diff 626631e..b34550f simplicial_volume_scalar_bound_v2.txt
diff --git a/simplicial_volume_scalar_bound_v2.txt b/simplicial_volume_scalar_bound_v2.txt
index 94b9b12..30a2670 100644
--- a/simplicial_volume_scalar_bound_v2.txt
+++ b/simplicial_volume_scalar_bound_v2.txt
@@ -1,48 +1,48 @@
-\documentclass[11pt]{article}
-\usepackage[margin=1in]{geometry}
-\usepackage{amsmath,amssymb,amsthm,mathtools}
-\usepackage[T1]{fontenc}
-\usepackage{lmodern}
-\usepackage{microtype}
-\usepackage{graphicx}
-\usepackage{tikz}
+
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\usetikzlibrary{arrows.meta,calc,positioning}
-\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
+
\hypersetup{
pdftitle={An integral scalar curvature bound for real simplicial volume},
}
-\numberwithin{equation}{section}
-\newtheorem{theorem}{Theorem}[section]
-\newtheorem{proposition}[theorem]{Proposition}
-\newtheorem{lemma}[theorem]{Lemma}
-\newtheorem{corollary}[theorem]{Corollary}
-\theoremstyle{definition}
-\newtheorem{definition}[theorem]{Definition}
-\newtheorem{remark}[theorem]{Remark}
-
-\DeclareMathOperator{\Scal}{Scal}
-\DeclareMathOperator{\Ric}{Ric}
-\DeclareMathOperator{\Vol}{Vol}
-\DeclareMathOperator{\supp}{supp}
-\DeclareMathOperator{\tr}{tr}
-\DeclareMathOperator{\diver}{div}
-\DeclareMathOperator{\Var}{Var}
-\DeclareMathOperator{\dist}{dist}
-\DeclareMathOperator{\id}{id}
-\newcommand{\R}{\mathbb R}
-\newcommand{\N}{\mathbb N}
-\newcommand{\Z}{\mathbb Z}
-\newcommand{\E}{\mathbb E}
-\newcommand{\wscal}{\mathcal S}
-\newcommand{\II}{\mathrm{II}}
-\newcommand{\HS}{\mathrm{HS}}
-\newcommand{\dd}{\,d}
-\newcommand{\cl}{\operatorname{cl}}
-\newcommand{\indic}{\mathbf 1}
-
-
-\begin{document}
+
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For every integer $n\ge3$ there is a constant $a_n>0$, depending only on
$n$, such that every closed connected oriented smooth $n$-manifold $M$
@@ -62,18 +62,18 @@ 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{MaWangXieYuZhu2026}, 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 $n$.
\par\noindent{\footnotesize\ttfamily src-ma2026.pdf}\par
-The \emph{real simplicial volume} of a closed connected oriented
+The real simplicial volume of a closed connected oriented
$n$-manifold $M$ is the $\ell^1$-seminorm of its real fundamental class:
\[
\|M\|=\inf\Bigl\{\sum_j|a_j|:\ \sum_j a_j\sigma_j
- \text{ is a finite real singular cycle representing }[M]_{\R}\Bigr\}.
+ is a finite real singular cycle representing [M]_{\R}\Bigr\}.
\]
It is the cheapest $\ell^1$ cost of writing the fundamental class as a
sum of singular simplices with real coefficients. It depends only on the
@@ -93,18 +93,18 @@ dimensionless; and it records both how negative the scalar curvature
becomes and over how much volume it does so.
-\par\noindent{\footnotesize\url{https://doi.org/10.1007/BFb0075649}}\par
+\par\noindent{\footnotesizehttps://doi.org/10.1007/BFb0075649}\par
+
+[Main theorem]
-\begin{theorem}[Main theorem]
-\label{thm:main}
For every integer $n\ge3$ 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
-\begin{equation}
+
\int_M(\Scal_g^-)^{n/2}\,dV_g\ \ge\ a_n\,\|M\|.
- \label{eq:main}
-\end{equation}
-\end{theorem}
+
+
+
The constant is uniform in both $M$ and $g$. As a representative case,
if $\Scal_g\ge-n(n-1)$, which is the natural normalization in dimension
@@ -117,15 +117,15 @@ 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 $\int_M(\Scal_g^-)^{n/2}dV_g$.
-Theorem~\ref{thm:main} is closely related to the pointwise-uniform
+Theorem~thm:main is closely related to the pointwise-uniform
assertion
-\begin{equation}
- \Scal_g\ge-\kappa^2\quad\Longrightarrow\quad
+
+ \Scal_g\ge-\kappa^2\quad⇒\quad
\|M\|\le C_n\kappa^n\Vol_g(M),\qquad \kappa\ge0 .
- \label{eq:pointwise-form}
-\end{equation}
+
+
For a fixed dimension $n\ge3$ the integral statement for all metrics on
-all such manifolds and \eqref{eq:pointwise-form} for all metrics on all
+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}.
@@ -135,26 +135,26 @@ such manifolds are equivalent by a Yamabe rescaling; see
Several related statements were known before the general case.
-\begin{itemize}
-\item Braun and Sauer \cite{BraunSauer2021} prove a macroscopic
+
+\item Braun and Sauer [cite] prove a macroscopic
counterpart when the volumes of unit balls in the universal cover are
uniformly bounded above; their constant depends also on that bound.
-\item Ma, Wang, Xie, Yu and Zhu \cite{MaWangXieYuZhu2026} prove that
+\item Ma, Wang, Xie, Yu and Zhu [cite] prove that
$\|M\|=0$ when the universal cover is spin and $\Scal_g\ge0$. This is
the qualitative vanishing result used below.
-\item Min, Zheng and Zhu \cite{MinZhengZhu2026} prove the integral
+\item Min, Zheng and Zhu [cite] prove the integral
estimate with coefficient $27/2$ for every metric on every closed
K\"ahler surface, and for an infinite family of non-K\"ahler
symplectic four-manifolds.
-\end{itemize}
+
\par\noindent{\footnotesize\ttfamily src-braunsauer2021.pdf}\par
The one external input is the qualitative vanishing
theorem of
-\cite{MaWangXieYuZhu2026}, stated precisely as
-Theorem~\ref{thm:vanishing-input}: a
+[cite], stated precisely as
+Theorem~thm:vanishing-input: a
closed connected oriented manifold of positive dimension admitting a
metric of nonnegative scalar curvature has zero real simplicial volume.
Everything else is proved here. The work is the passage from that
@@ -166,75 +166,75 @@ 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.
-\begin{enumerate}
-\item \emph{Conformal reduction to $\Scal_g=-1$}
- Proposition~\ref{prop:yamabe-reduction}.
+
+\item Conformal reduction to $\Scal_g=-1$
+ Proposition~prop:yamabe-reduction.
In every conformal class the Yamabe minimizer has constant scalar
curvature. If $\|M\|>0$, the vanishing theorem forces that constant to
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
- Theorem~\ref{thm:main} follows from the normalized estimate
- \begin{equation}
- \|M\|\le C_n\Vol_g(M)\qquad\text{whenever }\Scal_g\equiv-1,
- \label{eq:normalized-target}
- \end{equation}
- which is Theorem~\ref{thm:scalar-volume}.
-
-\item \emph{A detector for $\|M\|$.}
- Proposition~\ref{prop:detector}.
+ Theorem~thm:main follows from the normalized estimate
+
+ \|M\|\le C_n\Vol_g(M)\qquadwhenever \Scal_g\equiv-1,
+
+
+ which is Theorem~thm:scalar-volume.
+
+\item A detector for $\|M\|$.
+ Proposition~prop:detector.
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 \eqref{eq:normalized-target}
+ $\eta$ back with total integral $I$. So (eq:normalized-target)
follows once one bounds $I$ by $C_n\Vol_g(M)$.
-\item \emph{The graph engine.}
+\item The graph engine.
Replace a metric $h$ by the metric induced on the
graph of a function $f$, and shift the weight $\phi$ so that the
weighted volume $e^\phi dV_h$ does not change. The weighted graph
- identity, Lemma~\ref{lem:graph-identity}, writes the change of weighted
+ 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
for the prescribed weighted mean curvature equation
- (Lemma~\ref{lem:graph-prescription}, localized in
- Lemma~\ref{lem:graph-localization}) permits making that square
+ (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~\ref{prop:sweep}: one pass improves the energy of $F$
+ yields Proposition~prop:sweep: one pass improves the energy of $F$
at a cost bounded by an explicit nonnegative reserve $\Pi_p$.
-\item \emph{Iteration to an entropy metric.}
+\item Iteration to an entropy metric.
A conformal bootstrap raises the energy
coefficient from a very small value to a fixed dimensional one
- (Lemma~\ref{lem:entropy-bootstrap}). Phases of sweeps then raise it to
+ (Lemma~lem:entropy-bootstrap). Phases of sweeps then raise it to
an arbitrary prescribed $E$ while keeping the accumulated volume
- expansion controlled (Lemma~\ref{lem:entropy-accumulation}) and the
- residual gradient absolutely small (Lemma~\ref{lem:entropy-phase-exit}).
+ expansion controlled (Lemma~lem:entropy-accumulation) and the
+ residual gradient absolutely small (Lemma~lem:entropy-phase-exit).
The output is an entropy datum with
\[
\wscal(h,\phi)\ge-C+2W-n\log E+E\,e_h(F),\qquad
e_h(F)^{1/2}\le E^{-1/100},
\]
where $W=\log(dV_h/\rho)\ge0$ and $\rho\le dV_g$
- (Theorem~\ref{thm:entropy}).
+ (Theorem~thm:entropy).
-\item \emph{Degree and transfer.}
+\item Degree and transfer.
The measured degree
- estimate, Theorem~\ref{thm:degree}, bounds the signed degree of a map
+ 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
transfer builds such a map whose degree is the
detector pairing $I$ and whose tangential and normal lengths are
- $b_*\sqrt E$ and $b_*\sqrt{qE}$. The factor $E^{n/2}$ produced by the
+ $b_*\sqrt E$ and $b_*√(qE)$. The factor $E^{n/2}$ produced by the
degree bound cancels the $E^{-n/2}$ from the product volume, the
remaining $e^{-W}$ cancels the volume expansion, and what survives is
an integral of $\rho$, hence $\Vol_g(M)$
- (Theorem~\ref{thm:scalar-volume}).
-\end{enumerate}
+ (Theorem~thm:scalar-volume).
+
The last step contains the only place where the exponent of $E$ matters:
the tangent directions of the comparison family contribute the volume
@@ -253,8 +253,8 @@ simplicial volume or bounded cohomology.
Only the transfer uses all of them. A reader willing
to accept the three engines as black boxes can read
the reduction, the transfer and the statements
-of Theorem~\ref{thm:entropy} and Theorem~\ref{thm:degree} for a complete
-proof of Theorem~\ref{thm:main}.
+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
@@ -278,7 +278,7 @@ integral $I$. Graph deformations supply the identity
=D^2-2U(D)+|\II|^2+|d\log L|^2
\]
together with the fact that $D$ can be prescribed by solving the
-quasilinear equation $D=p(x,z,\nabla z)$ with $\partial_zp<0$; this is
+quasilinear equation $D=p(x,z,\nabla z)$ with $∂_zp<0$; this is
what makes the change of weighted scalar curvature positive on demand.
Geometric measure theory supplies the measured degree and its bound by a
weighted integral with gradient penalties, obtained by slicing, weighted
@@ -293,15 +293,15 @@ the metric and the probability map are deformed.
Use the following result, stated here in the precise form needed.
-\begin{theorem}
-\label{thm:vanishing-input}
+
+
Let \(M\) be a closed connected oriented smooth manifold of positive
dimension. If \(M\) admits a smooth Riemannian metric with nonnegative
scalar curvature, then its real simplicial volume is zero.
-\end{theorem}
+
In particular, the hypothesis permits scalar curvature to vanish
-identically. Theorem~\ref{thm:vanishing-input} is the only external result
+identically. Theorem~thm:vanishing-input is the only external result
invoked below without proof.
The Yamabe reformulation of the scalar curvature inequality is
@@ -309,40 +309,40 @@ standard; \cite[Lemma 2.1]{MinZhengZhu2026} records the equivalence of
its uniform pointwise and integral forms. Include the short
classwise reduction that identifies the use of the vanishing input.
-\begin{proposition}
-\label{prop:yamabe-reduction}
-Fix \(n\ge3\). Suppose that a constant \(C_n<\infty\) has the following
+
+
+Fix \(n\ge3\). Suppose that a constant \(C_n<∞\) has the following
property: every closed connected oriented \(n\)-manifold with a smooth
metric \(h\) of constant scalar curvature \(-1\) satisfies
-\begin{equation}
+
\|M\|\le C_n\Vol_h(M).
- \label{eq:scalar-minus-one-target}
-\end{equation}
+
+
Then every smooth Riemannian metric \(g\) on every such manifold
satisfies
-\begin{equation}
+
\|M\|\le C_n\int_M(\Scal_g^-)^{n/2}\,dV_g .
- \label{eq:yamabe-transfer}
-\end{equation}
-\end{proposition}
+
-\begin{proof}
+
+
+Proof.
The assertion is immediate when \(\|M\|=0\), so assume that
\(\|M\|>0\). Write \(Y(M,[g])\) for the Yamabe constant of the
conformal class of \(g\):
\[
Y(M,[g])=
-\inf_{u\in C^\infty(M),\,u>0}
+\inf_{u\in C^∞(M),\,u>0}
\frac{\displaystyle\int_M
- \left(\frac{4(n-1)}{n-2}|du|_g^2+\Scal_g u^2\right)dV_g}
-{\displaystyle\left(\int_Mu^{2n/(n-2)}\,dV_g\right)^{(n-2)/n}}.
+ ((4(n-1))/(n-2)|du|_g^2+\Scal_g u^2)dV_g}
+{\displaystyle(\int_Mu^{2n/(n-2)}\,dV_g)^{(n-2)/n}}.
\]
The Yamabe theorem, developed through the work of Yamabe, Trudinger,
-Aubin, and Schoen \cite{Yamabe1960,Trudinger1968,Aubin1976,Schoen1984},
+Aubin, and Schoen [cite],
gives a conformal metric attaining this value and having constant
scalar curvature; see also \cite[Section 11]{LeeParker1987}.
Its scalar curvature is negative by
-Theorem~\ref{thm:vanishing-input}. After a constant rescaling, it becomes a
+Theorem~thm:vanishing-input. After a constant rescaling, it becomes a
metric \(h\) with \(\Scal_h=-1\). Evaluating the Yamabe quotient at the
constant function for this minimizing metric gives
\[
@@ -352,29 +352,29 @@ For each positive \(u\), H\"older's inequality gives
\[
\int_M\Scal_g u^2\,dV_g
\ge
--\left(\int_M(\Scal_g^-)^{n/2}\,dV_g\right)^{2/n}
- \left(\int_Mu^{2n/(n-2)}\,dV_g\right)^{(n-2)/n}.
+-(\int_M(\Scal_g^-)^{n/2}\,dV_g)^{2/n}
+ (\int_Mu^{2n/(n-2)}\,dV_g)^{(n-2)/n}.
\]
The gradient term in the Yamabe quotient is nonnegative. Taking the
infimum therefore yields
\[
-Y(M,[g])
-\le \left(\int_M(\Scal_g^-)^{n/2}\,dV_g\right)^{2/n}.
