dH_i(G)}2)dV_h.
For $d=0$ the product is one, the sum is zero, and
-$\wscal(h,\phi)=0$; the assertion is the bound of a signed count by the
+$wscal(h,phi)=0$; the assertion is the bound of a signed count by the
unsigned count.
-The proof removes one coordinate at a time. For $r\ge1$, the gaps
-$T_r-T_{r-1}=\tau/q$ and $b_r-b_{r-1}=\tau/(4q^2)$ supply, respectively,
+The proof removes one coordinate at a time. For $rge1$, the gaps
+$T_r-T_{r-1}=tau/q$ and $b_r-b_{r-1}=tau/(4q^2)$ supply, respectively,
chart buffers and room for the residual value error when locating the
integration core, while the permitted scaled gradient of each retained
coordinate increases by $1/q$. A family of boundaries
-ordered by a real parameter supplies the factor $2\tau$ on the left.
+ordered by a real parameter supplies the factor $2tau$ on the left.
Its normal speed supplies the coarea factor on the right. In dimensions
where the boundaries can be singular one will construct a regular band
of the same degree; a scalar gain will pay exactly for the volume of the
@@ -3622,39 +3622,39 @@ graphs used to construct it. The following formulation separates that
construction from the final induction.
[One step of measured descent]
-Assume the hypotheses of Theorem~thm:degree with $d\ge1$, put
+Assume the hypotheses of Theorem~thm:degree with $dge1$, put
$p=d-1$, and put
-\[
- Θ=\exp(\frac{-\wscal(h,\phi)+\sum_{i=1}^dH_i(G)}2).
-\]
+[
+ Θ=exp(frac{-wscal(h,phi)+sum_{i=1}^dH_i(G)}2).
+]
There is a Borel family of oriented regular hypersurface portions
-$Σ_s$ in the leaves, $-\tau<s<\tau$, and, for almost every $s$, a
+$Σ_s$ in the leaves, $-tau<s<tau$, and, for almost every $s$, a
positive function $v_s$ on $Σ_s$, smooth along it and Borel in its
charts, with the following properties. Transverse null subsets and their
saturations may be removed for each such $s$.
-\item For every nonnegative Borel function $F$ supported in
-$\{|G|_∞<b_d\}$,
+item For every nonnegative Borel function $F$ supported in
+${|G|_∞<b_d}$,
- \int_{-\tau}^{\tau}\!ds\int_{Σ_s}v_sF\,dA_h
- \le\int_XF\,dV_h.
+ int_{-tau}^{tau}!dsint_{Σ_s}v_sF,dA_h
+ leint_XF,dV_h.
-\item For each such $s$ there is a controlled measured $p$-band
-$(X_s^+,h_s^+,\phi_s^+,J_s^+)$ over $Q_{T_p}^p$ of degree
-$\mathcal D_X(G)$. Its coordinates are indexed by $j=2,...,d$ and
+item For each such $s$ there is a controlled measured $p$-band
+$(X_s^+,h_s^+,phi_s^+,J_s^+)$ over $Q_{T_p}^p$ of degree
+$mathcal D_X(G)$.calD_X(G)$. Its coordinates are indexed by $j=2,...,d$ and
satisfy
-\[
- L_j|dJ_{s,j}^+|_{h_s^+}\le B-\frac pq.
-\]
+[
+ L_j|dJ_{s,j}^+|_{h_s^+}le B-frac pq.
+]
With $H_j^+=2(L_j|dJ_{s,j}^+|_{h_s^+}-1)_+$, there is an absolute constant
$C_*>0$ such that
- &\int_{\{|J_s^+|_∞<b_p\}}
- \exp(\frac{-\wscal(h_s^+,\phi_s^+)+\sum_{j=2}^dH_j^+}{2})
- dV_{h_s^+}\\
- &\hspace{25mm}\le \frac{e^{C_*/q}}{L_1}
- \int_{Σ_s\cap\{|G|_∞<b_d\}}v_sΘ\,dA_h.
+ &int_{{|J_s^+|_∞<b_p}}
+ exp(frac{-wscal(h_s^+,phi_s^+)+sum_{j=2}^dH_j^+}{2})
+ dV_{h_s^+}
+ &hspace{25mm}le frac{e^{C_*/q}}{L_1}
+ int_{Σ_scap{|G|_∞<b_d}}v_sΘ,dA_h.
In dimension $p=0$ the left side is the measured unsigned count.
@@ -3675,122 +3675,122 @@ charts, $t_3$ contains the boundary portions to be used, $t_2$ will bound
an auxiliary complete region, and $t_1$ will bound the child. Such
choices can depend on the given band.
-Choose once and for all a smooth even function $\gamma$ with
-$0\le\gamma(y)\le y^2$ and with $\gamma(y)$ bounded above and below by
-positive absolute multiples of $|y|^{3/2}$ for $|y|\ge1$. Let $P_0$ be the
+Choose once and for all a smooth even function $gamma$ with
+$0legamma(y)le y^2$ and with $gamma(y)$ bounded above and below by
+positive absolute multiples of $|y|^{3/2}$ for $|y|ge1$. Let $P_0$ be the
solution about zero of
- P_0(0)=0,\qquad P_0'=L_1\psi(P_0),\qquad
- \psi(y)=1+(\gamma(y))/(8(1+B)).
+ P_0(0)=0,qquad P_0'=L_1psi(P_0),qquad
+ psi(y)=1+(gamma(y))/(8(1+B)).
-It is odd and increasing and has poles at $\pm b$, where
-\[
- b=\frac1{L_1}\int_0^∞(dy)/(\psi(y))
- \le\frac{C(1+B)^{2/3}}{L_1}.
-\]
+It is odd and increasing and has poles at $pm b$, where
+[
+ b=frac1{L_1}int_0^∞(dy)/(psi(y))
+ lefrac{C(1+B)^{2/3}}{L_1}.
+]
For the inequality, split the integral at $(1+B)^{2/3}$ and use the
$|y|^{3/2}$ bound on the tail. Choose the absolute $C_0$ in
(eq:degree-horizon) at least this $C$. Thus
- b<(\tau)/(10q).
+ b<(tau)/(10q).
The local slice calculation.:
First record the analytic purpose of the pressure law. Let $Σ$
-be a smooth cooriented local hypersurface, with unit normal $\nu$, and set
-\[
+be a smooth cooriented local hypersurface, with unit normal $nu$, and set
+[
- D=H+∂_\nu\phi,\qquad
- H=trII,\qquad
- II(Y,Z)=<\nabla_Y\nu,Z>,\\
- \mathcal P=|II|^2+
- (Ric_h-\nabla^2\phi)(\nu,\nu).
+ D=H+∂_nuphi,qquad
+ H=trII,qquad
+ II(Y,Z)=<nabla_Ynu,Z>,
+ mathcal P=|II|^2+calP=|II|^2+
+ (Ric_h-nabla^2phi)(nu,nu).
-\]
-Put $\mathfrak a=L_1\psi(D)>0$. When $D=P_0(s-G_1)$, the derivatives
-of the raw pressure are $∂_sP_0(s-G_1)=\mathfrak a$ and
-$d_xP_0(s-G_1)=-\mathfrak a\,dG_1$. The following calculation uses
+]
+Put $mathfrak a=L_1fraka=L_1psi(D)>0$. When $D=P_0(s-G_1)$, the derivatives
+of the raw pressure are $∂_sP_0(s-G_1)=mathfrak a$fraka$ and
+$d_xP_0(s-G_1)=-mathfrak afraka,dG_1$. The following calculation uses
only a positive smooth function $v$ satisfying the local equation
- -(\Delta_{Σ,\phi}+\mathcal P
- -\mathfrak a\,dG_1(\nu))v=\mathfrak a.
+ -(Delta_{Σ,phi}+mathcal PcalP
+ -mathfrak afraka,dG_1(nu))v=mathfrak a.fraka.
-Here $\Delta_{Σ,\phi}$ is weighted by
-$e^{\phi|_Σ}dA_h$, and slice norms use the induced metric. The
+Here $Delta_{Σ,phi}$ is weighted by
+$e^{phi|_Σ}dA_h$, and slice norms use the induced metric. The
ordered construction below will produce this equation on almost every
regular slice.
-Put $\phi_Σ=\phi|_Σ+\ln v$ and
-$S_Σ=\wscal(h|_{TΣ},\phi_Σ)$. Gauss' equation and the
-normal/tangential decomposition of derivatives of $\phi$ give
-\[
- \wscal(h|_{TΣ},\phi|_Σ)
- =\wscal(h,\phi)+D^2-2\mathcal P+|II|^2.
-\]
-Adding $\ln v$ to the weight changes this by
-$-2\Delta_{Σ,\phi}v/v+|d\ln v|^2$. Substitution of
+Put $phi_Σ=phi|_Σ+ln v$ and
+$S_Σ=wscal(h|_{TΣ},phi_Σ)$. Gauss' equation and the
+normal/tangential decomposition of derivatives of $phi$ give
+[
+ wscal(h|_{TΣ},phi|_Σ)
+ =wscal(h,phi)+D^2-2mathcal P+|II|^2.calP+|II|^2.
+]
+Adding $ln v$ to the weight changes this by
+$-2Delta_{Σ,phi}v/v+|dln v|^2$. Substitution of
(eq:degree-local-lapse) yields the exact identity
- S_Σ=\wscal(h,\phi)+D^2-2\mathfrak a\,dG_1(\nu)
- +|II|^2+(2\mathfrak a)/(v)+|d\ln v|^2.
+ S_Σ=wscal(h,phi)+D^2-2mathfrak afraka,dG_1(nu)
+ +|II|^2+(2mathfrak a)/(v)+|dfraka)/(v)+|dln v|^2.
This is also the hypersurface calculation behind
Lemma~lem:graph-identity; it is valid for $p=0$ with left side zero.
-Set $t=L_1|dG_1|_h\le B$ and $x=v/L_1$. Since $\gamma(D)\le D^2$,
-\[
- -2\mathfrak a\,dG_1(\nu)
- \ge-2t(1+(\gamma(D))/(8(1+B))),
- \qquad (t\gamma(D))/(4(1+B))\le(D^2)/(4).
-\]
+Set $t=L_1|dG_1|_hle B$ and $x=v/L_1$. Since $gamma(D)le D^2$,
+[
+ -2mathfrak afraka,dG_1(nu)
+ ge-2t(1+(gamma(D))/(8(1+B))),
+ qquad (tgamma(D))/(4(1+B))le(D^2)/(4).
+]
Discard the remaining nonnegative part of $D^2$, and use
-$2\mathfrak a/v\ge2/x$. Finally,
-$t\le1+(t-1)_+$ and $x^{-1}+\ln x\ge1$ for $x>0$. Thus
+$2mathfrak a/vfraka/vge2/x$. Finally,
+$tle1+(t-1)_+$ and $x^{-1}+ln xge1$ for $x>0$. Thus
(eq:degree-slice-identity) implies
- S_Σ\ge\wscal(h,\phi)-H_1(G)-2\ln(v/L_1)+|d\ln v|^2.
+ S_Σgewscal(h,phi)-H_1(G)-2ln(v/L_1)+|dln v|^2.
The square term will be used later when the weight is changed near
singular points. For $j=2,...,d$, put
$H_j^Σ=2(L_j|d_Σ G_j|-1)_+$. Tangential restriction gives
-$H_j^Σ\le H_j(G)$, so the full pointwise density comparison is
+$H_j^Σle H_j(G)$, so the full pointwise density comparison is
- \exp(\frac{-S_Σ+\sum_{j=2}^dH_j^Σ}{2})
- \le\frac v{L_1}e^{-|d\ln v|^2/2}Θ
- \le\frac v{L_1}Θ.
+ exp(frac{-S_Σ+sum_{j=2}^dH_j^Σ}{2})
+ lefrac v{L_1}e^{-|dln v|^2/2}Θ
+ lefrac v{L_1}Θ.
For $p=0$ the sum and slice gradient are zero, and the left side is one.
The remaining work supplies the slices and speeds, and verifies the
controlled-band and degree conditions needed to use this comparison.
-For a small $\eta>0$, prolong $P_0$ smoothly and strictly increasingly
-past the points $\pm(b-\eta)$, keeping it unchanged on
-$[-b+\eta,b-\eta]$. Denote the resulting smooth function by
-$\widehat P$ and set
-\[
- \mathfrak h_s(x)=\widehat P(s-G_1(x)),\qquad -\tau<s<\tau.
-\]
-Choose $\eta$ sufficiently small for the given band and then keep
-it fixed. On every boundary portion used below, $\widehat P$ will agree
+For a small $eta>0$, prolong $P_0$ smoothly and strictly increasingly
+past the points $pm(b-eta)$, keeping it unchanged on
+$[-b+eta,b-eta]$. Denote the resulting smooth function by
+$widehatwidehat P$P$ and set
+[
+ mathfrak h_s(x)=frakh_s(x)=widehatwidehat P(s-G_1(x)),P(s-G_1(x)),qquad -tau<s<tau.
+]
+Choose $eta$ sufficiently small for the given band and then keep
+it fixed. On every boundary portion used below, $widehatwidehat P$P$ will agree
with $P_0$ and hence obey (eq:degree-pressure).
Least minimizers, including on noncompact leaves.:
-Write $P_\phi(E;U)=\int_{∂^*E\cap U}e^\phi\,dA_h$ for the weighted
+Write $P_phi(E;U)=int_{∂^*Ecap U}e^phi,dA_h$ for the weighted
perimeter of a set of locally finite perimeter. Use local differences
of the functional
- P_\phi(E)-\int_E\mathfrak h_s e^\phi\,dV_h.
+ P_phi(E)-int_Emathfrak h_sfrakh_s e^phi,dV_h.
-This is a weighted $\mu$-bubble functional. For soap bubbles in Gromov's
-terminology, see \cite[Section~5$\frac56$]{Gromov1996}; for the weighted
+This is a weighted $mu$-bubble functional. For soap bubbles in Gromov's
+terminology, see cite[Section~5$frac56$]{Gromov1996}; for the weighted
functional and its local first- and second-variation formulas, see also
-\cite[Section~3, Equation~(1), and Lemmas~13--14]{ChodoshLi2024}.
+cite[Section~3, Equation~(1), and Lemmas~13--14]{ChodoshLi2024}.
Neither term is required to be finite on the whole leaf. In particular,
this construction uses no global weighted-volume bound.
-\par\noindent{\footnotesizehttps://doi.org/10.1007/978-1-4612-4098-3_1}\par
-\par\noindent{\footnotesize\ttfamily src-chodoshli2024.pdf}\par
+parnoindent{footnotesizehttps://doi.org/10.1007/978-1-4612-4098-3_1}par
+parnoindent{footnotesizettfamily src-chodoshli2024.pdf}par
The output will be a canonical nested family $E_s$ on each open leaf,
independent of the exhaustion and jointly Borel in the plaque data and
@@ -3807,12 +3807,12 @@ The empty set is a competitor. On a neighborhood of $K$ the density is
smooth and positive and the forcing is bounded. A bound on the functional
therefore gives a perimeter bound, and BV compactness and lower
semicontinuity give a minimizer; see, for example,
-\cite[Theorem~12.26]{Maggi2012} for the compactness theorem. Among the
+cite[Theorem~12.26]{Maggi2012} for the compactness theorem. Among the
minimizers choose one of minimum
ordinary volume. The weighted perimeter satisfies
-\[
- P_\phi(E\cap F)+P_\phi(E\cup F)\le P_\phi(E)+P_\phi(F),
-\]
+[
+ P_phi(Ecap F)+P_phi(Ecup F)le P_phi(E)+P_phi(F),
+]
and the volume term for a fixed forcing is additive under union and
intersection. Thus the intersection of any two minimizers is again a
minimizer. Minimum volume forces the selected minimizer to be contained,
@@ -3820,26 +3820,26 @@ up to null sets, in every other one. It is consequently unique as an
$L^1$ set; call it the least minimizer in $K$.
-\par\noindent{\footnotesizehttps://doi.org/10.1017/CBO9781139108133}\par
+parnoindent{footnotesizehttps://doi.org/10.1017/CBO9781139108133}par
Order and exhaustion.
-The same inequality proves both monotonicities one needs. If $K\subset K'$
-and two forcings satisfy $\mathfrak h\le\mathfrak h'$, a minimizer for
-$(K,\mathfrak h)$ and one for $(K',\mathfrak h')$ obey
-\[
- \mathcal F_{\mathfrak h}(E\cap F)+\mathcal F_{\mathfrak h'}(E\cup F)
- \le \mathcal F_{\mathfrak h}(E)+\mathcal F_{\mathfrak h'}(F),
-\]
-where $\mathcal F$ denotes (eq:degree-functional); the inequality for
+The same inequality proves both monotonicities one needs. If $Ksubset K'$
+and two forcings satisfy $mathfrak hfrakhlemathfrak h'$,frakh'$, a minimizer for
+$(K,mathfrak h)$frakh)$ and one for $(K',mathfrak h')$frakh')$ obey
+[
+ mathcal F_calF_{mathfrak h}frakh(Ecap F)+mathcal F_calF_{mathfrak h'}frakh'(Ecup F)
+ le mathcal F_calF_{mathfrak h}frakh(E)+mathcal F_calF_{mathfrak h'}frakh'(F),
+]
+where $mathcal F$calF$ denotes (eq:degree-functional); the inequality for
the volume terms follows on $E∖ F$ from
-$\mathfrak h\le\mathfrak h'$. Minimality in
+$mathfrak hfrakhlemathfrak h'$.frakh'$. Minimality in
the two domains makes both inequalities equalities, and leastness in $K$
-implies $E\subset F$. Exhaust a leaf by increasing finite unions of
+implies $Esubset F$. Exhaust a leaf by increasing finite unions of
compact coordinate subboxes whose interiors exhaust it. The least
minimizers increase with this exhaustion. Their union, interpreted in
$L^1_{loc}$, is independent of the exhaustion, since each compact
subset of either exhaustion is eventually contained in one of the other.
-This defines $E_s$, and the forcing monotonicity makes $E_s\subset E_t$
+This defines $E_s$, and the forcing monotonicity makes $E_ssubset E_t$
for $s<t$.
Local comparison.
@@ -3871,26 +3871,26 @@ Explain why all these choices are Borel, including jointly in $s$.
For this coding, subdivide the original atlas into bounded coordinate
boxes with compact closure in the older boxes, and affinely normalize
each to one fixed cube $Q=(-1,1)^d$. Fix once a countable list
-$(R_\ell)_{\ell\ge1}$ of all open rational boxes with closure in $Q$.
+$(R_ell)_{ellge1}$ of all open rational boxes with closure in $Q$.
Starting at a plaque, enumerate its accessible normalized plaque
-occurrences Borel measurably in slots $j\ge1$, using dummy slots when
+occurrences Borel measurably in slots $jge1$, using dummy slots when
necessary. Use the fixed Polish space
-\[
- \mathcal Z=\prod_{j,\ell\ge1}L^1(R_\ell),
-\]
+[
+ mathcal Z=calZ=prod_{j,ellge1}L^1(R_ell),
+]
with Lebesgue $L^1$ norms and the product topology. A set is coded by its
-indicator on $R_\ell$ in occurrence $j$; dummy entries are zero.
+indicator on $R_ell$ in occurrence $j$; dummy entries are zero.
Compatibility is tested on compact exhaustions of all overlap domains,
including repeated occurrences. The closures of the first finitely
-many accessible $R_\ell$ in a fixed enumeration of the pairs $(j,\ell)$
+many accessible $R_ell$ in a fixed enumeration of the pairs $(j,ell)$
give the compact exhaustion domains. The conditions that the code is an
indicator, agrees on overlaps, and vanishes off the specified domain are
-Borel, and for fixed plaque data are closed in $\mathcal Z$.
+Borel, and for fixed plaque data are closed in $mathcal Z$.calZ$.
Integrals against chartwise Borel smooth densities are Borel by Fubini.
Perimeter is Borel as well: its supremum over divergence integrals may
be taken over fixed countable families of rational smooth test fields in
-the $R_\ell$, and finite sums of them. Partitions of unity in a leaf
+the $R_ell$, and finite sums of them. Partitions of unity in a leaf
show density among all compactly supported tests. Norm constraints are
checked on countably many dense coordinate points; overlap maps and a
partition by the first covering chart make the integrals Borel.
@@ -3898,77 +3898,77 @@ partition by the first covering chart make the integrals Borel.
Fix one plaque-data fiber and one compact exhaustion domain $K$. A
functional sublevel for sets supported in $K$ bounds their weighted
perimeter, since the forcing integral is uniformly bounded on this
-fixed compact set. In every active $R_\ell$ the metric and density
+fixed compact set. In every active $R_ell$ the metric and density
have a positive lower bound on its closure, so this gives a Euclidean BV
-bound there; the indicator's $L^1(R_\ell)$ norm is bounded by $|R_\ell|$.
-Diagonal BV compactness gives convergence in $\mathcal Z$ along a
+bound there; the indicator's $L^1(R_ell)$ norm is bounded by $|R_ell|$.
+Diagonal BV compactness gives convergence in $mathcal Z$calZ$ along a
subsequence. The fiberwise closed indicator, overlap, and support
conditions persist. The volume term is continuous, since $K$ is
covered by finitely many compact subboxes, and perimeter is lower
semicontinuous. Each sublevel is therefore compact in this fixed Polish
product. These compactness bounds may depend on the fiber and on
-$R_\ell$; the projection theorem requires compact fibers, not a common
+$R_ell$; the projection theorem requires compact fibers, not a common
bound across plaque data. Strict sublevels are countable unions of such
compact sections. The Borel projection theorem for sets with
-$\sigma$-compact sections therefore makes the minimum value a Borel
+$sigma$-compact sections therefore makes the minimum value a Borel
function of the chart data. Apply it again to minimize volume in the
compact set of minimizers. The selected section is a singleton in the
$L^1$ coding, by leastness, so the Borel singleton-section theorem gives
the selected minimizer as a Borel function. These are standard forms of
the Arsenin--Kunugui and Lusin--Souslin theorems; see
-\cite[Theorem~18.18 and Corollary~15.2]{Kechris1995}. The exhaustion limits agree
+cite[Theorem~18.18 and Corollary~15.2]{Kechris1995}. The exhaustion limits agree
for different starting plaques by exhaustion independence. They are therefore Borel
on the measured space. Use the density-one representative of
$E_s$; the support of its boundary measure is also Borel, by testing its
mass on a countable basis of coordinate balls.
Confinement.:
-For sufficiently small $\eta$, uniformly for $-\tau<s<\tau$, the boundary
-support in $\{|G|_∞\le t_4\}$ satisfies
+For sufficiently small $eta$, uniformly for $-tau<s<tau$, the boundary
+support in ${|G|_∞le t_4}$ satisfies
-supp∂[E_s]\cap\{|G|_∞\le t_4\}
- \subset\{|s-G_1|<b-\eta\}.
+supp∂[E_s]cap{|G|_∞le t_4}
+ subset{|s-G_1|<b-eta}.
-In this inner band the side $G_1\le s-b+\eta$ is full and the side
-$G_1\ge s+b-\eta$ is empty, up to null sets. To prove this uniform claim,
+In this inner band the side $G_1le s-b+eta$ is full and the side
+$G_1ge s+b-eta$ is empty, up to null sets. To prove this uniform claim,
the ODE at the positive pole gives
-\[
- b-x=\frac1{L_1}\int_{P_0(x)}^∞(dy)/(\psi(y))
- \asymp_{inst}P_0(x)^{-1/2}
- \quad(x↑ b).
-\]
-Thus at distance comparable to $\eta$ from a pole the absolute forcing is
-at least $c_{inst}\eta^{-2}$. The prolongation is monotone, so
+[
+ b-x=frac1{L_1}int_{P_0(x)}^∞(dy)/(psi(y))
+ asymp_{inst}P_0(x)^{-1/2}
+ quad(x↑ b).
+]
+Thus at distance comparable to $eta$ from a pole the absolute forcing is
+at least $c_{inst}eta^{-2}$. The prolongation is monotone, so
this remains a lower bound beyond that position. By the uniform
derivative bound on $G_1$, a chart ball of radius
-$r=c'_{inst}\eta$ centered at a point with $s-G_1\ge b-\eta$
-has forcing at least $H=c''_{inst}\eta^{-2}$ throughout it, if
+$r=c'_{inst}eta$ centered at a point with $s-G_1ge b-eta$
+has forcing at least $H=c''_{inst}eta^{-2}$ throughout it, if
$c'_{inst}$ is small. The balls are taken in a buffered atlas
for a slightly larger band than $t_4$.
Here is the phase-exclusion estimate in such a ball. Normalize the
weighted density by its value at the center; the derivative bounds for
-$\phi$ make this density uniformly comparable to coordinate volume in
+$phi$ make this density uniformly comparable to coordinate volume in
the ball. Let $V(t)$ be the normalized volume of the complementary
phase in a concentric coordinate ball $B_t$. Comparison with the set
which fills $B_t$, and perimeter slicing, give for almost every $t<r$
-\[
- P_{norm}(E_s;B_t)+H V(t)\le C V'(t).
-\]
+[
+ P_{norm}(E_s;B_t)+H V(t)le C V'(t).
+]
The chart isoperimetric inequality applied to that complementary phase,
extended by zero outside $B_t$, gives, when $V(t)>0$,
-\[
- V(t)^{(d-1)/d}\le C(P_{norm}(E_s;B_t)+V'(t)).
-\]
+[
+ V(t)^{(d-1)/d}le C(P_{norm}(E_s;B_t)+V'(t)).
+]
If the complementary phase occurs in $B_{r/4}$, the two inequalities first
-give $V'\ge cV^{(d-1)/d}$ and hence $V(r/2)\ge cr^d$. They also give
-$V'\ge cHV$ on $[r/2,r]$. It follows that
-$V(r)\ge cr^d\exp(cHr)$, contrary to $V(r)\le Cr^d$ when $Hr$ is large.
+give $V'ge cV^{(d-1)/d}$ and hence $V(r/2)ge cr^d$. They also give
+$V'ge cHV$ on $[r/2,r]$. It follows that
+$V(r)ge cr^dexp(cHr)$, contrary to $V(r)le Cr^d$ when $Hr$ is large.
This argument includes $d=1$, where the first differential inequality
-has exponent zero. Since $Hr\asymp_{inst}\eta^{-1}$, small
-$\eta$ excludes the complementary phase in the smaller ball. Negative
+has exponent zero. Since $Hrasymp_{inst}eta^{-1}$, small
+$eta$ excludes the complementary phase in the smaller ball. Negative
forcing gives the empty phase by the same argument. Applying these
-balls beginning at $|s-G_1|=b-\eta$, with a small fixed extra margin in
+balls beginning at $|s-G_1|=b-eta$, with a small fixed extra margin in
the forcing estimate, proves (eq:degree-confinement). In particular
the unprolonged Equation~(eq:degree-pressure) holds on all the
boundary portions used in the proof.
@@ -3976,7 +3976,7 @@ boundary portions used in the proof.
The local regularity and compactness inputs.:
Record precisely the local geometric measure theory used from now on.
Set $Σ_s^{all}=supp∂[E_s]$. For
-$d\ge2$, its regular portion is a smooth multiplicity-one hypersurface.
+$dge2$, its regular portion is a smooth multiplicity-one hypersurface.
The remaining set is closed in $Σ_s^{all}$ and has Hausdorff
dimension at most
@@ -3984,19 +3984,19 @@ dimension at most
It is empty if this number is negative. These conclusions hold in each
relatively compact chart. To place the functional under the standard
-codimension-one theorem, normalize $\phi$ by a constant value $\phi_0$
+codimension-one theorem, normalize $phi$ by a constant value $phi_0$
in the chart and use the smooth metric
-\[
- \widehat h=e^{2(\phi-\phi_0)/p}h.
