diff b1f9bc4..56bc728 simplicial_volume_scalar_bound_v2.txt
diff --git a/simplicial_volume_scalar_bound_v2.txt b/simplicial_volume_scalar_bound_v2.txt
index f55e98f..2e57d12 100644
--- a/simplicial_volume_scalar_bound_v2.txt
+++ b/simplicial_volume_scalar_bound_v2.txt
@@ -44,16 +44,16 @@ hypersetup{
-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
+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
-volume of $M$. The left side is a local analytic quantity: it is
+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
-$gmapstolambda^2g$, 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 +62,92 @@ 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.
-parnoindent{footnotesizettfamily src-ma2026.pdf}par
+parnoindent{footnotesizettfamily src-ma2026.pdf}
The real simplicial volume of a closed connected oriented
-$n$-manifold $M$ is the $ell^1$-seminorm of its real fundamental class:
+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
+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
+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),
-Equation~(8$'$)]{Gromov1986} that the integral of the $n/2$-power of the
-negative part of scalar curvature should bound $|M|$ above with a
-coefficient depending only on $n$. Writing
+Equation~(8')]{Gromov1986} that the integral of the n/2-power of the
+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 $gmapstolambda^2g$, so it is
+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.
-parnoindent{footnotesizehttps://doi.org/10.1007/BFb0075649}par
+parnoindent{footnotesizehttps://doi.org/10.1007/BFb0075649}
[Main theorem]
-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
+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|.
-The constant is uniform in both $M$ and $g$. As a representative case,
-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
+The constant is uniform in both M and g. As a representative case,
+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_gge-kappa^2quad⇒quad
- |M|le C_nkappa^nVol_g(M),qquad kappage0 .
+ Scal_gge-kappa^2quad⇒
+ |M|le C_nkappa^nVol_g(M), kappage0 .
-For a fixed dimension $nge3$ 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}.
-parnoindent{footnotesizettfamily src-min2026.pdf}par
+parnoindent{footnotesizettfamily src-min2026.pdf}
Several related statements were known before the general case.
-
-
-item Braun and Sauer [cite] prove a macroscopic
+ 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 $Scal_gge0$. This is
+ 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
- estimate with coefficient $27/2$ for every metric on every closed
+ 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
symplectic four-manifolds.
-parnoindent{footnotesizettfamily src-braunsauer2021.pdf}par
+parnoindent{footnotesizettfamily src-braunsauer2021.pdf}
The one external input is the qualitative vanishing
theorem of
@@ -165,83 +163,76 @@ uniform in the topology and in the metric.
The proof has five stages. Each is stated as a theorem or proposition
whose hypotheses are no stronger than what the previous stage delivers,
and the last stage composes them.
-
-
-
-item Conformal reduction to $Scal_g=-1$
+ 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
- be negative, and a constant rescaling normalizes it to $-1$.
+ 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
+ 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_nVol_g(M)qquadwhenever Scal_gequiv-1,
which is Theorem~thm:scalar-volume.
-
-item A detector for $|M|$.
+ 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,
- 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 $C_nVol_g(M)$.
-
-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
+ 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,
+ 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_nVol_g(M).
+ 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
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
+ scalar curvature as a square of a prescribed quantity D plus
+ nonnegative errors. Since D can be prescribed, a solvability theorem
for the prescribed weighted mean curvature equation
(Lemma~lem:graph-prescription, localized in
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 $Pi_p$.
-
-item Iteration to an entropy metric.
+ 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.
+ 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
- an arbitrary prescribed $E$ while keeping the accumulated volume
+ an arbitrary prescribed E while keeping the accumulated volume
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-nlog E+E,e_h(F),qquad
+ wscal(h,phi)ge-C+2W-nlog E+E,e_h(F),
e_h(F)^{1/2}le E^{-1/100},
]
- where $W=log(dV_h/rho)ge0$ and $rhole dV_g$
+ where W=log(dV_h/rho)ge0 and rhole dV_g
(Theorem~thm:entropy).
-
-item Degree and transfer.
+ 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
+ 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
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
- 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 $Vol_g(M)$
+ detector pairing I and whose tangential and normal lengths are
+ 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)
(Theorem~thm:scalar-volume).
-The last step contains the only place where the exponent of $E$ matters:
+The last step contains the only place where the exponent of E matters:
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.
+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.
The three technical engines are the graph engine, the entropy
@@ -257,7 +248,7 @@ of Theorem~thm:entropy and Theorem~thm:degree for a complete
proof of Theorem~thm:main.
Two further conventions are used throughout. Constants written
-$C_n,c_n,C$ are positive and depend only on the dimension unless a
+C_n,c_n,C are positive and depend only on the dimension unless a
subscript or an explicit dependence is indicated; their values may change
from line to line, and no effort is made to keep them sharp. Integrals
over measured families of leaves always include the transverse measure;
@@ -268,17 +259,17 @@ Four techniques carry the proof, and the logical spine above says where
each is used. Yamabe theory supplies, in every conformal class, a metric
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
+normalizes the constant to -1. Bounded cohomology supplies a bounded
cocycle of norm at most one, pairing with the fundamental class in an
-amount $Ige|M|/2$, 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
+integral I. Graph deformations supply the identity
[
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
+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
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
@@ -335,7 +326,7 @@ Y(M,[g])=
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}}.
+{(int_Mu^{2n/(n-2)},dV_g)^{(n-2)/n}}.
]
The Yamabe theorem, developed through the work of Yamabe, Trudinger,
Aubin, and Schoen [cite],
@@ -367,11 +358,11 @@ Consequently (Vol_h(M)leint_M(Scal_g^-)^{n/2},dV_g), and
-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
+parnoindent{footnotesize J. Math. Pures Appl. (9) 55 (1976), 269--296}
+parnoindent{footnotesizehttps://doi.org/10.4310/jdg/1214439291}
+parnoindent{footnotesizettfamily src-trudinger1968.pdf}
+parnoindent{footnotesizettfamily src-yamabe1960.pdf}
+parnoindent{footnotesizettfamily src-leeparker1987.pdf}
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).
@@ -397,8 +388,8 @@ cite[Section 3.1]{Gromov1982} and its canonical group formulation in
cite[Theorems 8.3 and 8.6]{IvanovNotes}.
-parnoindent{footnotesizettfamily src-gromov1982.pdf}par
-parnoindent{footnotesizettfamily src-ivanov2017.pdf}par
+parnoindent{footnotesizettfamily src-gromov1982.pdf}
+parnoindent{footnotesizettfamily src-ivanov2017.pdf}
Use the homogeneous model for group cochains. Thus an (n)-cochain
is a bounded function on (Gamma^{n+1}), invariant under simultaneous
@@ -411,7 +402,7 @@ 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, |c|_∞le1,
]
whose corresponding bounded singular cohomology class has value
@@ -425,7 +416,7 @@ and divide by (1+ε), with (0<εle1/2). Replacing
the cocycle by its negative first, if necessary, fixes the sign.
-parnoindent{footnotesizettfamily src-frigerio2016.pdf}par
+parnoindent{footnotesizettfamily src-frigerio2016.pdf}
Allow a finite number of copies of every group label. More
precisely, let (mathcal L=calL=Gammatimes{1,...,s}), with the free
@@ -459,8 +450,8 @@ coordinates (p_ell=F_ell^2) give the associated map to
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,
+ F:widetilde M→ell^2(mathcal L),calL),
+ F_ellge0, 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
@@ -599,9 +590,9 @@ map.
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
@@ -613,7 +604,7 @@ the graph and vertical Jacobi identities, see
calculation with the conventions needed here.
-parnoindent{footnotesizettfamily src-perelman2002.pdf}par
+parnoindent{footnotesizettfamily src-perelman2002.pdf}
[Weighted graph identity]
@@ -621,9 +612,9 @@ parnoindent{footnotesizettfamily src-perelman2002.pdf}par
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.
+ L=(1+N^2|df|_h^2)^{1/2}, v=L^{-1},
+ U=(Nnabla_h f)/(L),
+ h'=h+N^2df ⊗ df, phi'=phi-log L.
]
View (h') as the induced metric on the graph
(Σ={(x,Nf(x)):xin X}) in
@@ -632,7 +623,7 @@ View (h') as the induced metric on the graph
(nu=(-U,v)), put
(II(V,W)=<barnabla_Vnu,W>), and let
[
- H=tr_ΣII,qquad
+ H=tr_ΣII,
D=H+∂_nuphi=-diver_phi U .
]
Here (D), and its derivative (U(D)), are regarded as functions on
@@ -644,8 +635,8 @@ Here (D), and its derivative (U(D)), are regarded as functions on
Moreover,
- e^{phi'}dV_{h'}=e^phi dV_h,qquad
- N^2|df|_{h'}^2=1-v^2=|U|_h^2,qquad
+ e^{phi'}dV_{h'}=e^phi dV_h,
+ N^2|df|_{h'}^2=1-v^2=|U|_h^2,
|dlog L|_{h'}le L|U|_h|II|_{h'} .
@@ -727,7 +718,7 @@ 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
+ x=-II(tau,tau), X_{∥}=Nx, t=A^2/f,
B=1-|e|_{h_{in}}^2in[0,1].
]
At (A=0), choose any (h)-unit vector (e), put (tau=(e,0)),
@@ -740,8 +731,8 @@ 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
+ NU(f)=LA^2,
+ |II|_{h'}^2=x^2+|II^{rem}|_{h'}^2,
tr_{h'}II^{rem}=H+x,
x=e(A)quadwhere A>0.
@@ -778,7 +769,7 @@ energy. The (A^2/(2f)) term completes the longitudinal square:
&ε N^2|dF|_{h_{in}}^2
-4ε a(F/f)NAe(F)
- &quad=ε|N,dF
+ &=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).