+\le (\int_M(\Scal_g^-)^{n/2}\,dV_g)^{2/n}.
\]
Consequently \(\Vol_h(M)\le\int_M(\Scal_g^-)^{n/2}\,dV_g\), and
-\eqref{eq:scalar-minus-one-target} implies
-\eqref{eq:yamabe-transfer}.
-\end{proof}
+(eq:scalar-minus-one-target) implies
+(eq:yamabe-transfer).
+
\par\noindent{\footnotesize J. Math. Pures Appl. (9) 55 (1976), 269--296}\par
-\par\noindent{\footnotesize\url{https://doi.org/10.4310/jdg/1214439291}}\par
+\par\noindent{\footnotesizehttps://doi.org/10.4310/jdg/1214439291}\par
\par\noindent{\footnotesize\ttfamily src-trudinger1968.pdf}\par
\par\noindent{\footnotesize\ttfamily src-yamabe1960.pdf}\par
\par\noindent{\footnotesize\ttfamily src-leeparker1987.pdf}\par
For the rest of the proof one fixes \(M^n\) and \(g\) with
-\(\Scal_g=-1\), and prove \eqref{eq:scalar-minus-one-target}. Continue to assume \(\|M\|>0\).
+\(\Scal_g=-1\), and prove (eq:scalar-minus-one-target). Continue to assume \(\|M\|>0\).
Let \(\Gamma=\pi_1(M)\). This group is countable because \(M\) has a
@@ -382,12 +382,12 @@ finite triangulation. Use two standard facts about bounded
cohomology with trivial real coefficients. First, duality between the
\(\ell^1\)-seminorm on homology and the supremum seminorm on bounded
cohomology gives
-\begin{equation}
+
\|M\|=
-\sup\left\{\langle\beta,[M]_{\R}\rangle:
- \beta\in H_b^n(M;\R),\ \|\beta\|_\infty\le1\right\}.
-\label{eq:bounded-duality}
-\end{equation}
+\sup{<\beta,[M]_{\R}>:
+ \beta\in H_b^n(M;\R),\ \|\beta\|_∞\le1}.
+
+
See \cite[Section 1.1]{Gromov1982}.
Second, the isometric Mapping Theorem identifies \(H_b^k(M;\R)\) with
\(H_b^k(\Gamma;\R)\) for every \(k\ge2\), through the classifying map.
@@ -407,28 +407,28 @@ omitting one argument. Alternation is a cochain projection of norm at
most one and is equivariantly cochain homotopic to the identity through
maps bounded in each degree. It therefore preserves bounded cohomology
classes \cite[Section~4.10, Proposition~4.26]{Frigerio2016}.
-By \eqref{eq:bounded-duality}, the Mapping Theorem, and
+By (eq:bounded-duality), the Mapping Theorem, and
approximation of the seminorm by cocycle representatives, there is a
homogeneous alternating cocycle
\[
- c:\Gamma^{n+1}\longrightarrow\R,\qquad \|c\|_\infty\le1,
+ c:\Gamma^{n+1}→\R,\qquad \|c\|_∞\le1,
\]
whose corresponding bounded singular cohomology class has value
-\begin{equation}
- I:=\langle[c],[M]_{\R}\rangle\ge\frac12\|M\|.
- \label{eq:detector-value}
-\end{equation}
+
+ I:=<[c],[M]_{\R}>\ge\frac12\|M\|.
+
+
For completeness, one can choose a class in
-\eqref{eq:bounded-duality} with value at least \(3\|M\|/4\), choose a
-group representative of norm at most \(1+\varepsilon\), alternate it,
-and divide by \(1+\varepsilon\), with \(0<\varepsilon\le1/2\). Replacing
+(eq:bounded-duality) with value at least \(3\|M\|/4\), choose a
+group representative of norm at most \(1+ε\), alternate it,
+and divide by \(1+ε\), with \(0<ε\le1/2\). Replacing
the cocycle by its negative first, if necessary, fixes the sign.
\par\noindent{\footnotesize\ttfamily src-frigerio2016.pdf}\par
Allow a finite number of copies of every group label. More
-precisely, let \(\mathcal L=\Gamma\times\{1,\ldots,s\}\), with the free
+precisely, let \(\mathcal L=\Gamma\times\{1,...,s\}\), with the free
left \(\Gamma\)-action on the first factor, and evaluate \(c\) on
\(\mathcal L^{n+1}\) by projection to \(\Gamma^{n+1}\). The same
notation will be used when further finite copies of \(\mathcal L\) are
@@ -438,50 +438,50 @@ on \(\mathcal L\).
Let
\[
\Delta_{\mathcal L}=
-\left\{p=(p_\ell)_{\ell\in\mathcal L}:
- p_\ell\ge0,\ \sum_\ell p_\ell=1,\ p\text{ has finite support}\right\}.
+{p=(p_\ell)_{\ell\in\mathcal L}:
+ p_\ell\ge0,\ \sum_\ell p_\ell=1,\ p has finite support}.
\]
On each finite-dimensional face define the \(n\)-form
-\begin{equation}
-\eta_p(v_1,\ldots,v_n)=n!\,c(p,v_1,\ldots,v_n).
-\label{eq:probability-form}
-\end{equation}
+
+\eta_p(v_1,...,v_n)=n!\,c(p,v_1,...,v_n).
+
+
These definitions agree on common faces.
Call a map \(F=(F_\ell)\) with nonnegative coordinates and
-\(\sum_\ell F_\ell^2=1\) a \emph{root probability map}. For the maps
+\(\sum_\ell F_\ell^2=1\) a root probability map. For the maps
used here, whose coordinate supports are locally finite, the squared
coordinates \(p_\ell=F_\ell^2\) give the associated map to
\(\Delta_{\mathcal L}\).
-\begin{proposition}
-\label{prop:detector}
+
+
The form \(\eta\) is closed on every finite-dimensional face of
\(\Delta_{\mathcal L}\). There is a smooth \(\Gamma\)-equivariant map
\[
- F:\widetilde M\longrightarrow\ell^2(\mathcal L),\qquad
+ F:\widetilde M→\ell^2(\mathcal L),\qquad
F_\ell\ge0,\qquad \sum_\ell F_\ell^2=1,
\]
with finitely many label orbits and locally finite compact supports for
its coordinates. If \(p_\ell=F_\ell^2\), then the descended form
satisfies
-\begin{equation}
+
\int_M p^*\eta=I.
- \label{eq:detector-integral}
-\end{equation}
-\end{proposition}
+
+
-\begin{proof}
-Write \(\epsilon(v)=\sum_\ell v_\ell\) for the mass of a finitely
+
+Proof.
+Write \(ε(v)=\sum_\ell v_\ell\) for the mass of a finitely
supported vector. Multilinear extension of the cocycle equation gives
\[
-0=\sum_{j=0}^{n+1}(-1)^j\epsilon(w_j)
- c(w_0,\ldots,\widehat w_j,\ldots,w_{n+1}).
+0=\sum_{j=0}^{n+1}(-1)^jε(w_j)
+ c(w_0,...,\widehat w_j,...,w_{n+1}).
\]
-Choose \(w_0\) of mass one and let \(w_1,\ldots,w_{n+1}\) have mass
-zero. It follows that \(c(w_1,\ldots,w_{n+1})=0\). All tangent vectors
+Choose \(w_0\) of mass one and let \(w_1,...,w_{n+1}\) have mass
+zero. It follows that \(c(w_1,...,w_{n+1})=0\). All tangent vectors
to a probability simplex have mass zero. Differentiating
-\eqref{eq:probability-form} now shows \(d\eta=0\).
+(eq:probability-form) now shows \(d\eta=0\).
To construct \(F\), choose finitely many evenly covered open sets in
\(M\) and smooth nonnegative bumps subordinate to them whose nonzero
@@ -490,48 +490,48 @@ deck translates, denoting the resulting functions by \(b_\ell\).
Their supports are compact and locally finite on \(\widetilde M\).
Set
\[
- F_\ell=\frac{b_\ell}{\left(\sum_j b_j^2\right)^{1/2}}.
+ F_\ell=\frac{b_\ell}{(\sum_j b_j^2)^{1/2}}.
\]
The denominator is positive, and the stated properties follow.
Choose a finite oriented triangulation of \(M\). On its lift, assign
-one label to every vertex equivariantly, and let \(p^{\mathrm{aff}}\)
+one label to every vertex equivariantly, and let \(p^{aff}\)
be the probability map affine in barycentric coordinates on each
simplex. The homotopy
-\((1-t)p+t p^{\mathrm{aff}}\) is equivariant and has locally finite
+\((1-t)p+t p^{aff}\) is equivariant and has locally finite
support. Applying Stokes' theorem to each simplex times \([0,1]\), and
cancelling the common faces, shows that its pullback integral is
constant. On a simplex with ordered vertex labels
-\(\ell_0,\ldots,\ell_n\), write
+\(\ell_0,...,\ell_n\), write
\[
-p^{\mathrm{aff}}=e_{\ell_0}
+p^{aff}=e_{\ell_0}
+\sum_{j=1}^nt_j(e_{\ell_j}-e_{\ell_0}).
\]
Alternation makes the value of
-\(c(p^{\mathrm{aff}},dp^{\mathrm{aff}},\ldots,dp^{\mathrm{aff}})\)
+\(c(p^{aff},dp^{aff},...,dp^{aff})\)
on the coordinate frame constant and equal to
-\(c(\ell_0,\ldots,\ell_n)\). The coordinate simplex has volume
-\(1/n!\), so the factor \(n!\) in \eqref{eq:probability-form} makes its
+\(c(\ell_0,...,\ell_n)\). The coordinate simplex has volume
+\(1/n!\), so the factor \(n!\) in (eq:probability-form) makes its
integral exactly this cocycle value. Summing over the oriented
-fundamental triangulation gives \eqref{eq:detector-integral}. The
+fundamental triangulation gives (eq:detector-integral). The
identification with the chosen group class is the usual cochain
realization of the Mapping Theorem: an equivariant vertex labeling on
the universal cover evaluates the homogeneous group cochain, and two
such labelings give cochain-homotopic maps by the simplex prism
construction.
-\end{proof}
+
The deformations below use auxiliary probability choices. Record their integration convention so that the detector value remains
unambiguous.
-Let \((\Omega,\mu)\) be a standard Borel probability space with a
+Let \((Ω,\mu)\) be a standard Borel probability space with a
measure-preserving \(\Gamma\)-action. A measured equivariant family
over \(M\) consists of \(\Gamma\)-equivariant Borel data on
-\(\widetilde M\times\Omega\), smooth in the \(\widetilde M\) variable.
+\(\widetilde M\timesΩ\), smooth in the \(\widetilde M\) variable.
Geometrically one may regard the diagonal quotient
\[
- \mathcal M=(\widetilde M\times\Omega)/\Gamma
+ \mathcal M=(\widetilde M\timesΩ)/\Gamma
\]
as a space whose leaves are ordinary smooth manifolds. If
\(\Gamma_\omega\) is the stabilizer of \(\omega\), the corresponding
@@ -547,13 +547,13 @@ Choose a finite evenly covered
atlas \(U_a\) of \(M\), chosen lifts \(\widetilde U_a\), and a smooth
partition of unity \(\chi_a\) subordinate to the atlas. For an
integrable equivariant leafwise \(n\)-form \(\alpha\), set
-\begin{equation}
+
\int_{\mathcal M}\alpha
=
-\sum_a\int_\Omega\int_{\widetilde U_a}
+\sum_a\int_Ω\int_{\widetilde U_a}
(\chi_a\circ\pi)\,\alpha_\omega\,d\mu(\omega).
-\label{eq:measured-integral}
-\end{equation}
+
+
Equivariance and invariance of \(\mu\) show that this value is
independent of the lifts and partition. The same convention applies
to nonnegative densities. Write \(dV_h\) for the resulting measured
@@ -565,7 +565,7 @@ may depend on the instance: the original manifold and metric and the
finite sequence of constructions already made. For the forms used
here, these bounds ensure integrability over the compact base and
justify differentiation and Fubini. Leafwise Stokes' theorem follows from
-\eqref{eq:measured-integral}: apply ordinary Stokes' theorem in each
+(eq:measured-integral): apply ordinary Stokes' theorem in each
lifted chart to the partitioned form; the terms containing
\(d\chi_a\) cancel on overlaps after the measure-preserving changes of
transverse coordinate.
@@ -581,7 +581,7 @@ same properties. Stokes' theorem on the measured product with
\int_{\mathcal M}(p^1)^*\eta.
\]
Thus every probability map constructed below continues to detect the
-same number \(I\). Enlarging \(\Omega\) by an independent probability
+same number \(I\). Enlarging \(Ω\) by an independent probability
space also preserves the integral by Fubini. Discard invariant
null sets whenever needed. More general measured spaces of leaves
will enter the degree estimate in the degree estimate below; their chart and
@@ -609,196 +609,196 @@ is Perelman's \(R^m\), with his \(f=-\phi\)
\cite[Section~1.3]{Perelman2002}. The estimates below concern its
behavior under particular graph deformations. For antecedents of
the graph and vertical Jacobi identities, see
-\cite{MaWangXieYuZhu2026}; one includes the
+[cite]; one includes the
calculation with the conventions needed here.
\par\noindent{\footnotesize\ttfamily src-perelman2002.pdf}\par
-\begin{lemma}[Weighted graph identity]
-\label{lem:graph-identity}
+[Weighted graph identity]
+
Let \((X,h)\) be a smooth Riemannian manifold, let \(\phi,f\) be
smooth real functions, and let \(N>0\). Set
\[
L=(1+N^2|df|_h^2)^{1/2},\qquad v=L^{-1},\qquad
- U=\frac{N\nabla_h f}{L},\qquad
- h'=h+N^2df\otimes df,\qquad \phi'=\phi-\log L.
+ U=(N\nabla_h f)/(L),\qquad
+ h'=h+N^2df ⊗ df,\qquad \phi'=\phi-\log L.
\]
View \(h'\) as the induced metric on the graph
-\(\Sigma=\{(x,Nf(x)):x\in X\}\) in
+\(Σ=\{(x,Nf(x)):x\in X\}\) in
\((X\times\R,h+dz^2)\), and extend \(\phi\) constantly in the
\(z\)-direction. Orient the graph by
\(\nu=(-U,v)\), put
-\(\II(V,W)=\langle\bar\nabla_V\nu,W\rangle\), and let
+\(\II(V,W)=<\bar\nabla_V\nu,W>\), and let
\[
- H=\tr_\Sigma\II,\qquad
- D=H+\partial_\nu\phi=-\diver_\phi U .
+ H=\tr_Σ\II,\qquad
+ D=H+∂_\nu\phi=-\diver_\phi U .
\]
Here \(D\), and its derivative \(U(D)\), are regarded as functions on
\(X\) through the graph projection. Then
-\begin{equation}
+
\wscal(h',\phi')-\wscal(h,\phi)
=D^2-2U(D)+|\II|_{h'}^2+|d\log L|_{h'}^2.