-\]
+[
+ widehatwidehat h=e^h=e^{2(phi-phi_0)/p}h.
+]
Its area is the normalized weighted area. Its volume forcing is
-$\mathfrak h_s e^{-(\phi-\phi_0)/p}$, bounded smoothly in a smaller chart.
+$mathfrak h_sfrakh_s e^{-(phi-phi_0)/p}$, bounded smoothly in a smaller chart.
Minimality implies that its perimeter is minimizing up to a fixed
multiple of the volume of the symmetric difference for compact
variations. The regularity theorem for such almost-minimizing boundaries
-gives $C^{1,\alpha}$ regularity off a closed set of dimension
+gives $C^{1,alpha}$ regularity off a closed set of dimension
(eq:degree-singular-dimension); see
-\cite[Theorem A.1 and Remark A.3]{Eichmair2009}. The weak prescribed
+cite[Theorem A.1 and Remark A.3]{Eichmair2009}. The weak prescribed
mean-curvature equation on this regular portion and elliptic regularity
then make it smooth. The cited almost-minimizing formulation uses
integral-current variations. It applies to these set variations as
@@ -4007,13 +4007,13 @@ where the resulting set changes is at most the mass of the added current.
The set comparison therefore implies that formulation as well.
-\par\noindent{\footnotesize\ttfamily src-eichmair2009.pdf}\par
+parnoindent{footnotesizettfamily src-eichmair2009.pdf}par
Local compactness claim.
-Suppose $d\ge2$ and $E_j$ minimize the absorbed-perimeter functionals
+Suppose $dge2$ and $E_j$ minimize the absorbed-perimeter functionals
for all compact variations in a common buffered normalized outer chart.
Assume that their absorbed metrics and converted volume forcings
-converge smoothly on smaller boxes and that $E_j\to E$ in
+converge smoothly on smaller boxes and that $E_jto E$ in
$L^1_{loc}$. Then $E$ is locally minimizing, the normalized
perimeter measures and the associated boundary varifolds converge
weakly locally to those of $E$, and the supports converge locally.
@@ -4028,79 +4028,79 @@ subsequence satisfying these coefficient and set convergence hypotheses
after normalization.
Here is the proof, including the local perimeter assertion. In the
-common normalized coordinates subtract the value of $\phi$ at the chart
+common normalized coordinates subtract the value of $phi$ at the chart
center. The common bounds on all derivatives, and the metric and
inverse-metric bounds, make the normalized metric, weight, and forcing
coefficients precompact in $C^∞$ on every smaller fixed box, by
diagonal Arzelà--Ascoli. Pass to such a convergent subsequence; in
-particular the absorbed metrics $\widehat h_j$ converge smoothly to a
-positive metric $\widehat h_∞$. Ball comparison gives the local
+particular the absorbed metrics $widehatwidehat h_j$h_j$ converge smoothly to a
+positive metric $widehatwidehat h_∞$.h_∞$. Ball comparison gives the local
perimeter bounds used above, so BV compactness gives a further
subsequence converging in $L^1_{loc}$ to a set $E$. The belt
comparison with any compact variation, as proved above and now with the
smoothly converging coefficients, shows that $E$ is locally minimizing.
-Write $\mu_j=P_{\widehat h_j}(E_j;\,\cdot\,)$ and
-$\mu_∞=P_{\widehat h_∞}(E;\,\cdot\,)$. To prove convergence
+Write $mu_j=P_{widehatwidehat h_j}h_j(E_j;,cdot,)$ and
+$mu_∞=P_{widehatwidehat h_∞}h_∞(E;,cdot,)$. To prove convergence
of these measures, begin with a relatively compact coordinate domain
$V$ from a convergence-determining family, chosen so that
-$\mu_∞(∂ V)=0$. Surround it by a thin outer belt with level
+$mu_∞(∂ V)=0$. Surround it by a thin outer belt with level
domains $V_t$ contained in an outer enlargement $V_ε$. Coarea
chooses a level $t_j$ at which the trace mismatch between $E_j$ and $E$
tends to zero. Splice the unchanged $E$ inside $V_{t_j}$ to $E_j$
outside. The common coefficient bounds make the extra perimeter at the
splice tend to zero, and the bounded converted forcing times the
$L^1$ mismatch tends to zero as well. Minimality of $E_j$ gives
-\[
- limsup_{j\to∞}\mu_j(V)\le\mu_∞(V_ε).
-\]
+[
+ limsup_{jto∞}mu_j(V)lemu_∞(V_ε).
+]
Here the perimeter of the fixed set $E$ in the varying metric converges
by uniform convergence of the metric integrands on unit conormals.
-Shrinking the belt makes the right side tend to $\mu_∞(V)$, because
-$\mu_∞(∂ V)=0$. Lower semicontinuity gives the reverse
+Shrinking the belt makes the right side tend to $mu_∞(V)$, because
+$mu_∞(∂ V)=0$. Lower semicontinuity gives the reverse
inequality. Such coordinate continuity domains determine weak local
convergence of Radon measures; the local mass bounds give compactness,
and the equality just proved identifies every subsequential limit.
-Consequently $\mu_j$ converge weakly locally to $\mu_∞$, and their
+Consequently $mu_j$ converge weakly locally to $mu_∞$, and their
masses converge on every relatively compact continuity subdomain $U$.
For precision in passing to varifolds, restrict to a relatively compact
subdomain $U$ whose boundary has zero limit perimeter measure. Such
continuity subdomains can exhaust the smaller box. To check weak
-convergence of the restrictions of $D\mathbf1_{E_j}$ as vector measures
-on $\overline U$, extend a continuous vector test there to the outer
+convergence of the restrictions of $Dmathbf1_bf1_{E_j}$ as vector measures
+on $overlineoverline U$,U$, extend a continuous vector test there to the outer
chart and multiply it by a cutoff supported in $U$ and equal to one away
-from a $\delta$-collar of $∂ U$. Local weak BV convergence
+from a $delta$-collar of $∂ U$. Local weak BV convergence
handles this cutoff test. On the compact chart uniform ellipticity
-gives $|D\mathbf1_{E_j}|\le C\mu_j$. For a compact closed
-$\delta$-collar $F_\delta$, weak perimeter convergence and the closed-set
+gives $|Dmathbf1_bf1_{E_j}|le Cmu_j$. For a compact closed
+$delta$-collar $F_delta$, weak perimeter convergence and the closed-set
inequality give
-$limsup_j\mu_j(F_\delta)\le\mu_∞(F_\delta)$, which tends to zero
-as $\delta↓0$ because $\mu_∞(∂ U)=0$. The same
+$limsup_jmu_j(F_delta)lemu_∞(F_delta)$, which tends to zero
+as $delta↓0$ because $mu_∞(∂ U)=0$. The same
bound for the limiting vector measure lets the collar shrink and proves
the restriction convergence. The perimeter masses on $U$ also converge.
In coordinates the perimeter
integrand for a metric $k$ is
-$\mathfrak p_k(x,z)=√(\det k(x))\,|z|_{k^{-1}(x)}$.
+$mathfrak p_k(x,z)=√(frakp_k(x,z)=√(det k(x)),|z|_{k^{-1}(x)}$.
Uniform convergence of this integrand on unit conormals and the local
-mass bound replace $\widehat h_j$ by $\widehat h_∞$ in that mass
+mass bound replace $widehatwidehat h_j$h_j$ by $widehatwidehat h_∞$h_∞$ in that mass
convergence. This is convergence for one continuous positively
one-homogeneous integrand with Reshetnyak's strict convexity property:
equality in its triangle inequality occurs only for vectors on a common
nonnegative ray. Reshetnyak's continuity theorem
-\cite[Section~4, Theorem~3]{Reshetnyak1968} consequently gives convergence
+cite[Section~4, Theorem~3]{Reshetnyak1968} consequently gives convergence
for every continuous positively one-homogeneous integrand with linear
growth. For a continuous compactly supported varifold test $a(x,P)$, use
-$\mathfrak p_{\widehat h_∞}(x,z)a(x,\ker z)$ for $z\ne0$ and zero
+$mathfrak p_frakp_{widehatwidehat h_∞}h_∞(x,z)a(x,ker z)$ for $zne0$ and zero
at $z=0$. This gives precisely the local varifold convergence; the
uniform convergence of the metric integrands gives it also for the
original varying metrics.
-\par\noindent{\footnotesizehttps://doi.org/10.1007/BF02196453}\par
+parnoindent{footnotesizehttps://doi.org/10.1007/BF02196453}par
To prove the support and graph assertions, the first variation in
-the metric $\widehat h$ is, by BV Gauss--Green, integration of the bounded
+the metric $widehatwidehat h$h$ is, by BV Gauss--Green, integration of the bounded
volume forcing against the reduced-boundary conormal. It therefore has
bounded mean curvature and no singular part. The monotonicity formula
gives a common positive lower mass bound in a small ball about a support
@@ -4110,10 +4110,10 @@ the limit support. Conversely, every ball about a point of the limit
support has positive limit mass, so weak convergence forces nearby
support points. Near a fixed smooth multiplicity-one sheet of the
limit, Allard's interior regularity theorem gives single graphs with a
-common $C^{1,\alpha_{\mathrm A}}$ bound for some
-$\alpha_{\mathrm A}\in(0,1)$ on a smaller cylinder [cite].
+common $C^{1,alpha_{mathrm A}rmA}$ bound for some
+$alpha_{mathrm A}rmAin(0,1)$ on a smaller cylinder [cite].
Compactness and identification of the already known varifold limit give
-$C^{1,\beta}$ graph convergence for every $0<\beta<\alpha_{\mathrm A}$.
+$C^{1,beta}$ graph convergence for every $0<beta<alpha_{mathrm A}rmA$.
The prescribed equation, with the smoothly converging coefficients,
then gives $C^∞$ convergence on still smaller cylinders by
interior elliptic estimates. The cylinder here may depend on the fixed
@@ -4121,7 +4121,7 @@ regular point of the limit; uniform graph radii in later applications
will follow from their separate compactness contradictions.
-\par\noindent{\footnotesizehttps://doi.org/10.2307/1970868}\par
+parnoindent{footnotesizehttps://doi.org/10.2307/1970868}par
The varying-metric use of these local theorems reduces to one smooth
ambient metric as follows. Pass further in a smaller compact chart to a subsequence converging so
@@ -4132,14 +4132,14 @@ disjoint smooth interval cutoffs. The resulting product block-diagonal
metric is smooth; its slices have bounded second forms. After smoothly
extending from a still smaller box to a closed manifold, a smooth Nash
embedding puts this situation in a single Euclidean ambient space
-\cite[Theorem~2, p.~59]{Nash1956}.
+cite[Theorem~2, p.~59]{Nash1956}.
The preceding monotonicity and Allard statements apply there, with the
same local bounded-mean-curvature property. Applying this subsequence
argument to a sequence on which local convergence failed proves the
local support and graphical convergence asserted above.
-\par\noindent{\footnotesizehttps://doi.org/10.2307/1969989}\par
+parnoindent{footnotesizehttps://doi.org/10.2307/1969989}par
For $d=1$ the needed facts are elementary. A locally minimizing set has
locally finitely many jumps by the perimeter bound. Two consecutive
@@ -4157,32 +4157,32 @@ singular points in this case.
Now extract a smooth normal speed from the ordered family. Write
$Σ_s$ for the regular part of $Σ_s^{all}$ in
-$\{|G|_∞<t_3\}$. The boundary has its exterior coorientation and
+${|G|_∞<t_3}$. The boundary has its exterior coorientation and
the induced orientation. Its singular set has zero $p$-dimensional area,
so it never contributes to the integrals over the boundary current.
Differentiating the ordered family.:
On each leaf the sets $E_t$ converge to $E_s$ in $L^1_{loc}$ as
-$t\to s$ except at countably many $s$. In fact, on a compact chart set
-$K$ the function $t\mapsto\int_K\mathbf1_{E_t}$ is monotone, and nestedness
+$tto s$ except at countably many $s$. In fact, on a compact chart set
+$K$ the function $tmapstoint_Kmathbf1_bf1_{E_t}$ is monotone, and nestedness
identifies its increments with the $L^1(K)$ distance between the
indicators. It has at most countably many discontinuities. Use the
countable chart exhaustion to obtain the assertion on the leaf.
At such a continuity parameter, the local convergence result above makes
$Σ_t$ converge smoothly and graphically near each regular point of
-$Σ_s$ as $t\to s$.
+$Σ_s$ as $tto s$.
In a smaller graph cylinder choose coordinates $(y,z)$ with the sets
below the graphs, and write the graph heights as $z=r(t,y)$. They are
monotone in $t$. They are strictly ordered wherever two such regular
-graphs are defined: if $r(t,\cdot)\ge r(s,\cdot)$ for $t>s$ and they touch,
+graphs are defined: if $r(t,cdot)ge r(s,cdot)$ for $t>s$ and they touch,
their tangent planes agree at contact. The comparison of the graph mean
-curvatures at that minimum of $r(t,\cdot)-r(s,\cdot)$ gives
-$D_t\le D_s$, whereas their equations give
-$D_t=\mathfrak h_t>\mathfrak h_s=D_s$ at the same point. This is
+curvatures at that minimum of $r(t,cdot)-r(s,cdot)$ gives
+$D_tle D_s$, whereas their equations give
+$D_t=mathfrak h_t>frakh_t>mathfrak h_s=D_s$frakh_s=D_s$ at the same point. This is
impossible. Use the conventions for $D$ and $II$ from the
-local slice calculation, with $\nu$ the exterior unit normal. The
-equation $D=\mathfrak h_s$ is the first variation of
+local slice calculation, with $nu$ the exterior unit normal. The
+equation $D=mathfrak h_s$frakh_s$ is the first variation of
(eq:degree-functional) on a regular portion.
Choose countably many possible graph cylinders in the chart enumerations,
@@ -4190,52 +4190,52 @@ and in each choose a countable dense set of $y$-coordinates. At each of
these coordinates a monotone height has a finite derivative for almost
every parameter. The test is meaningful even outside an interval of
graphical representation: use instead the monotone column function
-\[
- t⟼\int_{z_-}^{z_+}\mathbf1_{E_t}(y,z)\,dz,
-\]
+[
+ t⟼int_{z_-}^{z_+}mathbf1_bf1_{E_t}(y,z),dz,
+]
which equals $r(t,y)-z_-$ in the single-graph situation. Consequently,
for almost every continuity parameter $s$, the nonnegative difference
quotients
-\[
- w_t(y)=(r(t,y)-r(s,y))/(t-s),\qquad t\ne s,
-\]
-converge to the finite column-derivative values as $t\to s$ at all these
+[
+ w_t(y)=(r(t,y)-r(s,y))/(t-s),qquad tne s,
+]
+converge to the finite column-derivative values as $tto s$ at all these
dense test coordinates. In particular they are bounded there.
In positive graph dimension the $w_t$ solve linear elliptic equations on
each smaller cylinder. This follows by subtracting the two prescribed
graph equations and integrating their linearizations between the two
graphs. Smooth graphical convergence gives common smooth coefficient
-bounds and ellipticity; the explicit $t$-dependence of $\mathfrak h_t$ gives a
+bounds and ellipticity; the explicit $t$-dependence of $mathfrak h_t$frakh_t$ gives a
smooth bounded source. Harnack's inequality with a bounded source,
applied using any one of the bounded test values, bounds $w_t$ on a
smaller patch. Interior elliptic estimates then give smoothly convergent
subsequences there; these are the standard local Harnack and interior
estimates for uniformly elliptic equations with bounded lower-order
-coefficients and source, as in \cite[Chapters~6 and~8]{GilbargTrudinger}.
+coefficients and source, as in cite[Chapters~6 and~8]{GilbargTrudinger}.
Every such limit takes the already determined
derivative values at the dense test coordinates, so the limits agree.
Thus the whole difference quotient has a smooth local limit. Its normal
-component is a smooth function $v\ge0$ on the regular portion. For
+component is a smooth function $vge0$ on the regular portion. For
$p=0$ the same conclusion follows from the ordinary almost-everywhere
derivative of a monotone height.
-Differentiation of $D=\mathfrak h_s$ now identifies this normal speed.
-Use $\mathcal P$ from the local slice calculation and put
-$\mathfrak a=∂_s\mathfrak h_s=L_1\psi(D)>0$; the last equality
+Differentiation of $D=mathfrak h_s$frakh_s$ now identifies this normal speed.
+Use $mathcal P$calP$ from the local slice calculation and put
+$mathfrak a=∂_sfraka=∂_smathfrak h_s=L_1frakh_s=L_1psi(D)>0$; the last equality
uses confinement, where the forcing agrees with the raw pressure.
The normal variation formula, with the tangential derivative of
-$D-\mathfrak h_s=0$ cancelling, is
+$D-mathfrak h_s=0$frakh_s=0$ cancelling, is
- -(\Delta_{Σ,\phi}+\mathcal P+∂_\nu\mathfrak h_s)v
- =\mathfrak a.
+ -(Delta_{Σ,phi}+mathcal P+∂_calP+∂_numathfrak h_s)vfrakh_s)v
+ =mathfrak a.fraka.
-The convention for $\Delta_{Σ,\phi}$ is weighted by
-$e^{\phi|_Σ}dA_h$. The variation of $∂_\nu\phi$ adds
-$v\nabla^2\phi(\nu,\nu)-< d\phi,dv>_Σ$ to the ordinary
+The convention for $Delta_{Σ,phi}$ is weighted by
+$e^{phi|_Σ}dA_h$. The variation of $∂_nuphi$ adds
+$vnabla^2phi(nu,nu)-< dphi,dv>_Σ$ to the ordinary
mean-curvature variation, giving the signs in
(eq:degree-lapse). A nonnegative $v$ cannot vanish at an interior
-point: there the left side is nonpositive, contrary to $\mathfrak a>0$.
+point: there the left side is nonpositive, contrary to $mathfrak a>0$.fraka>0$.
Thus $v>0$. In graph dimension zero the differential terms vanish and
the same equation rules out $v=0$.
@@ -4248,7 +4248,7 @@ representative is unique; the Borel singleton-section theorem applied to
its smooth-function code makes it Borel. The continuity and
differentiability tests above use countable sequences, by monotonicity.
The resulting local derivatives and $v$ are Borel. Fubini on each
-$\sigma$-finite transversal now gives, for almost every $s$, a null set
+$sigma$-finite transversal now gives, for almost every $s$, a null set
of exceptional plaques. There are countably many charts and overlap
moves, so their saturations can be discarded simultaneously. In
particular the family of regular portions and the function $v$ used in
@@ -4257,10 +4257,10 @@ Set $v=0$ on the exceptional parameters and plaques when writing an
integral over the whole parameter interval.
In a graph cylinder, normal speed times area has the expression
-\[
- v\,dA_h=J_h(y,r(s,y))\,∂_s r(s,y)\,dy,
-\]
-where $J_h(y,z)\,dy\,dz$ is ambient Riemannian volume in the coordinates.
+[
+ v,dA_h=J_h(y,r(s,y)),∂_s r(s,y),dy,
+]
+where $J_h(y,z),dy,dz$ is ambient Riemannian volume in the coordinates.
The one-dimensional change-of-variables inequality for a monotone
function says that integration against its almost-everywhere derivative
counts at most integration in $z$. A jump or singular part of the
@@ -4273,18 +4273,18 @@ Therefore no ambient point is counted more than once on the right.
Equivalently, the counted images can be taken Borel by the Borel
one-to-one image theorem. Transverse invariance on overlaps gives
- \int_{-\tau}^{\tau}\!ds\int_{Σ_s}vF\,dA_h
- \le\int_XF\,dV_h
+ int_{-tau}^{tau}!dsint_{Σ_s}vF,dA_h
+ leint_XF,dV_h
-for every nonnegative Borel $F$ supported in $\{|G|_∞<t_3\}$.
+for every nonnegative Borel $F$ supported in ${|G|_∞<t_3}$.
Since $b_d<t_3$, this proves (eq:degree-coarea).
Require one more consequence of (eq:degree-lapse). On a
family of regular graph patches with common buffers and graph estimates,
-all positive-order derivatives of $\ln v$ have uniform bounds on smaller
+all positive-order derivatives of $ln v$ have uniform bounds on smaller
patches. Indeed the coefficients in (eq:degree-lapse) and the
-derivatives of $\mathfrak a$ are uniformly bounded there, and
-$\mathfrak a$ has a uniform positive lower bound. Confinement provides
+derivatives of $mathfrak a$fraka$ are uniformly bounded there, and
+$mathfrak a$fraka$ has a uniform positive lower bound. Confinement provides
the finite margin from the pressure poles needed for these bounds. If
the value of $v$ could tend to zero at an interior point of such patches,
Harnack's inequality with bounded source would bound $v$ on a smaller
@@ -4293,56 +4293,56 @@ interior zero satisfying (eq:degree-lapse), an impossibility.
Thus $v$ has a positive local lower bound. Harnack then bounds its local
supremum by a constant times its value at an interior point, since the
bounded source can be absorbed using that lower bound. Interior
-estimates give $|\nabla^r v|\le C_r v$ on still smaller patches, and
-differentiating $\ln v$ gives the claim. This does not bound the value
-of $\ln v$, which is not required by Definition~def:degree-band.
+estimates give $|nabla^r v|le C_r v$ on still smaller patches, and
+differentiating $ln v$ gives the claim. This does not bound the value
+of $ln v$, which is not required by Definition~def:degree-band.
For $p=0$ the assertion about positive-order derivatives is empty.
The local slice calculation now applies to the actual boundaries.
-Confinement gives $D=\mathfrak h_s=P_0(s-G_1)$ and hence
-$∂_\nu\mathfrak h_s=-\mathfrak a\,dG_1(\nu)$ on every retained
+Confinement gives $D=mathfrak h_s=P_0(s-G_1)$frakh_s=P_0(s-G_1)$ and hence
+$∂_numathfrak h_s=-frakh_s=-mathfrak afraka,dG_1(nu)$ on every retained
regular portion. Equation~(eq:degree-lapse) therefore becomes
exactly (eq:degree-local-lapse). With
-$\phi_Σ=\phi|_Σ+\ln v$, the scalar bound
+$phi_Σ=phi|_Σ+ln v$, the scalar bound
(eq:degree-slice-bound) and the pointwise comparison
(eq:degree-raw-integrand) hold on these portions. The derivative
bounds just proved supply the required local controls for this weight.
Degree on the boundary.:
Let $J=(G_2,...,G_d)$. Fix a unit-integral one-form
-$a_-(z)\,dz$ supported in $(-1.9\tau,-1.6\tau)$ and one
-$a_+(z)\,dz$ supported in $(1.6\tau,1.9\tau)$. Let $\beta$ be a
+$a_-(z),dz$ supported in $(-1.9tau,-1.6tau)$ and one
+$a_+(z),dz$ supported in $(1.6tau,1.9tau)$. Let $beta$ be a
unit-integral smooth $p$-form supported in a sufficiently small cube
-about zero in the residual target; for $p=0$ let $\beta=1$. Set
-\[
- \chi_{deg}(z)=\int_{-∞}^{z}(a_-(u)-a_+(u))\,du.
-\]
-This function is compactly supported in $(-1.9\tau,1.9\tau)$ and is one
+about zero in the residual target; for $p=0$ let $beta=1$. Set
+[
+ chi_{deg}(z)=int_{-∞}^{z}(a_-(u)-a_+(u)),du.
+]
+This function is compactly supported in $(-1.9tau,1.9tau)$ and is one
between the two supports. By (eq:degree-pole-width) and
(eq:degree-confinement), the low support is in $E_s$, the high
-support is outside $E_s$, and $\chi_{deg}(G_1)=1$ on the confined
+support is outside $E_s$, and $chi_{deg}(G_1)=1$ on the confined
boundary where the residual form is supported. Measured current Stokes therefore gives
- \int_{∂[E_s]} \chi_{deg}(G_1)J^*\beta
- &=\int_{E_s}(a_-(G_1)-a_+(G_1))\,dG_1\wedge J^*\beta\\
- &=\int_Xa_-(G_1)\,dG_1\wedge J^*\beta
- =\mathcal D_X(G).
+ int_{∂[E_s]} chi_{deg}(G_1)J^*beta
+ &=int_{E_s}(a_-(G_1)-a_+(G_1)),dG_1wedge J^*beta
+ &=int_Xa_-(G_1),dG_1wedge J^*beta
+ =mathcal D_X(G).calD_X(G).
All forms here are supported in an inner band. The current $[E_s]$ has
bounded local volume and its boundary has the uniform local perimeter
bounds proved above, so this is precisely the current version of
(eq:degree-stokes). The singular set contributes no area, and
-the counted part of the boundary is in $\{|G|_∞<t_3\}$.
+the counted part of the boundary is in ${|G|_∞<t_3}$.
Consequently (eq:degree-boundary-degree) computes the degree by
-$J^*\beta$ on that confined regular branch, with the induced orientation.
+$J^*beta$ on that confined regular branch, with the induced orientation.
For $p=0$ it gives its measured signed count.
-If $2\le d\le7$, this regular branch already provides the child. To verify
+If $2le dle7$, this regular branch already provides the child. To verify
the required uniformity, fix a typical $s$. If graph radii and estimates
could not be chosen uniformly about the boundary in an inner region of
-$\{|G|_∞<t_3\}$, take offending points in buffered parent charts
+${|G|_∞<t_3}$, take offending points in buffered parent charts
and pass to normalized chart limits. Local BV and varifold compactness
gives a limiting boundary through the limiting point. There are no
singular points because $p-7<0$, so smooth graphical convergence near
@@ -4352,26 +4352,26 @@ now gives at most a bounded number of patches in each parent plaque.
Their transversals are the restricted parent transversals, counted with
the patch index. Their total transverse measure and plaque overlap
remain bounded. The graph estimates, together with the bounds for
-derivatives of $\ln v$, give the band controls for the induced metric and
-$\phi_Σ$.
+derivatives of $ln v$, give the band controls for the induced metric and
+$phi_Σ$.
Restrict this branch to
-\[
- |G|_∞<t_1,\qquad |J|_∞<T_p.
-\]
+[
+ |G|_∞<t_1,qquad |J|_∞<T_p.
+]
The $G_1$ sides of the first restriction cannot be reached by confinement,
-since $|G_1|<\tau+b<T_p$. Near any other side some residual coordinate
+since $|G_1|<tau+b<T_p$. Near any other side some residual coordinate
has absolute value near $t_1>T_p$. Hence every inner residual subcube
has buffered graph charts in this restriction. Its degree equals
-$\mathcal D_X(G)$ by (eq:degree-boundary-degree), and its coordinate gradients only
+$mathcal D_X(G)$calD_X(G)$ by (eq:degree-boundary-degree), and its coordinate gradients only
decrease on restriction to the tangent space. Take $J^+=J$ and use
(eq:degree-slice-bound).
If $d=1$, so $p=0$, let $X_s^+$ be the confined regular points in
-$Σ_s\cap\{|G_1|<t_1\}$, each with its boundary orientation, and let
+$Σ_scap{|G_1|<t_1}$, each with its boundary orientation, and let
$J^+$ be the unique map to the point $Q_{T_0}^0$. Give these point leaves
-the induced zero-dimensional metric and the weight $\phi_Σ$.
-Confinement gives $|G_1|<\tau+b<T_0<t_1$ on the counted branch, so this
+the induced zero-dimensional metric and the weight $phi_Σ$.
+Confinement gives $|G_1|<tau+b<T_0<t_1$ on the counted branch, so this
includes every point in (eq:degree-boundary-degree).