@@ -839,15 +830,15 @@ 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
+ L_z=(1+|dz|_h^2)^{1/2}, v_z=L_z^{-1},
+ U_z=(nabla_hz)/(L_z),
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).
+ 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
@@ -855,9 +846,7 @@ item Let (N,ε,a>0), (0<deltale1), and (H_*ge0).
p(x,z,v)=
(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
+ 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
@@ -878,8 +867,8 @@ the stated open prescription region. Its angle satisfies
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
+
+ |ez-f_0|<(pi w)/(2),
|z|le(|f_0|_∞+pi w/2)/(e).
]
The constants (v_0,eta) and the derivative bounds may depend on
@@ -894,7 +883,7 @@ 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,
|dz|_{h_z}^2=1-v_z^2<1.
]
For the family assertion, let (mathcal TcalT) be a standard Borel
@@ -934,7 +923,7 @@ of (eq:graph-prescription-positive) is nonnegative at
(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),
z_+=(tf_0+wtau)/(e)
]
have arguments (+tau) and (-tau), respectively. Their
@@ -1095,8 +1084,8 @@ uniqueness proves equivariance.
-parnoindent{footnotesizehttps://doi.org/10.1007/978-3-642-61798-0}par
-parnoindent{footnotesizehttps://doi.org/10.1007/978-1-4612-4190-4}par
+parnoindent{footnotesizehttps://doi.org/10.1007/978-3-642-61798-0}
+parnoindent{footnotesizehttps://doi.org/10.1007/978-1-4612-4190-4}
The positive solution in Lemma~lem:graph-prescription is
@@ -1119,8 +1108,8 @@ 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<lle1,qquad llethetaletfrac1{16}.
+ ε>0, N^2=lε,
+ a=tfrac12+theta, 0<lle1, llethetaletfrac1{16}.
]
There is a radius (R_{loc}), depending only on the
dimension and a fixed sufficiently small numerical constant
@@ -1130,7 +1119,7 @@ large (H_*), one can choose smooth functions
(beta=beta_s+alpha),
0lebetale H_*,
- qquad
+
N|dbeta_s|_h
le c_0(thetaε^{3/2}+beta_s^{3/2}).
@@ -1167,9 +1156,9 @@ 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|dlog L|_{h+N^2df ⊗ df}.
+ -wscal(h,phi)|,
+ A^2/f, |ND^0|, beta f,
+ |ND|, N|II|, N|dlog L|_{h+N^2df ⊗ df}.
@@ -1252,7 +1241,7 @@ smooth nondecreasing saturation
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=sigma_{H_*}(b+alpha),
beta_s=sigma_{H_*}(b+alpha)-alpha .
]
Take (H_*) much larger than the value and derivative bounds
@@ -1316,7 +1305,7 @@ for the uncut graph. In fact, at a recentered infimum with
(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_z|ge c_2(1+beta+widetilde f^{-2}),
|p_x|le C(widetilde f^{-1}
+widetilde f(1+beta^{3/2})).
]
@@ -1336,7 +1325,7 @@ On each of the outer plateau boxes the equation is
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 .
@@ -1385,8 +1374,8 @@ The global lower bound (eq:graph-global-plateau-angle)
makes a minimum with value below (1/2), if one exists,
interior, and at that minimum
[
- v<xi/2,qquad xige c/√(H_*),qquad
- nabla_Σ v=(v/xi)nabla_Σxi,qquad
+ v<xi/2, xige c/√(H_*),
+ nabla_Σ v=(v/xi)nabla_Σxi,
Delta_{Σ,phi}vge(v/xi)Delta_{Σ,phi}xi.
]
The formula for the restriction of an ambient function to the
@@ -1487,7 +1476,7 @@ On the uncut region the last term
|varpi|_∞
ledelta√(ε a(ε a+H_*))→0,
- qquad
+
-2NU(varpi)
=(2ε adelta^2 L A^2)/(f^2)ge0 .
@@ -1531,8 +1520,8 @@ For a root probability map, its energy is
Fix a smooth nonnegative nondecreasing function
(Psi:[0,∞)to[0,∞)) such that
- Psi(s)=0quad(sle8),qquad
- Psi(s)le Cs^2,qquad |Psi'(s)|^2le CPsi(s),qquad
+ Psi(s)=0quad(sle8),
+ Psi(s)le Cs^2, |Psi'(s)|^2le CPsi(s),
Psi(s)ge cs^2quad(sge R_Psi),
@@ -1543,8 +1532,8 @@ 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
+ psile Ckappa s^2,
+ |psi'|^2le Ckappapsi,
psi(s)ge ckappa s^2quad(sge R_Psi).
]
@@ -1553,9 +1542,9 @@ 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=tfrac12-2theta,qquad
- 0<lle1,quad llethetaletfrac1{16},
+ ε>0, N^2=lε,
+ a=tfrac12+theta, lambda=tfrac12-2theta,
+ 0<lle1, llethetaletfrac1{16},
with (N>0). On one leaf, start with ((h_0,phi_0,F)), where
@@ -1577,11 +1566,11 @@ reserve (Pi_p) below.
The graph identities give the output and its normalization:
- h_+=h_0+N^2sum_i df_i ⊗ df_i,qquad
+ h_+=h_0+N^2sum_i df_i ⊗ df_i,
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 .
+ phi_+=phi_0-W_p,
+ e^{phi_+}dV_{h_+}=e^{phi_0}dV_{h_0},
+ r:=sum_i f_i^2>0, Y_i=f_i/sqrt r .
@@ -1594,8 +1583,8 @@ 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
+ A_i=|U_i|_{h_{i-1}}, U_i=A_i e_i,
+ x_i=-II_i(tau_i,tau_i), X_i=Nx_i,
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
@@ -1605,7 +1594,7 @@ Here (tau_i=(L_i^{-1}e_i,A_i)) and (x_i=e_i(A_i)) when
II_i^{rem}
:=II_i+x_i,tau_i^flat ⊗ tau_i^flat,
- |II_i|^2=x_i^2+|II_i^{rem}|^2,qquad
+ |II_i|^2=x_i^2+|II_i^{rem}|^2,
trII_i^{rem}=H_i+x_i .
@@ -1615,7 +1604,7 @@ In particular, at (A_i=0) the combination
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,
C_i=
1-N^2|df_i|_{h_+}^2/A_i^2,&A_i>0,
@@ -1630,12 +1619,12 @@ 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)/(εnu_i),qquad
+ m=sum_iA_i^2,
+ u_i=(A_i^2)/(εnu_i),
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
+ sigma_i=(F_i^2)/(nu_i), E_nusigma_i=1,
+ M_0=a(1-r)-barbar u/2,u/2,
+ V_i=a(sigma_i-1)+lambda(u_i-barbar u),u),
Var_nu u=E_nu(u_i-barbar u)^2u)^2 .
@@ -1677,7 +1666,7 @@ 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)_{iinmathcal L}calL,qquad F_ige0,qquad
+ F=(F_i)_{iinmathcal L}calL, F_ige0,
sum_iF_i^2=1
]
has finitely many free label orbits and uniformly locally finite
@@ -1712,7 +1701,7 @@ root probability map with the same label orbits.
For each (i), the coefficient furnished by
Lemma~lem:graph-localization has the form
- beta_i=beta_{s,i}+εpsi(R_i),qquad
+ beta_i=beta_{s,i}+εpsi(R_i),
R_i=sum_{j<i}f_j^2 ,
@@ -1724,7 +1713,7 @@ Writing
]
on ({f_i>0}), extended by zero off the bump, one has
- ND_i=ND_i^0-beta_i f_i+err_i,qquad
+ ND_i=ND_i^0-beta_i f_i+err_i,
sum_i|err_i|^2leeta_0 .
@@ -1789,7 +1778,7 @@ 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,
+ 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
@@ -1858,7 +1847,7 @@ orthogonal projection in the target:
]
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_i}^2=A_i^2,
N^2|df_i|_{h_+}^2=A_i^2(1-C_i)le A_i^2 .
]
Consequently
@@ -1871,13 +1860,13 @@ Consequently
Put
[
K_i=(X_i-t_i)^2+N^2|II_i^{rem}|^2
- +N^2|dlog L_i|_{h_i}^2,qquad
+ +N^2|dlog L_i|_{h_i}^2,
g_i=a(sigma_i-r)+u_i/2 .
]
The definitions imply
[
- Z_i=(ε f_i)/(r)g_i,quad
- frac rεsum_iZ_i^2=εE_nu g_i^2,quad
+ Z_i=(ε f_i)/(r)g_i,
+ frac rεsum_iZ_i^2=εE_nu g_i^2,
frac rε(L_i-1)t_i^2=(L_i-1)A_i^2u_i .
]
Use these identities, (eq:graph-ideal-bound), and
@@ -1949,12 +1938,12 @@ For the other pair, divide by (ε^2f_i^2) and use
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
+ V_i=-a+lambda w_i,
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+u_ile2(|M_0|+|V_i|+|w_i|),
r^2+u_i^2le12(M_0^2+V_i^2+w_i^2).
]
After multiplication by (r/ε), the last cost in
@@ -1998,7 +1987,7 @@ 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_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
@@ -2012,7 +2001,7 @@ most
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
@@ -2027,7 +2016,7 @@ lower profile bound on each of these rows gives
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_*).
+ (rge C_*).
@@ -2095,7 +2084,7 @@ Write (exp^{[0]}(t)=t) and
(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).
+ mathcal B(q)calB(q)leq exp^{[H]}(B(1+q)^A)(qgeq1).
]
When the function is prescribed in a fixed dimension, its parameters
(H,A,B) are allowed to depend on that dimension.