- \label{eq:graph-identity}
-\end{equation}
+
+
Moreover,
-\begin{equation}
+
e^{\phi'}dV_{h'}=e^\phi dV_h,\qquad
N^2|df|_{h'}^2=1-v^2=|U|_h^2,\qquad
|d\log L|_{h'}\le L|U|_h|\II|_{h'} .
- \label{eq:graph-basic-bounds}
-\end{equation}
-\end{lemma}
+
+
-\begin{proof}
+
+Proof.
Write \(\bar h=h+dz^2\), and suppress pullback by the graph
projection. The formula for the graph mean curvature gives
-\(H=-\diver U\), while \(\partial_\nu\phi=-U(\phi)\); hence the stated
+\(H=-\diver U\), while \(∂_\nu\phi=-U(\phi)\); hence the stated
formula for \(D\) holds. The rank-one determinant formula gives
\(dV_{h'}=L\,dV_h\), proving the first assertion in
-\eqref{eq:graph-basic-bounds}. The inverse rank-one formula gives its
+(eq:graph-basic-bounds). The inverse rank-one formula gives its
second assertion.
Let
\[
P=|\II|^2+
- \bigl(\Ric_{\bar h}-\bar\nabla^2\phi\bigr)(\nu,\nu).
+ (\Ric_{\bar h}-\bar\nabla^2\phi)(\nu,\nu).
\]
The Gauss equation and the decompositions of the Laplacian and
gradient of \(\phi\) give
-\begin{align*}
+
\Scal_{h'}&=\Scal_{\bar h}-2\Ric_{\bar h}(\nu,\nu)
+H^2-|\II|^2,\\
\Delta_{\bar h}\phi
- &=\Delta_\Sigma(\phi|_\Sigma)
- +H\partial_\nu\phi+\bar\nabla^2\phi(\nu,\nu),\\
- |d(\phi|_\Sigma)|^2&=|d\phi|_{\bar h}^2-(\partial_\nu\phi)^2.
-\end{align*}
+ &=\Delta_Σ(\phi|_Σ)
+ +H∂_\nu\phi+\bar\nabla^2\phi(\nu,\nu),\\
+ |d(\phi|_Σ)|^2&=|d\phi|_{\bar h}^2-(∂_\nu\phi)^2.
+
Since \(\wscal(\bar h,\phi)=\wscal(h,\phi)\), their combination is
-\begin{equation}
- \wscal(h',\phi|_\Sigma)
+
+ \wscal(h',\phi|_Σ)
=\wscal(h,\phi)+D^2-2P+|\II|^2.
- \label{eq:graph-gauss-weighted}
-\end{equation}
+
+
Adding \(\log v\) to the restricted weight changes the left side by
\[
- -2\Delta_{\Sigma,\phi}\log v-|d\log v|^2
- =-\frac{2\Delta_{\Sigma,\phi}v}{v}+|d\log v|^2 .
+ -2\Delta_{Σ,\phi}\log v-|d\log v|^2
+ =-\frac{2\Delta_{Σ,\phi}v}{v}+|d\log v|^2 .
\]
The tangential part of vertical translation is
\[
- T=\partial_z-v\nu=(vU,1-v^2).
+ T=∂_z-v\nu=(vU,1-v^2).
\]
For a hypersurface variation with normal speed \(v\), the variation
of \(H\) is
-\(-\Delta_\Sigma v-(|\II|^2+\Ric_{\bar h}(\nu,\nu))v\).
-The additional variation of \(\partial_\nu\phi\) is
+\(-\Delta_Σ v-(|\II|^2+\Ric_{\bar h}(\nu,\nu))v\).
+The additional variation of \(∂_\nu\phi\) is
\[
v\bar\nabla^2\phi(\nu,\nu)
- -\langle d(\phi|_\Sigma),dv\rangle_\Sigma .
+ -< d(\phi|_Σ),dv>_Σ .
\]
Including the tangential velocity \(T\), and using that vertical
translation preserves the metric, weight, and \(D\), therefore gives
-\begin{equation}
- (\Delta_{\Sigma,\phi}+P)v=T(D).
- \label{eq:graph-translation}
-\end{equation}
+
+ (\Delta_{Σ,\phi}+P)v=T(D).
+
+
The horizontal part of \(T\) is \(vU\), so \(T(D)=vU(D)\).
-Substitution in \eqref{eq:graph-gauss-weighted} proves
-\eqref{eq:graph-identity}. Finally, differentiating
-\(v=\langle\nu,\partial_z\rangle\) tangentially gives
+Substitution in (eq:graph-gauss-weighted) proves
+(eq:graph-identity). Finally, differentiating
+\(v=<\nu,∂_z>\) tangentially gives
\(dv(V)=\II(V,T)\). Since \(|T|^2=1-v^2=|U|^2\), it follows that
\(|d\log v|\le v^{-1}|U||\II|\), which is the last bound in
-\eqref{eq:graph-basic-bounds}.
-\end{proof}
+(eq:graph-basic-bounds).
+
-\paragraph{A one-coordinate calculation.}
+A one-coordinate calculation.:
The square and differentiated terms in the graph identity suggest
an ideal target for \(ND\), the scaled weighted mean curvature.
Let \(F\) be one smooth input coordinate, restrict
-to a region where \(f>0\), and fix \(\epsilon,a>0\). Also let
-\(h_{\mathrm{in}}\le h\) be a comparison metric. In the later sweep,
-the incoming energy is measured in \(h_{\mathrm{in}}\), while \(h\)
+to a region where \(f>0\), and fix \(ε,a>0\). Also let
+\(h_{in}\le h\) be a comparison metric. In the later sweep,
+the incoming energy is measured in \(h_{in}\), while \(h\)
already includes the earlier graphs.
Put \(A=|U|_h\). Where \(A>0\), write \(U=Ae\) with \(|e|_h=1\).
The longitudinal unit tangent to the graph is
\(\tau=(ve,A)\). Define
\[
- x=-\II(\tau,\tau),\qquad X_{\parallel}=Nx,\qquad t=A^2/f,\qquad
- B=1-|e|_{h_{\mathrm{in}}}^2\in[0,1].
+ x=-\II(\tau,\tau),\qquad X_{∥}=Nx,\qquad t=A^2/f,\qquad
+ B=1-|e|_{h_{in}}^2\in[0,1].
\]
At \(A=0\), choose any \(h\)-unit vector \(e\), put \(\tau=(e,0)\),
and use the same definitions. With \(\tau^\flat\) denoting the dual
covector in the graph metric \(h'\), set
\[
- \II^{\mathrm{rem}}=\II+x\,\tau^\flat\otimes\tau^\flat .
+ \II^{rem}=\II+x\,\tau^\flat ⊗ \tau^\flat .
\]
-The component of \(\II^{\mathrm{rem}}\) in the
-\(\tau^\flat\otimes\tau^\flat\) direction vanishes, so
-\begin{equation}
- \begin{gathered}
+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^{\mathrm{rem}}|_{h'}^2,\qquad
- \tr_{h'}\II^{\mathrm{rem}}=H+x,\\
- x=e(A)\quad\text{where }A>0.
- \end{gathered}
- \label{eq:graph-longitudinal}
-\end{equation}
+ |\II|_{h'}^2=x^2+|\II^{rem}|_{h'}^2,\qquad
+ \tr_{h'}\II^{rem}=H+x,\\
+ x=e(A)\quadwhere A>0.
+
+
+
For the last identity, use \(\nu=(-Ae,v)\) and \(A^2+v^2=1\) to
compute
\[
\II(\tau,\tau)=-v^2e(A)+Av\,e(v)=-e(A).
\]
-The first identity in \eqref{eq:graph-longitudinal} follows directly
+The first identity in (eq:graph-longitudinal) follows directly
from the definitions of \(U,L,A\).
Define the ideal target for \(ND\) by
-\begin{equation}
- Z=\epsilon a(F^2/f-f)+A^2/(2f).
- \label{eq:graph-ideal-target}
-\end{equation}
-Since \(NU(A^2)=2A^2X_{\parallel}\), with both sides zero at \(A=0\),
+
+ Z=ε a(F^2/f-f)+A^2/(2f).
+
+
+Since \(NU(A^2)=2A^2X_{∥}\), with both sides zero at \(A=0\),
differentiation gives
-\begin{align}
+
-2NU(Z)
- ={}&-4\epsilon a(F/f)NAe(F)
- +2\epsilon a(F^2/f^2+1)LA^2 \notag\\
- &-2X_{\parallel}t+Lt^2.
- \label{eq:graph-ideal-derivative}
-\end{align}
+ ={}&-4ε a(F/f)NAe(F)
+ +2ε a(F^2/f^2+1)LA^2 \notag\\
+ &-2X_{∥}t+Lt^2.
+
+
The first term couples the prescribed \(F^2/f\) to the incoming
energy. The \(A^2/(2f)\) term completes the longitudinal square:
-\begin{equation}
- \begin{gathered}
- X_{\parallel}^2-2X_{\parallel}t+Lt^2
- =(X_{\parallel}-t)^2+(L-1)t^2,\\
- \begin{aligned}
- &\epsilon N^2|dF|_{h_{\mathrm{in}}}^2
- -4\epsilon a(F/f)NAe(F)\\
- &\quad=\epsilon\left|N\,dF
- -2a(F/f)A\,h_{\mathrm{in}}(e,\cdot)\right|_{h_{\mathrm{in}}}^2
- -4\epsilon a^2(F/f)^2A^2(1-B)\\
- &\quad\ge-4\epsilon a^2(F/f)^2A^2(1-B).
- \end{aligned}
- \end{gathered}
- \label{eq:graph-ideal-completions}
-\end{equation}
+
+
+ X_{∥}^2-2X_{∥}t+Lt^2
+ =(X_{∥}-t)^2+(L-1)t^2,\\
+
+ &ε N^2|dF|_{h_{in}}^2
+ -4ε a(F/f)NAe(F)\\
+ &\quad=ε|N\,dF
+ -2a(F/f)A\,h_{in}(e,\cdot)|_{h_{in}}^2
+ -4ε a^2(F/f)^2A^2(1-B)\\
+ &\quad\ge-4ε a^2(F/f)^2A^2(1-B).
+
+
+
+
Consequently
-\begin{align}
+
&Z^2-2NU(Z)+N^2|\II|_{h'}^2+N^2|d\log L|_{h'}^2
- +\epsilon N^2|dF|_{h_{\mathrm{in}}}^2 \notag\\
- \ge{}&Z^2+(X_{\parallel}-t)^2+N^2|\II^{\mathrm{rem}}|_{h'}^2
+ +ε N^2|dF|_{h_{in}}^2 \notag\\
+ \ge{}&Z^2+(X_{∥}-t)^2+N^2|\II^{rem}|_{h'}^2
+N^2|d\log L|_{h'}^2 \notag\\
- &+\epsilon A^2\bigl[(2a-4a^2)F^2/f^2+2a\bigr]
- +4\epsilon a^2(F^2/f^2)A^2B \notag\\
- &+(L-1)\bigl[2\epsilon a(F^2/f^2+1)A^2+t^2\bigr].
- \label{eq:graph-ideal-bound}
-\end{align}
+ &+ε A^2[(2a-4a^2)F^2/f^2+2a]
+ +4ε a^2(F^2/f^2)A^2B \notag\\
+ &+(L-1)[2ε a(F^2/f^2+1)A^2+t^2].
+
+
At \(A=0\), \(t=0\) and
-\((X_{\parallel}-t)^2+N^2|\II^{\mathrm{rem}}|^2=N^2|\II|^2\); every other
+\((X_{∥}-t)^2+N^2|\II^{rem}|^2=N^2|\II|^2\); every other
direction-dependent term vanishes. Thus the inequality is independent
of the arbitrary direction at such points.
@@ -810,7 +810,7 @@ interaction of the ordered graph metrics.
Specify the uniformity required for the analytic construction.
-A manifold has \emph{bounded geometry} here if it has coordinate
+A manifold has bounded geometry here if it has coordinate
charts of a common positive size whose smaller concentric domains
cover the manifold, with a common positive buffer to each chart
boundary, and with uniform bounds on the metric, its inverse, and
@@ -819,12 +819,12 @@ chart size, may depend on the instance. A function is uniformly
smooth if all its coordinate derivatives, including its value, are
uniformly bounded on these charts. For a weight \(\phi\), only
positive-order derivatives are required to be uniformly bounded.
-The \emph{uniform Hölder space} below means the Banach space
+The uniform Hölder space below means the Banach space
obtained by taking the supremum of the corresponding Hölder norms in
these charts, after a fixed shrinkage within the buffers.
The use of complete prescribed graphs has precedents in
-\cite{MaWangXieYuZhu2026}. The
+[cite]. The
following statement records the precise uniform and measurable
conclusions used here and in the hypersurface argument. Its positive
branch realizes the designed prescription with a positive floor and
@@ -832,55 +832,55 @@ will be localized for the sweep. Its tangent branch supplies the value
approximation and the intrinsic gradient bound used in
the degree estimate.
-\begin{lemma}[Prescribed graphs on a complete manifold]
-\label{lem:graph-prescription}
+[Prescribed graphs on a complete manifold]
+
Let \((X,h)\) be a complete smooth Riemannian manifold without
boundary, with bounded geometry as above, and suppose the
positive-order derivatives of \(\phi\) are uniformly bounded. For
a graph of height \(z\), set
\[
L_z=(1+|dz|_h^2)^{1/2},\quad v_z=L_z^{-1},\quad
- U_z=\frac{\nabla_hz}{L_z},\quad
+ U_z=(\nabla_hz)/(L_z),\quad
D_z=-\diver_\phi U_z .
\]
Consider either of the following equations \(D_z=p(x,z,v_z)\).
-\begin{enumerate}
-\renewcommand{\labelenumi}{\textup{(\roman{enumi})}}
-\item Let \(N,\epsilon,a>0\), \(0<\delta\le1\), and \(H_*\ge0\).
+
+\renewcommand{\labelenumi}{(\roman{enumi)}}
+\item Let \(N,ε,a>0\), \(0<\delta\le1\), and \(H_*\ge0\).
Suppose \(F_i,\beta\) are uniformly smooth,
\(|F_i|\le1\), and \(0\le\beta\le H_*\). On the region \(z>0\),
prescribe
- \begin{equation}
+
p(x,z,v)=
- \frac{\epsilon a(F_i(x)^2+\delta^2)+(1-v^2)/2}{z}
- -(\epsilon a+\beta(x))\frac{z}{N^2}.
- \label{eq:graph-prescription-positive}
- \end{equation}
+ (ε a(F_i(x)^2+\delta^2)+(1-v^2)/2)/(z)
+ -(ε a+\beta(x))(z)/(N^2).
+
+
\item Let \(e,w>0\) be constants. Suppose \(B,f_0\) are uniformly
smooth and \(B\ge B_{\min}>0\). On the region
\(|(f_0(x)-ez)/w|<\pi/2\), prescribe
- \begin{equation}
- p(x,z,v)=B(x)\tan\bigl((f_0(x)-ez)/w\bigr).
- \label{eq:graph-prescription-tangent}
- \end{equation}
-\end{enumerate}
+
+ p(x,z,v)=B(x)\tan((f_0(x)-ez)/w).