The local jump separation proved above bounds the number of such points
in each buffered parent plaque. Choose a fixed finite half-open mesh
@@ -4385,15 +4385,15 @@ plaque overlap remains bounded. Point boxes have the required common
buffers, and all metric, map, and positive-order
weight derivative conditions in Definition~def:degree-band are
vacuous in dimension zero, so this is a controlled $0$-band. Its measured
-signed count is $\mathcal D_X(G)$ by
+signed count is $mathcal D_X(G)$calD_X(G)$ by
(eq:degree-boundary-degree); its integral is the measured unsigned
count. At each point (eq:degree-raw-integrand) reads
-$1\le(v/L_1)Θ$, and confinement places every counted point in
+$1le(v/L_1)Θ$, and confinement places every counted point in
$|G_1|<b_d$. Return to the common integral estimate after treating
-the possible singularities when $p\ge7$.
+the possible singularities when $pge7$.
-Assume now $p\ge7$, and fix one of the typical parameters $s$ above.
+Assume now $pge7$, and fix one of the typical parameters $s$ above.
The singular set has zero $p$-dimensional area, so it does not change the
boundary-current integral. Deleting it can, however, leave ends of the
regular locus over interior residual values, where regular chart radii
@@ -4403,29 +4403,29 @@ boundary current.
Construct a controlled child carried by an open regular subset
of this slice, with the inherited orientation and transverse measure,
-and with degree $\mathcal D_X(G)$. For each residual row one requires $|dJ_j^+|_{h^+}\le |dG_j|_h+ε_{grad}$ and the
+and with degree $mathcal D_X(G)$.calD_X(G)$. For each residual row one requires $|dJ_j^+|_{h^+}le |dG_j|_h+ε_{grad}$ and the
one-sided value bound
-$|J_j^+|\ge |G_j|-ε_{val}$, with
+$|J_j^+|ge |G_j|-ε_{val}$, with
- \sum_{j=2}^d L_jε_{grad}<\frac1{10q},
- \qquad ε_{val}<b_d-b_p=(\tau)/(4q^2).
+ sum_{j=2}^d L_jε_{grad}<frac1{10q},
+ qquad ε_{val}<b_d-b_p=(tau)/(4q^2).
The gradient bound controls the residual penalties and the child's
coordinate bound.
The value bound, together with confinement of $G_1$, puts the child's
$b_p$ integration region over the parent's $b_d$ region. Write
-$\mathcal W=dV_{h^+}/dA_h$ for its ordinary leafwise volume density
+$mathcal W=dV_calW=dV_{h^+}/dA_h$ for its ordinary leafwise volume density
relative to the original induced metric. The scalar and volume target is
- dV_{h^+}=\mathcal W\,dA_h,\\
- \wscal(h^+,\phi^+)\ge\wscal(h,\phi)-H_1(G)-2\ln(v/L_1)
- +2\ln\mathcal W-\frac1q.
+ dV_{h^+}=mathcal WcalW,dA_h,
+ wscal(h^+,phi^+)gewscal(h,phi)-H_1(G)-2ln(v/L_1)
+ +2lnmathcal W-calW-frac1q.
The coefficient $2$ makes the volume factor cancel after exponentiation,
leaving the same lapse factor as in (eq:degree-raw-integrand).
-The density $\mathcal W$ includes neither $v$ nor the transverse measure.
+The density $mathcal W$calW$ includes neither $v$ nor the transverse measure.
Every constant and error chosen inside this subsection may depend on this
slice and on the band, but must be common across its retained plaques.
@@ -4439,8 +4439,8 @@ enter the scalar constants. The adjustable scalar error in
(eq:degree-child-scalar) is at most $1/q$, so chart and elliptic
constants will not enter the absolute loss in Theorem~thm:degree.
-Choose a parent atlas with buffers around $\{|G|_∞\le t_3\}$ still
-inside $\{|G|_∞<t_4\}$. One obtains it from an atlas for a larger
+Choose a parent atlas with buffers around ${|G|_∞le t_3}$ still
+inside ${|G|_∞<t_4}$. One obtains it from an atlas for a larger
inner cube by a bounded mesh subdivision, restricting transversals by
Borel tests as necessary. Its plaque overlaps, also for the enlarged
supports below, remain bounded. All operations in the parent made below
@@ -4469,33 +4469,33 @@ convergent graphical boundary. The graph coding already described shows
that recognition is Borel.
Let $D_k$ be the part of the boundary support in
-$\{|G|_∞\le t_3\}$ left after removing the open inner portions
+${|G|_∞le t_3}$ left after removing the open inner portions
recognized through level $k$ in any parent plaque. These are decreasing
Borel sets, closed along each leaf. Every regular point is eventually
removed. More precisely, in any converging sequence of controlled
charts with levels tending to infinity, the $D_k$ can accumulate on a
compact subbox only at singular points of the limiting boundary.
Outside the limit support this follows from support convergence. If
-points of $D_{k_j}$ with $k_j\to∞$ converged instead to a regular
-limit point, one test with a fixed index $\ell$ in the common list would
+points of $D_{k_j}$ with $k_jto∞$ converged instead to a regular
+limit point, one test with a fixed index $ell$ in the common list would
recognize that point with strict room. Smooth graphical convergence
makes the same test recognize the nearby points for large $j$, and then
-$k_j\ge\ell$, contradicting their membership in $D_{k_j}$. This is the
+$k_jgeell$, contradicting their membership in $D_{k_j}$. This is the
compactness fact used in both constructions that follow.
-First make an arbitrarily small smooth perturbation $\bar J$ of $J$ in
+First make an arbitrarily small smooth perturbation $barbar J$J$ of $J$ in
the parent band, small in value and first derivative, with all
higher derivatives uniformly bounded. Claim it can be made so that,
for some $a_0>0$ and some finite $k$,
- |\bar J|_∞>4a_0
- \quad\hbox{on a uniform neighborhood of }D_k.
+ |barbar J|_∞>4a_0J|_∞>4a_0
+ quadhbox{on a uniform neighborhood of }D_k.
After the initial homotopy on the full boundary current, this gap will
keep the homotopies over a small target cube inside a uniformly regular
region.
To prove the local assertion behind the claim, fix a positive budget for
-a constant shift in $\mathbb R^p$ and an inner compact plaque box.
+a constant shift in $mathbb R^p$bbR^p$ and an inner compact plaque box.
There are a level and a positive gap, uniform across the controlled
plaques, such that some shift in that budget gives this gap from zero on
the marked set in the box. Otherwise take a sequence of failing plaques,
@@ -4518,8 +4518,8 @@ family of Borel sets distinguishes each vertex from its finitely many
neighbors, and the first finite distinguishing pattern is a countable
color. The finite neighbor lists can be enumerated Borel measurably by
the Borel countable-section theorem. Processing these color classes
-successively and taking the first one of $\Delta+1$ colors avoided by
-all earlier neighbors, where $\Delta$ bounds the degree,
+successively and taking the first one of $Delta+1$ colors avoided by
+all earlier neighbors, where $Delta$ bounds the degree,
gives a finite Borel proper coloring. In each color the cutoff supports
are disjoint. Choose rational shifts satisfying the local gap test; the
test on a compact inner box is Borel by projection with compact sections.
@@ -4529,29 +4529,29 @@ sets decrease, so earlier gaps persist. There are only finitely many
colors. The budgets can also make the total value and first-derivative
change as small as prescribed; the cutoffs give uniform bounds for all
higher derivatives. Taking $a_0$ smaller than the final gaps and then
-using the derivative bound of $\bar J$ extends the gap to a uniform
+using the derivative bound of $barbar J$J$ extends the gap to a uniform
neighborhood. This proves (eq:degree-gap).
Second, set
- \alpha=p-(13)/(2)>0.
+ alpha=p-(13)/(2)>0.
The next cover will support a function large on the remaining marked
points while keeping the total kernel coefficients as small as needed.
For any prescribed positive budget and upper radius, a sufficiently
-large $k'\ge k$ permits a cover of $D_{k'}$ in each inner plaque box by
+large $k'ge k$ permits a cover of $D_{k'}$ in each inner plaque box by
finitely many small balls, of radii $r_i$, such that
- \sum_i r_i^\alpha<the prescribed budget.
+ sum_i r_i^alpha<the prescribed budget.
The radii can also have a common positive lower bound, depending on the
budget and the controlled family. To verify the uniform claim, suppose
it failed for levels tending to infinity and for requested lower radii
tending to zero. Pass to a limiting chart. Its compact singular set has
-zero $\alpha$-dimensional Hausdorff measure because
-$p-7<\alpha$. Cover it on a slightly larger compact box by finitely
+zero $alpha$-dimensional Hausdorff measure because
+$p-7<alpha$. Cover it on a slightly larger compact box by finitely
many balls with radii below the prescribed upper radius and with the
-$\alpha$-sum strictly below the budget, allowing room to enlarge the balls.
+$alpha$-sum strictly below the budget, allowing room to enlarge the balls.
This finite cover has a positive minimum radius. The concentration
observation about $D_k$ makes the enlarged balls cover the marked points
of the sequence eventually, contradicting failure. Rational chart balls
@@ -4566,48 +4566,48 @@ bound the number of balls in any plaque.
The two uses of (eq:degree-alpha) are distinct: its strict excess
over $p-7$ made this cover possible, while
-$\alpha+4=p-5/2<p-2$ will make the following potentials superharmonic.
+$alpha+4=p-5/2<p-2$ will make the following potentials superharmonic.
Potentials near the marked set.:
Choose a small parent normal-coordinate radius $R>0$ so that a ball of
that radius about any cover center stays in the gap region
-$|\bar J|_∞>3a_0$ and in the buffered atlas. It can also be chosen
+$|barbar J|_∞>3a_0$J|_∞>3a_0$ and in the buffered atlas. It can also be chosen
small with respect to the local metric, weight, and forcing bounds.
This radius is fixed before the final cover is chosen; require its ball
radii to be much smaller than $R$. For a center $c_i$ and radius $r_i$
let
-\[
+[
K_i(x)=(r_i^2+dist_h(x,c_i)^2)^{-1/2}
-\]
+]
on the smaller normal ball, and cut it off within radius $R$. Choose a
radial cutoff which is one on the smaller ball, is positive before the
outer boundary, and near that boundary has the tail
-$\exp(-1/(1-t))$ for $t<1$ and zero for $t\ge1$, where
+$exp(-1/(1-t))$ for $t<1$ and zero for $tge1$, where
$t=dist_h(x,c_i)^2/R^2$. Join this tail smoothly and
positively to the inner plateau. Every positive power of the cutoff
then has a smooth exponential flat tail, so each power of $K_i$ used
below is smooth. Denote its support by $V_i$.
-On the regular boundary in $V_i$, for $j=\alpha,\alpha+4$,
+On the regular boundary in $V_i$, for $j=alpha,alpha+4$,
- -\Delta_{Σ,\phi}K_i^j&\ge cK_i^{j+2}-C\mathbf1_{V_i},\\
- |dK_i^j|&\le C(K_i^{j+1}+\mathbf1_{V_i}).
+ -Delta_{Σ,phi}K_i^j&ge cK_i^{j+2}-Cmathbf1_bf1_{V_i},
+ |dK_i^j|&le C(K_i^{j+1}+mathbf1_bf1_{V_i}).
The constants may depend on the fixed local controls and on $p$, but are
independent of $r_i$ and of the eventual number of balls. To check the
first estimate before cutoff, put $u=dist_h(x,c_i)^2$.
The normal-coordinate Hessian bound, the bounded mean curvature
-$H=\mathfrak h_s-∂_\nu\phi$, and the bounded weight gradient give
-\[
- \Delta_{Σ,\phi}u\ge2p-C\sqrt u,\qquad |du|_Σ^2\le4u.
-\]
+$H=mathfrak h_s-∂_frakh_s-∂_nuphi$, and the bounded weight gradient give
+[
+ Delta_{Σ,phi}uge2p-Csqrt u,qquad |du|_Σ^2le4u.
+]
Differentiating $(r_i^2+u)^{-j/2}$ therefore gives
-\[
- -\Delta_{Σ,\phi}K_i^j
- \ge j(p-j-2-C\sqrt u)K_i^{j+2}.
-\]
+[
+ -Delta_{Σ,phi}K_i^j
+ ge j(p-j-2-Csqrt u)K_i^{j+2}.
+]
Both chosen exponents are less than $p-2$, so small $R$ makes the
coefficient positive. The gradient estimate follows by one
differentiation. On the cutoff annulus the distance from the center is
@@ -4615,93 +4615,93 @@ bounded below by a fixed multiple of $R$, and all cutoff derivatives are
bounded independently of $r_i$; this accounts for the displayed constant
losses. Outside $V_i$ they vanish, as the indicators record.
-For $j=\alpha,\alpha+2,\alpha+4,\alpha+6$ put
+For $j=alpha,alpha+2,alpha+4,alpha+6$ put
- u_j=ε_{base}^j+\sum_i M_*r_i^\alpha K_i^j,
- \qquad Λ=(\frac{u_{\alpha+4}}{u_\alpha})^{1/4},
- \qquad
- \mathcal L=|d\ln u_\alpha|^2+|d\ln u_{\alpha+4}|^2.
+ u_j=ε_{base}^j+sum_i M_*r_i^alpha K_i^j,
+ qquad Λ=(frac{u_{alpha+4}}{u_alpha})^{1/4},
+ qquad
+ mathcal L=|dcalL=|dln u_alpha|^2+|dln u_{alpha+4}|^2.
-These quantities have separate roles. Make $u_\alpha$ large on
+These quantities have separate roles. Make $u_alpha$ large on
$D_{k'}$ so that its logarithm locates the region to be excluded.
The term $Λ^2$ will supply the positive scalar contribution that
-pays for the later coordinate graphs, while $\mathcal L$ controls the
+pays for the later coordinate graphs, while $mathcal L$calL$ controls the
logarithmic derivatives introduced by the weight and the first graph.
Here $0<ε_{base}<1$ will be small; $M_*$ will be a
large finite number fixed before the final cover selection. At a point, the sum is over the
kernels active there. The overlap bound and
(eq:degree-cover-budget) allow the total active coefficient
-$\sum_i M_*r_i^\alpha$ to be made arbitrarily small, uniformly at every
-point. In particular, for any $\delta>0$, first taking
+$sum_i M_*r_i^alpha$ to be made arbitrarily small, uniformly at every
+point. In particular, for any $delta>0$, first taking
$ε_{base}$ small and then the cover budget small enough
-gives, for $j=\alpha,\alpha+4$,
+gives, for $j=alpha,alpha+4$,
- -\frac{\Delta_{Σ,\phi}u_j}{u_j}
- \ge c\frac{u_{j+2}}{u_j}-\delta.
+ -frac{Delta_{Σ,phi}u_j}{u_j}
+ ge cfrac{u_{j+2}}{u_j}-delta.
Indeed (eq:degree-kernel-bounds) leaves the errors
-$cε_{base}^{j+2}+C\sum_{i:x\in V_i}M_*r_i^\alpha$
-before division by $u_j\geε_{base}^j$. Choose
+$cε_{base}^{j+2}+Csum_{i:xin V_i}M_*r_i^alpha$
+before division by $u_jgeε_{base}^j$. Choose
$ε_{base}^2$ and then the active sum divided by
-$ε_{base}^{\alpha+4}$ small enough for the displayed
-$\delta$.
+$ε_{base}^{alpha+4}$ small enough for the displayed
+$delta$.
At each point the $u_j$ are the moments of positive atoms: one atom at
$ε_{base}$ of mass one, and atoms at $K_i$ of masses
-$M_*r_i^\alpha$.
+$M_*r_i^alpha$.
Make their total active mass at most two. Moment inequalities give
- \frac{u_{\alpha+6}}{u_{\alpha+4}}\geΛ^2
- \ge\frac{u_{\alpha+2}}{u_\alpha},\qquad
- |d\lnΛ|^2\le(\mathcal L)/(8),\qquad
- |d\ln u_\alpha|\le C(Λ+1),\qquad
- Λ\ge c' u_\alpha^{1/\alpha}.
+ frac{u_{alpha+6}}{u_{alpha+4}}geΛ^2
+ gefrac{u_{alpha+2}}{u_alpha},qquad
+ |dlnΛ|^2le(mathcal L)/(8),calL)/(8),qquad
+ |dln u_alpha|le C(Λ+1),qquad
+ Λge c' u_alpha^{1/alpha}.
For the first pair, use the probability weights proportional to the
-atoms' masses times their $\alpha$-powers. If $Z$ is the square of the
+atoms' masses times their $alpha$-powers. If $Z$ is the square of the
atom position, then
-$\mathbb E Z\le(\mathbb E Z^2)^{1/2}$ and
-$\mathbb E Z^3\ge(\mathbb E Z^2)^{3/2}$ give the two inequalities. The
+$mathbb EbbE Zle(mathbb EbbE Z^2)^{1/2}$ and
+$mathbb EbbE Z^3ge(mathbb EbbE Z^2)^{3/2}$ give the two inequalities. The
second estimate follows directly by differentiating
-$\lnΛ=(\ln u_{\alpha+4}-\ln u_\alpha)/4$. For the third, the
+$lnΛ=(ln u_{alpha+4}-ln u_alpha)/4$. For the third, the
gradient bound in (eq:degree-kernel-bounds) bounds
-$|du_\alpha|$ by $C\sum_iM_*r_i^\alpha(K_i^{\alpha+1}+1)$.
-The first part divided by $u_\alpha$ is at most $Λ$ by H\"older's
+$|du_alpha|$ by $Csum_iM_*r_i^alpha(K_i^{alpha+1}+1)$.
+The first part divided by $u_alpha$ is at most $Λ$ by H"older's
inequality for those probability weights; the second is at most one after
the same small active-mass choice. Finally, monotonicity of the
$L^r$ norms for a measure of mass at most two gives
-$u_{\alpha+4}/u_\alpha\ge(u_\alpha/2)^{4/\alpha}$, proving the last
+$u_{alpha+4}/u_alphage(u_alpha/2)^{4/alpha}$, proving the last
inequality. These constants are independent of $M_*$ and $k'$.
At a point of $D_{k'}$ some ball has distance at most its radius, so its
-term in $u_\alpha$ is at least $c''M_*$. Hence
+term in $u_alpha$ is at least $c''M_*$. Hence
- u_\alpha\ge c''M_*\quad\hbox{on }D_{k'}.
+ u_alphage c''M_*quadhbox{on }D_{k'}.
The positive lower bound on all radii and the bounded number of balls per
plaque make all $u_j$ smooth in the parent with uniform derivative bounds
for the final slice. These bounds are allowed to depend on that radius
lower bound.
-Add $\eta_0=(\ln u_\alpha+\ln u_{\alpha+4})/4$ to $\phi_Σ$ on the
-regular boundary. Put $Q_1=d\eta_0$. The weight-change formula used in
+Add $eta_0=(ln u_alpha+ln u_{alpha+4})/4$ to $phi_Σ$ on the
+regular boundary. Put $Q_1=deta_0$. The weight-change formula used in
(eq:degree-slice-identity), now with
-$\Delta_{Σ,\phi_Σ}=\Delta_{Σ,\phi}
-+< d\ln v,d\,\cdot\,>$, yields from
+$Delta_{Σ,phi_Σ}=Delta_{Σ,phi}
++< dln v,d,cdot,>$, yields from
(eq:degree-potential-laplacian) the scalar gains
-$cΛ^2+\mathcal L/2-2< d\ln v,Q_1>-|Q_1|^2-\delta$.
+$cΛ^2+mathcal L/2-2<calL/2-2< dln v,Q_1>-|Q_1|^2-delta$.
Here the high moment in (eq:degree-moments) supplies $Λ^2$.
-Since $|Q_1|^2\le\mathcal L/8$,
-\[
- (\mathcal L)/(2)+|d\ln v|^2
- -2< d\ln v,Q_1>-|Q_1|^2
- \ge(\mathcal L)/(2)-2|Q_1|^2\ge(\mathcal L)/(4).
-\]
+Since $|Q_1|^2lemathcal L/8$,calL/8$,
+[
+ (mathcal L)/(2)+|dcalL)/(2)+|dln v|^2
+ -2< dln v,Q_1>-|Q_1|^2
+ ge(mathcal L)/(2)-2|Q_1|^2calL)/(2)-2|Q_1|^2ge(mathcal L)/(4).calL)/(4).
+]
Combining with (eq:degree-slice-bound), and decreasing $c>0$ if
necessary, gives for the new weight
- \wscal(h|_{TΣ},\phi_Σ+\eta_0)
- \ge\wscal(h,\phi)-H_1(G)-2\ln(v/L_1)
- +cΛ^2+(\mathcal L)/(4)-\delta.
+ wscal(h|_{TΣ},phi_Σ+eta_0)
+ gewscal(h,phi)-H_1(G)-2ln(v/L_1)
+ +cΛ^2+(mathcal L)/(4)-calL)/(4)-delta.
A large scalar allowance has been obtained near the marked set. The next
construction uses it to move the residual map out of the target band
@@ -4710,146 +4710,146 @@ there, while paying for the graph volume created in doing so.
Two graph prescriptions and their costs.:
Describe the graph calculations first, and then give a complete region
on which the prescriptions can be solved. The first prescription will
-produce a function $f$ within two of a small multiple of $\ln u_\alpha$,
-with $|df|\le1$ in its graph metric. The coordinate prescriptions then
-use this controlled gradient to amplify $\bar J$ where $f$ is high;
+produce a function $f$ within two of a small multiple of $ln u_alpha$,
+with $|df|le1$ in its graph metric. The coordinate prescriptions then
+use this controlled gradient to amplify $barbar J$J$ where $f$ is high;
each correction adds at most $e$ to its coordinate gradient in the final
-graph metric, in addition to the earlier perturbation of $\bar J$.
+graph metric, in addition to the earlier perturbation of $barbar J$.J$.
Their scalar gains will pay the exact graph volume factors.
For a current metric and
-weight $(k,\varphi)$, a length-one graph of a function $f$ has
-\[
- L_f=(1+|df|_k^2)^{1/2},\quad U_f=(\nabla_k f)/(L_f),\quad
- k_f=k+df ⊗ df,\quad \varphi_f=\varphi-\ln L_f,
- \quad D_f=-div_{k,\varphi}U_f.
-\]
+weight $(k,varphi)$, a length-one graph of a function $f$ has
+[
+ L_f=(1+|df|_k^2)^{1/2},quad U_f=(nabla_k f)/(L_f),quad
+ k_f=k+df ⊗ df,quad varphi_f=varphi-ln L_f,
+ quad D_f=-div_{k,varphi}U_f.
+]
Lemma~lem:graph-identity says
- \wscal(k_f,\varphi_f)-\wscal(k,\varphi)
- =D_f^2-2U_f(D_f)+|II_f|^2+|d\ln L_f|_{k_f}^2.
+ wscal(k_f,varphi_f)-wscal(k,varphi)
+ =D_f^2-2U_f(D_f)+|II_f|^2+|dln L_f|_{k_f}^2.
Also
- |U_f|_k\le1,\qquad U_f(f)=L_f-L_f^{-1},\qquad
- |df|_{k_f}\le1,\\
- dV_{k_f}=L_f\,dV_k,\qquad L_f-L_f^{-1}\ge\ln L_f.
+ |U_f|_kle1,qquad U_f(f)=L_f-L_f^{-1},qquad
+ |df|_{k_f}le1,
+ dV_{k_f}=L_f,dV_k,qquad L_f-L_f^{-1}geln L_f.
-The last inequality holds for every $L_f\ge1$ by differentiation. These
+The last inequality holds for every $L_fge1$ by differentiation. These
relations will make the scalar cost and the volume change use the same
factor $L_f$.
For the first graph put
-\[
- f_0=\sigma\ln u_\alpha,\qquad C_f=B_0+Λ,
- \qquad D_f=C_f\tan(f_0-f),
-\]
-with the argument between its two poles. Here $B_0\ge2$ is fixed and
-$\sigma>0$ will be small. Pole avoidance gives $|f-f_0|<\pi/2<2$.
+[
+ f_0=sigmaln u_alpha,qquad C_f=B_0+Λ,
+ qquad D_f=C_ftan(f_0-f),
+]
+with the argument between its two poles. Here $B_0ge2$ is fixed and
+$sigma>0$ will be small. Pole avoidance gives $|f-f_0|<pi/2<2$.
Differentiating the prescribed $D_f$ gives the exact terms
D_f^2-2U_f(D_f)
- ={}&D_f^2-2D_fU_f(\ln C_f)
- -2(C_f+D_f^2/C_f)U_f(f_0)\\
+ ={}&D_f^2-2D_fU_f(ln C_f)
+ -2(C_f+D_f^2/C_f)U_f(f_0)
&+2(C_f+D_f^2/C_f)(L_f-L_f^{-1}).
-The final term is at least $2\ln L_f$ by
+The final term is at least $2ln L_f$ by
(eq:degree-graph-elementary). For the derivative of $C_f$, Young's
inequality and (eq:degree-moments) give
-\[
- 2|D_fU_f(\ln C_f)|
- \le0.8D_f^2+1.25|d\ln C_f|^2
- \le0.8D_f^2+(1.25)/(8)\mathcal L.
-\]
+[
+ 2|D_fU_f(ln C_f)|
+ le0.8D_f^2+1.25|dln C_f|^2
+ le0.8D_f^2+(1.25)/(8)mathcal L.calL.
+]
The $f_0$ term has absolute value at most
-\[
- 2\sigma C(C_f+D_f^2/C_f)(Λ+1).
-\]
-Since $(Λ+1)/C_f\le1$, a sufficiently small $\sigma$ spends less
+[
+ 2sigma C(C_f+D_f^2/C_f)(Λ+1).
+]
+Since $(Λ+1)/C_fle1$, a sufficiently small $sigma$ spends less
than the remaining $0.2D_f^2$ on its square term. Moreover
-$C_f(Λ+1)\le C_{B_0}(1+Λ^2)$, so a further small choice of
-$\sigma$ spends less than half the $cΛ^2$ in
+$C_f(Λ+1)le C_{B_0}(1+Λ^2)$, so a further small choice of
+$sigma$ spends less than half the $cΛ^2$ in
(eq:degree-weight-gain) and makes the remaining constant loss as
-small as prescribed. The available $\mathcal L/4$ covers
-$1.25\mathcal L/8$. Combining (eq:degree-weight-gain) and
+small as prescribed. The available $mathcal L/4$calL/4$ covers
+$1.25mathcal L/8$.calL/8$. Combining (eq:degree-weight-gain) and
(eq:degree-graph-identity), and discarding nonnegative terms, thus
gives
- \wscal(k_f,\varphi_f)\ge\wscal(h,\phi)-H_1(G)-2\ln(v/L_1)
- +c_1Λ^2+2\ln L_f-\delta_1,
+ wscal(k_f,varphi_f)gewscal(h,phi)-H_1(G)-2ln(v/L_1)
+ +c_1Λ^2+2ln L_f-delta_1,
-where $c_1>0$ is fixed, while $\delta_1$ can be made arbitrarily small. This
+where $c_1>0$ is fixed, while $delta_1$ can be made arbitrarily small. This
calculation is used only where the current base metric is the original
induced metric for which (eq:degree-weight-gain) was proved.
-For the remaining graphs, let $\zeta(f)$ be a smooth function valued in
-$[0,1]$, zero for $f\le K$ and one for $f\ge K+1$, with derivative bound
+For the remaining graphs, let $zeta(f)$ be a smooth function valued in
+$[0,1]$, zero for $fle K$ and one for $fge K+1$, with derivative bound
independent of the threshold $K$. Write
-$(k^{(1)},\varphi^{(1)})=(k_f,\varphi_f)$. For row $j=2,...,d$, its
+$(k^{(1)},varphi^{(1)})=(k_f,varphi_f)$. For row $j=2,...,d$, its
graph quantities use the metric and weight after all preceding rows:
-\[
+[
- L'_j=(1+|df'_j|_{k^{(j-1)}}^2)^{1/2},\qquad
- U'_j=\frac{\nabla_{k^{(j-1)}}f'_j}{L'_j},\\
- D'_j=-div_{k^{(j-1)},\varphi^{(j-1)}}U'_j,\\
- k^{(j)}=k^{(j-1)}+df'_j ⊗ df'_j,\qquad
- \varphi^{(j)}=\varphi^{(j-1)}-\ln L'_j.