@@ -2109,8 +2098,8 @@ 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
+ ell^2(mathcal L_0),calL_0),
+ F^{in}_ellgeq0,
sum_{ellinmathcal L_0}calL_0(F^{in}_ell)^2=1
]
is equivariant, Borel in the transverse variable, and smooth along
@@ -2136,11 +2125,11 @@ integer (qgeq n+1), and a constant (C=C(n,mathcal B)<∞calB)<∞) such
that, on an invariant conull set of transverse parameters,
- &rholeq dV_g,qquad
+ &rholeq dV_g,
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_h(F)^{1/2}leq E^{-1/100},
&E>mathcal B(q).calB(q).
@@ -2178,7 +2167,7 @@ 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+nlogεgeq-C+P,
- qquad Wgeq0,quad Pgeq0.
+ Wgeq0, Pgeq0.
]
First raise an initially small energy coefficient to a dimensional
positive value. Subsequent sweeps have a summable scalar loss and
@@ -2238,10 +2227,10 @@ Proposition~prop:sweep. In particular, its input is
((h_0,phi_0,F)), its output is ((h_+,phi_+,Y)), and
[
- N^2=lε,qquad a=tfrac12+theta,qquad
+ N^2=lε, a=tfrac12+theta,
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.
+ r=sum_i f_i^2>0, Y=f/sqrt r,
+ W_p=sum_ilog L_i, m=sum_i A_i^2.
]
The weights (nu_i=f_i^2/r) sum to one. Recall the probability
@@ -2249,11 +2238,11 @@ statistics and their centered combinations from
(eq:sweep-statistics):
[
- u_i=(A_i^2)/(εnu_i),qquad
- sigma_i=(F_i^2)/(nu_i),qquad
+ u_i=(A_i^2)/(εnu_i),
+ sigma_i=(F_i^2)/(nu_i),
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
+ M_0=a(1-r)-(barbar u)/(2),u)/(2),
+ V_i=a(sigma_i-1)+lambda(u_i-barbar u),u),
t_i=(A_i^2)/(f_i).
]
@@ -2263,8 +2252,8 @@ 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
+ frac{dV_{h_+}}{dV_{h_0}}=e^{W_p},
+ phi_+=phi_0-W_p,
N^2 e_{h_+}(Y)leq frac mr.
The added graph tensors are nonnegative. Thus, as quadratic forms,
@@ -2300,7 +2289,7 @@ boost, followed individually by conformal changes, produce a metric
(h_*), a root probability map (F_*), and
(ε_0in[ε_*,2ε_*]) such that
- rho:=dV_{h_*}leq dV_g,qquad
+ rho:=dV_{h_*}leq dV_g,
Scal_{h_*}geq-C_{boot}+ε_0e_{h_*}(F_*).
The metric and map have the uniformity and support properties in
@@ -2412,7 +2401,7 @@ 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)^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
@@ -2493,8 +2482,8 @@ 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.
+ 0<lleq l_0, theta=D_*lleqtfrac1{16},
+ N^2=lε.
]
The errors can be chosen so that
@@ -2606,7 +2595,7 @@ remains. Choose (gamma_n) below that fraction. The error term
It remains to prove the budget when
- m<m_0,qquad |1-r|<tfrac14.
+ m<m_0, |1-r|<tfrac14.
Here (T_0geq c_nm^2). Indeed, expanding the exponential in
(eq:entropy-trace) and using
@@ -2655,9 +2644,9 @@ 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
+ B_i=sum_{j<i}(Ndf_j(e_i))^2,
C_i=frac1{A_i^2}sum_{j>i}A_j^2(Ndf_i(e_j))^2
- quad(A_i>0).
+ (A_i>0).
]
The first identity follows from
(h_{i-1}=h_0+N^2sum_{j<i}df_j ⊗ df_j) and
@@ -2673,7 +2662,7 @@ 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^2xi_{ij}+O(m^2),qquad
+ B_i=sum_{j<i}A_j^2xi_{ij}+O(m^2),
C_i=sum_{j>i}A_j^2xi_{ij}+O(m^2).
@@ -2822,7 +2811,7 @@ term by (gamma_n/(lr)) proves the first inequality in
For the second, the retained derivative terms imply
- mathcal XcalXleq C_n N^2P_p,qquad
+ mathcal XcalXleq C_n N^2P_p,
(ε^2)/(rN^2)
(E V_i^2+thetaVar u)leq C_nP_p.
@@ -2841,7 +2830,7 @@ claimed bounds for (II_i) and (dlog L_i).
For (D_i), use
[
ND_i^0=(ε f_i)/(r)
- (a(sigma_i-r)+(u_i)/(2)),qquad
+ (a(sigma_i-r)+(u_i)/(2)),
E sigma_i^2leq C(1+E V_i^2+Var u).
]
Here the second inequality follows by solving the definition of (V_i)
@@ -2873,9 +2862,9 @@ 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
+ 0.26=tfrac14+0.01,
+ 0.26+0.04<(1+0.32)/(4),
+ 0.26+tfrac34(0.04)<0.32,
0.02>tfrac1{100}.
]
The estimates below show how each margin is used.
@@ -2885,7 +2874,7 @@ 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.
+ W=log(dV_h/rho)geq0, P_{in}geq0.
]
Successive sweeps of Lemma~lem:entropy-increment, with
(ε_{new}=(1+l(ε))ε), preserve this
@@ -2895,7 +2884,7 @@ form of inequality: at the final parameter (E) the surplus is
constant is (C_nsum_p l(ε_p)^2). Along the entire increasing
sequence of parameters,
- sum_p l(ε_p)^2leq C_n,qquad
+ sum_p l(ε_p)^2leq C_n,
#{p:ε_p<E}leq C_n(1+log^3(1+E)).
If the initial volume logarithm is zero, then, for all sufficiently
@@ -2959,7 +2948,7 @@ 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^0leq h^{(1)}leq...leq h_+,qquad
+ h^0leq h^{(1)}leq...leq h_+,
(h_+)^{-1}leq...leq(h^{(1)})^{-1}leq(h^0)^{-1}.
]
In particular, the starting metric is an admissible reference metric
@@ -2984,7 +2973,7 @@ surplus from that last sweep, that
wscal(h_+,phi_+)&geq-C+2W-nlog E+E e_{h_+}(Y)+P,
- W&=log(dV_{h_+}/rho)geq0,qquad
+ W&=log(dV_{h_+}/rho)geq0,
Wleq E^{0.26}(E+P)^{0.75}.
@@ -2996,7 +2985,7 @@ on two copies of the labels such that
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 W&=W&=log(dV_{widehatwidehat h}h/rho)geq0,
widehatwidehat WWleq E^{0.32}(E+P')^{0.75},
e_{widehatwidehat h}h(Z)^{1/2}&leq E^{-1/100}.
@@ -3011,7 +3000,7 @@ 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
+ c_E=frac1{sqrt E(1+K_0)},
Z=frac{(f,c_EY^0)}{√(r+c_E^2)}.
]
The two vectors in the numerator use disjoint copies of the labels.
@@ -3023,7 +3012,7 @@ 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^2sum_i|df_i|_{h_+}^2=m_fleq n,qquad
+ N^2sum_i|df_i|_{h_+}^2=m_fleq n,
|ds|_{h_+}^2leq n/N^2.
]
Monotonicity of the phase metrics also gives (e_{h_+}(Y^0)leq K_0^2).
@@ -3064,7 +3053,7 @@ Choose a smooth function (u=u(r)geq0), supported in
e^uasymp
min(E^{0.02},max(1,E^{-0.16}/sqrt r)),
- qquad |(du)/(dr)|leqfrac C r.
+ |(du)/(dr)|leqfrac C r.
For example, smooth, with bounded-width transitions on the logarithmic
scale, the function obtained by truncating
@@ -3074,15 +3063,15 @@ 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
[
- widehatwidehat h=e^h=e^{2u}h_+,qquad
- widehatphi=phi_+-(n-1)u,qquad
+ widehatwidehat h=e^h=e^{2u}h_+,
+ widehatphi=phi_+-(n-1)u,
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
+ leqfrac{2√(nr)}N,
|du|leq(C_n)/(Nsqrt r).
]
The coefficient (n-1) in the new weight cancels the Laplacian of
@@ -3179,7 +3168,7 @@ 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.
In the first bound use was made of (E+Pleq C_nP), valid for large (E)
@@ -3482,16 +3471,16 @@ 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
+underlying space X is standard Borel and has a countable collection of
flow boxes
[
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 $mathbb R^d$,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
+whose images cover X and which are Borel isomorphisms onto their images.
+Here Q_a is an open box
+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
Borel subsets, preserves the transverse measures, and the change of leaf
@@ -3501,9 +3490,9 @@ 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 $dmu_a(t),dV_h$ 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.
+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;
the equality on overlaps follows from invariance of the transverse
measures. From a plaque, all plaques on its leaf have a countable Borel
@@ -3512,48 +3501,47 @@ particular the saturation of a transverse null set is still null in every
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
+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
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 $G=(G_1,...,G_d):Xto Q_T^d$.
-For every $0<t<T$ the part ${|G|_∞le t}$ has a countable flow-box
+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):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:
+ 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
+ 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
+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
required.
-item The sum of the measures of the transversals of this cover is finite.
+ 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.
-The cover and every bound in this definition may depend on $t$ and on the
-given band. The leaves are not assumed complete, and $G$ is not assumed
+The cover and every bound in this definition may depend on t and on the
+given band. The leaves are not assumed complete, and G is not assumed
proper.
-For $dge1$, 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 $deta$ uniformly
+in geometric norms. Consequently, for a leafwise smooth (d-1)-form
+eta supported over a compact subcube, with eta and deta uniformly
bounded there, chartwise Stokes and Fubini give
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 +3551,58 @@ 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)=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 $dge1$, a compactly supported top form on a cube of integral
+derivatives of G there are bounded, and that band has finite measured
+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
-usual signed orientation convention. For $d=0$ it means the measured
-signed count of the oriented points. Abbreviate it to $mathcal D$calD$
+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 DcalD
when the band and map are understood.