+
+
+
In either case there is a smooth solution of bounded height with
uniform bounds on all its derivatives, and it lies uniformly inside
the stated open prescription region. Its angle satisfies
-\(v_z\ge v_0>0\). In Case \textup{(i)} it satisfies
-\begin{equation}
- c_{\min}:=\delta\sqrt{\frac{\epsilon a}{\epsilon a+H_*}}
+\(v_z\ge v_0>0\). In Case (i) it satisfies
+
+ c_{\min}:=\delta√((ε a)/(ε a+H_*))
\ \le\ f:=z/N\ \le 2.
- \label{eq:graph-height-positive}
-\end{equation}
-In Case \textup{(ii)} there is \(\eta>0\) such that
+
+
+In Case (ii) there is \(\eta>0\) such that
\[
- \left|\frac{f_0-ez}{w}\right|\le\frac{\pi}{2}-\eta;
+ |(f_0-ez)/(w)|\le(\pi)/(2)-\eta;
\qquad
- |ez-f_0|<\frac{\pi w}{2},\quad
- |z|\le\frac{\|f_0\|_\infty+\pi w/2}{e}.
+ |ez-f_0|<(\pi w)/(2),\quad
+ |z|\le(\|f_0\|_∞+\pi w/2)/(e).
\]
The constants \(v_0,\eta\) and the derivative bounds may depend on
all the indicated parameters and common bounds of the data.
@@ -890,7 +890,7 @@ for some \(0<\alpha<1\), and whose values have a uniform margin inside
the prescription region. Each such solution may have its own
exponent, bound, and margin.
-The graph metric \(h_z=h+dz\otimes dz\) is again complete and has
+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,
\[
@@ -903,39 +903,39 @@ Assume a common chart radius and buffers, common bounds for the
metric, its inverse, and all their chart derivatives, and common
bounds for the positive-order derivatives of \(\phi\). Require
common bounds, including order zero, for all chart derivatives of
-\(F_i,\beta\) in Case \textup{(i)} or \(B,f_0\) in Case
-\textup{(ii)}. In Case \textup{(i)}, let \(N,\epsilon,a,\delta\)
+\(F_i,\beta\) in Case (i) or \(B,f_0\) in Case
+(ii). In Case (i), let \(N,ε,a,\delta\)
range in fixed compact subsets of their permitted positive ranges,
and require a common finite upper bound on \(H_*\). In Case
-\textup{(ii)}, let \(e,w\) range in fixed compact subsets of
-\((0,\infty)\), and require \(B\ge B_{\min}\) for one common
+(ii), let \(e,w\) range in fixed compact subsets of
+\((0,∞)\), and require \(B\ge B_{\min}\) for one common
\(B_{\min}>0\). Suppose also that the data and chart overlaps are
Borel and that the charts accessible along each leaf admit a
countable Borel enumeration. Then the preceding estimates for the
solutions and their graph metrics and weights hold with constants common to the family,
-including a common positive height margin in Case \textup{(i)}
-and a common margin from the poles in Case \textup{(ii)}. The
+including a common positive height margin in Case (i)
+and a common margin from the poles in Case (ii). The
solutions are Borel in the chart representations and commute with
any equivariant identifications of the complete problems.
-\end{lemma}
-\begin{proof}
+
+Proof.
Give existence and the estimates together, since the estimates
also give uniqueness and measurable dependence.
-\emph{Height barriers and comparison.}
-In Case \textup{(i)}, replace \(F_i,\beta\) by \(tF_i,t\beta\),
+Height barriers and comparison.
+In Case (i), replace \(F_i,\beta\) by \(tF_i,t\beta\),
\(0\le t\le1\). At \(t=0\) the constant \(f=\delta\) is a
solution. For every \(t\), the constant graphs \(f=c_{\min}\) and
\(f=2\) are a lower and an upper barrier, respectively. Indeed their
weighted mean curvature is zero, and at angle \(v=1\) the right side
-of \eqref{eq:graph-prescription-positive} is nonnegative at
+of (eq:graph-prescription-positive) is nonnegative at
\(f=c_{\min}\) and strictly negative at \(f=2\). In Case
-\textup{(ii)}, replace \(f_0\) by \(tf_0\), starting with \(z=0\).
+(ii), replace \(f_0\) by \(tf_0\), starting with \(z=0\).
For \(\tau=\pi/2-\eta\), the graphs
\[
- z_-=\frac{tf_0-w\tau}{e},\qquad
- z_+=\frac{tf_0+w\tau}{e}
+ z_-=(tf_0-w\tau)/(e),\qquad
+ z_+=(tf_0+w\tau)/(e)
\]
have arguments \(+\tau\) and \(-\tau\), respectively. Their
weighted mean curvatures have a common bound independent of
@@ -946,16 +946,16 @@ They give the asserted height and pole margins.
Both comparisons use strict decrease of \(p\) in its height
argument, with the other arguments held fixed:
-\begin{align}
- p_z&=-\frac{\epsilon a(F_i^2+\delta^2)+(1-v^2)/2}{z^2}
- -\frac{\epsilon a+\beta}{N^2}
- \le-\frac{\epsilon a}{N^2}
- &&\text{in Case \textup{(i)}}, \notag\\
- p_z&=-\frac{eB}{w}\sec^2\bigl((f_0-ez)/w\bigr)
+
+ p_z&=-(ε a(F_i^2+\delta^2)+(1-v^2)/2)/(z^2)
+ -(ε a+\beta)/(N^2)
+ \le-(ε a)/(N^2)
+ &&in Case (i), \notag\\
+ p_z&=-(eB)/(w)\sec^2((f_0-ez)/w)
\le-\frac{eB_{\min}}w
- &&\text{in Case \textup{(ii)}}.
- \label{eq:graph-height-monotonicity}
-\end{align}
+ &&in Case (ii).
+
+
At an interior positive maximum of the difference between a
solution and an upper barrier, their gradients, hence their angles,
agree. The negative elliptic principal part of \(D_z\) gives a
@@ -971,59 +971,59 @@ give such a limit, which attains the extremum at the center of an
interior chart. This
justifies the same comparison without compactness.
-\emph{The angle bound.}
-For a solution, \eqref{eq:graph-translation} and the chain rule give
-\begin{equation}
- (\Delta_{\Sigma,\phi}+P)v_z
- =v_z\langle p_x,U_z\rangle
+The angle bound.
+For a solution, (eq:graph-translation) and the chain rule give
+
+ (\Delta_{Σ,\phi}+P)v_z
+ =v_z< p_x,U_z>
+p_z(1-v_z^2)+p_vT(v_z).
- \label{eq:graph-angle-equation}
-\end{equation}
+
+
Here \(p_x\) differentiates with \(z,v\) fixed, and use was made of \(T(z)=1-v_z^2\). The ambient curvature and weight bounds imply
\(P\ge-C\), since the remaining term \(|\II|^2\) is nonnegative.
At an infimum of \(v_z\), taken by the same recentering procedure,
-\(\Delta_{\Sigma,\phi}v_z\ge0\) and \(T(v_z)=0\). Consequently, if
+\(\Delta_{Σ,\phi}v_z\ge0\) and \(T(v_z)=0\). Consequently, if
\(v_z\le1/2\) there,
-\begin{equation}
+
(1-v_z^2)|p_z|\le v_z(C+|p_x|).
- \label{eq:graph-angle-infimum}
-\end{equation}
+
+
The height margins just proved bound \(p_x\) uniformly, and
-\eqref{eq:graph-height-monotonicity} bounds \(|p_z|\) below by a
+(eq:graph-height-monotonicity) bounds \(|p_z|\) below by a
positive constant. Since \(1-v_z^2\ge3/4\) in this case,
-\eqref{eq:graph-angle-infimum} gives a uniform positive lower bound
+(eq:graph-angle-infimum) gives a uniform positive lower bound
for \(v_z\). In using recentered limits at this stage, a solution
is first in an individual uniform \(C^{2,\alpha}\) class. Ordinary
local bootstrapping gives its individual higher regularity before
the common estimates are deduced, so the differentiations and
limits in this argument are justified.
-\emph{Higher estimates.}
+Higher estimates.
In a uniform chart let
-\(\omega=e^\phi\sqrt{\det h}\), and divide \(\omega\) by its value
+\(\omega=e^\phi√(\det h)\), and divide \(\omega\) by its value
at the chart center. The resulting density has a common upper
bound and a positive lower bound, and all its derivatives are
bounded. The equation is of the form
-\begin{equation}
- -\partial_j A^j(x,\partial z)=F(x,z,\partial z).
- \label{eq:graph-divergence-equation}
-\end{equation}
+
+ -∂_j A^j(x,∂ z)=F(x,z,∂ z).
+
+
For example \(A^j=\omega h^{jk}z_k/L_z\) after the normalization.
On the established height and gradient ranges the flux and the
right side are smooth with uniformly bounded derivatives, and
-\(\partial A^j/\partial z_k\) is uniformly elliptic. Differentiating
-\eqref{eq:graph-divergence-equation} gives linear divergence
-equations for \(\partial_kz\). Their leading coefficients are
+\(∂ A^j/∂ z_k\) is uniformly elliptic. Differentiating
+(eq:graph-divergence-equation) gives linear divergence
+equations for \(∂_kz\). Their leading coefficients are
bounded and uniformly elliptic. Gradient dependence of \(F\)
produces a bounded first-order coefficient; the remaining terms
are a bounded function plus the divergence of a bounded vector
-field. The term \(F_z\partial_kz\) is included in the bounded
+field. The term \(F_z∂_kz\) is included in the bounded
source, since the gradient is already bounded.
The interior De Giorgi--Nash estimate for these equations, with
bounded lower-order coefficients and source data in \(L^{s/2}\)
and \(L^s\) for \(s>\dim X\), gives a common Hölder bound for
-\(\partial z\); see \cite[Theorem~8.24]{GilbargTrudinger}.
+\(∂ z\); see \cite[Theorem~8.24]{GilbargTrudinger}.
The coefficients in the nondivergence form of the original
equation are now Hölder continuous. Interior Schauder estimates
give a uniform \(C^{2,\alpha}\) bound for some \(\alpha>0\), after
@@ -1032,7 +1032,7 @@ Schauder estimates again yields uniform bounds for every
derivative. All these estimates are local with common buffers, so
their constants are uniform throughout \(X\).
-\emph{Continuation.}
+Continuation.
Work in uniform \(C^{2,\alpha}\) spaces on the open set of functions
with a uniform height margin. The equation \(D_z-p(x,z,v_z)=0\)
defines a smooth map to uniform \(C^{0,\alpha}\). At a solution
@@ -1064,14 +1064,14 @@ specified above, so existence follows. Comparison gives the
claimed uniqueness.
The graph estimates now follow from
-\eqref{eq:graph-basic-bounds}. In particular \(h_z\ge h\), and
+(eq:graph-basic-bounds). In particular \(h_z\ge h\), and
the gradient bound also gives a uniform upper comparison in the
given charts. Thus \(h_z\) is complete and has uniform buffered
charts with bounds on all derivatives. The corresponding
derivative bounds on \(\log L_z\) give those for \(\phi_z\).
This also permits successive applications of the lemma.
-\emph{Borel dependence.}
+Borel dependence.
For completeness, fix a starting plaque and enumerate the charts
accessible along its leaf by words in the Borel overlap maps.
Represent a solution by its functions in this countable family
@@ -1092,14 +1092,14 @@ leaf. It applies independently of the transverse mass of the
chart family. Finally, applying an equivariant identification
to a solution produces a solution of the identified problem, so
uniqueness proves equivariance.
-\end{proof}
-\par\noindent{\footnotesize\url{https://doi.org/10.1007/978-3-642-61798-0}}\par
-\par\noindent{\footnotesize\url{https://doi.org/10.1007/978-1-4612-4190-4}}\par
+\par\noindent{\footnotesizehttps://doi.org/10.1007/978-3-642-61798-0}\par
+\par\noindent{\footnotesizehttps://doi.org/10.1007/978-1-4612-4190-4}\par
-The positive solution in Lemma~\ref{lem:graph-prescription} is
+
+The positive solution in Lemma~lem:graph-prescription is
defined on the complete manifold and need not have compact support.
For a probability coordinate one needs a graph supported near that
coordinate. Obtain one by forcing the complete solution close
@@ -1107,122 +1107,122 @@ to a small constant on an outer region, and cutting it off there.
The following lemma states the quantitative features of this
procedure that enter the sweep.
-\begin{lemma}[Localization of the positive graph]
-\label{lem:graph-localization}
+[Localization of the positive graph]
+
Let \(X,h,\phi\) satisfy the hypotheses of
-Lemma~\ref{lem:graph-prescription}, with \(X\) connected, and let
-\(h_{\mathrm{ref}}\le h\) be a complete second metric. Require the
-coefficients of \(h_{\mathrm{ref}}\), its inverse, and all their
+Lemma~lem:graph-prescription, with \(X\) connected, and let
+\(h_{ref}\le h\) be a complete second metric. Require the
+coefficients of \(h_{ref}\), its inverse, and all their
coordinate derivatives to have uniform bounds in the same buffered
charts used for \(h\).
Suppose \(F_i\) is uniformly smooth, \(|F_i|\le1\), with compact
support \(K\), and let \(\alpha\ge0\) be a uniformly smooth
function. Fix
\[
- \epsilon>0,\qquad N^2=l\epsilon,\qquad
+ ε>0,\qquad N^2=lε,\qquad
a=\tfrac12+\theta,\qquad 0<l\le1,\qquad l\le\theta\le\tfrac1{16}.
\]
-There is a radius \(R_{\mathrm{loc}}\), depending only on the
+There is a radius \(R_{loc}\), depending only on the
dimension and a fixed sufficiently small numerical constant
\(c_0>0\), with the following property. For every sufficiently
large \(H_*\), one can choose smooth functions
\(\beta_s\ge0\) and \(0\le\chi\le1\) such that, with
\(\beta=\beta_s+\alpha\),
-\begin{equation}
+
0\le\beta\le H_*,
\qquad
N|d\beta_s|_h
- \le c_0\bigl(\theta\epsilon^{3/2}+\beta_s^{3/2}\bigr).
- \label{eq:graph-spatial-cutoff}
-\end{equation}
+ \le c_0(\thetaε^{3/2}+\beta_s^{3/2}).
+
+
The function \(\beta_s\) vanishes on a neighborhood of \(K\).
The function \(\chi\) is one there, is supported in
-\(\{\dist_{h_{\mathrm{ref}}}(x,K)<R_{\mathrm{loc}}\}\), and changes
+\(\{\dist_{h_{ref}}(x,K)<R_{loc}\}\), and changes
from one to zero only within a region where \(F_i=0\) and
\(\beta=H_*\). This region and the support of \(d\chi\) have fixed
-positive buffers in \(h_{\mathrm{ref}}\), independent of \(H_*\).
+positive buffers in \(h_{ref}\), independent of \(H_*\).
The derivatives of \(\chi\) are bounded independently of \(H_*\),
\(\chi\) is flat on its zero set, and \(|d\chi|_h^2\le C\chi\).
Set \(\delta=H_*^{-2}\), and let \(\widetilde f=z/N\) be the
-complete solution of \eqref{eq:graph-prescription-positive} for
+complete solution of (eq:graph-prescription-positive) for
this \(\beta,\delta\). Define \(f=\chi\widetilde f\) and form the
graph of height \(Nf\). Denote its quantities by \(L,U,D,\II\),
put \(A=|U|_h\), and set, where \(f>0\),
-\begin{equation}
- ND^0=\epsilon a(F_i^2/f-f)+A^2/(2f).