+ L'_j=(1+|df'_j|_{k^{(j-1)}}^2)^{1/2},qquad
+ U'_j=frac{nabla_{k^{(j-1)}}f'_j}{L'_j},
+ D'_j=-div_{k^{(j-1)},varphi^{(j-1)}}U'_j,
+ k^{(j)}=k^{(j-1)}+df'_j ⊗ df'_j,qquad
+ varphi^{(j)}=varphi^{(j-1)}-ln L'_j.
-\]
+]
Prescribe successively
- \ell_j=A\zeta(f)\bar J_j,\qquad
- D'_j=B'\tan((\ell_j-e f'_j)/(w)),\qquad
- J_j^+=\bar J_j+e f'_j.
+ ell_j=Azeta(f)barbar J_j,J_j,qquad
+ D'_j=B'tan((ell_j-e f'_j)/(w)),qquad
+ J_j^+=barbar J_j+eJ_j+e f'_j.
The constants $A,w,e,B'$ are positive; their order of choice is given
-below. Put $h^+=k^{(d)}$, $\phi^+=\varphi^{(d)}$, and
-\[
- \mathcal W=L_f\prod_{j=2}^dL'_j.
-\]
-The successive volume identities give $dV_{h^+}=\mathcal W\,dV_k$
+below. Put $h^+=k^{(d)}$, $phi^+=varphi^{(d)}$, and
+[
+ mathcal W=L_fcalW=L_fprod_{j=2}^dL'_j.
+]
+The successive volume identities give $dV_{h^+}=mathcal WcalW,dV_k$
relative to the current base metric $k$. On the eventual child that
base will be the original induced metric, so this product will be the
-density in (eq:degree-child-scalar). In the metric after the first graph, $|df|\le1$ by
+density in (eq:degree-child-scalar). In the metric after the first graph, $|df|le1$ by
(eq:degree-graph-elementary), and later graph metrics only
-increase. The values and first derivatives of $\bar J$ are uniformly
+increase. The values and first derivatives of $barbar J$J$ are uniformly
bounded in the region in question. It follows that in each current
metric
- |d\ell_j|\le C_\ell\mathbf1_{\{f\ge K\}},
+ |dell_j|le C_ellmathbf1_bf1_{{fge K}},
-where $C_\ell$ depends on $A$ and the earlier band bounds but not on
+where $C_ell$ depends on $A$ and the earlier band bounds but not on
$K$, $M_*$, $k'$, or the final cover budget.
For a row graph the analogue of
(eq:degree-first-graph-differentiation) is
-\[
+[
(D'_j)^2-2U'_j(D'_j)
- =(D'_j)^2-\frac2w(B'+((D'_j)^2)/(B'))U'_j(\ell_j)
+ =(D'_j)^2-frac2w(B'+((D'_j)^2)/(B'))U'_j(ell_j)
+(2e)/(w)(B'+((D'_j)^2)/(B'))
(L'_j-(L'_j)^{-1}).
-\]
+]
Choose $B'$ after $w,e$ so that
- (B'e)/(w)\ge2,\qquad
- (2C_\ell)/(wB')\le\frac12.
+ (B'e)/(w)ge2,qquad
+ (2C_ell)/(wB')lefrac12.
-Then the differentiated $\ell_j$ spends at most half of $(D'_j)^2$ and
-the fixed amount $2B'C_\ell/w$ on $\{f\ge K\}$, while the last term pays
-$2\ln L'_j$ everywhere. The other terms of
+Then the differentiated $ell_j$ spends at most half of $(D'_j)^2$ and
+the fixed amount $2B'C_ell/w$ on ${fge K}$, while the last term pays
+$2ln L'_j$ everywhere. The other terms of
(eq:degree-graph-identity) are nonnegative. After all $p$ rows,
the sum of fixed errors is therefore at most
-$C_{rows}\mathbf1_{\{f\ge K\}}$, with
+$C_{rows}mathbf1_bf1_{{fge K}}$, with
$C_{rows}$ independent of $K,M_*,k'$ and the cover budget.
On this support, $f<f_0+2$ implies
-\[
- u_\alpha>\exp((K-2)/\sigma),\qquad
- Λ\ge c'\exp((K-2)/(\sigma\alpha))
-\]
+[
+ u_alpha>exp((K-2)/sigma),qquad
+ Λge c'exp((K-2)/(sigmaalpha))
+]
by (eq:degree-moments). Choose $K$ large enough that the remaining
$c_1Λ^2$ in (eq:degree-first-graph-gain) covers
$C_{rows}$ there. This proves the required payment for all row
@@ -4859,17 +4859,17 @@ Pole avoidance in (eq:degree-row-graph) also gives the output
estimates that will be used for the band sides and its gradients:
- |J_j^+-(1+A\zeta(f))\bar J_j|&<2w,\\
- |dJ_j^+|_{h^+}&\le |dG_j|_h+ε_{grad},\\
- |J_j^+|&\ge |G_j|-ε_{val}.
+ |J_j^+-(1+Azeta(f))barbar J_j|&<2w,J_j|&<2w,
+ |dJ_j^+|_{h^+}&le |dG_j|_h+ε_{grad},
+ |J_j^+|&ge |G_j|-ε_{val}.
For the gradient inequality,
-$|df'_j|\le1$ in its own graph metric and hence in $h^+$, and all these
+$|df'_j|le1$ in its own graph metric and hence in $h^+$, and all these
metrics dominate the original induced metric and the parent metric on
tangent vectors. Thus
-$|dJ_j^+|_{h^+}\le|d\bar J_j|_h+e$. For the value inequality use
-$1+A\zeta\ge1$ in the first line. The quantities
+$|dJ_j^+|_{h^+}le|dbarbar J_j|_h+e$.J_j|_h+e$. For the value inequality use
+$1+Azetage1$ in the first line. The quantities
$ε_{grad}$ and $ε_{val}$ can be as
small as prescribed by the first-derivative and value perturbation
budgets, respectively, and by $e,w$. Choose them to satisfy the
@@ -4877,36 +4877,36 @@ target budgets (eq:degree-tolerances) and also to be smaller than
all the fixed side margins used below.
The full dependency order is
-\[
+[
- &perturbation budgets\ →\ (\bar J,a_0,k)
- \ →\ R\\
- &\qquad→\ (scalar error,ε_{base})
- \ →\ B_0\ →\ \sigma\\
- &\qquad→\ A\ →\ (w,e)
- \ →\ B'\ →\ K\ →\ M_*\\
- &\qquad→\ (cover budget,k',radius lower bound).
+ &perturbation budgets → (barbar J,a_0,k)J,a_0,k)
+ → R
+ &qquad→ (scalar error,ε_{base})
+ → B_0 → sigma
+ &qquad→ A → (w,e)
+ → B' → K → M_*
+ &qquad→ (cover budget,k',radius lower bound).
-\]
+]
Each later choice may use the preceding ones. The intended scalar error
is much smaller than $1/q$, and $ε_{base}$ is small
enough for (eq:degree-potential-laplacian). The constants in
(eq:degree-moments) and the positive constant in
(eq:degree-weight-gain) are independent of the future cover, so
-$B_0$ and $\sigma$ can be fixed for
+$B_0$ and $sigma$ can be fixed for
(eq:degree-first-graph-gain) at this stage. Choose $A$ large compared with
$T_p/a_0$, then $w$ small compared with $a_0$ and all value margins,
and $e$ small for (eq:degree-tolerances). Choose $B'$ by
(eq:degree-row-B-choice), and only then $K$ to cover the row errors.
Also make $K$ larger than the baseline value
-$\sigma\ln(ε_{base}^\alpha)$ by a fixed margin.
+$sigmaln(ε_{base}^alpha)$ by a fixed margin.
Choose $M_*$ so large that
(eq:degree-potential-on-bad) implies
- f_0>K+100\quad\hbox{on }D_{k'}.
+ f_0>K+100quadhbox{on }D_{k'}.
Finally choose the cover budget sufficiently small for all the active-mass
-and error requirements, and obtain $k'\ge k$ and a cover with its positive
+and error requirements, and obtain $k'ge k$ and a cover with its positive
lower radius bound. This final selection is possible for every fixed
$M_*$ by (eq:degree-cover-budget). It also provides all finite
uniform derivative bounds needed for the actual equations. In
@@ -4925,46 +4925,46 @@ through the uniform derivative bounds for $f_0$.
$Σ_s$ need not be complete. Use the operations
only on the open region
- Ω=Σ_s\cap\{|G|_∞<t_2,\ f_0<K+12\}.
+ Ω=Σ_scap{|G|_∞<t_2, f_0<K+12}.
By (eq:degree-high-on-bad) it avoids $D_{k'}$ with a large margin in
$f_0$; by $t_2<t_3$ it also has a margin within the region of the tests.
The uniform derivative bounds on $f_0$ turn these into uniform spatial
margins. Thus the graph tests through level $k'$ give regular graph
charts with uniform buffers and all estimates on a neighborhood of this
-region. The estimates for derivatives of $\ln v$ apply there. The
-functions $u_j,Λ,f_0,\bar J$ have uniform derivatives because of
+region. The estimates for derivatives of $ln v$ apply there. The
+functions $u_j,Λ,f_0,barbar J$J$ have uniform derivatives because of
the radius lower bound. The weight
-$\phi_Σ+(\ln u_\alpha+\ln u_{\alpha+4})/4$ therefore has the
+$phi_Σ+(ln u_alpha+ln u_{alpha+4})/4$ therefore has the
positive-order derivative bounds required for a graph solve.
-Choose a smooth function $0<\xi\le1$ on $Ω$ which tends to zero
+Choose a smooth function $0<xile1$ on $Ω$ which tends to zero
at the sides in (eq:degree-auxiliary-region) and is one, with
-buffers, through the smaller bounds $|G|_∞\le t_1$ and
-$f_0\le K+7$. It can be made from smooth functions of
+buffers, through the smaller bounds $|G|_∞le t_1$ and
+$f_0le K+7$. It can be made from smooth functions of
$t_2^2-G_i^2$ and $K+12-f_0$, linear near zero and cut off to one away
from zero. Its derivatives in the original regular charts are bounded.
Replace the induced metric on $Ω$ by
-\[
- k_{aux}=\xi^{-2}h|_{TΣ}.
-\]
+[
+ k_{aux}=xi^{-2}h|_{TΣ}.
+]
This metric is complete with uniform buffered bounded geometry. Here is
the local verification, which also accounts for possible ends of the
original leaves. A ball in an original regular chart of radius a small
-fixed multiple of $\xi(x)$ remains in $Ω$, since $\xi(x)$ is bounded
+fixed multiple of $xi(x)$ remains in $Ω$, since $xi(x)$ is bounded
by a constant times each scalar distance to a side. The bound on
-$d\xi$ makes $\xi$ comparable to $\xi(x)$ on that ball. Rescale the
-coordinate ball by $\xi(x)$. The coefficients of $\xi^{-2}h$ and their
+$dxi$ makes $xi$ comparable to $xi(x)$ on that ball. Rescale the
+coordinate ball by $xi(x)$. The coefficients of $xi^{-2}h$ and their
inverse then have uniform bounds of every order: for the $m$-th derivative
of the conformal factor the new terms are products of ratios bounded by
-constants times $\xi(x)^m|\nabla^m\xi|/\xi$, and these are bounded for
-$m\ge1$.
+constants times $xi(x)^m|nabla^mxi|/xi$, and these are bounded for
+$mge1$.
The rescaled boxes have uniform inner buffers. They also prove
completeness. A curve of finite remaining length, or a Cauchy sequence,
eventually stays within the outer part of one such buffered box and
converges there; it cannot escape a leaf through a side or an uncharted
end. Equivalently, approach to a side already has infinite length
-because $|d\ln\xi|_{k_{aux}}$ is bounded. This argument used
+because $|dlnxi|_{k_{aux}}$ is bounded. This argument used
only the controlled charts near (eq:degree-auxiliary-region),
not completeness of a parent leaf.
@@ -4975,7 +4975,7 @@ the graph existence theorem. Apply Lemma~lem:graph-prescription
on these complete leaves. For the first prescription its positive
coefficient is $B_0+Λ$, its height target is $f_0$, and its
constants $e,w$ in that lemma are both one. For a row prescription the
-coefficient is the positive constant $B'$, the target is $\ell_j$, and
+coefficient is the positive constant $B'$, the target is $ell_j$, and
the constants are the chosen $e,w>0$. All coefficients have the
uniform smooth bounds just established. That lemma gives the bounded
solutions with a uniform margin from the poles and all derivative
@@ -4985,30 +4985,30 @@ metric dominates the complete base, is complete, and has uniform
geometry by the solution bounds; its changed weight again has uniform
positive-order derivative bounds. In applying the scalar estimates
(eq:degree-weight-gain)--(eq:degree-first-graph-gain), use
-only the smaller region where $\xi=1$. There the auxiliary metric and
+only the smaller region where $xi=1$. There the auxiliary metric and
the original induced metric agree with all derivatives.
Define the child to be the open measured band
- X_s^+=\{x\inΩ: |G|_∞<t_1,\ f_0<K+7,
- \ |J^+|_∞<T_p\},
+ X_s^+={xinΩ: |G|_∞<t_1, f_0<K+7,
+ |J^+|_∞<T_p},
-with the final graph metric $h^+$, final graph weight $\phi^+$, and
+with the final graph metric $h^+$, final graph weight $phi^+$, and
underlying boundary orientation. Its artificial sides stay away from
each inner residual subcube. For the $|G|=t_1$ sides, confinement gives
-$|G_1|<\tau+b<T_p$, so any approach to such a side has a residual row
+$|G_1|<tau+b<T_p$, so any approach to such a side has a residual row
$|G_j|$ near $t_1$. Choose
$ε_{val}<(t_1-T_p)/4$; then
(eq:degree-output-estimates) puts $|J_j^+|$ beyond $T_p$ already
on a uniform neighborhood of that side. For the $f_0=K+7$ side, if
-$f_0\ge K+4$ some kernel is active, since the baseline value of $f_0$
-is less than $K$. Its support lies in $|\bar J|_∞>3a_0$.
-Moreover $f>f_0-2\ge K+2$, so $\zeta(f)=1$. Choose $A$ so that
+$f_0ge K+4$ some kernel is active, since the baseline value of $f_0$
+is less than $K$. Its support lies in $|barbar J|_∞>3a_0$.J|_∞>3a_0$.
+Moreover $f>f_0-2ge K+2$, so $zeta(f)=1$. Choose $A$ so that
$(1+A)3a_0>T_p+4a_0$ and take $w<a_0/2$. The first line of
(eq:degree-output-estimates) then puts some $|J_j^+|$ beyond
$T_p$ with a positive uniform margin. This occurs before reaching
$f_0=K+7$. In particular the entire child lies in the region where
-$\xi=1$, with buffers for all its inner subcubes.
+$xi=1$, with buffers for all its inner subcubes.
These observations verify every part of
Definition~def:degree-band for the child. To be explicit about its
@@ -5025,15 +5025,15 @@ is precisely the restricted parent measure. It has no lapse factor and
no graph-volume factor; those belong to the leafwise integrations.
Preserving the degree through the modifications.:
-Use in (eq:degree-boundary-degree) a residual form $\beta$ supported
+Use in (eq:degree-boundary-degree) a residual form $beta$ supported
in $|z|_∞<a_0$, reducing $a_0$ if necessary. First homotope $J$ to
-$\bar J$ on the whole parent boundary current. The small value
+$barbar J$J$ on the whole parent boundary current. The small value
perturbation ensures that the preimages of this support throughout the
homotopy remain in a fixed old inner band. One can justify invariance
directly by the homotopy formula. The difference of the two pullbacks
-of $\beta$ is $d\mathcal H\beta$, where $\mathcal H\beta$ is its usual
+of $beta$ is $dmathcal HcalHbeta$, where $mathcal HcalHbeta$ is its usual
transgression along the homotopy. Multiply by the cutoff
-$\chi_{deg}(G_1)$ used
+$chi_{deg}(G_1)$ used
in (eq:degree-boundary-degree). The current $∂[E_s]$ is
closed, and the term differentiating this cutoff meets no boundary on
the small residual support, by confinement. Measured current Stokes,
@@ -5041,80 +5041,80 @@ with the local mass bounds already proved, therefore makes the integral
of the difference zero. This uses the boundary current across its
singular set and so does not assume regularity of the homotopy there.
-The resulting integral for $\bar J$ is entirely in the regular region
-used for the solves. Indeed $|\bar J|_∞<a_0$ avoids a neighborhood
+The resulting integral for $barbar J$J$ is entirely in the regular region
+used for the solves. Indeed $|barbar J|_∞<a_0$J|_∞<a_0$ avoids a neighborhood
of $D_k$ by (eq:degree-gap) and avoids every kernel support, which
-lies in $|\bar J|_∞>3a_0$. Thus $f_0$ has its baseline value there.
+lies in $|barbar J|_∞>3a_0$.J|_∞>3a_0$. Thus $f_0$ has its baseline value there.
The small perturbation also keeps $|G|$ in the smaller inner band.
-Now homotope on this regular region first from $\bar J$ to
-$(1+A\zeta(f))\bar J$ and then to $J^+$. Along the first homotopy the
+Now homotope on this regular region first from $barbar J$J$ to
+$(1+Azeta(f))barbar J$J$ and then to $J^+$. Along the first homotopy the
multiplier is at least one, so a preimage of the small target support
-has $|\bar J|_∞<a_0$. Along the second, the first line of
+has $|barbar J|_∞<a_0$.J|_∞<a_0$. Along the second, the first line of
(eq:degree-output-estimates) shows that it has
-$|\bar J|_∞<a_0+2w<2a_0$. Both supports thus avoid the marked
+$|barbar J|_∞<a_0+2w<2a_0$.J|_∞<a_0+2w<2a_0$. Both supports thus avoid the marked
region and the kernel supports, and retain the same spatial and
$G_1$ margins. Ordinary measured Stokes on the regular buffered charts
applies to these homotopies. The final support lies in
(eq:degree-child-domain), so its integral is the measured degree
-of the child. It equals $\mathcal D_X(G)$.
+of the child. It equals $mathcal D_X(G)$.calD_X(G)$.
On the child the initial auxiliary metric is the induced metric, so
(eq:degree-graph-elementary) verifies the volume identity in
-(eq:degree-child-scalar) for the product $\mathcal W$ defined above.
+(eq:degree-child-scalar) for the product $mathcal W$calW$ defined above.
Choose all the adjustable scalar errors above with sum at most $1/q$.
The absorption of the row errors and
(eq:degree-first-graph-gain) give the scalar inequality in
(eq:degree-child-scalar) on the child.
This completes the construction in high dimensions. In the case
-$d\le7$ treated above, use $\mathcal W=1$. The volume identity is then
+$dle7$ treated above, use $mathcal W=1$.calW=1$. The volume identity is then
immediate, and (eq:degree-slice-bound) gives the scalar inequality
in (eq:degree-child-scalar). In dimension $p=0$ the same statements
hold on the counted points with the zero-dimensional scalar convention.
Proof. [Completion of the proof of Lemma~lem:degree-descent]
The coarea assertion is (eq:degree-coarea-proof). The constructions
-above give a controlled child of degree $\mathcal D_X(G)$ for each typical
+above give a controlled child of degree $mathcal D_X(G)$calD_X(G)$ for each typical
$s$. Its coordinate bound follows from
(eq:degree-output-estimates) and (eq:degree-tolerances):
-\[
- L_j|dJ_j^+|_{h^+}\le B-\frac dq+L_jε_{grad}
- <B-\frac pq.
-\]
+[
+ L_j|dJ_j^+|_{h^+}le B-frac dq+L_jε_{grad}
+ <B-frac pq.
+]
For a low-dimensional child the same statement holds with zero error.
The horizon condition (eq:degree-horizon) for its rows is unchanged,
-since $q,B,\tau,L_2,...,L_d$ are unchanged.
+since $q,B,tau,L_2,...,L_d$ are unchanged.
Every point with $|J^+|_∞<b_p$ comes from
$|G|_∞<b_d$. For residual rows this follows from the last line of
(eq:degree-output-estimates) and
$ε_{val}<b_d-b_p$; in low dimensions it is immediate.
For the first row confinement gives
-\[
- |G_1|<\tau+b<\tau+(\tau)/(10q)<b_d.
-\]
+[
+ |G_1|<tau+b<tau+(tau)/(10q)<b_d.
+]
The same observation covers the counted case $p=0$.
-The function $x\mapsto(x-1)_+$ is one-Lipschitz, so
+The function $xmapsto(x-1)_+$ is one-Lipschitz, so
(eq:degree-output-estimates) gives
-\[
- \sum_{j=2}^dH_j^+\le\sum_{j=2}^dH_j(G)
- +2\sum_{j=2}^dL_jε_{grad}.
-\]
+[
+ sum_{j=2}^dH_j^+lesum_{j=2}^dH_j(G)
+ +2sum_{j=2}^dL_jε_{grad}.
+]
Use this and the scalar and volume relations in
(eq:degree-child-scalar) to compare the integrands on the child:
- &\exp(\frac{-\wscal(h^+,\phi^+)+\sum_{j=2}^dH_j^+}{2})
- \mathcal W\\
- &\quad\le
- \exp(\frac1{2q}+\sum_{j=2}^dL_jε_{grad})
- \frac v{L_1}
- \exp(\frac{-\wscal(h,\phi)+\sum_{i=1}^dH_i(G)}2)\\
- &\quad\le e^{3/(5q)}\frac v{L_1}Θ.
+ &exp(frac{-wscal(h^+,phi^+)+sum_{j=2}^dH_j^+}{2})
+ mathcal WcalW
+ &quadle
+ exp(frac1{2q}+sum_{j=2}^dL_jε_{grad})
+ frac v{L_1}
+ exp(frac{-wscal(h,phi)+sum_{i=1}^dH_i(G)}2)
+ &quadle e^{3/(5q)}frac v{L_1}Θ.
-The factor $\mathcal W$ cancels exactly against the term
-$2\ln\mathcal W$ in the scalar bound. The factor $v/L_1$ is the lapse
+The factor $mathcal W$calW$ cancels exactly against the term
+$2lnmathcal W$calW$ in the scalar bound. The factor $v/L_1$ is the lapse
factor in coarea. This calculation makes no bound on either factor.
For a low-dimensional child use (eq:degree-slice-bound), with no
row error; for $p=0$ the same estimate is the pointwise bound for a
@@ -5126,20 +5126,20 @@ the child and use its containment in the parent $b_d$ region. This is
Proof. [Proof of Theorem~thm:degree]
The case $d=0$ is the signed-count inequality in the statement. Suppose
-the theorem is known in dimension $p=d-1$, with the same $q,B,\tau$ and
+the theorem is known in dimension $p=d-1$, with the same $q,B,tau$ and
the residual lengths. Its hypotheses hold for the child of
Lemma~lem:degree-descent, including the unchanged strict horizon
condition. For almost every $s$ the induction hypothesis and
(eq:degree-child-integral) give
-\[
- |\mathcal D_X(G)|(2\tau)^p
- \le \frac{e^{C_1p/q+C_*/q}}{\prod_{i=1}^dL_i}
- \int_{Σ_s\cap\{|G|_∞<b_d\}}v_sΘ\,dA_h.
-\]
+[
+ |mathcal D_X(G)|(2calD_X(G)|(2tau)^p
+ le frac{e^{C_1p/q+C_*/q}}{prod_{i=1}^dL_i}
+ int_{Σ_scap{|G|_∞<b_d}}v_sΘ,dA_h.
+]
For $p=0$ this is the same use of the unsigned-count bound. Take
-$C_1\ge C_*$. Integrate this scalar inequality over $-\tau<s<\tau$
+$C_1ge C_*$. Integrate this scalar inequality over $-tau<s<tau$
and apply (eq:degree-coarea) with
-$F=\mathbf1_{\{|G|_∞<b_d\}}Θ$. The factor $2\tau$ on the
+$F=mathbf1_bf1_{{|G|_∞<b_d}}Θ$. The factor $2tau$ on the
left changes its power from $p$ to $d$, and the right is bounded by
the integral in (eq:degree-estimate). This proves the induction.
@@ -5156,8 +5156,8 @@ arbitrarily small slice errors were chosen. All geometry-dependent
constants were used only to make those choices.
-The entropy construction supplies a root probability map \(F\) whose
-energy \(e_h(F)\) has a large coefficient in the weighted scalar-curvature
+The entropy construction supplies a root probability map (F) whose
+energy (e_h(F)) has a large coefficient in the weighted scalar-curvature
inequality. Its coordinatewise square takes values in the probability
simplex. Now turn its pairing with the bounded cocycle into comparison
degrees to which Theorem~thm:degree applies. A common integer
@@ -5172,70 +5172,70 @@ The probability-map and auxiliary-degree strategy has antecedents in
quantitative measured transfer and its estimates here.
-For every integer \(n\geq 3\), there is a finite constant \(C_n>0\) such
-that every closed connected oriented smooth \(n\)-manifold \(M\) and every
-smooth Riemannian metric \(g\) with \(\Scal_g\equiv-1\) satisfy
-\[
- \|M\|\leq C_n\Vol_g(M).
-\]
+For every integer (ngeq 3), there is a finite constant (C_n>0) such
+that every closed connected oriented smooth (n)-manifold (M) and every
+smooth Riemannian metric (g) with (Scal_gequiv-1) satisfy
+[
+ |M|leq C_nVol_g(M).
+]
-Prepare the proof for a manifold with \(\|M\|>0\). Choose the
-homogeneous alternating bounded cocycle \(c\), the initial probability
-map, and the number \(I\geq\|M\|/2\) provided by
-Proposition~prop:detector; in particular \(\|c\|_∞\leq1\).
+Prepare the proof for a manifold with (|M|>0). Choose the
+homogeneous alternating bounded cocycle (c), the initial probability
+map, and the number (Igeq|M|/2) provided by
+Proposition~prop:detector; in particular (|c|_∞leq1).
First consider conditional data having the properties asserted by
Theorem~thm:entropy, without yet invoking their existence.
-Denote the scalar constant in this contract by \(C_{\rm ent}\).
+Denote the scalar constant in this contract by (C_{rm ent}).
All functions called elementary in this section have a fixed height of
-exponentiation on a polynomial, with parameters depending only on \(n\).
+exponentiation on a polynomial, with parameters depending only on (n).
-Write \(\mathcal M=(\widetilde M\timesΩ)/\Gamma\) for such a measured
+Write (mathcal M=(calM=(widetilde MtimesΩ)/Gamma) for such a measured
family. Denote its metric, weight, density, and
-root probability map by \(h,\phi,\rho,F\), and put
-\[
- \mathfrak e=e_h(F),\qquad W=\log(dV_h/\rho).
-\]
+root probability map by (h,phi,rho,F), and put
+[
+ mathfrak e=e_h(F),frake=e_h(F),qquad W=log(dV_h/rho).
+]
Its estimates are
- \rho\leq dV_g,\qquad W\geq0,\qquad
- \wscal(h,\phi)\geq-C_{\rm ent}+2W-n\log E+E\mathfrak e,
- \qquad \mathfrak e^{1/2}\leq E^{-1/100}.
+ rholeq dV_g,qquad Wgeq0,qquad
+ wscal(h,phi)geq-C_{rm ent}+2W-nlog E+Emathfrak e,frake,
+ qquad mathfrak e^frake^{1/2}leq E^{-1/100}.