[Measured degree estimate]
-There are absolute constants $C_0,C_1>0$ with the following property.
-Let $qge2$, let $d$ be an integer with $0le dle q$, and let $tau>0$.
-For $0le rle d$ set
+There are absolute constants C_0,C_1>0 with the following property.
+Let qge2, let d be an integer with 0le dle q, and let tau>0.
+For 0le rle d set
- T_r=2tau+(rtau)/(q),qquad
+ T_r=2tau+(rtau)/(q),
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 $Bge1$ and
-$L_1,...,L_d>0$. Suppose throughout $X$ that
+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|_hle B-frac dqqquad(1le ile d),
and suppose that
frac{C_0(1+B)^{2/3}}{L_i}<(tau)/(10q)
- qquad(1le ile d).
+ (1le ile d).
-Define $H_i(G)=2(L_i|dG_i|_h-1)_+$. Then
+Define H_i(G)=2(L_i|dG_i|_h-1)_+. Then
|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
+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
unsigned count.
-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,
+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 $2tau$ on the left.
+coordinate increases by 1/q. A family of boundaries
+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
@@ -3622,32 +3610,30 @@ 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 $dge1$, put
-$p=d-1$, and 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).
]
There is a Borel family of oriented regular hypersurface portions
-$Σ_s$ 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
+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
-${|G|_∞<b_d}$,
+saturations may be removed for each such s.
+ For every nonnegative Borel function F supported in
+{|G|_∞<b_d},
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
-$mathcal D_X(G)$.calD_X(G)$. Its coordinates are indexed by $j=2,...,d$ and
+ 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.
]
-With $H_j^+=2(L_j|dJ_{s,j}^+|_{h_s^+}-1)_+$, there is an absolute constant
-$C_*>0$ such that
+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}}
@@ -3657,10 +3643,10 @@ $C_*>0$ such that
int_{Σ_scap{|G|_∞<b_d}}v_sΘ,dA_h.
-In dimension $p=0$ the left side is the measured unsigned count.
+In dimension p=0 the left side is the measured unsigned count.
The child in the second assertion may be chosen separately for each
-typical $s$; no measurability of those choices in $s$ is asserted or needed.
+typical s; no measurability of those choices in s is asserted or needed.
@@ -3670,140 +3656,140 @@ subsections. Fix intermediate widths
T_p<t_1<t_2<t_3<t_4<T_d.
Whenever a chart buffer is needed, choose further intermediate widths
-between these. The width $t_4$ controls confinement and the parent
-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
+between these. The width t_4 controls confinement and the parent
+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
-$0legamma(y)le y^2$ and with $gamma(y)$ bounded above and below by
-positive absolute multiples of $|y|^{3/2}$ for $|y|ge1$. Let $P_0$ be the
+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_1psi(P_0),qquad
+ P_0(0)=0, P_0'=L_1psi(P_0),
psi(y)=1+(gamma(y))/(8(1+B)).
-It is odd and increasing and has poles at $pm b$, where
+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
+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).
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
+First record the analytic purpose of the pressure law. Let Σ
+be a smooth cooriented local hypersurface, with unit normal nu, and set
[
- D=H+∂_nuphi,qquad
- H=trII,qquad
+ D=H+∂_nuphi,
+ H=trII,
II(Y,Z)=<nabla_Ynu,Z>,
mathcal P=|II|^2+calP=|II|^2+
(Ric_h-nabla^2phi)(nu,nu).
]
-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
+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 afraka 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 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
+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
+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^2-2mathfrak 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.
+Lemma~lem:graph-identity; it is valid for p=0 with left side zero.
-Set $t=L_1|dG_1|_hle B$ and $x=v/L_1$. Since $gamma(D)le D^2$,
+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).
+ (tgamma(D))/(4(1+B))le(D^2)/(4).
]
-Discard the remaining nonnegative part of $D^2$, and use
-$2mathfrak a/vfraka/vge2/x$. Finally,
-$tle1+(t-1)_+$ and $x^{-1}+ln xge1$ for $x>0$. Thus
+Discard the remaining nonnegative part of D^2, and use
+2mathfrak a/vfraka/vge2/x. Finally,
+tle1+(t-1)_+ and x^{-1}+ln xge1 for x>0. Thus
(eq:degree-slice-identity) implies
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
-$H_j^Σle H_j(G)$, so the full pointwise density comparison is
+singular points. For j=2,...,d, put
+H_j^Σ=2(L_j|d_Σ G_j|-1)_+. Tangential restriction gives
+H_j^Σle H_j(G), so the full pointwise density comparison is
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.
+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
-$widehatwidehat P$P$ and set
+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 PP and set
[
- mathfrak h_s(x)=frakh_s(x)=widehatwidehat P(s-G_1(x)),P(s-G_1(x)),qquad -tau<s<tau.
+ mathfrak h_s(x)=frakh_s(x)=widehatwidehat P(s-G_1(x)),P(s-G_1(x)), -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).
+Choose eta sufficiently small for the given band and then keep
+it fixed. On every boundary portion used below, widehatwidehat PP will agree
+with P_0 and hence obey (eq:degree-pressure).
Least minimizers, including on noncompact leaves.:
-Write $P_phi(E;U)=int_{∂^*Ecap U}e^phi,dA_h$ 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_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~5frac56]{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}.
Neither term is required to be finite on the whole leaf. In particular,
this construction uses no global weighted-volume bound.
-parnoindent{footnotesizehttps://doi.org/10.1007/978-1-4612-4098-3_1}par
-parnoindent{footnotesizettfamily src-chodoshli2024.pdf}par
+parnoindent{footnotesizehttps://doi.org/10.1007/978-1-4612-4098-3_1}
+parnoindent{footnotesizettfamily src-chodoshli2024.pdf}
-The output will be a canonical nested family $E_s$ on each open leaf,
+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
-$s$. Each $E_s$ will minimize these local differences for compact
+s. Each E_s will minimize these local differences for compact
variations. The local comparison argument also supplies the perimeter
compactness needed later to turn convergence of sets into graphical
convergence at regular limit points. Establish these properties in
turn, before confining the boundaries to the raw-pressure region.
Compact domains and leastness.
-For a compact subset $K$ of a leaf, minimize (eq:degree-functional)
-over sets which vanish off $K$, using their full perimeter in the leaf.
-The empty set is a competitor. On a neighborhood of $K$ the density is
+For a compact subset K of a leaf, minimize (eq:degree-functional)
+over sets which vanish off K, using their full perimeter in the leaf.
+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,
@@ -3817,30 +3803,30 @@ 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,
up to null sets, in every other one. It is consequently unique as an
-$L^1$ set; call it the least minimizer in $K$.
+L^1 set; call it the least minimizer in K.
-parnoindent{footnotesizehttps://doi.org/10.1017/CBO9781139108133}par
+parnoindent{footnotesizehttps://doi.org/10.1017/CBO9781139108133}
Order and exhaustion.
-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
+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
-$mathfrak hfrakhlemathfrak h'$.frakh'$. Minimality in
-the two domains makes both inequalities equalities, and leastness in $K$
-implies $Esubset F$. Exhaust a leaf by increasing finite unions of
+where mathcal FcalF denotes (eq:degree-functional); the inequality for
+the volume terms follows on E∖ F from
+mathfrak hfrakhlemathfrak h'.frakh'. Minimality in
+the two domains makes both inequalities equalities, and leastness in K
+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
+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 $E_ssubset E_t$
-for $s<t$.
+This defines E_s, and the forcing monotonicity makes E_ssubset E_t
+for s<t.
Local comparison.
The limit must still minimize for compact variations, since all later
@@ -3850,257 +3836,257 @@ perimeter bounds for the exhaustion minimizers: choose a radius with
controlled slicing area, replace the set inside that ball by the full or
empty ball, and bound the forcing term by its local supremum times the
ball volume. Thus the increasing sets converge locally in BV weakly and
-in $L^1$. To pass a comparison to the limit, surround the compact region
+in L^1. To pass a comparison to the limit, surround the compact region
to be changed by a thin outer belt on which the proposed competitor
-agrees with $E_s$. Choose a Lipschitz coordinate level function across
+agrees with E_s. Choose a Lipschitz coordinate level function across
the belt. Coarea bounds the integral, over its levels, of the trace
-mismatch between an exhaustion minimizer and $E_s$ by their $L^1$
+mismatch between an exhaustion minimizer and E_s by their L^1
mismatch in the belt. One may therefore splice the proposed competitor
inside a level to the exhaustion minimizer outside it so that the extra
trace area tends to zero. The levels can also be chosen with zero
limiting perimeter on their boundaries, since this excludes only a null
set of levels. Minimality before passage to the limit, convergence of
the bounded forcing integrals, and lower semicontinuity give the compact
-comparison for $E_s$. The same argument applies under smooth local
+comparison for E_s. The same argument applies under smooth local
convergence of the metric, density, and forcing coefficients. Below, apply it with the unchanged limit as competitor to prove convergence of
the local perimeter measures. Multiplying a density by a positive
constant has no effect on any of these comparisons.
Borel dependence.
-Explain why all these choices are Borel, including jointly in $s$.
+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
-$(R_ell)_{ellge1}$ of all open rational boxes with closure in $Q$.
+each to one fixed cube Q=(-1,1)^d. Fix once a countable list
+(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 $jge1$, using dummy slots when
+occurrences Borel measurably in slots jge1, using dummy slots when
necessary. Use the fixed Polish space
[
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.
+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.
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 $mathcal Z$.calZ$.
+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.
-Fix one plaque-data fiber and one compact exhaustion domain $K$. A
-functional sublevel for sets supported in $K$ bounds their weighted
+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 $mathcal Z$calZ$ along a
+bound there; the indicator's L^1(R_ell) norm is bounded by |R_ell|.