- \label{eq:graph-ideal-prescription}
-\end{equation}
+
+ ND^0=ε a(F_i^2/f-f)+A^2/(2f).
+
+
The expressions \(A^2/f\) and \(ND^0\) are extended by zero
where \(f=0\). The bump \(f\) is smooth,
\(0\le f\le2\), is positive on \(K\), and is flat on its zero set.
On the region \(\chi=1\) it satisfies
-\begin{equation}
- ND=ND^0-\beta f+\epsilon a\delta^2/f .
- \label{eq:graph-core-prescription}
-\end{equation}
+
+ ND=ND^0-\beta f+ε a\delta^2/f .
+
+
On the region where the cutoff changes, the following quantities
-tend uniformly to zero as \(H_*\to\infty\), with the preceding
+tend uniformly to zero as \(H_*\to∞\), with the preceding
data and parameters fixed:
-\begin{equation}
- \begin{gathered}
- N^2\bigl|\wscal(h+N^2df\otimes df,\phi-\log L)
- -\wscal(h,\phi)\bigr|,\quad
+
+
+ N^2|\wscal(h+N^2df ⊗ df,\phi-\log L)
+ -\wscal(h,\phi)|,\quad
A^2/f,\quad |ND^0|,\quad \beta f,\\
- |ND|,\quad N|\II|,\quad N|d\log L|_{h+N^2df\otimes df}.
- \end{gathered}
- \label{eq:graph-cutoff-small}
-\end{equation}
+ |ND|,\quad N|\II|,\quad N|d\log L|_{h+N^2df ⊗ df}.
+
+
+
In particular, on all of \(X\) the function
-\(\mathrm{err}:=ND-(ND^0-\beta f)\) tends uniformly to zero.
+\(err:=ND-(ND^0-\beta f)\) tends uniformly to zero.
All data retain the corresponding uniform bounds for each fixed
choice of \(H_*\), with the positive-order convention for weights.
For the family assertion, let a standard Borel parameter space index
such problems of a fixed dimension. Require common sizes and buffers
for these simultaneous charts, and common bounds there for
-\(h,h^{-1},h_{\mathrm{ref}},h_{\mathrm{ref}}^{-1}\) and all their
+\(h,h^{-1},h_{ref},h_{ref}^{-1}\) and all their
coordinate derivatives. Require common bounds for all derivatives,
including values, of \(F_i,\alpha\), and for the positive-order
-derivatives of \(\phi\). Let \((\epsilon,l,\theta)\) range in a fixed
+derivatives of \(\phi\). Let \((ε,l,\theta)\) range in a fixed
compact subset of the permitted parameter set. Require the data,
-including \(h_{\mathrm{ref}}\), to
+including \(h_{ref}\), to
be Borel in the charts and preserved by the equivariant identifications
under consideration. Require also the Borel chart overlaps and
countable Borel enumerations of accessible charts in
-Lemma~\ref{lem:graph-prescription}. Then the choices are Borel and
+Lemma~lem:graph-prescription. Then the choices are Borel and
commute with those identifications; their bounds for each fixed
-\(H_*\), and the limits above as \(H_*\to\infty\), are uniform over
+\(H_*\), and the limits above as \(H_*\to∞\), are uniform over
the family. When \(K\) is empty one may take \(f=0\) and omit the
operation.
-\end{lemma}
-\begin{proof}
+
+Proof.
First construct the outer coefficient while keeping its
transition width bounded. Starting at a fixed positive distance
from \(K\), a scalar profile \(b\) can increase from zero to
\(H_*\) at speed at most
\[
- \frac{c_0}{C N}\bigl(\theta\epsilon^{3/2}+b^{3/2}\bigr),
+ (c_0)/(C N)(\thetaε^{3/2}+b^{3/2}),
\]
where the fixed factor \(C\) leaves room for the estimates below.
Choose the speed comparable to this expression during the
increase, with smooth flattening at either end. The width needed
is at most
-\begin{equation}
- C N\int_0^\infty\frac{dt}
- {\theta\epsilon^{3/2}+t^{3/2}}
- =C\frac{N}{\theta^{1/3}\sqrt\epsilon}
+
+ C N\int_0^∞\frac{dt}
+ {\thetaε^{3/2}+t^{3/2}}
+ =C\frac{N}{\theta^{1/3}\sqrtε}
=C\frac{\sqrt l}{\theta^{1/3}}\le C .
- \label{eq:graph-cutoff-width}
-\end{equation}
+
+
For example one can use the primitive of the reciprocal of a smooth
positive function comparable to
-\(\theta\epsilon^{3/2}+t^{3/2}\) as the new profile coordinate,
+\(\thetaε^{3/2}+t^{3/2}\) as the new profile coordinate,
and compose with fixed smooth flattenings. The last inequality
uses \(l\le\theta\le1/16\).
One may use this profile in a smooth approximation of
-\(\dist_{h_{\mathrm{ref}}}(x,K)\) with gradient at most \(2\)
+\(\dist_{h_{ref}}(x,K)\) with gradient at most \(2\)
and a uniformly small error. Here is a construction with the
needed uniformity. Shrinking the controlled charts gives normal
charts at a fixed positive radius, by the uniform geodesic
@@ -1239,16 +1239,16 @@ bound follows. Higher derivatives have uniform bounds from the
smooth local kernels: in differentiating the normalized kernel
one may subtract the distance at the center, leaving a uniformly
bounded oscillation on its support. These derivative bounds hold in
-the common buffered charts, since the \(h_{\mathrm{ref}}\) exponential
+the common buffered charts, since the \(h_{ref}\) exponential
maps and kernels have uniformly bounded derivatives there. The inverse metric
-inequality \(h^{-1}\le h_{\mathrm{ref}}^{-1}\) then bounds the
+inequality \(h^{-1}\le h_{ref}^{-1}\) then bounds the
gradient of the composed profile in \(h\).
Choose this raw profile \(b\) to vanish near \(K\), to equal
-\(H_*\) past the width in \eqref{eq:graph-cutoff-width}, and to
+\(H_*\) past the width in (eq:graph-cutoff-width), and to
leave fixed outer buffers. To accommodate \(\alpha\), take a
smooth nondecreasing saturation
-\(\sigma_{H_*}:[0,\infty)\to[0,H_*]\) that equals the identity
+\(\sigma_{H_*}:[0,∞)\to[0,H_*]\) that equals the identity
up to \(H_*/3\), equals \(H_*\) from \(H_*\) on, and has a
derivative bounded by a numerical constant. Put
\[
@@ -1262,9 +1262,9 @@ the identity, both \(\beta_s\) and the scale of \(b\) are
comparable to \(H_*\). The formula
\[
d\beta_s=\sigma_{H_*}'(b+\alpha)\,db
- +\bigl(\sigma_{H_*}'(b+\alpha)-1\bigr)d\alpha
+ +(\sigma_{H_*}'(b+\alpha)-1)d\alpha
\]
-therefore proves \eqref{eq:graph-spatial-cutoff}: choose the
+therefore proves (eq:graph-spatial-cutoff): choose the
initial fixed speed factor large enough to handle the first
term, and then choose \(H_*\) large enough for the fixed
\(d\alpha\) term to be a small fraction of
@@ -1272,7 +1272,7 @@ term, and then choose \(H_*\) large enough for the fixed
the raw profile bound. Where \(b=H_*\), the total \(\beta\)
equals \(H_*\). A cutoff supported beyond fixed additional
buffers can consequently be chosen once at a radius
-\(R_{\mathrm{loc}}\) as in the statement. Take it to be the
+\(R_{loc}\) as in the statement. Take it to be the
square of a smooth nonnegative cutoff that is flat at its outer
end. This gives \(|d\chi|^2\le C\chi\) and the other asserted
cutoff properties. If some outer regions are empty, the
@@ -1282,89 +1282,89 @@ Next prove that the complete solution can be cut on this
plateau. All constants in the rest of the proof may use the
fixed data and parameters, but not \(H_*\). Write
\[
- c_H=H_*^{-2}\sqrt{\frac{\epsilon a}{\epsilon a+H_*}}
+ c_H=H_*^{-2}√((ε a)/(ε a+H_*))
\asymp H_*^{-5/2}.
\]
-By \eqref{eq:graph-height-positive},
+By (eq:graph-height-positive),
\(c_H\le\widetilde f\le2\). Cover the cutoff region by inner chart
boxes, each with a fixed succession of larger buffered boxes whose
closures lie in the plateau \(F_i=0,\ \beta=H_*\). Their sizes may
use the current \(h\)-chart bounds inside the fixed
-\(h_{\mathrm{ref}}\) buffers; all these bounds were fixed before
+\(h_{ref}\) buffers; all these bounds were fixed before
\(H_*\), so the boxes and their buffers have sizes independent of
\(H_*\). Our target is that, for every integer \(j\ge0\) and every
\(M>0\), there is \(C_{j,M}\) such that, for all sufficiently large
\(H_*\),
-\begin{equation}
+
\|\widetilde f-c_H\|_{C^j}
\le C_{j,M}H_*^{-M},
- \label{eq:graph-plateau-jets}
-\end{equation}
+
+
on the inner boxes, with norms in these fixed coordinates.
The constants use only the indicated data and parameters fixed
before \(H_*\). The family assertion of
-Lemma~\ref{lem:graph-prescription} is applied separately for each
+Lemma~lem:graph-prescription is applied separately for each
fixed \(H_*\); the estimates that follow establish this
superpolynomial decay uniformly as \(H_*\) grows.
There is first a global angle bound
-\begin{equation}
- v\ge\frac{c_1}{1+\sqrt{H_*}}
- \label{eq:graph-global-plateau-angle}
-\end{equation}
+
+ v\ge\frac{c_1}{1+√(H_*)}
+
+
for the uncut graph. In fact, at a recentered infimum with
\(v\le1/2\), the formulas for \(p_x,p_z\) and
-\eqref{eq:graph-spatial-cutoff} give
+(eq:graph-spatial-cutoff) give
\[
|p_z|\ge c_2(1+\beta+\widetilde f^{-2}),\qquad
- |p_x|\le C\bigl(\widetilde f^{-1}
- +\widetilde f(1+\beta^{3/2})\bigr).
+ |p_x|\le C(\widetilde f^{-1}
+ +\widetilde f(1+\beta^{3/2})).
\]
For the first bound use \(1-v^2\ge3/4\) in \(p_z\); for the
second use the fixed bounds on \(F_i,\alpha\) and their
derivatives. Since \(\widetilde f\le2\) and \(\beta\le H_*\),
the ratio of the right side of the second estimate, plus a
constant, to the right side of the first is at most
-\(C(1+\sqrt{H_*})\). Equation
-\eqref{eq:graph-angle-infimum} proves
-\eqref{eq:graph-global-plateau-angle}. This recentering is done
+\(C(1+√(H_*))\). Equation
+(eq:graph-angle-infimum) proves
+(eq:graph-global-plateau-angle). This recentering is done
for each fixed \(H_*\), using the individual regularity supplied
-by Lemma~\ref{lem:graph-prescription}.
+by Lemma~lem:graph-prescription.
On each of the outer plateau boxes the equation is
-\begin{equation}
- ND=\frac{\epsilon a\delta^2}{\widetilde f}
- +\frac{1-v^2}{2\widetilde f}
- -(\epsilon a+H_*)\widetilde f,
+
+ ND=(ε a\delta^2)/(\widetilde f)
+ +(1-v^2)/(2\widetilde f)
+ -(ε a+H_*)\widetilde f,
\qquad
- (\epsilon a+H_*)c_H^2=\epsilon a\delta^2 .
- \label{eq:graph-plateau-equation}
-\end{equation}
+ (ε a+H_*)c_H^2=ε a\delta^2 .
+
+
There is an exponential value estimate on an inner box:
-\begin{equation}
- 0\le\widetilde f-c_H\le C\exp(-c_3\sqrt{H_*}).
- \label{eq:graph-plateau-value}
-\end{equation}
-To prove it, let \(d_{\mathrm{face}}\) be ordinary coordinate
+
+ 0\le\widetilde f-c_H\le C\exp(-c_3√(H_*)).
+
+
+To prove it, let \(d_{face}\) be ordinary coordinate
distance to a face of the outer box and put
\[
- w_1=\sum_{\mathrm{faces}}3
- \exp(-\gamma\sqrt{H_*}\,d_{\mathrm{face}}).
+ w_1=\sum_{faces}3
+ \exp(-\gamma√(H_*)\,d_{face}).
\]
On the boundary, \(c_H+w_1\ge3\ge\widetilde f\).
Its first and second coordinate derivatives are bounded by
-\(C\gamma\sqrt{H_*}w_1\) and \(C\gamma^2H_*w_1\),
+\(C\gamma√(H_*)w_1\) and \(C\gamma^2H_*w_1\),
respectively. Direct differentiation of the weighted graph
divergence therefore bounds its mean curvature from below by
\[
- -C(\gamma^2H_*+\gamma\sqrt{H_*})w_1 .
+ -C(\gamma^2H_*+\gamma√(H_*))w_1 .
\]
For this graph, the height terms in its prescription satisfy
\[
- \frac{\epsilon a\delta^2}{N(c_H+w_1)}
- -\frac{(\epsilon a+H_*)(c_H+w_1)}N
- =-\frac{\epsilon a+H_*}{N}w_1
- \left(1+\frac{c_H}{c_H+w_1}\right).
+ (ε a\delta^2)/(N(c_H+w_1))
+ -\frac{(ε a+H_*)(c_H+w_1)}N
+ =-(ε a+H_*)/(N)w_1
+ (1+(c_H)/(c_H+w_1)).
\]
The remaining angle term is at most
\(C\gamma^2H_*w_1\), because
@@ -1373,43 +1373,43 @@ The remaining angle term is at most
Choosing \(\gamma>0\) sufficiently small and then \(H_*\)
sufficiently large makes \(c_H+w_1\) an upper barrier.
Comparison with this barrier and the lower constant \(c_H\)
-proves \eqref{eq:graph-plateau-value} on any inner box a fixed
+proves (eq:graph-plateau-value) on any inner box a fixed
distance from the faces.
On a smaller box the angle is bounded below independently of
\(H_*\). To see this, choose a smooth chart cutoff
\(0\le\xi\le1\), supported in a region where
-\eqref{eq:graph-plateau-value} holds and equal to one on the
+(eq:graph-plateau-value) holds and equal to one on the
smaller box. Minimize \(v/\xi\) over \(\{\xi>0\}\).
-The global lower bound \eqref{eq:graph-global-plateau-angle}
+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
\[
- v<\xi/2,\qquad \xi\ge c/\sqrt{H_*},\qquad
- \nabla_\Sigma v=(v/\xi)\nabla_\Sigma\xi,\qquad
- \Delta_{\Sigma,\phi}v\ge(v/\xi)\Delta_{\Sigma,\phi}\xi.
+ v<\xi/2,\qquad \xi\ge c/√(H_*),\qquad
+ \nabla_Σ v=(v/\xi)\nabla_Σ\xi,\qquad
+ \Delta_{Σ,\phi}v\ge(v/\xi)\Delta_{Σ,\phi}\xi.