There is a fixed invariant graph on the countable label set, with finitely
-many free \(\Gamma\)-orbits, containing every cooccurrence of nonzero
-coordinates of \(F\). The integer \(q\geq n+1\) bounds the number of
+many free (Gamma)-orbits, containing every cooccurrence of nonzero
+coordinates of (F). The integer (qgeq n+1) bounds the number of
orbits and the maximum graph degree plus one. The equivariant coloring
-by \(q\) colors separates any two labels at graph distance at most two.
-Its dependence on \(\omega\inΩ\) is Borel. Set
-\(m=q-n>0\). Integrals over \(\mathcal M\) and over the measured spaces
+by (q) colors separates any two labels at graph distance at most two.
+Its dependence on (omegainΩ) is Borel. Set
+(m=q-n>0). Integrals over (mathcal McalM) and over the measured spaces
constructed below include their transverse measures.
-\medskip\noindentConditional transfer target.
-For every \(n\geq3\), there are a constant \(K_n>0\) and an elementary
-function \(\Psi_n:[1,∞)\to(0,∞)\), chosen from \(n\) alone,
+medskipnoindentConditional transfer target.
+For every (ngeq3), there are a constant (K_n>0) and an elementary
+function (Psi_n:[1,∞)to(0,∞)), chosen from (n) alone,
such that every choice of detector and conditional entropy data just
-specified, including its constant \(C_{\rm ent}\), satisfies
+specified, including its constant (C_{rm ent}), satisfies
- E\geq\Psi_n(q)\quad⇒\quad
- I\leq K_n e^{C_{\rm ent}/2}\int_{\mathcal M}\rho.
+ EgeqPsi_n(q)quad⇒quad
+ Ileq K_n e^{C_{rm ent}/2}int_{mathcal M}calMrho.
-Thus \(K_n\) and \(\Psi_n\) are fixed before the manifold, cocycle,
-instance geometry, and \(C_{\rm ent}\) are chosen. Carry out the
-estimates for each integer \(q\geq n+1\), uniformly over conditional
+Thus (K_n) and (Psi_n) are fixed before the manifold, cocycle,
+instance geometry, and (C_{rm ent}) are chosen. Carry out the
+estimates for each integer (qgeq n+1), uniformly over conditional
data satisfying the stated label and coloring bounds.
The grid, the common rounding, and an upper density bound for the
resolution come first. That density bound permits a small collar;
the ensuing curvature and rotation bounds then determine one comparison
width and the required energy threshold. Only after the conditional
target is closed will Theorem~thm:entropy supply an actual pair
-\((q,E)\) and make \(C_{\rm ent}\) dimension-only.
+((q,E)) and make (C_{rm ent}) dimension-only.
-For a fixed \(\omega\), let \(K=K_\omega\) consist of all nonnegative
+For a fixed (omega), let (K=K_omega) consist of all nonnegative
unit root vectors whose support is a clique in the label graph. Thus
-\(K\) is the union of the spherical orthants on those cliques. A vector
-\(y\in K\) has a color image \(z_y\in\mathbb R^q\), obtained by placing
+(K) is the union of the spherical orthants on those cliques. A vector
+(yin K) has a color image (z_yinmathbb R^qbbR^q), obtained by placing
each of its root coordinates at the color of its label. Distinct labels
in a clique have distinct colors. Distant cliques can reuse colors, so
the flat neighborhood below keeps distinct label configurations with the
@@ -5243,46 +5243,46 @@ same color image on separate sheets for comparison, while retaining
Euclidean coordinates for the grid averages.
[Flat sheet neighborhood]
-Put \(\delta=(1000\sqrt q)^{-1}\). There is an oriented flat smooth
-\(q\)-manifold \(\mathcal U_\omega\) with a local isometry
-\(z:\mathcal U_\omega\to\mathbb R^q\), and an inclusion of \(K_\omega\)
-into it, such that the image of every \(y\in K_\omega\) has a flat ball
-chart of radius \(10\delta\) centered at \(z_y\). This construction is
-a standard Borel family in \(\omega\), with countably many charts and an
-equivariant \(\Gamma\)-action. The map \(F\) lifts smoothly to this
-family, and, writing \(z_F=z\circ F\),
-\[
- e_h(z_F)=e_h(F)=\mathfrak e.
-\]
-There is a closed smooth \(n\)-form, again denoted \(\eta\), obtained
+Put (delta=(1000sqrt q)^{-1}). There is an oriented flat smooth
+(q)-manifold (mathcal U_calU_omega) with a local isometry
+(z:mathcal U_calU_omegatomathbb R^qbbR^q), and an inclusion of (K_omega)
+into it, such that the image of every (yin K_omega) has a flat ball
+chart of radius (10delta) centered at (z_y). This construction is
+a standard Borel family in (omega), with countably many charts and an
+equivariant (Gamma)-action. The map (F) lifts smoothly to this
+family, and, writing (z_F=zcirc F),
+[
+ e_h(z_F)=e_h(F)=mathfrak e.frake.
+]
+There is a closed smooth (n)-form, again denoted (eta), obtained
in these flat charts by pulling back the detector form through the cutoff
probabilities constructed below. It is Borel and equivariant in the
family, and satisfies
- \int_{\mathcal M}F^*\eta=I,\qquad |\eta|_{\HS}\leq C_n.
+ int_{mathcal M}calMF^*eta=I,qquad |eta|_{HS}leq C_n.
Here the exterior Hilbert norm uses the Euclidean metric on the charts.
For every fixed derivative order its derivatives admit an elementary
-bound in \(q\).
+bound in (q).
Proof.
-Take one copy of \(B(z_y,10\delta)\subset\mathbb R^q\) for each witness
-\(y\in K\). At the same point \(z\) in two such balls, identify the two
+Take one copy of (B(z_y,10delta)subsetmathbb R^qbbR^q) for each witness
+(yin K). At the same point (z) in two such balls, identify the two
copies if their witnesses share a label in their nonzero supports. Check that these identifications give ball charts, rather than a branched
-space. For every point under consideration, some coordinate of \(z\)
-is at least \(q^{-1/2}-10\delta\). Every witness for that point has a
+space. For every point under consideration, some coordinate of (z)
+is at least (q^{-1/2}-10delta). Every witness for that point has a
nonzero label at this color. If two witnesses share any label, their
labels at every common color agree: each of the two labels at that color
is adjacent to the shared label, so distance-two separation of colors
forces them to be the same. Conversely, agreement at the indicated
-large color is already a shared label. Thus, at \(z\), the proposed
+large color is already a shared label. Thus, at (z), the proposed
identification is precisely agreement on the label at that large color,
which is an equivalence relation.
The same color remains positive in all witnesses at every sufficiently
-nearby point; indeed the margin \(q^{-1/2}-21\delta>0\) suffices after
-a further displacement of at most \(\delta\). In that neighborhood
+nearby point; indeed the margin (q^{-1/2}-21delta>0) suffices after
+a further displacement of at most (delta). In that neighborhood
the equivalence classes are still distinguished by its label. Each ball
therefore projects injectively as a local chart. Points with distinct
Euclidean projections can be separated in Euclidean space, and points
@@ -5292,36 +5292,36 @@ property. Rational dense choices of centers in each spherical orthant
give a countable subatlas: there are countably many labels and cliques.
The descriptions by labels, clique coordinates, and color assignments
are Borel, and are unchanged by simultaneous translation of labels and
-\(\omega\). They prove all the claimed family and equivariance
-properties. At points of \(K\) this also shows that its inclusion is
+(omega). They prove all the claimed family and equivariance
+properties. At points of (K) this also shows that its inclusion is
unambiguous.
-The lift of \(F\) is locally in one of these charts: its label at a
+The lift of (F) is locally in one of these charts: its label at a
chosen large color persists in a neighborhood. A nonnegative smooth
root coordinate has zero differential wherever it is zero. All the
nonzero coordinates at a point have different colors. It follows that
-the nonzero rows of \(dF\) are merely moved to different color positions
-by \(dz_F\), proving the asserted energy equality.
-
-To construct this cutoff pullback, choose a nonnegative smooth function \(\chi\)
-which is zero for \(t\leq12\delta\), equals \(t\) for
-\(t\geq25\delta\), lies between zero and \(\max\{t,0\}\), and satisfies the usual rescaled derivative bounds
-\(|\chi^{(j)}|\leq C_j\delta^{1-j}\) for \(j\geq1\). In a ball put
-\[
- w(z)=\frac{(\chi(z_j))_{j=1}^q}{|(\chi(z_j))_{j=1}^q|},
- \qquad p_j=w_j^2.
-\]
+the nonzero rows of (dF) are merely moved to different color positions
+by (dz_F), proving the asserted energy equality.
+
+To construct this cutoff pullback, choose a nonnegative smooth function (chi)
+which is zero for (tleq12delta), equals (t) for
+(tgeq25delta), lies between zero and (max{t,0}), and satisfies the usual rescaled derivative bounds
+(|chi^{(j)}|leq C_jdelta^{1-j}) for (jgeq1). In a ball put
+[
+ w(z)=frac{(chi(z_j))_{j=1}^q}{|(chi(z_j))_{j=1}^q|},
+ qquad p_j=w_j^2.
+]
The denominator is bounded below by an absolute positive constant.
-In fact \(z\) is within \(10\delta\) of a nonnegative unit vector, and
+In fact (z) is within (10delta) of a nonnegative unit vector, and
the change in each coordinate caused by cutting it off is at most
-\(25\delta\); hence its total change is at most \(25\delta\sqrt q\).
-If \(\chi(z_j)\ne0\), every witness in this ball has a positive
-coordinate at color \(j\), because its distance from \(z\) is less
-than \(10\delta\). The label for every active coordinate is therefore
+(25delta); hence its total change is at most (25deltasqrt q).
+If (chi(z_j)ne0), every witness in this ball has a positive
+coordinate at color (j), because its distance from (z) is less
+than (10delta). The label for every active coordinate is therefore
unambiguous on chart overlaps. Evaluate
-\[
- \eta=n!\,c(p,dp,...,dp)
-\]
+[
+ eta=n!,c(p,dp,...,dp)
+]
using those labels. The cocycle law makes this form closed, as in
Proposition~prop:detector, and a straight probability homotopy
from the old probabilities to these new ones preserves the measured
@@ -5330,75 +5330,75 @@ instance, so the measured Stokes argument of that proposition applies.
This gives the first identity in (eq:transfer-form).
For the dimension-only norm bound, first regard the form on nonnegative
-unit root vectors \(w\). In the Euclidean coordinate directions
-\(i_1,...,i_n\), its component is
-\[
- n!2^n(\prod_{j=1}^n w_{i_j})
- \sum_a w_a^2 c(a,i_1,...,i_n).
-\]
-Its absolute value is at most \(n!2^n\prod_j w_{i_j}\). The sum of the
-squares of these products over all increasing \(n\)-tuples is at most
-\((\sum_i w_i^2)^n=1\). Thus its exterior Hilbert norm is at most a
-constant depending only on \(n\). The map \(z\mapsto w(z)\) has
+unit root vectors (w). In the Euclidean coordinate directions
+(i_1,...,i_n), its component is
+[
+ n!2^n(prod_{j=1}^n w_{i_j})
+ sum_a w_a^2 c(a,i_1,...,i_n).
+]
+Its absolute value is at most (n!2^nprod_j w_{i_j}). The sum of the
+squares of these products over all increasing (n)-tuples is at most
+((sum_i w_i^2)^n=1). Thus its exterior Hilbert norm is at most a
+constant depending only on (n). The map (zmapsto w(z)) has
absolutely bounded operator norm of its differential: its denominator
is bounded below, normalization is an orthogonal projection divided by
-that denominator, and \(\chi'\) is absolutely bounded. Pullback
-multiplies the exterior Hilbert norm by at most the \(n\)-th power of
+that denominator, and (chi') is absolutely bounded. Pullback
+multiplies the exterior Hilbert norm by at most the (n)-th power of
this operator bound. This proves the second assertion in
(eq:transfer-form). Differentiating the same formulas a fixed
-number of times gives elementary bounds (polynomial bounds in \(q\)
-already suffice), since \(\delta^{-1}=1000\sqrt q\).
+number of times gives elementary bounds (polynomial bounds in (q)
+already suffice), since (delta^{-1}=1000sqrt q).
-Choose a rotation \(V\in SO(q)\) with Haar probability, independently
-of the entropy data, and put \(\widehat z=Vz\). In these coordinates
-take the cubical grid of step \(r\) and shift \(s\), where \(s\) is
-uniform in \([0,r)^q\). Fix \(r=r(q)>0\) so that
-\(r\sqrt q\ll\delta\) and so that every \(n\)-component of \(\eta\)
-in every rotated frame changes by at most \(q^{-n}\) over a ball of
-radius \(10r\sqrt q\) in the inner sheet charts. The derivative bounds
+Choose a rotation (Vin SO(q)) with Haar probability, independently
+of the entropy data, and put (widehatwidehat z=Vzz=Vz). In these coordinates
+take the cubical grid of step (r) and shift (s), where (s) is
+uniform in ([0,r)^q). Fix (r=r(q)>0) so that
+(rsqrt qlldelta) and so that every (n)-component of (eta)
+in every rotated frame changes by at most (q^{-n}) over a ball of
+radius (10rsqrt q) in the inner sheet charts. The derivative bounds
of Lemma~lem:transfer-flat permit such a choice with an elementary
-bound on \(r^{-1}\), uniformly in \(V,s,\omega\).
+bound on (r^{-1}), uniformly in (V,s,omega).
-A primal \(n\)-cell has an axis set \(\mathcal I\subset\{1,...,q\}\)
-of size \(n\), varies along those grid intervals, and has the remaining
+A primal (n)-cell has an axis set (mathcal IcalIsubset{1,...,q})
+of size (n), varies along those grid intervals, and has the remaining
coordinates at grid vertices. Orient it in increasing axis order.
-Include each lift \(C\) whose center is at distance less than \(4\delta\)
-from some witness \(z_y\) in its sheet chart. The full cell fits in that
-chart. Its dual \(m\)-cube \(C^\#\) has the same center, varies along
-the complementary axes by \([-r/2,r/2]\), and has the other coordinates
+Include each lift (C) whose center is at distance less than (4delta)
+from some witness (z_y) in its sheet chart. The full cell fits in that
+chart. Its dual (m)-cube (C^#) has the same center, varies along
+the complementary axes by ([-r/2,r/2]), and has the other coordinates
fixed. It too fits in the chart. Orient it so that the primal oriented
basis, followed by the negatives of its dual oriented basis, is a
-positive basis of \(\mathbb R^q\). Give the primal cell the coefficient
+positive basis of (mathbb R^qbbR^q). Give the primal cell the coefficient
- u_C=\int_C\eta.
+ u_C=int_Ceta.
These coefficients satisfy the cubical cocycle equations where they
-are needed. If a primal \((n+1)\)-cell has its center within
-\(2\delta\) of a witness, all its boundary \(n\)-faces are included
+are needed. If a primal ((n+1))-cell has its center within
+(2delta) of a witness, all its boundary (n)-faces are included
with coherent lifts, and their signed integrals sum to zero by Stokes.
A facet of a dual cube is centered at just such a primal
-\((n+1)\)-cell. The incidence signs of its incident dual cubes agree,
+((n+1))-cell. The incidence signs of its incident dual cubes agree,
up to one common sign, with the boundary signs of the primal cell:
this follows by placing the varying primal axes first and the negative
dual axes last in the ambient oriented basis. Consequently the dual
-\(m\)-cubes weighted by \(u_C\) have balanced oriented boundary near
-\(K\), in particular at every facet meeting the tube of radius
-\(\delta\). Indeed the corresponding centers differ from those
-facet points by at most \(r\sqrt q\), so they have the stipulated
-\(2\delta\) witnesses. Projected grid cells on other sheets give no
+(m)-cubes weighted by (u_C) have balanced oriented boundary near
+(K), in particular at every facet meeting the tube of radius
+(delta). Indeed the corresponding centers differ from those
+facet points by at most (rsqrt q), so they have the stipulated
+(2delta) witnesses. Projected grid cells on other sheets give no
incidence: all these statements use lifts in the same flat chart.
[Finite-type unbiased rounding]
-There is an integer \(D_0\geq1\), bounded by an elementary function of
-\(q\) and constant in \(V,s,\omega\), and a Borel equivariant random
-choice of coefficients \(u'_C\in D_0^{-1}\mathbb Z\) such that
+There is an integer (D_0geq1), bounded by an elementary function of
+(q) and constant in (V,s,omega), and a Borel equivariant random
+choice of coefficients (u'_Cin D_0^{-1}mathbb ZbbZ) such that
- \E_{\rm rd}u'_C=u_C,\qquad |u'_C-u_C|\leq r^nq^{-n}.
+ E_{rm rd}u'_C=u_C,qquad |u'_C-u_C|leq r^nq^{-n}.
The expectation is conditional on the grid and entropy data. Every
-cocycle equation just described is satisfied exactly by the \(u'_C\).
+cocycle equation just described is satisfied exactly by the (u'_C).
Proof.
@@ -5408,29 +5408,29 @@ same rounded vector. First establish that the vector has elementary
size despite the infinitely many lifted cells.
All projected centers lie
-in a fixed bounded Euclidean ball. For fixed \(V,s\), their grid
-indices and axis sets therefore have at most \((C/r)^{Cq}\) choices.
+in a fixed bounded Euclidean ball. For fixed (V,s), their grid
+indices and axis sets therefore have at most ((C/r)^{Cq}) choices.
At a selected center choose, breaking ties by color order, its largest
unrotated color coordinate. This coordinate is at least
-\(q^{-1/2}-4\delta\), and its label occurs in every witness for the
+(q^{-1/2}-4delta), and its label occurs in every witness for the
center. It determines the lift uniquely, by the proof of
Lemma~lem:transfer-flat; call it the anchor label.
-The closed graph one-neighborhood of an anchor has at most \(q\)
+The closed graph one-neighborhood of an anchor has at most (q)
labels. Every center-test witness, meaning a witness at distance less
-than \(4\delta\) from the center, lies in this neighborhood because it
+than (4delta) from the center, lies in this neighborhood because it
contains the anchor. Every point of the full cell is within
-\(4\delta+r\sqrt n/2<5\delta\) of each such witness. Thus every
-active color, whose coordinate is greater than \(12\delta\), occurs
+(4delta+rsqrt n/2<5delta) of each such witness. Thus every
+active color, whose coordinate is greater than (12delta), occurs
in each center-test witness. For an arbitrary ball-chart witness of
the center the corresponding bound is instead
-\(10\delta+r\sqrt n/2<11\delta\), which is still below the active
+(10delta+rsqrt n/2<11delta), which is still below the active
cutoff and gives the same conclusion. Thus the
-coefficient \(u_C\) is determined by its projected grid index, its
+coefficient (u_C) is determined by its projected grid index, its
anchor, and the color assignment on this one-neighborhood. In
particular, no labels farther out in the graph enter its calculation.
-Choose and order representatives of the at most \(q\) anchor orbits,
+Choose and order representatives of the at most (q) anchor orbits,
and order each of their finite one-neighborhoods. An anchor has a
unique translate to its representative because the label action is
free. After this translate, its neighborhood is a fixed finite set
@@ -5439,103 +5439,103 @@ ordered list of slots consisting of an anchor representative, an
injective color assignment on its neighborhood, an axis set and grid
index, and a witness clique in that neighborhood containing the anchor.
It is enough to use the fixed index box
-\(|k_j|\leq\lceil3/r\rceil+3\), which contains every possible selected
+(|k_j|leqlceil3/rrceil+3), which contains every possible selected
center for all shifts. The slot is valid when the anchor color is the
canonically chosen largest unrotated color of its center and that center
-is at distance less than \(4\delta\) from the spherical orthant of its
+is at distance less than (4delta) from the spherical orthant of its
witness clique. Give a valid slot the integral of the explicit form
over its primal cell, using the labels specified by the slot, and give
an invalid slot value zero. The preceding active-color observation
makes this integral unambiguous. The validity tests and integrals are
-Borel in \(V,s\).
+Borel in (V,s).
-This is one vector \(v=v(V,s)\in\mathbb R^{N_0}\) for the fixed graph
-and cocycle, common to all \(\omega\). Only its size and the bounds
+This is one vector (v=v(V,s)inmathbb R^bbR^{N_0}) for the fixed graph
+and cocycle, common to all (omega). Only its size and the bounds
below are uniform over all graphs and cocycles of the given complexity:
-there are at most \(q^q\) neighborhood color assignments, \(2^q\)
-witness cliques, and \((C/r)^{Cq}\) projected indices, so \(N_0\) is
-elementarily bounded in \(q\). An actual cell receives the slot
-\(t(C,\omega)\) obtained by normalizing its canonical anchor, restricting
+there are at most (q^q) neighborhood color assignments, (2^q)
+witness cliques, and ((C/r)^{Cq}) projected indices, so (N_0) is
+elementarily bounded in (q). An actual cell receives the slot
+(t(C,omega)) obtained by normalizing its canonical anchor, restricting
its actual coloring, and choosing the first valid witness clique in the
-fixed order. Its value is \(u_C=v_{t(C,\omega)}\). This type is Borel
-and is unchanged by translating \(C\) and \(\omega\) together. Slots
+fixed order. Its value is (u_C=v_{t(C,omega)}). This type is Borel
+and is unchanged by translating (C) and (omega) together. Slots
with the same anchor, restricted coloring, axes, and grid index but
different valid witness cliques have the same value: the
-\(5\delta\) observation gives the identical active-label formula
+(5delta) observation gives the identical active-label formula
throughout the primal cell.
-Impose on this list every integer row \(b\) with
-\(\|b\|_∞\leq2(n+1)\) for which \(b\cdot v=0\). There are
+Impose on this list every integer row (b) with
+(|b|_∞leq2(n+1)) for which (bcdot v=0). There are
finitely many candidate rows, so the resulting matrix depends Borelly
-on \(v\). In particular it includes the difference row for any two
+on (v). In particular it includes the difference row for any two
duplicate valid slots. It also includes every needed cubical equation,
even when its faces have anchors normalized by different deck elements:
-all their values are entries of this same vector \(v\), and Stokes gives
+all their values are entries of this same vector (v), and Stokes gives
the signed zero sum of those entries. After repeated slots in its
-\(2(n+1)\) faces have been combined, each coefficient has the stated
+(2(n+1)) faces have been combined, each coefficient has the stated
bound. The kernel of this
-integer matrix has a real basis of integer vectors \(b_1,...,b_k\)
+integer matrix has a real basis of integer vectors (b_1,...,b_k)
with
-\[
- k\leq N_0,\qquad
- \max_j\|b_j\|_∞\leq K_0:=(C_nN_0)^{C N_0}.
-\]
+[
+ kleq N_0,qquad
+ max_j|b_j|_∞leq K_0:=(C_nN_0)^{C N_0}.
+]
To see this, choose a nonsingular maximal minor and solve for its pivot
coordinates in terms of the free coordinates; multiplying each basis
vector by its nonzero determinant makes it integral. Hadamard's
-bound for minors of a matrix with entries at most \(2(n+1)\) gives
+bound for minors of a matrix with entries at most (2(n+1)) gives
the displayed estimate. Lexicographically choosing the pivot minor
makes the basis Borel. The case of a zero-dimensional kernel simply
-has \(v=0\) and needs no rounding.
-
-Write \(v=\sum_j t_jb_j\), using the pivot solution, which is Borel.
-Independently round each \(t_j\) to the two adjacent multiples of
-\(D_0^{-1}\), choosing their probabilities so that its conditional
-expectation is \(t_j\). Choose a single integer
-\(D_0\geq N_0K_0r^{-n}q^n\), using uniform elementary upper bounds
-for \(N_0,K_0\). The resulting vector \(v'\) lies in the same
-kernel, has entries in \(D_0^{-1}\mathbb Z\), has expectation \(v\),
-and satisfies \(\|v'-v\|_∞\leq N_0K_0/D_0\leq r^nq^{-n}\).
+has (v=0) and needs no rounding.
+
+Write (v=sum_j t_jb_j), using the pivot solution, which is Borel.
+Independently round each (t_j) to the two adjacent multiples of
+(D_0^{-1}), choosing their probabilities so that its conditional
+expectation is (t_j). Choose a single integer
+(D_0geq N_0K_0r^{-n}q^n), using uniform elementary upper bounds
+for (N_0,K_0). The resulting vector (v') lies in the same
+kernel, has entries in (D_0^{-1}mathbb ZbbZ), has expectation (v),
+and satisfies (|v'-v|_∞leq N_0K_0/D_0leq r^nq^{-n}).
All operations, including taking floors and using independent uniform
parameters for the adjacent choices, are Borel. The fixed slot order
-is used for every pivot choice. Thus one parameter \(\xi\), independent
-of \(Ω\), gives one rounded vector \(v'(V,s,\xi)\) for all cells.
-Set \(u'_C=v'_{t(C,\omega)}\). The invariance of the type under
+is used for every pivot choice. Thus one parameter (xi), independent
+of (Ω), gives one rounded vector (v'(V,s,xi)) for all cells.
+Set (u'_C=v'_{t(C,omega)}). The invariance of the type under
translation proves equivariance, and the common zero rows prove all
the asserted incidence equations.
The rounded coefficients define the original oriented dual current
- Z_{\rm rd}:=\sum_C D_0u'_C[C^\#].
+ Z_{rm rd}:=sum_C D_0u'_C[C^#].
-Here \([C^\#]\) is integration over the oriented cube, and the
+Here ([C^#]) is integration over the oriented cube, and the
coefficients are integers. Verify the local finiteness implicit
in this notation. The form bound and rounding give
-\(|u_C|\leq C_nr^n\) and an elementary bound for each \(|D_0u'_C|\).
-Let \(B\) be a small flat ball in one sheet chart and \(Q\) one full
-projected dual cube. The intersection \(Q\cap z(B)\) is convex. Any
-lift of \(Q\) meeting \(B\) agrees with the chart inverse on this
+(|u_C|leq C_nr^n) and an elementary bound for each (|D_0u'_C|).
+Let (B) be a small flat ball in one sheet chart and (Q) one full
+projected dual cube. The intersection (Qcap z(B)) is convex. Any
+lift of (Q) meeting (B) agrees with the chart inverse on this
connected intersection, by uniqueness of lifts for the local
-diffeomorphism \(z\). Thus two lifts meeting \(B\), even at different
+diffeomorphism (z). Thus two lifts meeting (B), even at different
projected points, agree there and then agree on the connected cube
-\(Q\). At most one geometric lift of each projected cube meets \(B\).
-Only finitely many projected grid cubes meet \(z(B)\), with bounded
+(Q). At most one geometric lift of each projected cube meets (B).
+Only finitely many projected grid cubes meet (z(B)), with bounded
coefficients, proving neighborhood local finiteness.
-Let \(\mathcal T_\delta\subset\mathcal U_\omega\) be the strict tube
-of points at distance less than \(\delta\) from a witness in its ball
+Let (mathcal T_calT_deltasubsetmathcal U_calU_omega) be the strict tube
+of points at distance less than (delta) from a witness in its ball
chart. The exact incidence equations in
Lemma~lem:transfer-rounding give
- ∂ Z_{\rm rd}\big|_{\mathcal T_\delta}=0.
+ ∂ Z_{rm rd}big|_{mathcal T_calT_delta}=0.
This is a local assertion: the boundary current vanishes on test forms
-compactly supported in \(\mathcal T_\delta\).
+compactly supported in (mathcal T_calT_delta).