+Diagonal BV compactness gives convergence in mathcal ZcalZ along a
subsequence. The fiberwise closed indicator, overlap, and support
-conditions persist. The volume term is continuous, since $K$ is
+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
+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
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
+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}.
-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,
+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).
+ (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
+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 $s-G_1ge b-eta$
-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 $t_4$.
+derivative bound on G_1, a chart ball of radius
+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
-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 $t<r$
+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).
]
The chart isoperimetric inequality applied to that complementary phase,
-extended by zero outside $B_t$, gives, when $V(t)>0$,
+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)).
]
-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
-$V(r)ge cr^dexp(cHr)$, 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 $Hrasymp_{inst}eta^{-1}$, small
-$eta$ excludes the complementary phase in the smaller ball. Negative
+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^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 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.
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
-$dge2$, its regular portion is a smooth multiplicity-one hypersurface.
-The remaining set is closed in $Σ_s^{all}$ and has Hausdorff
+Set Σ_s^{all}=supp∂[E_s]. For
+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
p-7.
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
[
widehatwidehat h=e^h=e^{2(phi-phi_0)/p}h.
]
Its area is the normalized weighted area. Its volume forcing is
-$mathfrak h_sfrakh_s e^{-(phi-phi_0)/p}$, 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
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
follows. Add a compactly supported ambient-dimensional integral current
-to $[E_s]$ and truncate its integer BV multiplicity to the interval
-$[0,1]$. BV truncation does not increase boundary mass, and the volume
+to [E_s] and truncate its integer BV multiplicity to the interval
+[0,1]. BV truncation does not increase boundary mass, and the volume
where the resulting set changes is at most the mass of the added current.
The set comparison therefore implies that formulation as well.
-parnoindent{footnotesizettfamily src-eichmair2009.pdf}par
+parnoindent{footnotesizettfamily src-eichmair2009.pdf}
Local compactness claim.
-Suppose $dge2$ 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 $E_jto E$ in
-$L^1_{loc}$. Then $E$ is locally minimizing, the normalized
+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.
+weakly locally to those of E, and the supports converge locally.
Near every regular point of the limiting boundary, the boundaries are
-single smooth graphs converging in $C^∞$ on a cylinder which may
+single smooth graphs converging in C^∞ on a cylinder which may
depend on that point. In particular, for every relatively compact
-continuity subdomain $U$, meaning that the limiting perimeter of
-$∂ U$ is zero, the perimeter masses on $U$ converge. For the
+continuity subdomain U, meaning that the limiting perimeter of
+∂ U is zero, the perimeter masses on U converge. For the
minimizing boundaries considered here, every sequence of controlled
plaques from the fixed band and fixed forcing construction has a
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
+coefficients precompact in C^∞ on every smaller fixed box, by
diagonal Arzelà--Ascoli. Pass to such a convergent subsequence; in
-particular the absorbed metrics $widehatwidehat h_j$h_j$ converge smoothly to a
-positive metric $widehatwidehat h_∞$.h_∞$. Ball comparison gives the local
+particular the absorbed metrics widehatwidehat h_jh_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
+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.
+smoothly converging coefficients, shows that E is locally minimizing.
-Write $mu_j=P_{widehatwidehat h_j}h_j(E_j;,cdot,)$ and
-$mu_∞=P_{widehatwidehat h_∞}h_∞(E;,cdot,)$. 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
-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 $E_j$
+V from a convergence-determining family, chosen so that
+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
+L^1 mismatch tends to zero as well. Minimality of E_j gives
[
limsup_{jto∞}mu_j(V)lemu_∞(V_ε).
]
-Here the perimeter of the fixed set $E$ in the varying metric converges
+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
-masses converge on every relatively compact continuity subdomain $U$.
+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
+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 $Dmathbf1_bf1_{E_j}$ 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
+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
handles this cutoff test. On the compact chart uniform ellipticity
-gives $|Dmathbf1_bf1_{E_j}|le Cmu_j$. 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
-$limsup_jmu_j(F_delta)lemu_∞(F_delta)$, 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.
+the restriction convergence. The perimeter masses on U also converge.
In coordinates the perimeter
-integrand for a metric $k$ is
-$mathfrak p_k(x,z)=√(frakp_k(x,z)=√(det k(x)),|z|_{k^{-1}(x)}$.
+integrand for a metric k is
+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 $widehatwidehat h_j$h_j$ by $widehatwidehat h_∞$h_∞$ in that mass
+mass bound replace widehatwidehat h_jh_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
for every continuous positively one-homogeneous integrand with linear
-growth. For a continuous compactly supported varifold test $a(x,P)$, use
-$mathfrak p_frakp_{widehatwidehat h_∞}h_∞(x,z)a(x,ker z)$ for $zne0$ and zero
-at $z=0$. This gives precisely the local varifold convergence; the
+growth. For a continuous compactly supported varifold test a(x,P), use
+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.
-parnoindent{footnotesizehttps://doi.org/10.1007/BF02196453}par
+parnoindent{footnotesizehttps://doi.org/10.1007/BF02196453}
To prove the support and graph assertions, the first variation in
-the metric $widehatwidehat h$h$ is, by BV Gauss--Green, integration of the bounded
+the metric widehatwidehat hh 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
@@ -4110,24 +4096,24 @@ 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 $C^{1,alpha_{mathrm A}rmA}$ bound for some
-$alpha_{mathrm A}rmAin(0,1)$ 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 $0<beta<alpha_{mathrm A}rmA$.
+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
+then gives C^∞ convergence on still smaller cylinders by
interior elliptic estimates. The cylinder here may depend on the fixed
regular point of the limit; uniform graph radii in later applications
will follow from their separate compactness contradictions.
-parnoindent{footnotesizehttps://doi.org/10.2307/1970868}par
+parnoindent{footnotesizehttps://doi.org/10.2307/1970868}
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
-rapidly in $C^∞$ that its metric is realized on slices $2^{-j}$ of
+rapidly in C^∞ that its metric is realized on slices 2^{-j} of
one extra coordinate, with the limiting metric on slice zero. Indeed,
-choose the error through order $j$ below $2^{-j^2}$ and interpolate by
+choose the error through order j below 2^{-j^2} and interpolate by
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
@@ -4139,9 +4125,9 @@ argument to a sequence on which local convergence failed proves the
local support and graphical convergence asserted above.
-parnoindent{footnotesizehttps://doi.org/10.2307/1969989}par
+parnoindent{footnotesizehttps://doi.org/10.2307/1969989}
-For $d=1$ the needed facts are elementary. A locally minimizing set has
+For d=1 the needed facts are elementary. A locally minimizing set has
locally finitely many jumps by the perimeter bound. Two consecutive
jumps cannot be arbitrarily close in a controlled chart, since one can
replace the interval between them by the surrounding phase: delete it if
@@ -4156,59 +4142,59 @@ 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
-the induced orientation. Its singular set has zero $p$-dimensional area,
+Σ_s for the regular part of Σ_s^{all} in
+{|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
-$tto s$ 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
+On each leaf the sets E_t converge to E_s in L^1_{loc} as
+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 $tto s$.
+Σ_t converge smoothly and graphically near each regular point of
+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,
+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,
their tangent planes agree at contact. The comparison of the graph mean
-curvatures at that minimum of $r(t,cdot)-r(s,cdot)$ gives
-$D_tle D_s$, 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 $D=mathfrak h_s$frakh_s$ is the first variation of
+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_sfrakh_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_sfrakh_s is the first variation of
(eq:degree-functional) on a regular portion.
Choose countably many possible graph cylinders in the chart enumerations,
-and in each choose a countable dense set of $y$-coordinates. At each of
+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_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
+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 tne s,
+ w_t(y)=(r(t,y)-r(s,y))/(t-s), tne s,
]
-converge to the finite column-derivative values as $tto s$ at all these
+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
+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 $mathfrak h_t$frakh_t$ gives a
+bounds and ellipticity; the explicit t-dependence of mathfrak h_tfrakh_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
+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
@@ -4216,56 +4202,56 @@ 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 $vge0$ on the regular portion. For
-$p=0$ the same conclusion follows from the ordinary almost-everywhere
+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 $D=mathfrak h_s$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
+Differentiation of D=mathfrak h_sfrakh_s now identifies this normal speed.
+Use mathcal PcalP 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
-$D-mathfrak h_s=0$frakh_s=0$ cancelling, is
+D-mathfrak h_s=0frakh_s=0 cancelling, is
-(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 $∂_nuphi$ adds
-$vnabla^2phi(nu,nu)-< dphi,dv>_Σ$ 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 $mathfrak a>0$.fraka>0$.
-Thus $v>0$. In graph dimension zero the differential terms vanish and
-the same equation rules out $v=0$.
+(eq:degree-lapse). A nonnegative v cannot vanish at an interior
+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.
These assertions hold jointly in the measured charts for almost every
-$s$. Here are the measurable details needed for that conclusion.
+s. Here are the measurable details needed for that conclusion.
Single-graph regions with buffers can be recognized by countably many
-cylinder tests on the $L^1$ subgraph codes, with slope and derivative
+cylinder tests on the L^1 subgraph codes, with slope and derivative
bounds and with room at the top and bottom. Their smooth graph
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
+The resulting local derivatives and v are Borel. Fubini on each
+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
+particular the family of regular portions and the function v used in
the integral below have the asserted Borel descriptions.
-Set $v=0$ on the exceptional parameters and plaques when writing an
+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.
+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
+counts at most integration in z. A jump or singular part of the
monotone function only removes mass from the former integral. Apply this
-to each column and then integrate in $y$ and the transversal. Partition
+to each column and then integrate in y and the transversal. Partition
the counted portions among the countably many graph cylinders so they
are not counted twice within one parameter. They cannot give the same
ambient point at two parameters, by the noncontact argument above.