\]
The formula for the restriction of an ambient function to the
graph gives
-\(\Delta_{\Sigma,\phi}\xi\ge-C(1+|\II|)\). The positive
+\(\Delta_{Σ,\phi}\xi\ge-C(1+|\II|)\). The positive
\(|\II|^2\) term in \(P\), and completion of its square against
\(C|\II|/\xi\), show that the left side of
-\eqref{eq:graph-angle-equation} is at least \(-CH_*v\).
+(eq:graph-angle-equation) is at least \(-CH_*v\).
On this plateau \(p_x=0\). Since \(v<1/2\), the \(p_z\) term
on its right side is at most \(-c/\widetilde f^2\).
Also \(p_v=-v/(N\widetilde f)\) and
\(|T(v)|\le Cv/\xi\), so that right side is at most
\[
- -\frac{c}{\widetilde f^2}
- +\frac{C\sqrt{H_*}\,v}{\widetilde f}.
+ -(c)/(\widetilde f^2)
+ +\frac{C√(H_*)\,v}{\widetilde f}.
\]
Combining the two bounds and multiplying by \(\widetilde f^2\)
would give
\[
c\le CH_*\widetilde f^2+
- C\sqrt{H_*}\,\widetilde f .
+ C√(H_*)\,\widetilde f .
\]
-By \eqref{eq:graph-plateau-value},
+By (eq:graph-plateau-value),
\(\widetilde f\le2c_H\) for large \(H_*\), and the last right
side tends to zero. This contradiction proves \(v\ge1/2\) on
the smaller box.
@@ -1419,25 +1419,25 @@ using any growth rate for the global elliptic constants.
In a still smaller box, rescale coordinates \(x=x_0+c_Hy\) and
write \(\widetilde f=c_Hg\). The weighted graph curvature in
the rescaled chart is \(c_HD\), and its right side becomes
-\begin{align*}
+
c_Hp(x,Nc_Hg,v)
- &=\frac{\epsilon a\delta^2+(1-v^2)/2}{Ng}
- -\frac{c_H^2(\epsilon a+H_*)g}{N}\\
- &=\frac{\epsilon a\delta^2+(1-v^2)/2}{Ng}
- -\frac{\epsilon a\delta^2g}{N}.
-\end{align*}
+ &=(ε a\delta^2+(1-v^2)/2)/(Ng)
+ -(c_H^2(ε a+H_*)g)/(N)\\
+ &=(ε a\delta^2+(1-v^2)/2)/(Ng)
+ -(ε a\delta^2g)/(N).
+
Here \(1\le g\le2\) for large \(H_*\), and the angle bound gives
a uniform gradient bound. The rescaled metric and normalized
weight have uniformly bounded derivatives; the displayed
right side and all its derivatives are uniformly bounded on
these ranges. The local higher estimates in the proof of
-Lemma~\ref{lem:graph-prescription}, applied on unit boxes in
+Lemma~lem:graph-prescription, applied on unit boxes in
these coordinates, consequently give uniform bounds for every
derivative of \(g\). Scaling back gives bounds of the form
\(C_jc_H^{1-j}\) for the \(j\)-th derivatives of
\(\widetilde f\). Interior interpolation between these polynomial
-bounds and the exponential bound \eqref{eq:graph-plateau-value}, on
-further fixed buffered boxes, proves \eqref{eq:graph-plateau-jets}.
+bounds and the exponential bound (eq:graph-plateau-value), on
+further fixed buffered boxes, proves (eq:graph-plateau-jets).
Indeed the usual finite-difference interpolation of a
\(j\)-th derivative between the supremum norm and a higher
derivative norm gives an exponential factor times a power of
@@ -1448,64 +1448,64 @@ Now cut off. On the transition region,
smaller than every power of \(H_*^{-1}\), and the derivatives
of \(\chi\) are fixed. On \(\{\chi>0\}\),
\[
- \frac{|d(\chi\widetilde f)|^2}{\chi\widetilde f}
+ (|d(\chi\widetilde f)|^2)/(\chi\widetilde f)
\le
- 2\widetilde f\,\frac{|d\chi|^2}{\chi}
- +2\chi\,\frac{|d\widetilde f|^2}{\widetilde f}
+ 2\widetilde f\,(|d\chi|^2)/(\chi)
+ +2\chi\,(|d\widetilde f|^2)/(\widetilde f)
=O(c_H).
\]
It follows that \(A^2/f\le N^2|df|^2/f\) tends to zero.
All fixed derivatives of \(f\) are \(O(c_H)\) there.
The graph formulas therefore give \(ND,N|\II|\), and
\(N|d\log L|\) tending to zero. They also control
-\(\nabla D\), so \eqref{eq:graph-identity} gives the first
-limit in \eqref{eq:graph-cutoff-small}. Finally \(F_i=0\) in
-this region, \(ND^0=-\epsilon af+A^2/(2f)\), and
+\(\nabla D\), so (eq:graph-identity) gives the first
+limit in (eq:graph-cutoff-small). Finally \(F_i=0\) in
+this region, \(ND^0=-ε af+A^2/(2f)\), and
\(\beta f\le H_*\,O(c_H)\to0\). This proves every limit in
-\eqref{eq:graph-cutoff-small}. To verify the extensions at the zero
+(eq:graph-cutoff-small). To verify the extensions at the zero
set, write \(\chi=\zeta^2\), where \(\zeta\) is smooth and flat there.
On \(\{\zeta>0\}\),
\[
- \frac{|df|_h^2}{f}
+ (|df|_h^2)/(f)
=4\widetilde f\,|d\zeta|_h^2
- +4\zeta\langle d\zeta,d\widetilde f\rangle_h
- +\zeta^2\frac{|d\widetilde f|_h^2}{\widetilde f}.
+ +4\zeta< d\zeta,d\widetilde f>_h
+ +\zeta^2(|d\widetilde f|_h^2)/(\widetilde f).
\]
The right side extends smoothly and flatly across \(\{\zeta=0\}\),
since \(\widetilde f>0\). The same is therefore true of
\[
- \frac{A^2}{f}
- =\frac{N^2}{1+N^2|df|_h^2}\,\frac{|df|_h^2}{f}.
+ (A^2)/(f)
+ =(N^2)/(1+N^2|df|_h^2)\,(|df|_h^2)/(f).
\]
Moreover, \(F_i\) vanishes on a neighborhood of the zero set of
\(f\). Thus \(ND^0\) extends smoothly by zero there, and all jets
of \(f=\zeta^2\widetilde f\) vanish on its zero set.
On the uncut region the last term
-\(\varpi=\epsilon a\delta^2/f\) in
-\eqref{eq:graph-core-prescription} has
-\begin{equation}
- \|\varpi\|_\infty
- \le\delta\sqrt{\epsilon a(\epsilon a+H_*)}\longrightarrow0,
+\(\varpi=ε a\delta^2/f\) in
+(eq:graph-core-prescription) has
+
+ \|\varpi\|_∞
+ \le\delta√(ε a(ε a+H_*))→0,
\qquad
-2NU(\varpi)
- =\frac{2\epsilon a\delta^2 L A^2}{f^2}\ge0 .
- \label{eq:graph-floor-derivative}
-\end{equation}
-Since \(ND^0-\beta f\ge-(\epsilon a+\beta)f\), its contribution
+ =(2ε a\delta^2 L A^2)/(f^2)\ge0 .
+
+
+Since \(ND^0-\beta f\ge-(ε a+\beta)f\), its contribution
inside the curvature square satisfies
-\begin{equation}
+
(ND^0-\beta f+\varpi)^2
\ge (ND^0-\beta f)^2
- -2\epsilon a\delta^2(\epsilon a+H_*).
- \label{eq:graph-floor-square}
-\end{equation}
+ -2ε a\delta^2(ε a+H_*).
+
+
The last loss tends to zero. These estimates on the core and
-\eqref{eq:graph-cutoff-small} on the transition prove the
-claimed bound for \(\mathrm{err}\). The curvature estimates
-use the derivative sign in \eqref{eq:graph-floor-derivative}
+(eq:graph-cutoff-small) on the transition prove the
+claimed bound for \(err\). The curvature estimates
+use the derivative sign in (eq:graph-floor-derivative)
and the full transition estimates, as well as that bound on
-the value of \(\mathrm{err}\).
+the value of \(err\).
Every construction has uniform bounds for fixed \(H_*\).
For Borel data, the distance to the open nonzero set of \(F_i\)
@@ -1513,10 +1513,10 @@ is Borel: using the enumerated charts it is the infimum of
lengths of a countable dense collection of chart paths ending
where \(F_i\ne0\). The normalized smoothing kernels and all
subsequent scalar operations are Borel and equivariant.
-Lemma~\ref{lem:graph-prescription} supplies these same
+Lemma~lem:graph-prescription supplies these same
properties for the complete solution. This proves the final
assertions.
-\end{proof}
+
Now apply the localized graph to each coordinate of a root
@@ -1529,13 +1529,13 @@ For a root probability map, its energy is
\]
Fix a smooth nonnegative nondecreasing function
-\(\Psi:[0,\infty)\to[0,\infty)\) such that
-\begin{equation}
+\(\Psi:[0,∞)\to[0,∞)\) such that
+
\Psi(s)=0\quad(s\le8),\qquad
\Psi(s)\le Cs^2,\qquad |\Psi'(s)|^2\le C\Psi(s),\qquad
\Psi(s)\ge cs^2\quad(s\ge R_\Psi),
- \label{eq:sweep-boost-profile}
-\end{equation}
+
+
where \(R_\Psi\) and the constants are numerical. Such a
function can be obtained as the square of a smooth
nondecreasing function that is zero up to \(8\), has bounded
@@ -1548,16 +1548,16 @@ constant \(\kappa>0\). In the latter case,
\psi(s)\ge c\kappa s^2\quad(s\ge R_\Psi).
\]
-\paragraph{Notation for an ordered sweep.}
+Notation for an ordered sweep.:
Introduce the pointwise quantities for any candidate sequence of
graphs before asserting that the sequence can be constructed. The
pass parameters are
-\begin{equation}
- \epsilon>0,\qquad N^2=l\epsilon,\qquad
+
+ ε>0,\qquad N^2=lε,\qquad
a=\tfrac12+\theta,\qquad \lambda=\tfrac12-2\theta,\qquad
0<l\le1,\quad l\le\theta\le\tfrac1{16},
- \label{eq:sweep-parameters}
-\end{equation}
+
+
with \(N>0\). On one leaf, start with \((h_0,\phi_0,F)\), where
\(F_i\ge0\) and \(\sum_iF_i^2=1\). Suppose nonnegative smooth
bumps \(f_i\) form a locally finite family ordered in finitely many
@@ -1575,24 +1575,24 @@ quantities for \(f_i\), and let \(h_+,\phi_+\) be the final pair.
The subscript \(p\) denotes the current pass in \(W_p\) and in the
reserve \(\Pi_p\) below.
The graph identities give the output and its normalization:
-\begin{equation}
- \begin{gathered}
- h_+=h_0+N^2\sum_i df_i\otimes df_i,\qquad
+
+
+ h_+=h_0+N^2\sum_i df_i ⊗ df_i,\qquad
W_p:=\sum_i\log L_i=\log(dV_{h_+}/dV_{h_0})\ge0,\\
\phi_+=\phi_0-W_p,\qquad
e^{\phi_+}dV_{h_+}=e^{\phi_0}dV_{h_0},\qquad
r:=\sum_i f_i^2>0,\qquad Y_i=f_i/\sqrt r .
- \end{gathered}
- \label{eq:sweep-output}
-\end{equation}
+
+
+
The sums are pointwise finite, with at most one nonzero bump per
-color. The implication \(F_i\ne0\Rightarrow f_i>0\) ensures
+color. The implication \(F_i\ne0⇒ f_i>0\) ensures
\(r>0\). In all the quantities below, indices are restricted to
\(f_i>0\).
For each such index, use the longitudinal notation of the
one-coordinate calculation, with \(h=h_{i-1}\),
-\(h'=h_i\), and \(h_{\mathrm{in}}=h_0\):
+\(h'=h_i\), 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
@@ -1600,27 +1600,27 @@ one-coordinate calculation, with \(h=h_{i-1}\),
\]
Here \(\tau_i=(L_i^{-1}e_i,A_i)\) and \(x_i=e_i(A_i)\) when
\(A_i>0\). At \(A_i=0\) use any unit direction as in
-\eqref{eq:graph-longitudinal}. In the graph metric \(h_i\), define
-\begin{equation}
- \begin{gathered}
- \II_i^{\mathrm{rem}}
- :=\II_i+x_i\,\tau_i^\flat\otimes\tau_i^\flat,\\
- |\II_i|^2=x_i^2+|\II_i^{\mathrm{rem}}|^2,\qquad
- \tr\II_i^{\mathrm{rem}}=H_i+x_i .
- \end{gathered}
- \label{eq:sweep-longitudinal}
-\end{equation}
+(eq:graph-longitudinal). In the graph metric \(h_i\), define
+
+
+ \II_i^{rem}
+ :=\II_i+x_i\,\tau_i^\flat ⊗ \tau_i^\flat,\\
+ |\II_i|^2=x_i^2+|\II_i^{rem}|^2,\qquad
+ \tr\II_i^{rem}=H_i+x_i .
+
+
+
In particular, at \(A_i=0\) the combination
-\((X_i-t_i)^2+N^2|\II_i^{\mathrm{rem}}|^2=N^2|\II_i|^2\)
+\((X_i-t_i)^2+N^2|\II_i^{rem}|^2=N^2|\II_i|^2\)
is independent of the chosen direction. The losses from comparing
the current graph with the input and final metrics are
\[
B_i=1-|e_i|_{h_0}^2,\qquad
C_i=
- \begin{cases}
+
1-N^2|df_i|_{h_+}^2/A_i^2,&A_i>0,\\
0,&A_i=0.
- \end{cases}
+
\]
Metric monotonicity and the inverse graph formula give
\(0\le B_i,C_i\le1\).
@@ -1628,32 +1628,32 @@ Metric monotonicity and the inverse graph formula give
The normalization gives probability weights \(\nu_i=f_i^2/r\).
Write \(\E_\nu b_i=\sum_i\nu_i b_i\). The mean and centered
quantities used in the estimate are
-\begin{equation}
- \begin{gathered}
+
+
m=\sum_iA_i^2,\qquad
- u_i=\frac{A_i^2}{\epsilon\nu_i},\qquad
- \bar u=\E_\nu u_i=\frac m\epsilon,\\
- \sigma_i=\frac{F_i^2}{\nu_i},\qquad \E_\nu\sigma_i=1,\\
+ u_i=(A_i^2)/(ε\nu_i),\qquad
+ \bar u=\E_\nu u_i=\frac mε,\\
+ \sigma_i=(F_i^2)/(\nu_i),\qquad \E_\nu\sigma_i=1,\\
M_0=a(1-r)-\bar u/2,\qquad
V_i=a(\sigma_i-1)+\lambda(u_i-\bar u),\qquad
\Var_\nu u=\E_\nu(u_i-\bar u)^2 .
- \end{gathered}
- \label{eq:sweep-statistics}
-\end{equation}
+
+
+
The identity for \(\E_\nu\sigma_i\) uses
-\(F_i\ne0\Rightarrow f_i>0\); the individual ratios \(\sigma_i\)
+\(F_i\ne0⇒ f_i>0\); the individual ratios \(\sigma_i\)
need not be at most one.