-To resolve \(Z_{\rm rd}\) smoothly, realize each coefficient by
-\(|D_0u'_C|\) oriented copies of its cube, reversing orientation when
-\(u'_C<0\). At every facet where boundary balance is required, pair
+To resolve (Z_{rm rd}) smoothly, realize each coefficient by
+(|D_0u'_C|) oriented copies of its cube, reversing orientation when
+(u'_C<0). At every facet where boundary balance is required, pair
positive and negative incident copies and glue them by their common
face. Leave other facets free, and double this diagram with an
oppositely oriented mirror, joining each free facet to its mirror.
@@ -5543,8 +5543,8 @@ All codimension-one faces of the doubled diagram are now paired.
Local axis, grid, and copy indices order the incident copies, so these
choices may be Borel and equivariant.
-The doubling does not join the two sides near \(K\), even through a
-lower face. If a point of any face maps into \(\mathcal T_\delta\),
+The doubling does not join the two sides near (K), even through a
+lower face. If a point of any face maps into (mathcal T_calT_delta),
every path of facet identifications relating chambers incident to that
face preserves its image point. Each facet on the path meets the tube,
so was balanced before doubling and was not a free mirror facet. The
@@ -5555,14 +5555,14 @@ identifications preserve that point and its sheet, so even after
doubling they relate only this bounded collection of incident chambers.
Take barycentric subdivision of the doubled cubes. Each top simplex
-\(\sigma\) has vertices colored \(0,...,m\) by the dimensions of
+(sigma) has vertices colored (0,...,m) by the dimensions of
its barycentric faces, and every facet pairing respects these colors.
-Write \(ε_\sigma\in\{1,-1\}\) for its sign relative to that
+Write (ε_sigmain{1,-1}) for its sign relative to that
color order, and call this doubled signed simplicial diagram
-\(\mathcal P\). It maps facewise to \(\mathcal U_\omega\), with
+(mathcal PcalP). It maps facewise to (mathcal U_calU_omega), with
elementary subdivision and incidence bounds. The original current
-\(Z_{\rm rd}\) remains the object to be represented on
-\(\mathcal T_\delta\); the doubling supplies closed facet pairings
+(Z_{rm rd}) remains the object to be represented on
+(mathcal T_calT_delta); the doubling supplies closed facet pairings
for the resolution.
@@ -5575,215 +5575,215 @@ will be controlled by its coordinate volume.
The reflected gluing below uses Gaifullin's explicit permutohedral
resolution, including the face-preserving sign-changing pairings,
their conjugation under reflections, and the auxiliary bit toggles
-\cite[pp.~562--563]{Gaifullin2008}. Replace his rank-indexed bits
+cite[pp.~562--563]{Gaifullin2008}. Replace his rank-indexed bits
by independent subset-indexed bits, put invariant probability laws on
the pairing data, and supply smooth measured charts and quantitative
metric bounds.
-\par\noindent{\footnotesize\ttfamily src-gaifullin2008.pdf}\par
+parnoindent{footnotesizettfamily src-gaifullin2008.pdf}par
[Measured resolution]
-For the diagram \(\mathcal P\) there is a standard Borel measured
-family \(\mathcal A\) of oriented smooth \(m\)-manifolds and a smooth
-map \(X:\mathcal A\to\mathcal U_\omega\), equivariant and Borel in
+For the diagram (mathcal PcalP) there is a standard Borel measured
+family (mathcal AcalA) of oriented smooth (m)-manifolds and a smooth
+map (X:mathcal AcalAtomathcal U_calU_omega), equivariant and Borel in
all the data. The integration identities below are for fixed grid,
-entropy, and rounding data, including \((V,s,\omega,\xi)\); their
+entropy, and rounding data, including ((V,s,omega,xi)); their
integrals use the conditional state measure and depend Borelly on those
-data. The points over \(\mathcal T_\delta\) represented by
+data. The points over (mathcal T_calT_delta) represented by
original chamber tiles form a Borel leaf-open subset
-\(\mathcal A_{\rm orig}\subset X^{-1}(\mathcal T_\delta)\). For every
-compactly supported smooth \(m\)-form \(\alpha\) on \(\mathcal T_\delta\),
+(mathcal A_calA_{rm orig}subset X^{-1}(mathcal T_calT_delta)). For every
+compactly supported smooth (m)-form (alpha) on (mathcal T_calT_delta),
- \int_{\mathcal A_{\rm orig}}X^*\alpha=Z_{\rm rd}(\alpha).
+ int_{mathcal A_calA_{rm orig}}X^*alpha=Z_{rm rd}(alpha).
For the full doubled diagram, measured oriented integration also gives
- X_*[\mathcal A]=[\mathcal P].
+ X_*[mathcal A]=[calA]=[mathcal P].calP].
-For a parameter \(0<\lambda<1/2\) and a choice
+For a parameter (0<lambda<1/2) and a choice
- 0<2\gamma_0<(m+1)^{-(m+1)}\lambda^{m(m+1)},
+ 0<2gamma_0<(m+1)^{-(m+1)}lambda^{m(m+1)},
-there are a smooth metric \(k\) and a closed seam set in each tile such
-that \(\|dX\|_k\leq1\), and \(k=X^*k_{\rm flat}\) off the seam.
+there are a smooth metric (k) and a closed seam set in each tile such
+that (|dX|_kleq1), and (k=X^*k_{rm flat}) off the seam.
There are buffered smooth charts with elementary overlap, metric,
inverse-metric, and fixed-order derivative bounds whenever
-\(\lambda^{-1}\) and \(\gamma_0^{-1}\) are elementary. In the sorted
+(lambda^{-1}) and (gamma_0^{-1}) are elementary. In the sorted
charts for each tile, the seam is contained in the union of the slabs
-\(|Z_j|\leq\lambda\), \(1\leq j\leq m\). The \(k\)-volume density
+(|Z_j|leqlambda), (1leq jleq m). The (k)-volume density
in those charts has an elementary upper bound chosen before
-\(\lambda,\gamma_0\); in particular this bound does not depend on
+(lambda,gamma_0); in particular this bound does not depend on
these two parameters.
One may prescribe for each chamber over a dual cube a rotation
-\(O_\sigma\in SO(q)\). There is also a smooth Borel equivariant
-\(\mathcal R:\mathcal A\to SO(q)\) equal to \(O_\sigma\) off the
+(O_sigmain SO(q)). There is also a smooth Borel equivariant
+(mathcal R:calR:mathcal AcalAto SO(q)) equal to (O_sigma) off the
seam in that tile, equal to the identity near all tile facets, and
homotopic to the identity. Its fixed-order chart derivatives and
-\(\|d\mathcal R\|_k\) are elementarily bounded under the same
+(|dmathcal RcalR|_k) are elementarily bounded under the same
hypotheses.
Proof.
Invariant reflection states.
For each nonempty proper subset
-\(S\subset\{0,...,m\}\) and each face with exactly those vertex
+(Ssubset{0,...,m}) and each face with exactly those vertex
colors, pair the incident positive chambers with the incident negative
chambers. The two counts agree. Indeed choose a color omitted by
-\(S\); the given codimension-one matching in \(\mathcal P\) across
+(S); the given codimension-one matching in (mathcal PcalP) across
facets omitting this color pairs the chambers at that face with opposite
signs. All incident sets are
finite by the incidence bound above. Choose uniformly among their
-possible pairings, independently for each face and each \(S\), and
-let \(p_S\) denote the resulting involution on chambers. Use also
-independent uniform bits \(t_S\in\{0,1\}\), one for each \(S\).
-The state space consists of \(\zeta=(\sigma,p,t)\), with counting measure
-on \(\sigma\) and these product probabilities on \(p,t\). These
+possible pairings, independently for each face and each (S), and
+let (p_S) denote the resulting involution on chambers. Use also
+independent uniform bits (t_Sin{0,1}), one for each (S).
+The state space consists of (zeta=(sigma,p,t)), with counting measure
+on (sigma) and these product probabilities on (p,t). These
are countable products of finite probability spaces. As the original
data vary, the incidence lists and finite pairing choices are Borel,
so the product laws are Borel probability kernels.
-Define a move \(s_S\) by
-\[
- \sigma⟼ p_S\sigma,\qquad t_S⟼1-t_S,
- \qquad p_R⟼ p_Sp_Rp_S\quad(R⊊ S),
-\]
-leaving the other data unchanged. The conjugated \(p_R\) is again
-a pairing at each \(R\)-face: \(p_S\) preserves every such face and
+Define a move (s_S) by
+[
+ sigma⟼ p_Ssigma,qquad t_S⟼1-t_S,
+ qquad p_R⟼ p_Sp_Rp_Squad(R⊊ S),
+]
+leaving the other data unchanged. The conjugated (p_R) is again
+a pairing at each (R)-face: (p_S) preserves every such face and
flips the chamber sign, so conjugation preserves its face and its
-sign-reversing property. Each \(s_S\) is an involution. If
-\(R⊊ S\), the two moves commute. On chambers, for example,
-their two orders both give \(p_Sp_R\sigma\), since after \(s_S\) the
-\(R\)-pairing is \(p_Sp_Rp_S\). On a pairing indexed by a subset of
-\(R\), both orders conjugate by \(p_Sp_R\); on the other indices the
+sign-reversing property. Each (s_S) is an involution. If
+(R⊊ S), the two moves commute. On chambers, for example,
+their two orders both give (p_Sp_Rsigma), since after (s_S) the
+(R)-pairing is (p_Sp_Rp_S). On a pairing indexed by a subset of
+(R), both orders conjugate by (p_Sp_R); on the other indices the
same assertion follows directly from which of the two moves changes
that index. The bits commute as well. Thus all moves in a nested
flag commute, and they act freely: each changes its own bit, so no
nonempty product of these moves can fix a state.
These moves preserve the state measure. They permute the counting
-chamber index, flip a uniform bit, and, conditional on \(p_S\),
+chamber index, flip a uniform bit, and, conditional on (p_S),
conjugate each relevant uniformly distributed finite pairing by a
permutation of its incident set. The latter operation permutes its
finite set of possible pairings and preserves its uniform law. It
also preserves the independent product of those laws. All moves are
Borel and commute with translating the original label data.
-\medskip\noindentSmooth leaves and the measured atlas.
+medskipnoindentSmooth leaves and the measured atlas.
Attach the following tile to each state. Start with the positive
-simplex \(x_0,...,x_m>0\), \(\sum_i x_i=1\), and take the closure
-of its collection of pair ratios \((x_i/x_j)_{i\ne j}\) in the
-product of copies of \([0,∞]\). For each permutation
-\(i_0,...,i_m\) use the coordinate chart
+simplex (x_0,...,x_m>0), (sum_i x_i=1), and take the closure
+of its collection of pair ratios ((x_i/x_j)_{ine j}) in the
+product of copies of ([0,∞]). For each permutation
+(i_0,...,i_m) use the coordinate chart
- u_j=x_{i_j}/x_{i_{j-1}}\in[0,4),\qquad 1\leq j\leq m.
+ u_j=x_{i_j}/x_{i_{j-1}}in[0,4),qquad 1leq jleq m.
All pair ratios in the chart are products of these or their reciprocals,
-so the \(u_j\) are independent coordinates. Sorted orders have
-\(u_j\leq1\) and cover the closure. The facet \(u_j=0\) has label
-\(S_j=\{i_0,...,i_{j-1}\}\); the labels of a corner are nested.
+so the (u_j) are independent coordinates. Sorted orders have
+(u_jleq1) and cover the closure. The facet (u_j=0) has label
+(S_j={i_0,...,i_{j-1}}); the labels of a corner are nested.
Glue each facet to the identical resolved coordinates in the state
-obtained by \(s_{S_j}\). The commutation just proved makes the gluing
+obtained by (s_{S_j}). The commutation just proved makes the gluing
consistent at corners.
-Use signed square root coordinates \(u_j=Z_j^2\), with
-\(Z\in(-2,2)^m\), starting from any state. For \(Z_j<0\) use the
-tile obtained by \(s_{S_j}\); use the product of these commuting moves
+Use signed square root coordinates (u_j=Z_j^2), with
+(Zin(-2,2)^m), starting from any state. For (Z_j<0) use the
+tile obtained by (s_{S_j}); use the product of these commuting moves
when several coordinates are negative. At zero use the starting tile
as a convention for representing the same glued point.
Figure~fig:resolution illustrates the tile and the four signed
-quadrants at a two-facet corner when \(m=2\).
-
-\par\medskip
-\noindent{\textwidth}
-\makeatletter\def\@captype{figure}\makeatother
-\centering
-[t]{0.53\textwidth}
-\centering
+quadrants at a two-facet corner when (m=2).
+
+parmedskip
+noindent{textwidth}
+makeatletterdef@captype{figure}makeatother
+centering
+[t]{0.53textwidth}
+centering
[scale=1.15, line cap=round, line join=round]
- \coordinate (v0) at (-2,1.05);
- \coordinate (v1) at (-2.95,-0.65);
- \coordinate (v2) at (-1.05,-0.65);
- \draw[thick,fill=blue!3] (v0)--(v1)--(v2)--cycle;
- \node[above] at (v0) {\(0\)};
- \node[below left] at (v1) {\(1\)};
- \node[below right] at (v2) {\(2\)};
- \node at (-2,-0.12) {\(\sigma\)};
- \node[font=\footnotesize,align=center] at (-2,-1.42)
- {one barycentric\\chamber \(\sigma\simeq\Delta^2\)};
- \draw[-{Stealth[length=2mm]},thick] (-0.25,0.12)--(-0.90,0.12)
- node[midway,above] {\(X\)};
+ coordinate (v0) at (-2,1.05);
+ coordinate (v1) at (-2.95,-0.65);
+ coordinate (v2) at (-1.05,-0.65);
+ draw[thick,fill=blue!3] (v0)--(v1)--(v2)--cycle;
+ node[above] at (v0) {(0)};
+ node[below left] at (v1) {(1)};
+ node[below right] at (v2) {(2)};
+ node at (-2,-0.12) {(sigma)};
+ node[font=footnotesize,align=center] at (-2,-1.42)
+ {one barycentricchamber (sigmasimeqDelta^2)};
+ draw[-{Stealth[length=2mm]},thick] (-0.25,0.12)--(-0.90,0.12)
+ node[midway,above] {(X)};
[shift={(1,0)}]
- \path[fill=gray!22,draw=black,thick]
+ path[fill=gray!22,draw=black,thick]
(30:1.40)--(90:1.40)--(150:1.40)--(210:1.40)--
(270:1.40)--(330:1.40)--cycle;
- \path[fill=white,draw=none]
+ path[fill=white,draw=none]
(30:0.76)--(90:0.76)--(150:0.76)--(210:0.76)--
(270:0.76)--(330:0.76)--cycle;
- \node[font=\small] at (60:0.97) {\(\{0\}\)};
- \node[font=\small] at (120:0.97) {\(\{0,1\}\)};
- \node[font=\small] at (180:0.97) {\(\{1\}\)};
- \node[font=\small] at (240:0.97) {\(\{1,2\}\)};
- \node[font=\small] at (300:0.97) {\(\{2\}\)};
- \node[font=\small] at (0:0.87) {\(\{0,2\}\)};
- \node[font=\small] at (0,0) {tile};
+ node[font=small] at (60:0.97) {({0})};
+ node[font=small] at (120:0.97) {({0,1})};
+ node[font=small] at (180:0.97) {({1})};
+ node[font=small] at (240:0.97) {({1,2})};
+ node[font=small] at (300:0.97) {({2})};
+ node[font=small] at (0:0.87) {({0,2})};
+ node[font=small] at (0,0) {tile};
- \node[font=\small,align=center] at (-0.55,-2.15)
+ node[font=small,align=center] at (-0.55,-2.15)
{(a) A chamber and its resolved tile};
-\hfill
-[t]{0.43\textwidth}
-\centering
+hfill
+[t]{0.43textwidth}
+centering
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- \draw[-{Stealth[length=1.6mm]}] (-1.4,0)--(1.5,0)
- node[right,font=\small] {\(Z_S\)};
- \draw[-{Stealth[length=1.6mm]}] (0,-1.4)--(0,1.5)
- node[above,font=\small] {\(Z_T\)};
- \node at (0.68,0.67) {\(\zeta\)};
- \node at (-0.68,0.67) {\(s_S\zeta\)};
- \node at (0.68,-0.67) {\(s_T\zeta\)};
- \node at (-0.68,-0.67) {\(s_Ss_T\zeta\)};
- \node[font=\small] at (1.03,1.04) {\(+\)};
- \node[font=\small] at (-1.03,1.04) {\(-\)};
- \node[font=\small] at (1.03,-1.04) {\(-\)};
- \node[font=\small] at (-1.03,-1.04) {\(+\)};
- \node[font=\small,align=center] at (0,-2.15)
- {(b) \(u_S=Z_S^2,\quad u_T=Z_T^2\)};
-
-
-\caption{The reflected resolution in dimension two, after Gaifullin's
+ fill[blue!12] (0,0) rectangle (1.25,1.25);
+ fill[orange!15] (-1.25,0) rectangle (0,1.25);
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+ draw[-{Stealth[length=1.6mm]}] (-1.4,0)--(1.5,0)
+ node[right,font=small] {(Z_S)};
+ draw[-{Stealth[length=1.6mm]}] (0,-1.4)--(0,1.5)
+ node[above,font=small] {(Z_T)};
+ node at (0.68,0.67) {(zeta)};
+ node at (-0.68,0.67) {(s_Szeta)};
+ node at (0.68,-0.67) {(s_Tzeta)};
+ node at (-0.68,-0.67) {(s_Ss_Tzeta)};
+ node[font=small] at (1.03,1.04) {(+)};
+ node[font=small] at (-1.03,1.04) {(-)};
+ node[font=small] at (1.03,-1.04) {(-)};
+ node[font=small] at (-1.03,-1.04) {(+)};
+ node[font=small,align=center] at (0,-2.15)
+ {(b) (u_S=Z_S^2,quad u_T=Z_T^2)};
+
+
+caption{The reflected resolution in dimension two, after Gaifullin's
permutohedral construction [cite]. (a) A barycentric
-chamber \(\sigma\simeq\Delta^2\) and its resolved tile. The six facets
-correspond to the nonempty proper subsets of \(\{0,1,2\}\), nested when
-the labels are adjacent, and \(X\) maps the tile interior
+chamber (sigmasimeqDelta^2) and its resolved tile. The six facets
+correspond to the nonempty proper subsets of ({0,1,2}), nested when
+the labels are adjacent, and (X) maps the tile interior
diffeomorphically onto the chamber interior. (b) At a corner with
-labels \(S\subset T\), a state \(\zeta=(\sigma,p,t)\) and its three
+labels (Ssubset T), a state (zeta=(sigma,p,t)) and its three
reflections fill the four quadrants of signed coordinates; the
-commutation \(s_Ss_T=s_Ts_S\) and the independent bit toggles make these
+commutation (s_Ss_T=s_Ts_S) and the independent bit toggles make these
four quadrants one smooth chart. Signs record whether the chamber
-orientation agrees with that of \(\zeta\) (blue) or opposes it (orange).
+orientation agrees with that of (zeta) (blue) or opposes it (orange).
The collar and strips indicate the seam schematically, with illustrative
-widths, and \(X\) is not affine.}
+widths, and (X) is not affine.}
-\par\medskip
+parmedskip
Verify that these are ordinary smooth charts. On an overlap of two permutations,
-a cut \(S_j\) which is not a cut of the other permutation has
-\(u_j\geq4^{-2m}\). To see this, choose a pair of indices on opposite
+a cut (S_j) which is not a cut of the other permutation has
+(u_jgeq4^{-2m}). To see this, choose a pair of indices on opposite
sides of that cut whose order is inverted in the other permutation.
-Their ratio in the first ordering is at most \(4^m u_j\); its inverse
-in the second ordering is at most \(4^m\). The asserted lower bound
+Their ratio in the first ordering is at most (4^m u_j); its inverse
+in the second ordering is at most (4^m). The asserted lower bound
follows. At a cut common to both permutations, the ratio for the
second chart is the first ratio at that cut multiplied by a product
of ratios within the blocks between common cuts. Those within-block
@@ -5798,8 +5798,8 @@ at the noncommon cuts. The remaining state relation is a product of
the common reflections, and its product is unique by their free
action on the bits. Commutation makes this sign rule persist across
the zero coordinates. Thus the transitions are smooth and invertible
-across every corner. A point incident to \(j\) facets has exactly the
-finite star of \(2^j\) states under these reflections, and hence has
+across every corner. A point incident to (j) facets has exactly the
+finite star of (2^j) states under these reflections, and hence has
one neighborhood of this form rather than several branches.
The signed charts give smooth leaf neighborhoods. Next check the
@@ -5829,54 +5829,54 @@ The chamber signs orient the leaves. In a flag chart the Jacobian
sign from positive ratio coordinates to oriented barycentric
coordinates depends only on the permutation. Changing one signed
coordinate from positive to negative changes the Jacobian sign of
-\(u_j=Z_j^2\) once, while the move changes the chamber sign once.
+(u_j=Z_j^2) once, while the move changes the chamber sign once.
The induced orientation is therefore the same across each facet.
The transverse measures here are nonnegative; the signs enter only
through this orientation in integrals of forms. The value of
-\(\omega\) is constant along each tile leaf, and translation by
-\(\Gamma\) reindexes the states and preserves these measures.
+(omega) is constant along each tile leaf, and translation by
+(Gamma) reindexes the states and preserves these measures.
-\medskip\noindentThe map and the local current.
-Recover the raw \(x_i\) from (eq:transfer-ratios) by normalizing
-the successive products \(1,u_1,u_1u_2,...,u_1... u_m\).
-If \(v_{\sigma,i}\) are the vertices of the indicated chamber in its
+medskipnoindentThe map and the local current.
+Recover the raw (x_i) from (eq:transfer-ratios) by normalizing
+the successive products (1,u_1,u_1u_2,...,u_1... u_m).
+If (v_{sigma,i}) are the vertices of the indicated chamber in its
flat chart, map its tile by
- X=\sum_{i=0}^m b_i v_{\sigma,i},\qquad
- b_i=(\exp(-1/x_i))/(\sum_l\exp(-1/x_l)),
+ X=sum_{i=0}^m b_i v_{sigma,i},qquad
+ b_i=(exp(-1/x_i))/(sum_lexp(-1/x_l)),
-where \(\exp(-1/x_i)\) is extended by zero at \(x_i=0\). On a facet
-with label \(S\), all coefficients outside \(S\) vanish to infinite
-order in its signed coordinate. The vertices in \(S\) are the same
+where (exp(-1/x_i)) is extended by zero at (x_i=0). On a facet
+with label (S), all coefficients outside (S) vanish to infinite
+order in its signed coordinate. The vertices in (S) are the same
on the paired chambers, and their coefficients are smooth functions
-of the squares of the signed coordinates. Thus \(X\) is smooth
-through the gluing. Its denominator is at least \(e^{-(m+1)}\), since
-some \(x_i\geq1/(m+1)\), which also gives elementary upper derivative
+of the squares of the signed coordinates. Thus (X) is smooth
+through the gluing. Its denominator is at least (e^{-(m+1)}), since
+some (x_igeq1/(m+1)), which also gives elementary upper derivative
bounds in the tile charts.
-On the positive simplex the map \(x\mapsto b\) is a diffeomorphism.
-Given positive \(b_i\) of sum one, its inverse is obtained from the
-unique \(t\in(0,1/\max_i b_i)\) satisfying
-\[
- \sum_i\frac1{-\log(tb_i)}=1;
-\]
+On the positive simplex the map (xmapsto b) is a diffeomorphism.
+Given positive (b_i) of sum one, its inverse is obtained from the
+unique (tin(0,1/max_i b_i)) satisfying
+[
+ sum_ifrac1{-log(tb_i)}=1;
+]
the left side increases smoothly from zero to infinity. The inverse
function theorem also gives the smooth inverse. This diffeomorphism
preserves orientation. Indeed it extends to the closed simplex
preserving every face, and the linear homotopy to the identity has
-the same property, so its relative topological degree is \(+1\).
+the same property, so its relative topological degree is (+1).
It follows that the tile interior maps diffeomorphically, with sign
-\(ε_\sigma\), onto the interior of its chamber. Each
+(ε_sigma), onto the interior of its chamber. Each
chamber's state probability integrates to one and the tile boundaries
have leafwise measure zero. Fubini and change of variables therefore
-give, for every compactly supported smooth \(m\)-form \(\alpha\),
-\[
- \int_{\mathcal A}X^*\alpha
- =\sum_\sigmaε_\sigma\int_\sigma\alpha.
-\]
+give, for every compactly supported smooth (m)-form (alpha),
+[
+ int_{mathcal A}calAX^*alpha
+ =sum_sigmaε_sigmaint_sigmaalpha.
+]
This proves (eq:transfer-push). To obtain the local identity,
-define \(\mathcal A_{\rm orig}\) as the image in
-\(X^{-1}(\mathcal T_\delta)\) of tile points belonging to original
+define (mathcal A_calA_{rm orig}) as the image in
+(X^{-1}(mathcal T_calT_delta)) of tile points belonging to original
chambers. At an interior point this subset is leaf-open. At a lower
face, the incidence argument preceding the lemma shows that every state
in its finite reflection star is on the same side. A neighborhood in
@@ -5885,190 +5885,190 @@ avoid nonincident facets, so it stays on that side. This proves
leaf-openness and makes the definition unambiguous at glued points;
the finite indexed charts also make it Borel. Applying the same
change-of-variables calculation only to original tile interiors, for
-a form supported in \(\mathcal T_\delta\), gives
+a form supported in (mathcal T_calT_delta), gives
(eq:transfer-local-push): each original cube is counted with
-its coefficient \(D_0u'_C\), and each chamber's state fiber has mass one.
+its coefficient (D_0u'_C), and each chamber's state fiber has mass one.
-\medskip\noindentThe metric and upper density.
-The signed charts admit a background metric \(k_0\) with elementary
+medskipnoindentThe metric and upper density.
+The signed charts admit a background metric (k_0) with elementary
bounds for its coefficients, inverse, and every fixed number of
derivatives in buffered charts. To construct it, take smooth product
-cutoffs supported in the signed boxes \((-2,2)^m\) and equal to one
-on \([-1,1]^m\), and average their Euclidean chart metrics with the
+cutoffs supported in the signed boxes ((-2,2)^m) and equal to one
+on ([-1,1]^m), and average their Euclidean chart metrics with the
normalized cutoff weights. The smaller boxes cover, by sorted flags.
The overlap bound above bounds the number of terms per plaque. The
transition formulas and their inverses are products of at most
-\(O(m)\) factors, and every required reciprocal at a noncommon cut
-is at most \(4^{O(m)}\). Their fixed-order derivatives consequently
+(O(m)) factors, and every required reciprocal at a noncommon cut
+is at most (4^{O(m)}). Their fixed-order derivatives consequently
have elementary bounds. These facts give the claimed bounds for the
average and its inverse, and make the construction Borel and
equivariant. The same transition and product estimates, applied to
(eq:transfer-tile-map), give an elementary upper bound for
-\(dX\) and its fixed-order derivatives before any choice of
-\(\lambda\). The affine simplexes have diameter at most \(r\sqrt m\).
+(dX) and its fixed-order derivatives before any choice of
+(lambda). The affine simplexes have diameter at most (rsqrt m).
-Put \(Θ=\prod_i x_i\). Choose a smooth \(j=j(Θ)\) equal
-to one for \(Θ\leq\gamma_0\), zero for
-\(Θ\geq2\gamma_0\), and between zero and one elsewhere, with
+Put (Θ=prod_i x_i). Choose a smooth (j=j(Θ)) equal
+to one for (Θleqgamma_0), zero for
+(Θgeq2gamma_0), and between zero and one elsewhere, with
the rescaled smooth cutoff estimates. It is smooth in the signed
charts and the same on both sides of every facet. Set
- k=X^*k_{\rm flat}+j^2 k_0,
- \qquad seam=\{Θ\leq2\gamma_0\}.