@@ -4276,51 +4262,51 @@ one-to-one image theorem. Transverse invariance on overlaps gives
int_{-tau}^{tau}!dsint_{Σ_s}vF,dA_h
leint_XF,dV_h
-for every nonnegative Borel $F$ supported in ${|G|_∞<t_3}$.
-Since $b_d<t_3$, this proves (eq:degree-coarea).
+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 $mathfrak a$fraka$ are uniformly bounded there, and
-$mathfrak a$fraka$ has a uniform positive lower bound. Confinement provides
+derivatives of mathfrak afraka are uniformly bounded there, and
+mathfrak afraka 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
+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
patch, and elliptic compactness would give a nonnegative limit with an
interior zero satisfying (eq:degree-lapse), an impossibility.
-Thus $v$ has a positive local lower bound. Harnack then bounds its local
+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.
-For $p=0$ the assertion about positive-order derivatives is empty.
+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 $D=mathfrak h_s=P_0(s-G_1)$frakh_s=P_0(s-G_1)$ and hence
-$∂_numathfrak h_s=-frakh_s=-mathfrak afraka,dG_1(nu)$ 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 $(-1.9tau,-1.6tau)$ 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
+Let J=(G_2,...,G_d). Fix a unit-integral one-form
+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 $(-1.9tau,1.9tau)$ and is one
+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
+(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
boundary where the residual form is supported. Measured current Stokes therefore gives
@@ -4330,48 +4316,48 @@ boundary where the residual form is supported. Measured current Stokes therefor
=mathcal D_X(G).calD_X(G).
-All forms here are supported in an inner band. The current $[E_s]$ has
+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.
-For $p=0$ it gives its measured signed count.
+J^*beta on that confined regular branch, with the induced orientation.
+For p=0 it gives its measured signed count.
-If $2le dle7$, this regular branch already provides the child. To verify
-the required uniformity, fix a typical $s$. If graph radii and estimates
+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
+singular points because p-7<0, so smooth graphical convergence near
that point supplies a common graph radius and all interior estimates,
contradicting the choice of points. A fixed fine mesh of graph cylinders
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, |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
-has absolute value near $t_1>T_p$. Hence every inner residual subcube
+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
+has absolute value near t_1>T_p. Hence every inner residual subcube
has buffered graph charts in this restriction. Its degree equals
-$mathcal D_X(G)$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
+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
-$Σ_scap{|G_1|<t_1}$, 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
+If d=1, so p=0, let X_s^+ be the confined regular points in
+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
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
@@ -4384,17 +4370,17 @@ counted with the mesh index. Their total transverse measure is finite and their
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 $mathcal D_X(G)$calD_X(G)$ by
+vacuous in dimension zero, so this is a controlled 0-band. Its measured
+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
-$1le(v/L_1)Θ$, and confinement places every counted point in
-$|G_1|<b_d$. Return to the common integral estimate after treating
-the possible singularities when $pge7$.
+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 pge7.
-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
+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
have no common lower bound. Those ends must be kept away from every
@@ -4403,18 +4389,18 @@ 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 $mathcal D_X(G)$.calD_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).
+ 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
-$mathcal W=dV_calW=dV_{h^+}/dA_h$ for its ordinary leafwise volume density
+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_calW=dV_{h^+}/dA_h for its ordinary leafwise volume density
relative to the original induced metric. The scalar and volume target is
@@ -4423,9 +4409,9 @@ relative to the original induced metric. The scalar and volume target is
+2lnmathcal W-calW-frac1q.
-The coefficient $2$ makes the volume factor cancel after exponentiation,
+The coefficient 2 makes the volume factor cancel after exponentiation,
leaving the same lapse factor as in (eq:degree-raw-integrand).
-The density $mathcal W$calW$ includes neither $v$ nor the transverse measure.
+The density mathcal WcalW 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.
@@ -4436,11 +4422,11 @@ data and that gap. Finally one chooses the marked-set cover small enough
for all scalar errors. Its positive radius lower bound controls the
number and derivatives of the kernels for the graph solves; it does not
enter the scalar constants. The adjustable scalar error in
-(eq:degree-child-scalar) is at most $1/q$, so chart and elliptic
+(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
@@ -4451,8 +4437,8 @@ Use finitely many graph tests at a time to obtain a regular region
with common chart controls. The points not yet recognized will be
excluded from the child, and the target gap below will keep the later
homotopies on the regular locus away from them.
-Use one common normalized outer cube $Q$ for the buffered parent plaques.
-Fix in $Q$ one ordered countable list of tests for graph cylinders, using
+Use one common normalized outer cube Q for the buffered parent plaques.
+Fix in Q one ordered countable list of tests for graph cylinders, using
rational centers, radii, bounds, and a fixed countable dense set of
rotations. Transport this same list, in this same order, by every parent
plaque chart; the list is independent of the plaque and its transverse
@@ -4460,7 +4446,7 @@ parameter. A test recognizes that the boundary
in its outer cylinder is exactly one smooth graph, with a specified slope
bound and strict room at its top, bottom, and sides, and that the set is
on its specified side. Its inner cylinder is then an open regular
-region. At level $k$ use the first $k$ tests of this fixed list in every
+region. At level k use the first k tests of this fixed list in every
plaque, with their specified positive buffers. Elliptic estimates on these
graphs give all interior derivative bounds for a fixed level. Make the
tests with strict margins, so a regular limit point recognized by one of
@@ -4468,25 +4454,25 @@ them is recognized by that same test on every sufficiently close smoothly
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
-recognized through level $k$ in any parent plaque. These are decreasing
+Let D_k be the part of the boundary support in
+{|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
+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 $k_jto∞$ 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
-$k_jgeell$, contradicting their membership in $D_{k_j}$. This is the
+makes the same test recognize the nearby points for large j, and then
+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 $barbar J$J$ of $J$ in
+First make an arbitrarily small smooth perturbation barbar JJ 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 $k$,
+for some a_0>0 and some finite k,
|barbar J|_∞>4a_0J|_∞>4a_0
quadhbox{on a uniform neighborhood of }D_k.
@@ -4495,7 +4481,7 @@ 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 $mathbb R^p$bbR^p$ and an inner compact plaque box.
+a constant shift in mathbb R^pbbR^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,
@@ -4503,8 +4489,8 @@ with levels tending to infinity and requested gaps tending to zero.
Pass to normalized chart limits, including a smooth limit of the maps.
On a slightly larger compact box the marked sets concentrate at the
singular set of the limit. By (eq:degree-singular-dimension) this
-set has dimension at most $p-7$, so its image under the smooth limit map
-has $p$-dimensional measure zero. That image is compact. A shift in the
+set has dimension at most p-7, so its image under the smooth limit map
+has p-dimensional measure zero. That image is compact. A shift in the
given open budget can therefore avoid it with a positive distance.
Uniform convergence of the maps and concentration of the marked sets
contradict the alleged failure. This proves the local uniform assertion.
@@ -4518,8 +4504,8 @@ 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.
@@ -4528,8 +4514,8 @@ next shift budget smaller than the preceding positive gaps. The marked
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 $barbar J$J$ extends the gap to a uniform
+higher derivatives. Taking a_0 smaller than the final gaps and then
+using the derivative bound of barbar JJ extends the gap to a uniform
neighborhood. This proves (eq:degree-gap).
Second, set
@@ -4539,8 +4525,8 @@ Second, set
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
-finitely many small balls, of radii $r_i$, such that
+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.
@@ -4548,155 +4534,155 @@ 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
+observation about D_k makes the enlarged balls cover the marked points
of the sequence eventually, contradicting failure. Rational chart balls
and fixed comparison factors give Riemannian balls with the same
conclusion. Keep only balls meeting the marked set, allowing a small
change of centers and the reserved enlargement. The centers then also
-lie near $D_k$. Such finite covers can be selected Borel: enumerate
+lie near D_k. Such finite covers can be selected Borel: enumerate
finite tuples of rational balls, and test whether the compact marked
part outside their union is empty. This is again a Borel compact-section
test. The positive lower radius and (eq:degree-cover-budget)
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.
+over p-7 made this cover possible, while
+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
+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
-$|barbar J|_∞>3a_0$J|_∞>3a_0$ and in the buffered atlas. It can also be chosen
+|barbar J|_∞>3a_0J|_∞>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 $r_i$
+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
+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 $tge1$, where
-$t=dist_h(x,c_i)^2/R^2$. Join this tail smoothly and
+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 $V_i$.
+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}-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 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
-$H=mathfrak h_s-∂_frakh_s-∂_nuphi$, and the bounded weight gradient give
+H=mathfrak h_s-∂_frakh_s-∂_nuphi, and the bounded weight gradient give
[
- Delta_{Σ,phi}uge2p-Csqrt u,qquad |du|_Σ^2le4u.
+ Delta_{Σ,phi}uge2p-Csqrt u, |du|_Σ^2le4u.
]
-Differentiating $(r_i^2+u)^{-j/2}$ therefore gives
+Differentiating (r_i^2+u)^{-j/2} therefore gives
[
-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
+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
-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.
+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
+ Λ=(frac{u_{alpha+4}}{u_alpha})^{1/4},
+
mathcal L=|dcalL=|dln u_alpha|^2+|dln u_{alpha+4}|^2.
-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 $mathcal L$calL$ controls the
+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 LcalL controls the
logarithmic derivatives introduced by the weight and the first graph.
-Here $0<ε_{base}<1$ will be small; $M_*$ will be a
+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
-$ε_{base}$ small and then the cover budget small enough
-gives, for $j=alpha,alpha+4$,
+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,
-frac{Delta_{Σ,phi}u_j}{u_j}
ge cfrac{u_{j+2}}{u_j}-delta.
Indeed (eq:degree-kernel-bounds) leaves the errors
-$cε_{base}^{j+2}+Csum_{i:xin V_i}M_*r_i^alpha$
-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
-$delta$.