Finally, define two nonnegative quantities:
-\begin{align}
- T_0={}&\sum_i A_i^2\left[
- C_i+4a^2\sigma_iB_i+(L_i-1)\{2a(\sigma_i+r)+u_i\}\right],
- \label{eq:sweep-T}\\
- \mathcal X={}&\sum_i\left[
- (X_i-t_i)^2+N^2|\II_i^{\mathrm{rem}}|^2
- +N^2|d\log L_i|_{h_i}^2+\beta_i^2f_i^2\right].
- \label{eq:sweep-X}
-\end{align}
+
+ T_0={}&\sum_i A_i^2[
+ C_i+4a^2\sigma_iB_i+(L_i-1)\{2a(\sigma_i+r)+u_i\}],
+ \\
+ \mathcal X={}&\sum_i[
+ (X_i-t_i)^2+N^2|\II_i^{rem}|^2
+ +N^2|d\log L_i|_{h_i}^2+\beta_i^2f_i^2].
+
+
The quantity \(T_0\) records the gains from the metric comparisons
and from each graph's stretching. Its full coefficient will yield
the sharp quadratic term \(m^2/n\) in the entropy iteration.
@@ -1662,17 +1662,17 @@ applied coefficients. Together with the mean and variance squares
below, it controls graph curvature; those moment squares also control
the normalization.
-\begin{proposition}[One sweep]
-\label{prop:sweep}
+[One sweep]
+
There is a numerical \(c>0\) with the following property.
For a sweep without boost, set \(\psi=0\). For a boosted sweep,
-given \(\epsilon_*>0\), there are constants
-\(\kappa>0\), \(c_*>0\), and \(C_*<\infty\), depending only on
-\(\epsilon_*\), for which one may use \(\psi=\kappa\Psi\) whenever
-\(\epsilon\ge\epsilon_*\).
+given \(ε_*>0\), there are constants
+\(\kappa>0\), \(c_*>0\), and \(C_*<∞\), depending only on
+\(ε_*\), for which one may use \(\psi=\kappa\Psi\) whenever
+\(ε\geε_*\).
Let \(h_0,\phi_0,F\) be a smooth equivariant Borel family on
-\(\widetilde M\times\Omega\), with bounds uniform in the lifted
+\(\widetilde M\timesΩ\), with bounds uniform in the lifted
charts for the instance. Suppose \(h_0\) is complete, its
geometry and the positive-order derivatives of \(\phi_0\) have
the uniform bounds above, and
@@ -1683,25 +1683,25 @@ the uniform bounds above, and
has finitely many free label orbits and uniformly locally finite
compact coordinate supports. For each orbit representative,
suppose all its transverse supports lie in a fixed compact
-subset of \(\widetilde M\). Let \(h_{\mathrm{ref}}\le h_0\)
+subset of \(\widetilde M\). Let \(h_{ref}\le h_0\)
be a complete equivariant Borel leafwise metric whose coefficients,
inverse, and all coordinate derivatives have uniform bounds in the
same lifted buffered charts as \(h_0\). Suppose a Borel
equivariant coloring
-\(\chi:\mathcal L\times\Omega\to\{1,\ldots,k\}\), \(k<\infty\),
+\(\chi:\mathcal L\timesΩ\to\{1,...,k\}\), \(k<∞\),
has been chosen such that, for each retained \(\omega\) and distinct
labels \(i,j\) with \(\chi(i,\omega)=\chi(j,\omega)\), the
-\(R_{\mathrm{loc}}\)-neighborhoods in \(h_{\mathrm{ref}}(\omega)\)
+\(R_{loc}\)-neighborhoods in \(h_{ref}(\omega)\)
of \(\supp F_i(\cdot,\omega)\) and \(\supp F_j(\cdot,\omega)\)
are disjoint. Fix the pass parameters in
-\eqref{eq:sweep-parameters}, with \(\epsilon\ge\epsilon_*\) in
+(eq:sweep-parameters), with \(ε\geε_*\) in
the boosted case.
For any \(\delta_0,\eta_0>0\), there is a candidate sweep as above
with smooth nonnegative bumps \(f_i\le2\), supported in the
indicated enlargements, positive on the supports of \(F_i\), and
flat on their zero sets. Its output is given by
-\eqref{eq:sweep-output}. The lower bound for \(r\) is uniform
+(eq:sweep-output). The lower bound for \(r\) is uniform
for the instance. The bumps, coefficients, all intermediate and
final graph metrics and weights, their smooth graph quantities, and
\(Y\) are equivariant and Borel and retain the corresponding uniform
@@ -1710,57 +1710,57 @@ The map \(Y\) is again a
root probability map with the same label orbits.
For each \(i\), the coefficient furnished by
-Lemma~\ref{lem:graph-localization} has the form
-\begin{equation}
- \beta_i=\beta_{s,i}+\epsilon\psi(R_i),\qquad
+Lemma~lem:graph-localization has the form
+
+ \beta_i=\beta_{s,i}+ε\psi(R_i),\qquad
R_i=\sum_{j<i}f_j^2 ,
- \label{eq:sweep-beta}
-\end{equation}
+
+
Here \(\beta_{s,i}\) vanishes near \(\supp F_i\), and the coefficients
-satisfy \eqref{eq:graph-spatial-cutoff} in \(h_{i-1}\).
+satisfy (eq:graph-spatial-cutoff) in \(h_{i-1}\).
Writing
\[
- ND_i^0=\epsilon a(F_i^2/f_i-f_i)+A_i^2/(2f_i)
+ ND_i^0=ε a(F_i^2/f_i-f_i)+A_i^2/(2f_i)
\]
on \(\{f_i>0\}\), extended by zero off the bump, one has
-\begin{equation}
- ND_i=ND_i^0-\beta_i f_i+\mathrm{err}_i,\qquad
- \sum_i|\mathrm{err}_i|^2\le\eta_0 .
- \label{eq:sweep-prescription-error}
-\end{equation}
+
+ ND_i=ND_i^0-\beta_i f_i+err_i,\qquad
+ \sum_i|err_i|^2\le\eta_0 .
+
+
Define the nonnegative reserve
-\begin{equation}
- \begin{split}
+
+
\Pi_p:={}&
- T_0+c\epsilon\bigl(M_0^2+\E_\nu V_i^2+\theta\Var_\nu u\bigr)
+ T_0+cε(M_0^2+\E_\nu V_i^2+\theta\Var_\nu u)
\\
- &+c\frac r\epsilon\mathcal X
- +c_*\epsilon r^6\indic_{\{r\ge C_*\}},
- \end{split}
- \label{eq:sweep-positive-reserve}
-\end{equation}
+ &+c\frac rε\mathcal X
+ +c_*ε r^6\indic_{\{r\ge C_*\}},
+
+
+
where the final term is omitted in a sweep without boost. Then,
pointwise,
-\begin{align}
- Q:=\frac{rN^2}{\epsilon}\bigl[
+
+ Q:=(rN^2)/(ε)[
&\wscal(h_+,\phi_+)-\wscal(h_0,\phi_0)
- +\epsilon e_{h_0}(F)-\epsilon e_{h_+}(Y)\bigr]
+ +ε e_{h_0}(F)-ε e_{h_+}(Y)]
\notag\\
- &\ge\Pi_p-4\theta^2m-\frac r\epsilon\delta_0 .
- \label{eq:sweep-inequality}
-\end{align}
+ &\ge\Pi_p-4\theta^2m-\frac rε\delta_0 .
+
+
The numerical constant \(c\) is independent of the instance, the
number of colors, and the pass parameters. The errors
\(\delta_0,\eta_0\) can be prescribed independently before the pass.
-\end{proposition}
-\begin{proof}
+
+Proof.
Process the colors in their prescribed order. At a label in
the current color, the sum \(R_i\) of earlier bumps and all its
derivatives have common bounds for that color, so the function
-\(\alpha_i=\epsilon\psi(R_i)\) is fixed and uniformly smooth.
-Apply Lemma~\ref{lem:graph-localization} with this
+\(\alpha_i=ε\psi(R_i)\) is fixed and uniformly smooth.
+Apply Lemma~lem:graph-localization with this
\(\alpha_i\), using a common sufficiently large \(H_*\) for
the color. Uniqueness of the complete solutions and the
equivariant cutoff construction make all choices equivariant.
@@ -1770,7 +1770,7 @@ individual graph at each point of their supports. The graph
metric and weight bounds hold in the same lifted charts as the input
bounds. The reference metric is unchanged during the pass, so the
simultaneous chart control required by
-Lemma~\ref{lem:graph-localization} persists for the next color.
+Lemma~lem:graph-localization persists for the next color.
At any point some \(F_i\) is nonzero, and its cutoff is one
there. The lower barrier for that color gives
@@ -1778,83 +1778,83 @@ there. The lower barrier for that color gives
many colors gives the stated instance-uniform lower bound for
\(r\); also \(r\le4k\). Thus \(Y\) is smooth with uniform
bounds and \(\sum_iY_i^2=1\). The graph determinant and
-density identities prove \eqref{eq:sweep-output}.
+density identities prove (eq:sweep-output).
By making the error in each color small enough that its square
is at most \(\eta_0/k\), the last conclusion of
-Lemma~\ref{lem:graph-localization} gives
-\eqref{eq:sweep-prescription-error}.
+Lemma~lem:graph-localization gives
+(eq:sweep-prescription-error).
Prove the curvature estimate in the units before
-multiplication by \(r/\epsilon\). Write
+multiplication by \(r/ε\). Write
\[
- G=N^2\bigl[\wscal(h_+,\phi_+)-\wscal(h_0,\phi_0)
- +\epsilon e_{h_0}(F)-\epsilon e_{h_+}(Y)\bigr],
+ G=N^2[\wscal(h_+,\phi_+)-\wscal(h_0,\phi_0)
+ +ε e_{h_0}(F)-ε e_{h_+}(Y)],
\qquad Z_i=ND_i^0,
\]
-so \(Q=(r/\epsilon)G\). For each \(i\), the one-coordinate calculation
+so \(Q=(r/ε)G\). For each \(i\), the one-coordinate calculation
applies with \(h=h_{i-1}\), \(h'=h_i\), and
-\(h_{\mathrm{in}}=h_0\). First account for the difference between
+\(h_{in}=h_0\). First account for the difference between
the ideal target \(Z_i\) for \(ND_i\) and the realized prescription.
For the moment omit the floor in
-\eqref{eq:graph-core-prescription}. Replacing \(Z_i\) by
+(eq:graph-core-prescription). Replacing \(Z_i\) by
\(Z_i-\beta_i f_i\) in the first two terms of
-\eqref{eq:graph-identity}, after multiplying by \(N^2\),
+(eq:graph-identity), after multiplying by \(N^2\),
adds exactly
-\begin{align*}
+
&(Z_i-\beta_i f_i)^2-2NU_i(Z_i-\beta_i f_i)
-Z_i^2+2NU_i(Z_i)\\
&\hspace{1cm}=\beta_i^2f_i^2
- +2\epsilon a\beta_i(f_i^2-F_i^2)
+ +2ε a\beta_i(f_i^2-F_i^2)
+(2L_i-1)\beta_iA_i^2
+2NA_if_i e_i(\beta_i).
-\end{align*}
+
Since \(L_i\ge1\), this is at least
-\begin{equation}
+
\mathcal B_i:=
- \beta_i^2f_i^2+2\epsilon a\beta_i f_i^2+\beta_i A_i^2
- -2\epsilon a\beta_iF_i^2
+ \beta_i^2f_i^2+2ε a\beta_i f_i^2+\beta_i A_i^2
+ -2ε a\beta_iF_i^2
-2NA_if_i|d\beta_i|_{h_{i-1}}.
- \label{eq:sweep-beta-correction}
-\end{equation}
+
+
On the uncut region the actual floor has nonnegative
differentiated contribution by
-\eqref{eq:graph-floor-derivative}, and its square loses at
-most the quantity in \eqref{eq:graph-floor-square}.
+(eq:graph-floor-derivative), and its square loses at
+most the quantity in (eq:graph-floor-square).
On a cutoff transition \(F_i=0\) and \(d\beta_i=0\).
-There \eqref{eq:graph-cutoff-small} makes the actual scalar
+There (eq:graph-cutoff-small) makes the actual scalar
change small. It also makes every term on the right of the
single-step expansion just obtained small:
\(\beta_i A_i^2=(\beta_i f_i)t_i\),
the longitudinal terms are controlled by \(N|\II_i|\) and
\(t_i\), and the other terms are controlled by
\(Z_i,\beta_i f_i,A_i\). The first term in
-\eqref{eq:graph-ideal-derivative} is zero. Thus this
+(eq:graph-ideal-derivative) is zero. Thus this
expanded lower bound still holds on a cutoff transition with
an arbitrarily small additive error in these units. It is
also valid off the bumps, where all its terms vanish.
Choose the floor and transition errors for each color so their
sum is at most \(\delta_0\) at every point. Telescoping
-\eqref{eq:graph-identity} over the colors gives
-\begin{align}
- G\ge \sum_i\bigl[&
+(eq:graph-identity) over the colors gives
+
+ G\ge \sum_i[&
Z_i^2-2NU_i(Z_i)+N^2|\II_i|^2
+N^2|d\log L_i|_{h_i}^2
- +\epsilon N^2|dF_i|_{h_0}^2+\mathcal B_i\bigr]
+ +ε N^2|dF_i|_{h_0}^2+\mathcal B_i]
\notag\\
- &-\epsilon N^2e_{h_+}(Y)-\delta_0 .
- \label{eq:sweep-raw}
-\end{align}
+ &-ε N^2e_{h_+}(Y)-\delta_0 .
+
+
This argument uses the full differentiated floor and cutoff
estimates, rather than differentiating the value error in
-\eqref{eq:sweep-prescription-error}.
+(eq:sweep-prescription-error).
The input energy is already included in
-\eqref{eq:graph-ideal-bound}. For the output, normalization is an
+(eq:graph-ideal-bound). For the output, normalization is an
orthogonal projection in the target:
\[
- dY=r^{-1/2}(I-Y\otimes Y)\,df .
+ dY=r^{-1/2}(I-Y ⊗ Y)\,df .
\]
The rank-one inverse identity and metric monotonicity give
\[
@@ -1862,66 +1862,66 @@ The rank-one inverse identity and metric monotonicity give
N^2|df_i|_{h_+}^2=A_i^2(1-C_i)\le A_i^2 .
\]
Consequently
-\begin{equation}
+
N^2e_{h_+}(Y)
\le\frac1r\sum_i A_i^2(1-C_i).
- \label{eq:sweep-output-energy}
-\end{equation}
+
+
Put
\[
- K_i=(X_i-t_i)^2+N^2|\II_i^{\mathrm{rem}}|^2
+ K_i=(X_i-t_i)^2+N^2|\II_i^{rem}|^2
+N^2|d\log L_i|_{h_i}^2,\qquad
g_i=a(\sigma_i-r)+u_i/2 .
\]
The definitions imply
\[
- Z_i=\frac{\epsilon f_i}{r}g_i,\quad
- \frac r\epsilon\sum_iZ_i^2=\epsilon\E_\nu g_i^2,\quad
- \frac r\epsilon(L_i-1)t_i^2=(L_i-1)A_i^2u_i .