+ k=X^*k_{rm flat}+j^2 k_0,
+ qquad seam={Θleq2gamma_0}.
Call the complement of this closed seam the straight part.
-The metric is positive definite near the tile boundary because \(j=1\) there. Where
-\(j<1\), one has \(Θ>\gamma_0\), hence \(x_i\geq\gamma_0\) for
-every \(i\), and \(X\) is an immersion. The formula proves
-\(\|dX\|_k\leq1\) and equality of metrics off the seam.
+The metric is positive definite near the tile boundary because (j=1) there. Where
+(j<1), one has (Θ>gamma_0), hence (x_igeqgamma_0) for
+every (i), and (X) is an immersion. The formula proves
+(|dX|_kleq1) and equality of metrics off the seam.
The upper density bound requires no inverse estimates. As quadratic forms,
-\(k\leq X^*k_{\rm flat}+k_0\), so its volume density in every sorted
-box is bounded by the upper bounds for \(dX,k_0\) already chosen.
-Call a common elementary bound \(H_0(q)\). In a sorted box
-\(|Z_j|\leq1\). If all \(|Z_j|>\lambda\), normalization of the
+(kleq X^*k_{rm flat}+k_0), so its volume density in every sorted
+box is bounded by the upper bounds for (dX,k_0) already chosen.
+Call a common elementary bound (H_0(q)). In a sorted box
+(|Z_j|leq1). If all (|Z_j|>lambda), normalization of the
successive products gives
-\[
- Θ\geq(m+1)^{-(m+1)}
- \lambda^{2(1+2+...+m)}
- =(m+1)^{-(m+1)}\lambda^{m(m+1)}.
-\]
+[
+ Θgeq(m+1)^{-(m+1)}
+ lambda^{2(1+2+...+m)}
+ =(m+1)^{-(m+1)}lambda^{m(m+1)}.
+]
Condition (eq:transfer-gamma) shows that the seam lies in the
stated union of slabs. Its coordinate volume in such a box is at
-most an elementary factor times \(\lambda\). The bound \(H_0(q)\)
+most an elementary factor times (lambda). The bound (H_0(q))
was fixed before this factor was made small.
-The remaining metric estimates may depend on \(\gamma_0\). On the part
-with \(j<1\), all positive ratios are bounded below by powers of
-\(\gamma_0\), so the product formulas control the inverse from raw
+The remaining metric estimates may depend on (gamma_0). On the part
+with (j<1), all positive ratios are bounded below by powers of
+(gamma_0), so the product formulas control the inverse from raw
barycentric to signed coordinates. The normalized weights in
(eq:transfer-tile-map) obey
-\(b_i\geq e^{-1/\gamma_0}/(m+1)\). On a tangent vector
-\(w=(w_i)\), \(\sum_i w_i=0\), their derivative is
-\[
+(b_igeq e^{-1/gamma_0}/(m+1)). On a tangent vector
+(w=(w_i)), (sum_i w_i=0), their derivative is
+[
db_i(w)=b_i((w_i)/(x_i^2)
- -\sum_l b_l(w_l)/(x_l^2)).
-\]
-These differences control \(w\): if their common subtracted mean is
-\(a\), the relation \(\sum_iw_i=0\) expresses \(a\) as a weighted
-average, with weights \(x_i^2\), of the negatives of those differences.
-Thus \(|w|\) is bounded by a polynomial in \(m\) times
-\(e^{1/\gamma_0}\max_i|db_i(w)|\). Finally, the affine map from the
+ -sum_l b_l(w_l)/(x_l^2)).
+]
+These differences control (w): if their common subtracted mean is
+(a), the relation (sum_iw_i=0) expresses (a) as a weighted
+average, with weights (x_i^2), of the negatives of those differences.
+Thus (|w|) is bounded by a polynomial in (m) times
+(e^{1/gamma_0}max_i|db_i(w)|). Finally, the affine map from the
barycentric simplex to a simplex in a subdivided cube has elementary
-inverse bounds. Its matrix has \(r\)-scaled rational entries with
+inverse bounds. Its matrix has (r)-scaled rational entries with
bounded denominators, and its nonzero determinant and cofactors give
this by minors. These estimates control the inverse metric where
-\(j<1\); where \(j=1\), \(k_0\) already supplies that control.
-Differentiating the formula for \(k\) a fixed number of times now gives
+(j<1); where (j=1), (k_0) already supplies that control.
+Differentiating the formula for (k) a fixed number of times now gives
elementary bounds, including the curvature bound, when
-\(\gamma_0^{-1}\) is elementary.
+(gamma_0^{-1}) is elementary.
-\medskip\noindentFrame interpolation.
+medskipnoindentFrame interpolation.
The comparison needs a constant frame ordering tangent and normal axes
on each straight tile, a frame that glues smoothly near facets, and a homotopy to the
identity for the degree calculation. Choose a Borel skew-symmetric logarithm
-\(A_\sigma\) of each \(O_\sigma\) with operator norm at most \(\pi\).
+(A_sigma) of each (O_sigma) with operator norm at most (pi).
Such a choice follows by decomposing an orthogonal rotation into its
two-dimensional angle blocks with principal angles and, on its
-\(-1\)-eigenspace, pairing a Borel choice of orthonormal basis; the
+(-1)-eigenspace, pairing a Borel choice of orthonormal basis; the
even dimension there follows from determinant one. Define on that
tile
-\[
- \mathcal R=\exp((1-j)A_\sigma).
-\]
+[
+ mathcal R=calR=exp((1-j)A_sigma).
+]
It is the identity in a neighborhood of each facet and thus glues
-smoothly despite the change of \(\sigma\). It equals \(O_\sigma\)
-where \(j=0\), and replacing \(1-j\) by \(t(1-j)\), \(0\leq t\leq1\),
+smoothly despite the change of (sigma). It equals (O_sigma)
+where (j=0), and replacing (1-j) by (t(1-j)), (0leq tleq1),
gives the asserted homotopy. The cutoff and metric inverse estimates
give the derivative bounds. The choices can be made on the indexed
chamber data, so they are Borel and equivariant as asserted.
-For the present purpose, choose \(O_\sigma\) to take \(z\)-increments to rotated
-grid increments, putting the \(m\) axes tangent to the dual cube first
-and the \(n\) complementary axes last. Use a signed permutation if
-needed to have determinant one. Thus \(O_\sigma=Q_\sigma V\) for a
-signed permutation \(Q_\sigma\in SO(q)\) with this order. Signs will
+For the present purpose, choose (O_sigma) to take (z)-increments to rotated
+grid increments, putting the (m) axes tangent to the dual cube first
+and the (n) complementary axes last. Use a signed permutation if
+needed to have determinant one. Thus (O_sigma=Q_sigma V) for a
+signed permutation (Q_sigmain SO(q)) with this order. Signs will
not affect any row norm or any cubical test.
-The parameter selection below fixes \(\lambda(q)\) and \(\gamma_0(q)\)
+The parameter selection below fixes (lambda(q)) and (gamma_0(q))
deterministically. For those choices the bounds in
Lemma~lem:transfer-resolution give fixed elementary numerical
-envelopes \(K_R(q)\) and \(K_{\rm scal}(q)\) such that
-\[
- \|d\mathcal R\|_{∞,k}\leq K_R(q),\qquad
- (-\Scal_k)_+\leq K_{\rm scal}(q)
-\]
-for every allowed diagram and every \(V,s,\omega\), rounding parameter,
+envelopes (K_R(q)) and (K_{rm scal}(q)) such that
+[
+ |dmathcal RcalR|_{∞,k}leq K_R(q),qquad
+ (-Scal_k)_+leq K_{rm scal}(q)
+]
+for every allowed diagram and every (V,s,omega), rounding parameter,
and state. These envelopes come from the uniform chart derivative
-bounds and do not use \(C_{\rm ent}\). Fix the single number
+bounds and do not use (C_{rm ent}). Fix the single number
- \tau=\tau(q)=
- \frac{1}{300\sqrt q\max\{r^{-1},\delta^{-1},1+K_R(q)\}}.
+ tau=tau(q)=
+ frac{1}{300sqrt qmax{r^{-1},delta^{-1},1+K_R(q)}}.
It has an elementary reciprocal and is common to all those parameters.
In particular
-\(3\tau\sqrt q\leq\min\{r/100,\delta/100,(1+K_R(q))^{-1}\}\).
-
-Fix \(V,s\) and the rounding parameter for the moment. Take the
-product of \(\widetilde M\) and the tile family \(\mathcal A\) over
-their shared \(\omega\), and divide by the diagonal \(\Gamma\)-action.
-Orient the product with the \(\widetilde M\) directions first and the
-\(\mathcal A\) directions second. The resulting measured leaves have
-dimension \(n+m=q\). Using the leaf-open set
-\(\mathcal A_{\rm orig}\) from Lemma~lem:transfer-resolution,
-work only on the open set of pairs \((x,a)\) with
-\(a\in\mathcal A_{\rm orig}\) for which \(X(a)\) is in the lift of the ball chart of
-\(F(x)\), at distance less than \(\delta/4\). Define there
-
- G(x,a)=\mathcal R(a)(z_F(x)-z_{X(a)}).
-
-Here \(z_{X(a)}=z\circ X(a)\). The requirement on the lifts is part of
+(3tausqrt qleqmin{r/100,delta/100,(1+K_R(q))^{-1}}).
+
+Fix (V,s) and the rounding parameter for the moment. Take the
+product of (widetilde M) and the tile family (mathcal AcalA) over
+their shared (omega), and divide by the diagonal (Gamma)-action.
+Orient the product with the (widetilde M) directions first and the
+(mathcal AcalA) directions second. The resulting measured leaves have
+dimension (n+m=q). Using the leaf-open set
+(mathcal A_calA_{rm orig}) from Lemma~lem:transfer-resolution,
+work only on the open set of pairs ((x,a)) with
+(ainmathcal A_calA_{rm orig}) for which (X(a)) is in the lift of the ball chart of
+(F(x)), at distance less than (delta/4). Define there
+
+ G(x,a)=mathcal R(a)(z_F(x)-z_calR(a)(z_F(x)-z_{X(a)}).
+
+Here (z_{X(a)}=zcirc X(a)). The requirement on the lifts is part of
the domain; coincident color coordinates on incompatible sheets are
never compared. The map is invariant under the diagonal action.
-Use the band \(\{|G|_∞<3\tau\}\) in that open domain, and put
-\(b_q=\tau(1+3/(4q))\), the inner width in Theorem~thm:degree
-when \(d=q\). Let \(\mathcal D\) be its measured degree.
+Use the band ({|G|_∞<3tau}) in that open domain, and put
+(b_q=tau(1+3/(4q))), the inner width in Theorem~thm:degree
+when (d=q). Let (mathcal DcalD) be its measured degree.
[Comparison degree]
The band in (eq:transfer-comparison) satisfies the chart and
integration control assumptions of Theorem~thm:degree, for each
-fixed instance, the product metric \(h+L^2k\) with any fixed \(L>0\),
-and the pullback of the entropy weight \(\phi\). Its degree is Borel
+fixed instance, the product metric (h+L^2k) with any fixed (L>0),
+and the pullback of the entropy weight (phi). Its degree is Borel
in the auxiliary parameters and satisfies
- \E_{\rm rd}\mathcal D=D_0I
+ E_{rm rd}mathcal D=D_0IcalD=D_0I
-for every fixed \(V,s\).
+for every fixed (V,s).
Proof.
The local pushforward (eq:transfer-local-push) will express the
-degree using a smoothed form of \(Z_{\rm rd}\). Its primal-cell periods
-are the integer coefficients \(D_0u'_C\). Their expectations match the periods of
-\(D_0\eta\); a compatible piecewise cubical primitive and mollification
-then give the same measured pairing with \(F\). First verify the
+degree using a smoothed form of (Z_{rm rd}). Its primal-cell periods
+are the integer coefficients (D_0u'_C). Their expectations match the periods of
+(D_0eta); a compatible piecewise cubical primitive and mollification
+then give the same measured pairing with (F). First verify the
band control that makes these integrations valid.
The controlled comparison band.
Before taking the quotient, the product flow boxes use the entropy
-transversal \(\omega\) and the
-state transversals of \(\mathcal A\). Variation along the tile keeps
-\(\omega\) constant. These transversals carry the invariant measures
+transversal (omega) and the
+state transversals of (mathcal AcalA). Variation along the tile keeps
+(omega) constant. These transversals carry the invariant measures
constructed above. A group element that stabilizes a product leaf
-acts freely and properly because it acts so on the \(\widetilde M\)
+acts freely and properly because it acts so on the (widetilde M)
factor. Hence the quotient leaves are ordinary oriented Hausdorff
second countable manifolds. The actions reindex labels and states,
leave the color and flag coordinates unchanged, and preserve the
transverse measures and orientations.
For an inner closed comparison cube there is a uniform margin from
-the boundary of the allowed sheet domain. Indeed \(\mathcal R\) is
+the boundary of the allowed sheet domain. Indeed (mathcal RcalR) is
orthogonal and (eq:transfer-tau) gives
-\(|z_F-z_X|_2\leq3\tau\sqrt q\ll\delta/4\). The large witness
-color at \(F\) remains present with a uniform small buffer. In a
+(|z_F-z_X|_2leq3tausqrt qlldelta/4). The large witness
+color at (F) remains present with a uniform small buffer. In a
flat ball chart a lifted path whose projected coordinates stay in
the ball stays in that same lift, by uniqueness of lifts for a local
diffeomorphism. The nearby points therefore remain in coherent
@@ -6077,7 +6077,7 @@ The available smooth bounds for the instance give fixed buffered
coordinate charts for all the data in these regions.
To check the finiteness condition on those charts, choose lifts of a
-finite buffered atlas on \(M\). On their compact closures \(F\) uses
+finite buffered atlas on (M). On their compact closures (F) uses
only uniformly finitely many possible labels, by the lifted support
bounds in Theorem~thm:entropy. Any nearby cell lift has its
anchor in a closed one-neighborhood of one of these labels. The
@@ -6095,91 +6095,91 @@ metric and data bounds in this verification may depend on the
instance. It also justifies Fubini and measured Stokes for the
integrations that follow.
-\medskip\noindentThe local convolution form.
-Choose a smooth unit-integral top form \(\beta\) on \(\mathbb R^q\)
+medskipnoindentThe local convolution form.
+Choose a smooth unit-integral top form (beta) on (mathbb R^qbbR^q)
supported in a Euclidean ball about zero of radius less than
-\(\min\{\tau,r/10\}\). In calculating the degree using \(\beta\),
-homotope \(\mathcal R\) to the identity as in
+(min{tau,r/10}). In calculating the degree using (beta),
+homotope (mathcal RcalR) to the identity as in
Lemma~lem:transfer-resolution. All maps in this homotopy are
-orthogonal on the difference variable. Their pullbacks of \(\beta\)
+orthogonal on the difference variable. Their pullbacks of (beta)
are thus supported where the difference has this same small norm,
strictly inside both the sheet domain and the original band. Measured
Stokes for this supported homotopy shows that it does not change the
degree.
After removing the rotation, fiber integration of
-\((z,w)\mapsto z-w\) against \(\beta\) and the original current
-\(Z_{\rm rd}\) gives a smooth \(n\)-form \(\alpha_{\rm rd}\) near \(K\).
-More explicitly, in a flat chart put \(\mathsf d(z,w)=z-w\). Integrate
-\(\mathsf d^*\beta\) along each compatible \(C^\#\) in the second
-variable, multiply by \(D_0u'_C\), and sum. The sum is locally finite
+((z,w)mapsto z-w) against (beta) and the original current
+(Z_{rm rd}) gives a smooth (n)-form (alpha_{rm rd}) near (K).
+More explicitly, in a flat chart put (mathsf d(z,w)=z-wsfd(z,w)=z-w). Integrate
+(mathsf d^*sfd^*beta) along each compatible (C^#) in the second
+variable, multiply by (D_0u'_C), and sum. The sum is locally finite
and uses the original side. By the local identity
(eq:transfer-local-push), Fubini, and change of variables on tile
interiors, the degree is
- \mathcal D=\int_{\mathcal M}F^*\alpha_{\rm rd}.
+ mathcal D=calD=int_{mathcal M}calMF^*alpha_{rm rd}.
Boundary cancellation of the dual cubes shows that
-\(d\alpha_{\rm rd}=0\) on, for example, the tube of radius
-\(\delta/8\) around \(K\). To verify this directly, apply fiber
+(dalpha_{rm rd}=0) on, for example, the tube of radius
+(delta/8) around (K). To verify this directly, apply fiber
Stokes to each cube. Its boundary terms only involve facets within
-the support distance of the point \(z\); these lie in the balanced
+the support distance of the point (z); these lie in the balanced
tube and cancel with their paired incidence signs.
-For every primal \(n\)-cell sufficiently near \(K\) in the same
+For every primal (n)-cell sufficiently near (K) in the same
sheet,
- \int_C\alpha_{\rm rd}=D_0u'_C.
+ int_Calpha_{rm rd}=D_0u'_C.
-Indeed a primal \(n\)-cell and a complementary dual \(m\)-cell of
+Indeed a primal (n)-cell and a complementary dual (m)-cell of
this grid meet only when their indices match, and then at their common
center. If their axis sets differ, a coordinate fixed for both is a
grid vertex for the primal cell and a half-step center for the dual
-cell, so their difference in that coordinate is at least \(r/2\).
+cell, so their difference in that coordinate is at least (r/2).
If their axis sets agree but their grid indices differ, a differing
coordinate gives the same separation. Such pairs cannot meet the
-support of \(\beta\). For the matching pair, the difference map
-from \(C\times C^\#\) contains the support ball in its image with
+support of (beta). For the matching pair, the difference map
+from (Ctimes C^#) contains the support ball in its image with
multiplicity one. Our fiber integration convention is
-\(\int_C\int_{C^\#}\mathsf d^*\beta=
-\int_{C\times C^\#}\mathsf d^*\beta\) with the base-first product
-orientation. The differential of \(\mathsf d(z,w)=z-w\) sends this
+(int_Cint_{C^#}mathsf d^*sfd^*beta=
+int_{Ctimes C^#}mathsf d^*sfd^*beta) with the base-first product
+orientation. The differential of (mathsf d(z,w)=z-wsfd(z,w)=z-w) sends this
ordered frame to the primal basis followed by the negative dual basis,
so its orientation is positive by the choice of the dual orientation.
Its integral is therefore
one, proving (eq:transfer-convolution-cells). The small support
and the chart buffers ensure that only the indicated lifts contribute.
-\medskip\noindentThe detector pairing.
-Set \(\alpha=\E_{\rm rd}\alpha_{\rm rd}-D_0\eta\). It is a smooth
+medskipnoindentThe detector pairing.
+Set (alpha=E_{rm rd}alpha_{rm rd}-D_0eta). It is a smooth
closed form on that tube, and (eq:transfer-rounding),
(eq:transfer-cell-value), and
-(eq:transfer-convolution-cells) give \(\int_C\alpha=0\) on all
-the primal \(n\)-cells there. Explain why its measured pairing
-with \(F\) vanishes, including when \(F\) meets grid faces.
-
-Use all lifted full \(q\)-cells needed to cover the tube of radius
-\(\delta/32\) around \(K\). Taking \(r\sqrt q\) sufficiently small
-as above, these cells lie in the tube of radius \(\delta/8\), all
-their \(n\)-faces are selected, and their shared faces fit in
-coherent sheet charts. On one grid interval \([a,a+r]\), project a
+(eq:transfer-convolution-cells) give (int_Calpha=0) on all
+the primal (n)-cells there. Explain why its measured pairing
+with (F) vanishes, including when (F) meets grid faces.
+
+Use all lifted full (q)-cells needed to cover the tube of radius
+(delta/32) around (K). Taking (rsqrt q) sufficiently small
+as above, these cells lie in the tube of radius (delta/8), all
+their (n)-faces are selected, and their shared faces fit in
+coherent sheet charts. On one grid interval ([a,a+r]), project a
function to its linear interpolation between the endpoints and
-project a one-form \(f(t)dt\) to the constant one-form with the same
+project a one-form (f(t)dt) to the constant one-form with the same
integral. The homotopy for a one-form is the function
-\[
- \mathsf h(f(t)dt)(t)=\int_a^t
- (f(v)-r^{-1}\int_a^{a+r}f(s)\,ds)dv,
-\]
+[
+ mathsf h(f(t)dt)(t)=sfh(f(t)dt)(t)=int_a^t
+ (f(v)-r^{-1}int_a^{a+r}f(s),ds)dv,
+]
which vanishes at both endpoints; the homotopy on functions is zero.
-Direct differentiation gives \(id-\mathsf P
-=d\mathsf h+\mathsf h d\). Apply these operations one coordinate
+Direct differentiation gives (id-mathsf PsfP
+=dmathsf h+sfh+mathsf hsfh d). Apply these operations one coordinate
at a time, with the ordinary exterior differential signs. The
tensor-product homotopy still commutes with tangential restriction
to a face, since the one-dimensional primitive vanishes at the
-endpoints. The full projection of an \(n\)-form has, for each
-axis set, coefficients obtained by integrating on its \(n\)-faces
+endpoints. The full projection of an (n)-form has, for each
+axis set, coefficients obtained by integrating on its (n)-faces
and interpolating linearly in the remaining axes. Hence the full
-projection of \(\alpha\) is zero. Since \(d\alpha=0\), the
+projection of (alpha) is zero. Since (dalpha=0), the
homotopy supplies a primitive on each full cube, and its tangential
restrictions agree on shared faces.
@@ -6191,11 +6191,11 @@ This can be done with the same small Euclidean convolution in every
chart: the buffered sheets give a unique lift of the convolution ball,
and on overlaps the piecewise primitive and flat coordinates agree.
Its smooth exterior derivatives are the mollifications of the smooth
-form \(\alpha\), and converge to \(\alpha\) there. The construction
+form (alpha), and converge to (alpha) there. The construction
is Borel and equivariant and has bounded coefficients for the instance.
Measured Stokes applied to the pullback of each smooth primitive gives
zero; convergence and the uniform chart bounds give
-\(\int_{\mathcal M}F^*\alpha=0\). All expectations and integrals
+(int_{mathcal M}calMF^*alpha=0). All expectations and integrals
above are measurable by their locally finite chart formulas, and
their absolute bounds for the instance justify their interchange.
Using (eq:transfer-form) in
@@ -6205,218 +6205,218 @@ Using (eq:transfer-form) in
Finish by estimating these degrees. It will be seen separately how
-the straight part uses the dimension-only Hilbert norm of \(\eta\)
+the straight part uses the dimension-only Hilbert norm of (eta)
and how the small coordinate volume of the seam pays for its rougher
bound.
Proof. [Proof of Theorem~thm:scalar-volume]
-The case \(\|M\|=0\) is immediate. Otherwise, first prove the
-conditional target for the data above, choosing \(\Psi_n\) from the
-requirements on \(E\) derived below, and then apply it to the entropy
-output. Choose a positive constant \(b_*\), depending only on \(n\),
+The case (|M|=0) is immediate. Otherwise, first prove the
+conditional target for the data above, choosing (Psi_n) from the
+requirements on (E) derived below, and then apply it to the entropy
+output. Choose a positive constant (b_*), depending only on (n),
whose smallness will be fixed in the estimate below, and put
- L_T=b_*\sqrt E,\qquad L_N=b_*√(qE),\qquad
- h_{\rm tot}=h+L_T^2k.
+ L_T=b_*sqrt E,qquad L_N=b_*√(qE),qquad
+ h_{rm tot}=h+L_T^2k.
Once the degree hypotheses are verified, its length denominator, the
product volume, and the entropy contribution will combine as
- \frac{L_T^mE^{n/2}}{L_T^mL_N^n}\,e^{-W}dV_h
- =b_*^{-n}q^{-n/2}\rho.
+ frac{L_T^mE^{n/2}}{L_T^mL_N^n},e^{-W}dV_h
+ =b_*^{-n}q^{-n/2}rho.
This is the cancellation of the tangent volume and all powers of
-\(E\). The estimates below justify the degree application and bound
-the straight angular sum by a dimensional multiple of \(q^{n/2}\),
+(E). The estimates below justify the degree application and bound
+the straight angular sum by a dimensional multiple of (q^{n/2}),
while the collar volume controls the seam.
-Use the weight \(\phi\), constant in the tile directions. The
-product scalar formula and the definition of \(\wscal\) give
-\(\wscal(h_{\rm tot},\phi)=\wscal(h,\phi)+L_T^{-2}\Scal_k\).
-By the definition of \(K_{\rm scal}(q)\), the lower bound
-\(E\geq b_*^{-2}K_{\rm scal}(q)\) ensures
-\(L_T^{-2}\Scal_k\geq-1\). By (eq:transfer-entropy),
+Use the weight (phi), constant in the tile directions. The
+product scalar formula and the definition of (wscal) give
+(wscal(h_{rm tot},phi)=wscal(h,phi)+L_T^{-2}Scal_k).
+By the definition of (K_{rm scal}(q)), the lower bound
+(Egeq b_*^{-2}K_{rm scal}(q)) ensures
+(L_T^{-2}Scal_kgeq-1). By (eq:transfer-entropy),
- \wscal(h_{\rm tot},\phi)\geq-C_{\rm ent}-1+2W-n\log E+E\mathfrak e.
+ wscal(h_{rm tot},phi)geq-C_{rm ent}-1+2W-nlog E+Emathfrak e.frake.
-Assign \(L_i=L_T\) to the first \(m\) comparison rows and
-\(L_i=L_N\) to the last \(n\) rows. On the band, the derivative in
-tile directions of each row of \(G\), measured with \(k\), has norm
-at most two. The \(dX\) term has norm at most one, and the
-\(d\mathcal R\) term has norm at most
-\(3\tau\sqrt q\|d\mathcal R\|_k\leq1\) by
+Assign (L_i=L_T) to the first (m) comparison rows and
+(L_i=L_N) to the last (n) rows. On the band, the derivative in
+tile directions of each row of (G), measured with (k), has norm
+at most two. The (dX) term has norm at most one, and the
+(dmathcal RcalR) term has norm at most
+(3tausqrt q|dmathcal RcalR|_kleq1) by
(eq:transfer-tau). The horizontal derivative is a row of a
-rotation of \(dz_F\). It follows that
-\[
- L_i|dG_i|_{h_{\rm tot}}
- \leq C_n(\sqrt q+√(qE)\,E^{-1/100}).
-\]
-Take \(B\) to be this upper bound increased by two. Then
-\(L_i|dG_i|\leq B-1=B-q/q\). The horizon condition of
-Theorem~thm:degree holds after an elementary increase of \(E\):
-it suffices to test the smallest length \(L_T\), and, for \(E\geq1\),
-\[
- \frac{C(1+B)^{2/3}}{L_T}
- \leq c_{{\rm hor},n}q^{1/3}E^{-13/75}
- <(\tau)/(10q).
-\]
-Here \(c_{{\rm hor},n}\) depends only on the absolute horizon constant
-and the dimensional choice of \(b_*\), and
-\(13/75=1/2-(2/3)(1/2-1/100)>0\). Thus the displayed strict horizon
+rotation of (dz_F). It follows that
+[
+ L_i|dG_i|_{h_{rm tot}}
+ leq C_n(sqrt q+√(qE),E^{-1/100}).
+]
+Take (B) to be this upper bound increased by two. Then
+(L_i|dG_i|leq B-1=B-q/q). The horizon condition of
+Theorem~thm:degree holds after an elementary increase of (E):
+it suffices to test the smallest length (L_T), and, for (Egeq1),
+[
+ frac{C(1+B)^{2/3}}{L_T}
+ leq c_{{rm hor},n}q^{1/3}E^{-13/75}
+ <(tau)/(10q).