-
-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
-$M_*r_i^alpha$.
+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.
+
+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.
Make their total active mass at most two. Moment inequalities give
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
+ gefrac{u_{alpha+2}}{u_alpha},
+ |dlnΛ|^2le(mathcal L)/(8),calL)/(8),
+ |dln u_alpha|le C(Λ+1),
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
-$mathbb EbbE Zle(mathbb EbbE Z^2)^{1/2}$ and
-$mathbb EbbE Z^3ge(mathbb EbbE Z^2)^{3/2}$ 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 $Csum_iM_*r_i^alpha(K_i^{alpha+1}+1)$.
-The first part divided by $u_alpha$ is at most $Λ$ by H"older's
+|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
-$u_{alpha+4}/u_alphage(u_alpha/2)^{4/alpha}$, 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
+L^r norms for a measure of mass at most two gives
+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
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
+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 $Q_1=deta_0$. 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
-$Delta_{Σ,phi_Σ}=Delta_{Σ,phi}
-+< dln v,d,cdot,>$, yields from
+Delta_{Σ,phi_Σ}=Delta_{Σ,phi}
++< dln v,d,cdot,>, yields from
(eq:degree-potential-laplacian) the scalar gains
-$cΛ^2+mathcal L/2-2<calL/2-2< dln v,Q_1>-|Q_1|^2-delta$.
-Here the high moment in (eq:degree-moments) supplies $Λ^2$.
-Since $|Q_1|^2lemathcal L/8$,calL/8$,
+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|^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
+Combining with (eq:degree-slice-bound), and decreasing c>0 if
necessary, gives for the new weight
wscal(h|_{TΣ},phi_Σ+eta_0)
@@ -4710,19 +4696,19 @@ 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 $barbar J$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 $barbar J$.J$.
+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 JJ 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 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
+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.
+ L_f=(1+|df|_k^2)^{1/2}, U_f=(nabla_k f)/(L_f),
+ k_f=k+df ⊗ df, varphi_f=varphi-ln L_f,
+ D_f=-div_{k,varphi}U_f.
]
Lemma~lem:graph-identity says
@@ -4732,23 +4718,23 @@ Lemma~lem:graph-identity says
Also
- |U_f|_kle1,qquad U_f(f)=L_f-L_f^{-1},qquad
+ |U_f|_kle1, U_f(f)=L_f-L_f^{-1},
|df|_{k_f}le1,
- dV_{k_f}=L_f,dV_k,qquad L_f-L_f^{-1}geln L_f.
+ dV_{k_f}=L_f,dV_k, L_f-L_f^{-1}geln L_f.
-The last inequality holds for every $L_fge1$ 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 $L_f$.
+factor L_f.
For the first graph put
[
- f_0=sigmaln u_alpha,qquad C_f=B_0+Λ,
- qquad D_f=C_ftan(f_0-f),
+ f_0=sigmaln u_alpha, C_f=B_0+Λ,
+ 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
+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)
@@ -4757,73 +4743,73 @@ Differentiating the prescribed $D_f$ gives the exact terms
&+2(C_f+D_f^2/C_f)(L_f-L_f^{-1}).
-The final term is at least $2ln L_f$ by
-(eq:degree-graph-elementary). For the derivative of $C_f$, Young's
+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|dln C_f|^2
le0.8D_f^2+(1.25)/(8)mathcal L.calL.
]
-The $f_0$ term has absolute value at most
+The f_0 term has absolute value at most
[
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
+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
(eq:degree-weight-gain) and makes the remaining constant loss as
-small as prescribed. The available $mathcal L/4$calL/4$ covers
-$1.25mathcal L/8$.calL/8$. Combining (eq:degree-weight-gain) and
+small as prescribed. The available mathcal L/4calL/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)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 $fle K$ 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
+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
graph quantities use the metric and weight after all preceding rows:
[
- L'_j=(1+|df'_j|_{k^{(j-1)}}^2)^{1/2},qquad
+ L'_j=(1+|df'_j|_{k^{(j-1)}}^2)^{1/2},
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
+ k^{(j)}=k^{(j-1)}+df'_j ⊗ df'_j,
varphi^{(j)}=varphi^{(j-1)}-ln L'_j.
]
Prescribe successively
- ell_j=Azeta(f)barbar J_j,J_j,qquad
- D'_j=B'tan((ell_j-e f'_j)/(w)),qquad
+ ell_j=Azeta(f)barbar J_j,J_j,
+ D'_j=B'tan((ell_j-e f'_j)/(w)),
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
+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_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
+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 $barbar J$J$ are uniformly
+increase. The values and first derivatives of barbar JJ are uniformly
bounded in the region in question. It follows that in each current
metric
|dell_j|le C_ellmathbf1_bf1_{{fge K}},
-where $C_ell$ depends on $A$ and the earlier band bounds but not on
-$K$, $M_*$, $k'$, or the final cover budget.
+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
@@ -4833,26 +4819,26 @@ For a row graph the analogue of
+(2e)/(w)(B'+((D'_j)^2)/(B'))
(L'_j-(L'_j)^{-1}).
]
-Choose $B'$ after $w,e$ so that
+Choose B' after w,e so that
- (B'e)/(w)ge2,qquad
+ (B'e)/(w)ge2,
(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 ${fge K}$, while the last term pays
-$2ln L'_j$ everywhere. The other terms of
-(eq:degree-graph-identity) are nonnegative. After all $p$ rows,
+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
-$C_{rows}mathbf1_bf1_{{fge K}}$, with
-$C_{rows}$ independent of $K,M_*,k'$ and the cover budget.
-On this support, $f<f_0+2$ implies
+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
+ u_alpha>exp((K-2)/sigma),
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
+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
graphs without any estimate on their individual volume expansions.
Pole avoidance in (eq:degree-row-graph) also gives the output
@@ -4865,14 +4851,14 @@ estimates that will be used for the band sides and its gradients:
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
-$|dJ_j^+|_{h^+}le|dbarbar J_j|_h+e$.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
+|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
+budgets, respectively, and by e,w. Choose them to satisfy the
target budgets (eq:degree-tolerances) and also to be smaller than
all the fixed side margins used below.
@@ -4881,90 +4867,90 @@ The full dependency order is
&perturbation budgets → (barbar J,a_0,k)J,a_0,k)
→ R
- &qquad→ (scalar error,ε_{base})
+ &→ (scalar error,ε_{base})
→ B_0 → sigma
- &qquad→ A → (w,e)
+ &→ A → (w,e)
→ B' → K → M_*
- &qquad→ (cover budget,k',radius lower bound).
+ &→ (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
+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
-(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
-$sigmaln(ε_{base}^alpha)$ by a fixed margin.
-Choose $M_*$ so large that
+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
+sigmaln(ε_{base}^alpha) by a fixed margin.
+Choose M_* so large that
(eq:degree-potential-on-bad) implies
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
+M_* by (eq:degree-cover-budget). It also provides all finite
uniform derivative bounds needed for the actual equations. In
-particular, none of the constants that forced the choices of $K$ or $M_*$
+particular, none of the constants that forced the choices of K or M_*
was allowed to depend on the final level or cover.
Solving on a complete regular region.:
-The thresholds have distinct jobs: $K+100$ places the marked set beyond
+The thresholds have distinct jobs: K+100 places the marked set beyond
the auxiliary cutoff,
-$K+12$ bounds the auxiliary region for solving, and $K+7$ bounds the
+K+12 bounds the auxiliary region for solving, and K+7 bounds the
child inside the part where its base metric is unchanged. The lower
-threshold $K+4$ will already put the output outside the target near the
+threshold K+4 will already put the output outside the target near the
potential cutoff side. Their gaps provide the spatial buffers below
-through the uniform derivative bounds for $f_0$.
+through the uniform derivative bounds for f_0.
-$Σ_s$ need not be complete. Use the operations
+Σ_s need not be complete. Use the operations
only on the open region
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
+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 $u_j,Λ,f_0,barbar J$J$ have uniform derivatives because of
+region. The estimates for derivatives of ln v apply there. The
+functions u_j,Λ,f_0,barbar JJ 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 $0<xile1$ 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
-$f_0le K+7$. 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
+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
+Replace the induced metric on Ω by
[
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
-$dxi$ 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
+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 $xi(x)^m|nabla^mxi|/xi$, and these are bounded for
-$mge1$.
+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 $|dlnxi|_{k_{aux}}$ 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.
@@ -4973,10 +4959,10 @@ have a countable Borel enumeration. The auxiliary atlas may use further
countable subdivisions; its total transverse measure is irrelevant for
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
-the constants are the chosen $e,w>0$. All coefficients have the
+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
+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
bounds, and gives their Borel dependence from uniqueness in the
@@ -4985,7 +4971,7 @@ 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
@@ -4993,29 +4979,29 @@ Define the child to be the open measured band
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
-$|G_j|$ 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
-$f_0ge K+4$ 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.
+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_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_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.
These observations verify every part of
Definition~def:degree-band for the child. To be explicit about its
-atlas, use the recognized regular graph patches through level $k'$ in
+atlas, use the recognized regular graph patches through level k' in
each parent plaque, restricting their transversals as necessary. There
are only boundedly many such tests per plaque. The derivative bounds
-for $G,f_0,J^+$ and the side margins just proved permit a further bounded
+for G,f_0,J^+ and the side margins just proved permit a further bounded
subdivision covering any chosen inner residual subcube with buffers.
The solved graph metrics and weights have uniform derivatives in these
charts. Holonomy is inherited from the parent overlap maps, with the
@@ -5025,56 +5011,56 @@ 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
-in $|z|_∞<a_0$, reducing $a_0$ if necessary. First homotope $J$ to
-$barbar J$J$ on the whole parent boundary current. The small value
+Use in (eq:degree-boundary-degree) a residual form beta supported
+in |z|_∞<a_0, reducing a_0 if necessary. First homotope J to
+barbar JJ 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 $dmathcal HcalHbeta$, where $mathcal HcalHbeta$ 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
-in (eq:degree-boundary-degree). The current $∂[E_s]$ is
+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,
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 $barbar J$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 $|barbar J|_∞>3a_0$.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 $barbar J$J$ to
-$(1+Azeta(f))barbar J$J$ and then to $J^+$. Along the first homotopy the
+The resulting integral for barbar JJ is entirely in the regular region
+used for the solves. Indeed |barbar J|_∞<a_0J|_∞<a_0 avoids a neighborhood
+of D_k by (eq:degree-gap) and avoids every kernel support, which
+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 barbar JJ to
+(1+Azeta(f))barbar JJ and then to J^+. Along the first homotopy the
multiplier is at least one, so a preimage of the small target support
-has $|barbar J|_∞<a_0$.J|_∞<a_0$. Along the second, the first line of
+has |barbar J|_∞<a_0.J|_∞<a_0. Along the second, the first line of
(eq:degree-output-estimates) shows that it has
-$|barbar J|_∞<a_0+2w<2a_0$.J|_∞<a_0+2w<2a_0$. 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
+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 $mathcal D_X(G)$.calD_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 $mathcal W$calW$ defined above.
-Choose all the adjustable scalar errors above with sum at most $1/q$.
+(eq:degree-child-scalar) for the product mathcal WcalW 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
-$dle7$ treated above, use $mathcal W=1$.calW=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
+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 $mathcal D_X(G)$calD_X(G)$ for each typical
-$s$. Its coordinate bound follows from
+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}
@@ -5082,19 +5068,19 @@ $s$. Its coordinate bound follows from
]
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
+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.
+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.
]
-The same observation covers the counted case $p=0$.
+The same observation covers the counted case p=0.
-The function $xmapsto(x-1)_+$ is one-Lipschitz, so
+The function xmapsto(x-1)_+ is one-Lipschitz, so
(eq:degree-output-estimates) gives
[
sum_{j=2}^dH_j^+lesum_{j=2}^dH_j(G)
@@ -5113,45 +5099,45 @@ Use this and the scalar and volume relations in
&quadle e^{3/(5q)}frac v{L_1}Θ.
-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
+The factor mathcal WcalW cancels exactly against the term
+2lnmathcal WcalW 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
+row error; for p=0 the same estimate is the pointwise bound for a
counted point. Integrate (eq:degree-integrand-cancellation) over
-the child and use its containment in the parent $b_d$ region. This is
-(eq:degree-child-integral), for example with $C_*=1$.
+the child and use its containment in the parent b_d region. This is
+(eq:degree-child-integral), for example with C_*=1.
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 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 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
+condition. For almost every s the induction hypothesis and
(eq:degree-child-integral) give
[
|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
-$C_1ge C_*$. Integrate this scalar inequality over $-tau<s<tau$
+For p=0 this is the same use of the unsigned-count bound. Take
+C_1ge C_*. Integrate this scalar inequality over -tau<s<tau
and apply (eq:degree-coarea) with
-$F=mathbf1_bf1_{{|G|_∞<b_d}}Θ$. The factor $2tau$ on the
-left changes its power from $p$ to $d$, and the right is bounded by
+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.
There is no implicit measurable selection of children in this last step.
-For each typical $s$ a child proves the displayed scalar inequality
+For each typical s a child proves the displayed scalar inequality
against the same parent integral. That parent integral is a measurable
-function of $s$ by the lapse construction and is governed by
+function of s by the lapse construction and is governed by
(eq:degree-coarea); it is the only function being integrated.
-Its integral is finite because $Θ$ is bounded on the controlled
-$b_d$ region, which has finite measured volume.
-The constants $C_0$ and $C_1$ are absolute: $C_0$ came from the fixed
-pressure profile, and each descent costs at most $e^{C_*/q}$ after the
+Its integral is finite because Θ is bounded on the controlled
+b_d region, which has finite measured volume.
+The constants C_0 and C_1 are absolute: C_0 came from the fixed
+pressure profile, and each descent costs at most e^{C_*/q} after the
arbitrarily small slice errors were chosen. All geometry-dependent
constants were used only to make those choices.
@@ -5194,13 +5180,13 @@ 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),frake=e_h(F),qquad W=log(dV_h/rho).
+ mathfrak e=e_h(F),frake=e_h(F), W=log(dV_h/rho).
]
Its estimates are
- rholeq dV_g,qquad Wgeq0,qquad
+ rholeq dV_g, Wgeq0,
wscal(h,phi)geq-C_{rm ent}+2W-nlog E+Emathfrak e,frake,
- qquad mathfrak e^frake^{1/2}leq E^{-1/100}.
+ 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
@@ -5217,7 +5203,7 @@ 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
- EgeqPsi_n(q)quad⇒quad
+ EgeqPsi_n(q)⇒
Ileq K_n e^{C_{rm ent}/2}int_{mathcal M}calMrho.
Thus (K_n) and (Psi_n) are fixed before the manifold, cocycle,
@@ -5259,7 +5245,7 @@ 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}calMF^*eta=I,qquad |eta|_{HS}leq C_n.
+ int_{mathcal M}calMF^*eta=I, |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
@@ -5309,7 +5295,7 @@ which is zero for (tleq12delta), equals (t) for
(|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.
+ p_j=w_j^2.
]
The denominator is bounded below by an absolute positive constant.
In fact (z) is within (10delta) of a nonnegative unit vector, and
@@ -5395,7 +5381,7 @@ 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, |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).
@@ -5477,7 +5463,7 @@ bound. The kernel of this
integer matrix has a real basis of integer vectors (b_1,...,b_k)
with
[
- kleq N_0,qquad
+ kleq N_0,
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
@@ -5581,7 +5567,7 @@ the pairing data, and supply smooth measured charts and quantitative
metric bounds.
-parnoindent{footnotesizettfamily src-gaifullin2008.pdf}par
+parnoindent{footnotesizettfamily src-gaifullin2008.pdf}
[Measured resolution]
For the diagram (mathcal PcalP) there is a standard Borel measured
@@ -5646,8 +5632,8 @@ so the product laws are Borel probability kernels.
Define a move (s_S) by
[
- sigma⟼ p_Ssigma,qquad t_S⟼1-t_S,
- qquad p_R⟼ p_Sp_Rp_Squad(R⊊ S),
+ sigma⟼ p_Ssigma, t_S⟼1-t_S,
+ 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
@@ -5677,7 +5663,7 @@ 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 1leq jleq m.
+ u_j=x_{i_j}/x_{i_{j-1}}in[0,4), 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
@@ -5697,7 +5683,7 @@ Figure~fig:resolution illustrates the tile and the four signed
quadrants at a two-facet corner when (m=2).
parmedskip
-noindent{textwidth}
+{textwidth}
makeatletterdef@captype{figure}makeatother
centering
[t]{0.53textwidth}
@@ -5758,7 +5744,7 @@ centering
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)};
+ {(b) (u_S=Z_S^2, u_T=Z_T^2)};
caption{The reflected resolution in dimension two, after Gaifullin's
@@ -5842,7 +5828,7 @@ 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
+ X=sum_{i=0}^m b_i v_{sigma,i},
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
@@ -5914,7 +5900,7 @@ 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={Θleq2gamma_0}.
+ 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
@@ -5997,7 +5983,7 @@ 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
[
- |dmathcal RcalR|_{∞,k}leq K_R(q),qquad
+ |dmathcal RcalR|_{∞,k}leq K_R(q),
(-Scal_k)_+leq K_{rm scal}(q)
]
for every allowed diagram and every (V,s,omega), rounding parameter,
@@ -6216,7 +6202,7 @@ 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
+ L_T=b_*sqrt E, L_N=b_*√(qE),
h_{rm tot}=h+L_T^2k.
Once the degree hypotheses are verified, its length denominator, the
@@ -6277,7 +6263,7 @@ 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.
+ 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.
@@ -6299,7 +6285,7 @@ shows that
(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)).
+ c_n=(n b_*^2)/(2(1-b_*^2)).
Choose (b_*) sufficiently small that
@@ -6338,7 +6324,7 @@ rows,
Completing the square once more gives
(sum_iH_i-Emathfrak e)/(2)frake)/(2)leq c_{{rm seam},n}q^2,
- qquad c_{{rm seam},n}=2+(b_*^2)/(2).
+ c_{{rm seam},n}=2+(b_*^2)/(2).
Apply the degree inequality, then use
@@ -6521,7 +6507,7 @@ over all permitted graphs and cocycles. Fix (b_*) using
(C_{rm ent}). The displayed formula for (C_*(q)) is consequently
available at this stage. Set
[
- lambda(q)=frac1{4C_*(q)},qquad
+ lambda(q)=frac1{4C_*(q)},
gamma_0(q)=frac14(m+1)^{-(m+1)}lambda(q)^{m(m+1)}.
]
These deterministic choices have elementary reciprocals and satisfy
@@ -6591,63 +6577,63 @@ the stated integral bound.
{99}
-bibitem{Allard1972}
+
W. K. Allard, On the first variation of a varifold,
Annals of Mathematics (2) 95 (1972), no. 3, 417--491.
doi:10.2307/1970868.
-bibitem{Aubin1976}
+
T. Aubin, 'Equations diff'erentielles non lin'eaires et
probl`eme de Yamabe concernant la courbure scalaire,
Journal de Math'ematiques Pures et Appliqu'ees (9) 55
(1976), no. 3, 269--296.
-bibitem{BraunSauer2021}
+
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-bibitem{ChodoshLi2024}
+
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+
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+
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+
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-bibitem{GilbargTrudinger}
+
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-bibitem{Gromov1982}
+
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