+]
+Here (c_{{rm hor},n}) depends only on the absolute horizon constant
+and the dimensional choice of (b_*), and
+(13/75=1/2-(2/3)(1/2-1/100)>0). Thus the displayed strict horizon
inequality follows if
-\(E\geq(20c_{{\rm hor},n}q^{4/3}/\tau)^{75/13}\).
+(Egeq(20c_{{rm hor},n}q^{4/3}/tau)^{75/13}).
Therefore apply Theorem~thm:degree
-with \(d=q\).
-
-Write \(H_i=2(L_i|dG_i|-1)_+\), as in that theorem. At a straight
-point in a tile, let \(\mathcal I\) be the \(n\)-set of grid axes
-normal to its dual cube. Put \(A_i=(Vdz_F)_i\), with covector norms
-measured by \(h\), and define
-
- U_{\mathcal I}^2=
- \frac q{\mathfrak e}\sum_{i\in\mathcal I}|A_i|_h^2
- \quadif \mathfrak e>0,
- \qquad U_{\mathcal I}=0\quadif \mathfrak e=0.
-
-Here \(\sum_i|A_i|^2=\mathfrak e\). At a straight point
-\(k=X^*k_{\rm flat}\) and \(\mathcal R=Q_\sigma V\) is constant.
-For a tangent comparison row corresponding to axis \(i\), the squared
-row norm is exactly \(L_T^{-2}+|A_i|^2\). Thus
-\[
- \sum_{\rm tangent}H_i
- \leq L_T^2\mathfrak e=b_*^2E\mathfrak e,
-\]
-by \(2(√(1+t)-1)\leq t\). Normal rows have no tile derivative,
+with (d=q).
+
+Write (H_i=2(L_i|dG_i|-1)_+), as in that theorem. At a straight
+point in a tile, let (mathcal IcalI) be the (n)-set of grid axes
+normal to its dual cube. Put (A_i=(Vdz_F)_i), with covector norms
+measured by (h), and define
+
+ U_{mathcal I}calI^2=
+ frac q{mathfrak e}frakesum_{iinmathcal I}calI|A_i|_h^2
+ quadif mathfrak e>0,frake>0,
+ qquad U_{mathcal I}calI=0quadif mathfrak e=0.frake=0.
+
+Here (sum_i|A_i|^2=mathfrak efrake). At a straight point
+(k=X^*k_{rm flat}) and (mathcal R=Q_calR=Q_sigma V) is constant.
+For a tangent comparison row corresponding to axis (i), the squared
+row norm is exactly (L_T^{-2}+|A_i|^2). Thus
+[
+ sum_{rm tangent}H_i
+ leq L_T^2mathfrak e=b_*^2Efrake=b_*^2Emathfrak e,frake,
+]
+by (2(√(1+t)-1)leq t). Normal rows have no tile derivative,
so Cauchy--Schwarz gives
-\[
- \sum_{\rm normal}H_i
- \leq2L_N\sum_{i\in\mathcal I}|A_i|
- \leq2b_*\sqrt n√(E\mathfrak e)\,U_{\mathcal I}.
-\]
-Completing the square in \(√(E\mathfrak e)\), for \(b_*<1\),
+[
+ sum_{rm normal}H_i
+ leq2L_Nsum_{iinmathcal I}calI|A_i|
+ leq2b_*sqrt n√(Emathfrak e)frake),U_{mathcal I}calI.
+]
+Completing the square in (√(Emathfrak e)frake)), for (b_*<1),
shows that
- (\sum_iH_i-E\mathfrak e)/(2)
- \leq c_n U_{\mathcal I}^2,
- \qquad c_n=(n b_*^2)/(2(1-b_*^2)).
+ (sum_iH_i-Emathfrak e)/(2)frake)/(2)
+ leq c_n U_{mathcal I}calI^2,
+ qquad c_n=(n b_*^2)/(2(1-b_*^2)).
-Choose \(b_*\) sufficiently small that
+Choose (b_*) sufficiently small that
- \E_V\exp(2c_nU_{\mathcal I}^2)\leq C_n
+ E_Vexp(2c_nU_{mathcal I}calI^2)leq C_n
for every fixed horizontal differential and every fixed
-\(\mathcal I\). Here is a direct verification of this choice. If
-\(u\) is one coordinate of a uniform point on \(S^{q-1}\), its sphere
+(mathcal IcalI). Here is a direct verification of this choice. If
+(u) is one coordinate of a uniform point on (S^{q-1}), its sphere
moments satisfy
-\[
- \E(q u^2)^j
- =(q^j(1\cdot3...(2j-1)))/(q(q+2)...(q+2j-2))
- \leq2^j j!.
-\]
-Consequently \(\E e^{tqu^2}\leq(1-2t)^{-1}\) for \(0\leq t<1/2\).
-The singular-value decomposition of \(dz_F\) expresses each
-\(q|A_i|^2/\mathfrak e\) as a convex combination of such random
+[
+ E(q u^2)^j
+ =(q^j(1cdot3...(2j-1)))/(q(q+2)...(q+2j-2))
+ leq2^j j!.
+]
+Consequently (E e^{tqu^2}leq(1-2t)^{-1}) for (0leq t<1/2).
+The singular-value decomposition of (dz_F) expresses each
+(q|A_i|^2/mathfrak efrake) as a convex combination of such random
variables, with its singular-value weights. Jensen's inequality
gives the same exponential bound for that convex combination.
-Another application of Jensen to the sum of the \(n\) rows in
+Another application of Jensen to the sum of the (n) rows in
(eq:transfer-U) bounds its exponential moment by
-\((1-2nt)^{-1}\). Choosing, for instance, \(2nc_n<1/4\) proves
-(eq:transfer-sphere). No independence of the rows of \(V\)
+((1-2nt)^{-1}). Choosing, for instance, (2nc_n<1/4) proves
+(eq:transfer-sphere). No independence of the rows of (V)
is used.
At a seam point the uniform tile row bound gives a rougher estimate
which is still independent of the instance. By the triangle inequality
-for each row, \((a-1)_+\leq a\), and Cauchy--Schwarz on the horizontal
+for each row, ((a-1)_+leq a), and Cauchy--Schwarz on the horizontal
rows,
-\[
- \sum_iH_i\leq 2L_N√(q\mathfrak e)
+[
+ sum_iH_ileq 2L_N√(qmathfrak e)frake)
+4q L_N/L_T
- \leq2b_*q√(E\mathfrak e)+4q^{3/2}.
-\]
+ leq2b_*q√(Emathfrak e)+4q^frake)+4q^{3/2}.
+]
Completing the square once more gives
- (\sum_iH_i-E\mathfrak e)/(2)\leq c_{{\rm seam},n}q^2,
- \qquad c_{{\rm seam},n}=2+(b_*^2)/(2).
+ (sum_iH_i-Emathfrak e)/(2)frake)/(2)leq c_{{rm seam},n}q^2,
+ qquad c_{{rm seam},n}=2+(b_*^2)/(2).
Apply the degree inequality, then use
-(eq:transfer-total-scalar). Its loss is \(\exp(Cq/q)\), an
+(eq:transfer-total-scalar). Its loss is (exp(Cq/q)), an
absolute constant. The product volume form is
-\(dV_{h_{\rm tot}}=L_T^m dV_h\,dV_k\), so the measure factor is
+(dV_{h_{rm tot}}=L_T^m dV_h,dV_k), so the measure factor is
exactly (eq:transfer-prefactor). Together with
-(eq:transfer-degree) and \(|\E\mathcal D|\leq\E|\mathcal D|\),
+(eq:transfer-degree) and (|Emathcal D|calD|leqE|mathcal D|calD|),
this gives the following inequality, with
-\(C_{\rm out}=\exp(C_{\rm deg}+(C_{\rm ent}+1)/2)b_*^{-n}\), where
-\(C_{\rm deg}\) is the absolute exponent in Theorem~thm:degree:
+(C_{rm out}=exp(C_{rm deg}+(C_{rm ent}+1)/2)b_*^{-n}), where
+(C_{rm deg}) is the absolute exponent in Theorem~thm:degree:
- D_0I(2\tau)^q\leq C_{\rm out}q^{-n/2}
- \int_{\mathcal M}\rho\,
- \E_{V,s,\rm rd}
- \int_{\{a:\ |G(x,a)|_∞<b_q\}}
- \mathcal B(x,a)\,dV_k(a),
+ D_0I(2tau)^qleq C_{rm out}q^{-n/2}
+ int_{mathcal M}calMrho,
+ E_{V,s,rm rd}
+ int_{{a: |G(x,a)|_∞<b_q}}
+ mathcal B(x,a)calB(x,a),dV_k(a),
where the fiber integral includes its state transversals and only the
compatible original-side band, and
-\[
- \mathcal B(x,a)=
+[
+ mathcal B(x,a)=calB(x,a)=
- \exp(c_nU_{\mathcal I}^2),&a straight over axes \mathcal I,\\
- \exp(c_{{\rm seam},n}q^2),&a in the seam.
+ exp(c_nU_{mathcal I}calI^2),&a straight over axes mathcal I,calI,
+ exp(c_{{rm seam},n}q^2),&a in the seam.
-\]
+]
The entropy constant enters only through this final prefactor, not the
geometric bounds used below. The length and volume prefactor has no
-exponential loss in \(m\); the seam exponential will be absorbed by
-the choice of \(\lambda\).
+exponential loss in (m); the seam exponential will be absorbed by
+the choice of (lambda).
-Fix \(x,\omega\), and put \(y=F(x)\). At a coupled pair in
+Fix (x,omega), and put (y=F(x)). At a coupled pair in
(eq:transfer-average), a fixed projected grid index has at most
-one possible cell lift in the ball chart of \(y\). Its full primal
+one possible cell lift in the ball chart of (y). Its full primal
and dual cells fit in that chart by (eq:transfer-tau) and the
buffers, and the lift there is unique. Moreover each point of its
-primal cell is within \(10r\sqrt q\) of \(y\): the dual point is
-within \(b_q\sqrt q\) of \(y\), and the dual and primal extents are
-at most \(r\sqrt m/2\) and \(r\sqrt n/2\). The variation choice
-for \(r\) and the rounding error now give the copy-count estimate
+primal cell is within (10rsqrt q) of (y): the dual point is
+within (b_qsqrt q) of (y), and the dual and primal extents are
+at most (rsqrt m/2) and (rsqrt n/2). The variation choice
+for (r) and the rounding error now give the copy-count estimate
- D_0|u'_C|\leq D_0r^n
- (|\eta_{\mathcal I}(y)|+2q^{-n}),
+ D_0|u'_C|leq D_0r^n
+ (|eta_{mathcal I}calI(y)|+2q^{-n}),
-where the component is in the \(V\)-rotated frame.
+where the component is in the (V)-rotated frame.
-On a straight tile the \(k\)-volume is exactly the flat volume of
-its image in the chamber. The signed permutation in \(G\) does not
+On a straight tile the (k)-volume is exactly the flat volume of
+its image in the chamber. The signed permutation in (G) does not
change its cube test, which is
-\(|\widehat z_y-\widehat z_X|_∞<b_q\). Summing the chamber
-pieces in a copy is thus bounded by the \(m\)-area of its whole dual
+(|widehatwidehat z_y-z_y-widehatwidehat z_X|_∞<b_qz_X|_∞<b_q). Summing the chamber
+pieces in a copy is thus bounded by the (m)-area of its whole dual
cube inside that test; the state probabilities for each chamber
-integrate to one. For a fixed axis set \(\mathcal I\), summing over
+integrate to one. For a fixed axis set (mathcal IcalI), summing over
all projected centers and averaging the shift gives exactly
- r^{-q}\int_{\mathbb R^q}\int_{[-r/2,r/2]^m}
- \indic_{\{|\widehat z_y-c-t|_∞<b_q\}}\,dt\,dc
+ r^{-q}int_{mathbb R^q}bbR^qint_{[-r/2,r/2]^m}
+ indic_{{|widehatwidehat z_y-c-t|_∞<b_qz_y-c-t|_∞<b_q}},dt,dc
=r^{-q}r^m(2b_q)^q.
-Here \(t\) is placed in the complementary axes. This identity is
-just the density \(r^{-q}\) of centers under uniform shift; extending
+Here (t) is placed in the complementary axes. This identity is
+just the density (r^{-q}) of centers under uniform shift; extending
the sum to the full grid is an upper bound for the selected lifted
cells.
The form bound holds in every rotated frame. Using Cauchy--Schwarz
-jointly in \(V\) and the \(\binom qn\) axis sets, and then
+jointly in (V) and the (binom qn) axis sets, and then
(eq:transfer-sphere), one obtains
- \E_V\sum_{|\mathcal I|=n}
- (|\eta_{\mathcal I}(y)|+2q^{-n})e^{c_nU_{\mathcal I}^2}
- &\leq
- (\E_V\sum_{\mathcal I}|\eta_{\mathcal I}(y)|^2)^{1/2}
- (\E_V\sum_{\mathcal I}e^{2c_nU_{\mathcal I}^2})^{1/2}
- +C_nq^{-n}\binom qn\\
- &\leq C_nq^{n/2}.
+ E_Vsum_{|mathcal I|=n}calI|=n
+ (|eta_{mathcal I}calI(y)|+2q^{-n})e^{c_nU_{mathcal I}calI^2}
+ &leq
+ (E_Vsum_{mathcal I}calI|eta_{mathcal I}calI(y)|^2)^{1/2}
+ (E_Vsum_{mathcal I}calIe^{2c_nU_{mathcal I}calI^2})^{1/2}
+ +C_nq^{-n}binom qn
+ &leq C_nq^{n/2}.
This use of Cauchy--Schwarz allows the form components and row norms
@@ -6424,230 +6424,230 @@ to depend on the same rotation. Combining
(eq:transfer-copies)--(eq:transfer-form-average), the
straight part of the inner expectation in
(eq:transfer-average) is at most
-\(C_nD_0q^{n/2}(2b_q)^q\), because \(r^{n+m-q}=1\). Multiplying
-by its prefactor \(q^{-n/2}\) bounds that part, for a fixed
-dimension-only constant \(C_{{\rm str},n}\), by
+(C_nD_0q^{n/2}(2b_q)^q), because (r^{n+m-q}=1). Multiplying
+by its prefactor (q^{-n/2}) bounds that part, for a fixed
+dimension-only constant (C_{{rm str},n}), by
- C_{{\rm str},n}D_0(2\tau)^q,
+ C_{{rm str},n}D_0(2tau)^q,
-since \((b_q/\tau)^q=(1+3/(4q))^q\leq e^{3/4}\).
+since ((b_q/tau)^q=(1+3/(4q))^qleq e^{3/4}).
For the seam use only the consequence
-\(|z_X-z_y|_2\leq b_q\sqrt q\) of the cube test. Discard the tile
+(|z_X-z_y|_2leq b_qsqrt q) of the cube test. Discard the tile
boundaries, which have leafwise measure zero. The remaining set is
the disjoint union of tile interiors indexed by their actual
-chamber \(\sigma\) and its state arrays. Partition each such interior
+chamber (sigma) and its state arrays. Partition each such interior
into positive sorted-coordinate pieces, for example by taking the
first sorted permutation in a fixed order. On these pieces
-\(0<Z_j\leq1\), so no signed-chart reflection changes the chamber.
+(0<Z_jleq1), so no signed-chart reflection changes the chamber.
The conditional state measure over each actual chamber has mass one,
-and its \(k\)-density is at most the elementary \(H_0(q)\) chosen
-before \(\lambda\).
+and its (k)-density is at most the elementary (H_0(q)) chosen
+before (lambda).
Retain the compatible-sheet restriction at this point. For a fixed
projected grid index the uniqueness proved before
(eq:transfer-copies) allows at most one compatible geometric
lift. Apply the rough copy bound
-\(D_0|u'_C|\leq C_{\eta,n}D_0r^n\) from that equation, where
-\(C_{\eta,n}\) depends only on the form norm, and use the uniform
+(D_0|u'_C|leq C_{eta,n}D_0r^n) from that equation, where
+(C_{eta,n}) depends only on the form norm, and use the uniform
density bound. Only after these bounds may the restrictions be dropped
on available sheets, roundings, and chambers and sum over all projected
centers and chamber shapes.
-For a fixed actual chamber shape, normal axis set \(\mathcal I\), and
+For a fixed actual chamber shape, normal axis set (mathcal IcalI), and
positive sorted flag, (eq:transfer-tile-map) has rotated position
-\[
- \widehat z_X=s+r(k+\tfrac12\indic_{\mathcal I})
- +d_{\mathcal I,shape,flag}(Z).
-\]
-Here \(\indic_{\mathcal I}\) is the coordinate indicator vector of
-\(\mathcal I\).
-The displacement \(d\) is independent of the pairing and bit arrays,
-of \(\lambda,\gamma_0\), and of the shift; it is determined by the
-fixed chamber shape and the barycentric formula. For each fixed \(Z\)
+[
+ widehatwidehat z_X=s+r(k+z_X=s+r(k+tfrac12indic_{mathcal I}calI)
+ +d_{mathcal I,shape,flag}calI,shape,flag(Z).
+]
+Here (indic_{mathcal I}calI) is the coordinate indicator vector of
+(mathcal IcalI).
+The displacement (d) is independent of the pairing and bit arrays,
+of (lambda,gamma_0), and of the shift; it is determined by the
+fixed chamber shape and the barycentric formula. For each fixed (Z)
the shift integral of the rough test is consequently
-\[
+[
- \frac1{r^q}\int_{[0,r)^q}\sum_{k\in\mathbb Z^q}
- \indic_{\{|s+r(k+\frac12\indic_{\mathcal I})+d(Z)-\widehat z_y|_2
- \leq b_q\sqrt q\}}\,ds
- &=r^{-q}\Vol_{\mathbb R^q}(B_{b_q\sqrt q})\\
- &\leq r^{-q}(4\tau\sqrt q)^q.
+ frac1{r^q}int_{[0,r)^q}sum_{kinmathbb Z^q}bbZ^q
+ indic_{{|s+r(k+frac12indic_{mathcal I}calI)+d(Z)-widehatwidehat z_y|_2z_y|_2
+ leq b_qsqrt q}},ds
+ &=r^{-q}Vol_{mathbb R^q}bbR^q(B_{b_qsqrt q})
+ &leq r^{-q}(4tausqrt q)^q.
-\]
+]
This calculation is valid despite the shift dependence of the actual
diagram, because the copy, lift, and density bounds were imposed before
the enlargement to all centers. Tonelli's theorem applies to these
nonnegative bounds.
In each positive sorted piece the seam lies in
-\(∪_j\{Z_j\leq\lambda\}\), of coordinate volume at most
-\(m\lambda\). A cube has \(2^m m!\) barycentric top chambers and
-at most \((m+1)!\) sorted flags per chamber. Thus the seam part of
+(∪_j{Z_jleqlambda}), of coordinate volume at most
+(mlambda). A cube has (2^m m!) barycentric top chambers and
+at most ((m+1)!) sorted flags per chamber. Thus the seam part of
the inner expectation is bounded by
- C_*(q)\lambda D_0(2\tau)^q,
+ C_*(q)lambda D_0(2tau)^q,
where one may take the elementary envelope
-\[
- C_*(q)=\max{1,\,
- C_{\eta,n}H_0(q)\binom qn(2^m m!)(m+1)!\,m\,
- r^{n-q}(2\sqrt q)^q\exp(c_{{\rm seam},n}q^2)}.
-\]
-Every factor here was bounded before choosing \(\lambda\). Thus
-\(C_*\) is independent of \(\lambda,\gamma_0,\tau\), of the bounds
-\(K_R,K_{\rm scal}\), and of \(C_{\rm ent}\). The powers \(\tau^q\) in the shift
-count have already canceled those in \((2\tau)^q\).
+[
+ C_*(q)=max{1,,
+ C_{eta,n}H_0(q)binom qn(2^m m!)(m+1)!,m,
+ r^{n-q}(2sqrt q)^qexp(c_{{rm seam},n}q^2)}.
+]
+Every factor here was bounded before choosing (lambda). Thus
+(C_*) is independent of (lambda,gamma_0,tau), of the bounds
+(K_R,K_{rm scal}), and of (C_{rm ent}). The powers (tau^q) in the shift
+count have already canceled those in ((2tau)^q).
The two fiber estimates now have the same normalization. Insert
(eq:transfer-straight-average) and
(eq:transfer-seam-average) into (eq:transfer-average),
-using \(q^{-n/2}\leq1\) for the seam, and cancel the positive factor
-\(D_0(2\tau)^q\). For the conditional data and parameter bounds used
+using (q^{-n/2}leq1) for the seam, and cancel the positive factor
+(D_0(2tau)^q). For the conditional data and parameter bounds used
so far, this gives
- I\leq C_{\rm out}(C_{{\rm str},n}+C_*(q)\lambda)
- \int_{\mathcal M}\rho.
+ Ileq C_{rm out}(C_{{rm str},n}+C_*(q)lambda)
+ int_{mathcal M}calMrho.
Now fix these bounds before invoking the entropy theorem. For each
-integer \(q\geq n+1\), the form derivative bound first gives a
-deterministic \(r(q)\) with elementary reciprocal. The size bound for
-the universal type list, \(D_0\), incidence counts, and the bounds for
-\(k_0,dX\), including \(H_0(q)\), have elementary envelopes uniform
-over all permitted graphs and cocycles. Fix \(b_*\) using
+integer (qgeq n+1), the form derivative bound first gives a
+deterministic (r(q)) with elementary reciprocal. The size bound for
+the universal type list, (D_0), incidence counts, and the bounds for
+(k_0,dX), including (H_0(q)), have elementary envelopes uniform
+over all permitted graphs and cocycles. Fix (b_*) using
(eq:transfer-sphere); this determines the dimensional constants
-\(c_{{\rm seam},n}\) and \(c_{{\rm hor},n}\), independently of
-\(C_{\rm ent}\). The displayed formula for \(C_*(q)\) is consequently
+(c_{{rm seam},n}) and (c_{{rm hor},n}), independently of
+(C_{rm ent}). The displayed formula for (C_*(q)) is consequently
available at this stage. Set
-\[
- \lambda(q)=\frac1{4C_*(q)},\qquad
- \gamma_0(q)=\frac14(m+1)^{-(m+1)}\lambda(q)^{m(m+1)}.
-\]
+[
+ lambda(q)=frac1{4C_*(q)},qquad
+ gamma_0(q)=frac14(m+1)^{-(m+1)}lambda(q)^{m(m+1)}.
+]
These deterministic choices have elementary reciprocals and satisfy
(eq:transfer-gamma). The resolution estimates now give common
elementary envelopes for its fixed-order chart derivatives, hence
-\(K_R(q),K_{\rm scal}(q)\), and then the common \(\tau(q)\) in
+(K_R(q),K_{rm scal}(q)), and then the common (tau(q)) in
(eq:transfer-tau). None of these choices uses the entropy constant.
-For integer \(j\geq n+1\), define
-\[
- \Psi_n^\circ(j)=\max{1,\ b_*^{-2}K_{\rm scal}(j),
- (\frac{20c_{{\rm hor},n}j^{4/3}}{\tau(j)})^{75/13}
+For integer (jgeq n+1), define
+[
+ Psi_n^circ(j)=max{1, b_*^{-2}K_{rm scal}(j),
+ (frac{20c_{{rm hor},n}j^{4/3}}{tau(j)})^{75/13}
},
-\]
-and define a majorant on all \(t\in[1,∞)\) by
-\[
- \Psi_n(t)=1+\max_{\substack{j\in\mathbb Z\\
- n+1\leq j\leq\max\{n+1,\lceil t\rceil\}}}
- \Psi_n^\circ(j).
-\]
+]
+and define a majorant on all (tin[1,∞)) by
+[
+ Psi_n(t)=1+max_{substack{jinmathbb ZbbZ
+ n+1leq jleqmax{n+1,lceil trceil}}}
+ Psi_n^circ(j).
+]
Every expression used here has an elementary upper bound; the finite
maximum is bounded by an increasing elementary envelope at
-\(\max\{n+1,\lceil t\rceil\}\). Thus \(\Psi_n\) is an elementary
-majorant fixed solely from \(n\), and it dominates both the curvature
+(max{n+1,lceil trceil}). Thus (Psi_n) is an elementary
+majorant fixed solely from (n), and it dominates both the curvature
and strict horizon requirements. Hence every conditional datum with
-\(E\geq\Psi_n(q)\) satisfies all hypotheses used in
+(EgeqPsi_n(q)) satisfies all hypotheses used in
(eq:transfer-before-choices). Our choice
-\(C_*(q)\lambda(q)=1/4\) and the definition of \(C_{\rm out}\) prove
+(C_*(q)lambda(q)=1/4) and the definition of (C_{rm out}) prove
(eq:transfer-conditional-target) with
-\[
- K_n=e^{C_{\rm deg}+1/2}b_*^{-n}
- (C_{{\rm str},n}+1/4).
-\]
+[
+ K_n=e^{C_{rm deg}+1/2}b_*^{-n}
+ (C_{{rm str},n}+1/4).
+]
This closes the conditional target for all the specified data before
any entropy family is chosen.
-Now apply Theorem~thm:entropy with \(\mathcal B=\Psi_n\),
-\(E_{\min}=1\), the detector root map as \(F^{in}\), and the
-one-point transverse space \(Ω_0\). The detector construction uses
+Now apply Theorem~thm:entropy with (mathcal B=calB=Psi_n),
+(E_{min}=1), the detector root map as (F^{in}), and the
+one-point transverse space (Ω_0). The detector construction uses
finitely many compactly supported bump representatives and their deck
translates, giving the required uniform support and derivative bounds in
a lifted atlas and the equivariant compact support sets. After pulling
-the detector map to \(Ω\) and embedding its labels equivariantly
+the detector map to (Ω) and embedding its labels equivariantly
among the output's finite copies, the measured affine probability homotopy
-of Section~2 preserves \(I\); the output therefore supplies the conditional
-data with \(E>\Psi_n(q)\). Its constant
-\(C_{\rm ent}=C_{\rm ent}(n,\Psi_n)\) depends only on \(n\). Since
-\(\rho\leq dV_g\) and the transverse measure over \(M\) is a
+of Section~2 preserves (I); the output therefore supplies the conditional
+data with (E>Psi_n(q)). Its constant
+(C_{rm ent}=C_{rm ent}(n,Psi_n)) depends only on (n). Since
+(rholeq dV_g) and the transverse measure over (M) is a
probability measure,
-\[
- I\leq K_ne^{C_{\rm ent}/2}\int_{\mathcal M}\rho
- \leq K_ne^{C_{\rm ent}/2}\Vol_g(M).
-\]
-Finally \(I\geq\|M\|/2\), so the constant
-\(C_n=2K_ne^{C_{\rm ent}/2}\) proves the theorem.
+[
+ Ileq K_ne^{C_{rm ent}/2}int_{mathcal M}calMrho
+ leq K_ne^{C_{rm ent}/2}Vol_g(M).
+]
+Finally (Igeq|M|/2), so the constant
+(C_n=2K_ne^{C_{rm ent}/2}) proves the theorem.
Proof. [Proof of Theorem~thm:main]
-Fix \(n\geq3\), and let \(C_n>0\) be the constant for the normalized
+Fix (ngeq3), and let (C_n>0) be the constant for the normalized
scalar-curvature estimate in Theorem~thm:scalar-volume.
Proposition~prop:yamabe-reduction gives
-\[
- \|M\|\leq C_n\int_M(\Scal_g^-)^{n/2}\,dV_g
-\]
-for every metric in the theorem. Taking \(a_n=C_n^{-1}>0\) proves
+[
+ |M|leq C_nint_M(Scal_g^-)^{n/2},dV_g
+]
+for every metric in the theorem. Taking (a_n=C_n^{-1}>0) proves
the stated integral bound.
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