-\]
-Use these identities, \eqref{eq:graph-ideal-bound}, and
-\eqref{eq:sweep-output-energy} in \eqref{eq:sweep-raw}.
-After multiplication by \(r/\epsilon\), the result is
-\begin{align}
- Q\ge{}&T_0+\frac r\epsilon\sum_iK_i
- +\epsilon\E_\nu g_i^2
- +\sum_i A_i^2\bigl[(2a-4a^2)\sigma_i+2ar-1\bigr]
- +\frac r\epsilon\left(\sum_i\mathcal B_i-\delta_0\right).
- \label{eq:sweep-expanded}
-\end{align}
+ Z_i=(ε f_i)/(r)g_i,\quad
+ \frac rε\sum_iZ_i^2=ε\E_\nu g_i^2,\quad
+ \frac rε(L_i-1)t_i^2=(L_i-1)A_i^2u_i .
+\]
+Use these identities, (eq:graph-ideal-bound), and
+(eq:sweep-output-energy) in (eq:sweep-raw).
+After multiplication by \(r/ε\), the result is
+
+ Q\ge{}&T_0+\frac rε\sum_iK_i
+ +ε\E_\nu g_i^2
+ +\sum_i A_i^2[(2a-4a^2)\sigma_i+2ar-1]
+ +\frac rε(\sum_i\mathcal B_i-\delta_0).
+
+
Indeed \(rF_i^2/f_i^2=\sigma_i\). The terms with \(B_i\) and
-\(L_i-1\) in \eqref{eq:graph-ideal-bound} become the corresponding
+\(L_i-1\) in (eq:graph-ideal-bound) become the corresponding
terms of \(T_0\), while the output contributes
\(-A_i^2+A_i^2C_i\). The remaining height terms give the displayed
quadratic expression.
The remaining quadratic expression has an exact mean and
variance decomposition. Use \(\E_\nu\sigma_i=1\) and
-\(\E_\nu u_i=\bar u\) from \eqref{eq:sweep-statistics}, and write
+\(\E_\nu u_i=\bar u\) from (eq:sweep-statistics), and write
\(w_i=u_i-\bar u\). Dividing the quadratic expression in
-\eqref{eq:sweep-expanded} by \(\epsilon\), its contribution
+(eq:sweep-expanded) by \(ε\), its contribution
from the means is
-\begin{align*}
+
[a(1-r)+\bar u/2]^2+\bar u(2a-4a^2+2ar-1)
&= [a(1-r)-\bar u/2]^2-(2a-1)^2\bar u\\
&=M_0^2-4\theta^2\bar u .
-\end{align*}
+
The coefficient of
\(\E_\nu[w_i(\sigma_i-1)]\) in the centered expression is
\[
a+(2a-4a^2)=3a-4a^2=2a\lambda .
\]
Completing its square therefore gives the identity
-\begin{align}
- \epsilon\E_\nu g_i^2
- &+\sum_i A_i^2\bigl[(2a-4a^2)\sigma_i+2ar-1\bigr] \notag\\
+
+ ε\E_\nu g_i^2
+ &+\sum_i A_i^2[(2a-4a^2)\sigma_i+2ar-1] \notag\\
&=
- \epsilon\left(M_0^2+\E_\nu V_i^2+
- (\tfrac14-\lambda^2)\Var_\nu u\right)
+ ε(M_0^2+\E_\nu V_i^2+
+ (\tfrac14-\lambda^2)\Var_\nu u)
-4\theta^2m .
- \label{eq:sweep-mean-identity}
-\end{align}
+
+
Here
\(\tfrac14-\lambda^2=2\theta-4\theta^2\ge7\theta/4\).
This supplies a reserve for all three squares in the
@@ -1929,23 +1929,23 @@ proposition. Finish by paying the terms in
\(\mathcal B_i\) from fixed fractions of this reserve and of
the nonnegative \(\beta_i\) terms.
-Write \(s_i=\beta_{s,i}\) and \(b_i=\epsilon\psi(R_i)\), so
+Write \(s_i=\beta_{s,i}\) and \(b_i=ε\psi(R_i)\), so
\(\beta_i=s_i+b_i\) with both summands nonnegative.
In particular \(\beta_i^2\ge s_i^2+b_i^2\).
The negative \(F_i^2\) term in
-\eqref{eq:sweep-beta-correction} has no \(s_i\) part, because
+(eq:sweep-beta-correction) has no \(s_i\) part, because
\(s_i\) vanishes near the support of \(F_i\).
The spatial derivative term satisfies
-\begin{align}
+
2NA_if_i|ds_i|
- &\le2c_0 A_if_i(\theta\epsilon^{3/2}+s_i^{3/2}) \notag\\
+ &\le2c_0 A_if_i(\thetaε^{3/2}+s_i^{3/2}) \notag\\
&\le c_0(s_i^2f_i^2+s_iA_i^2)
- +Cc_0\theta(\epsilon^2f_i^2+A_i^4/f_i^2).
- \label{eq:sweep-spatial-cost}
-\end{align}
+ +Cc_0\theta(ε^2f_i^2+A_i^4/f_i^2).
+
+
For the first pair of terms use \(2xy\le x^2+y^2\).
-For the other pair, divide by \(\epsilon^2f_i^2\) and use
-\(t\le1+t^4\) with \(t=A_i/(\sqrt\epsilon f_i)\).
+For the other pair, divide by \(ε^2f_i^2\) and use
+\(t\le1+t^4\) with \(t=A_i/(\sqrtε f_i)\).
This derivative cost occurs only in rows with \(F_i=0\), hence
\(\sigma_i=0\). In those rows,
\[
@@ -1957,94 +1957,94 @@ Since \(a\ge1/2\) and \(|\lambda|\le1/2\), these equations give
r+u_i\le2(|M_0|+|V_i|+|w_i|),\qquad
r^2+u_i^2\le12(M_0^2+V_i^2+w_i^2).
\]
-After multiplication by \(r/\epsilon\), the last cost in
-\eqref{eq:sweep-spatial-cost}, summed over these rows, is
+After multiplication by \(r/ε\), the last cost in
+(eq:sweep-spatial-cost), summed over these rows, is
\[
- Cc_0\epsilon\theta
+ Cc_0ε\theta
\sum_{\{\sigma_i=0\}}\nu_i(r^2+u_i^2).
\]
The preceding bound shows that it is a small fraction of the
-three squares in \eqref{eq:sweep-mean-identity} when \(c_0\)
+three squares in (eq:sweep-mean-identity) when \(c_0\)
is a sufficiently small numerical constant. The first cost
-in \eqref{eq:sweep-spatial-cost} is paid by a small fraction
+in (eq:sweep-spatial-cost) is paid by a small fraction
of \(\beta_i^2f_i^2+\beta_iA_i^2\).
This proves the required absorption when \(\psi=0\), leaving
a fixed fraction of every asserted square.
Suppose now that \(\psi=\kappa\Psi\) and
-\(\epsilon\ge\epsilon_*\). Metric addition over earlier
+\(ε\geε_*\). Metric addition over earlier
colors implies, as quadratic forms,
\[
- h_{i-1}\ge N^2\sum_{j<i}df_j\otimes df_j .
+ h_{i-1}\ge N^2\sum_{j<i}df_j ⊗ df_j .
\]
For a unit vector in \(h_{i-1}\), the vector
\((Ndf_j)_{j<i}\) consequently has Euclidean norm at most one.
Cauchy--Schwarz applied to
\(dR_i=2\sum_{j<i}f_jdf_j\) proves
-\begin{equation}
- |dR_i|_{h_{i-1}}\le2\sqrt{R_i}/N .
- \label{eq:sweep-prefix-gradient}
-\end{equation}
+
+ |dR_i|_{h_{i-1}}\le2√(R_i)/N .
+
+
For any fixed small \(\zeta>0\), the derivative bound in
-\eqref{eq:sweep-boost-profile} and Young's inequality give
+(eq:sweep-boost-profile) and Young's inequality give
\[
2NA_if_i|db_i|
- \le4\epsilon A_if_i\sqrt{R_i}\,|\psi'(R_i)|
- \le\zeta\epsilon\psi(R_i)A_i^2
- +C_\zeta\epsilon\kappa f_i^2R_i .
+ \le4ε A_if_i√(R_i)\,|\psi'(R_i)|
+ \le\zetaε\psi(R_i)A_i^2
+ +C_\zetaε\kappa f_i^2R_i .
\]
The first term is a small fraction of \(\beta_iA_i^2\).
For the other costs, use
\[
\sum_i f_i^2R_i
- =\tfrac12\left(r^2-\sum_i f_i^4\right)\le r^2/2,
+ =\tfrac12(r^2-\sum_i f_i^4)\le r^2/2,
\qquad
\sum_i\psi(R_i)F_i^2\le C\kappa r^2\sum_iF_i^2=C\kappa r^2.
\]
The profile and its derivative vanish up to \(8\). Thus the
sum of all remaining boost costs in the units of \(G\) is at
most
-\begin{equation}
- C(\epsilon+\epsilon^2)\kappa r^2\indic_{\{r\ge8\}}.
- \label{eq:sweep-boost-cost}
-\end{equation}
+
+ C(ε+ε^2)\kappa r^2\indic_{\{r\ge8\}}.
+
+
Two positive terms cover this cost. First, for \(r\ge8\),
\[
-M_0=a(r-1)+\bar u/2\ge7r/16,
\qquad
- \frac{\epsilon^2}{r}M_0^2\ge c\epsilon^2r .
+ (ε^2)/(r)M_0^2\ge cε^2r .
\]
The latter is the mean square in
-\eqref{eq:sweep-mean-identity} expressed in the units of \(G\).
+(eq:sweep-mean-identity) expressed in the units of \(G\).
Second, the increments \(f_i^2\) in \(R_i\) are at most \(4\).
For \(r\) beyond a fixed threshold, the rows with
\(R_i\ge r/2\) carry total increment at least
\(r/2-4\ge r/4\). In fact the first prefix crossing \(r/2\)
overshoots it by at most \(4\). Once \(r/2\ge R_\Psi\), the
lower profile bound on each of these rows gives
-\begin{equation}
+
\sum_i f_i^2 b_i^2
- =\epsilon^2\sum_i f_i^2\psi(R_i)^2
- \ge c\epsilon^2\kappa^2 r^5
+ =ε^2\sum_i f_i^2\psi(R_i)^2
+ \ge cε^2\kappa^2 r^5
\qquad(r\ge C_*).
- \label{eq:sweep-boost-square}
-\end{equation}
+
+
To make the parameter choice explicit, let
-\(K_*=C(1+\epsilon_*^{-1})\). The cost
-\eqref{eq:sweep-boost-cost} is at most
-\(K_*\epsilon^2\kappa r^2\) on \(r\ge8\).
+\(K_*=C(1+ε_*^{-1})\). The cost
+(eq:sweep-boost-cost) is at most
+\(K_*ε^2\kappa r^2\) on \(r\ge8\).
Choose \(\kappa\) so small that
\(\kappa^{-1/2}\ge C_*\) and \(K_*\sqrt\kappa\) is smaller
than the fixed fractions of the preceding two reserves
available for this purpose. If
\(8\le r\le\kappa^{-1/2}\), this cost is at most
-\(K_*\sqrt\kappa\,\epsilon^2r\). If
+\(K_*\sqrt\kappa\,ε^2r\). If
\(r\ge\kappa^{-1/2}\), it is at most
-\(K_*\sqrt\kappa\,\epsilon^2\kappa^2r^5\).
+\(K_*\sqrt\kappa\,ε^2\kappa^2r^5\).
It is therefore covered in both ranges. Fixed fractions of
-the mean square and of \eqref{eq:sweep-boost-square} remain.
+the mean square and of (eq:sweep-boost-square) remain.
One can simultaneously retain a fixed fraction of
\(\sum_i\beta_i^2f_i^2\) for \(\mathcal X\): reserve, for
example, one half of this sum at the outset and use
@@ -2052,21 +2052,21 @@ example, one half of this sum at the outset and use
spatial and boost estimates. Take \(c_0,\zeta,\kappa\) small
enough for these fixed allocations.
-Insert the absorptions into \eqref{eq:sweep-expanded} and
-\eqref{eq:sweep-mean-identity}. A fixed fraction of
+Insert the absorptions into (eq:sweep-expanded) and
+(eq:sweep-mean-identity). A fixed fraction of
\(\sum_iK_i+\sum_i\beta_i^2f_i^2=\mathcal X\) and of the
three mean and variance squares survives. The remaining
-part of \eqref{eq:sweep-boost-square}, after multiplication
-by \(r/\epsilon\), is
-\(c_*\epsilon r^6\indic_{\{r\ge C_*\}}\), with
+part of (eq:sweep-boost-square), after multiplication
+by \(r/ε\), is
+\(c_*ε r^6\indic_{\{r\ge C_*\}}\), with
\(c_*\) a positive multiple of \(\kappa^2\).
The terms \(T_0\) and \(-4\theta^2m\) have not been spent.
-Choose the constant \(c\) in \eqref{eq:sweep-positive-reserve}
+Choose the constant \(c\) in (eq:sweep-positive-reserve)
below the fixed numerical fractions that survive for \(\mathcal X\)
and the mean and variance squares. These fractions are independent
-of \(\epsilon_*\), because \(\kappa\) was chosen to fit the fixed
+of \(ε_*\), because \(\kappa\) was chosen to fit the fixed
allocations.
-This proves \eqref{eq:sweep-inequality}.
+This proves (eq:sweep-inequality).
Finally, the choices of errors impose no estimate on the
growth of analytic constants from one color to the next.
@@ -2078,9 +2078,9 @@ be made at most \(\delta_0/k\) per color and the squared
value error at most \(\eta_0/k\) per color. Later choices
of \(H_*\) may be arbitrarily larger. The summed errors
are therefore precisely those asserted in
-\eqref{eq:sweep-prescription-error} and
-\eqref{eq:sweep-inequality}.
-\end{proof}
+(eq:sweep-prescription-error) and
+(eq:sweep-inequality).
+
This section turns a root probability map for a metric of scalar curvature
\(-1\) into a map with a large energy coefficient in a weighted scalar
@@ -2092,26 +2092,26 @@ to be chosen larger than the combinatorial quantities used later.
Write \(\exp^{[0]}(t)=t\) and
\(\exp^{[j+1]}(t)=\exp(\exp^{[j]}(t))\). Call a function
-\(\mathcal B:[1,\infty)\to[0,\infty)\) \emph{elementary} if, for some fixed
+\(\mathcal B:[1,∞)\to[0,∞)\) elementary if, for some fixed
\(H\in\N\cup\{0\}\) and \(A,B>0\),
\[
- \mathcal B(q)\leq \exp^{[H]}\bigl(B(1+q)^A\bigr)\qquad(q\geq1).
+ \mathcal B(q)\leq \exp^{[H]}(B(1+q)^A)\qquad(q\geq1).
\]
When the function is prescribed in a fixed dimension, its parameters
\(H,A,B\) are allowed to depend on that dimension.
-\begin{theorem}[Entropy metrics]\label{thm:entropy}
+[Entropy metrics]
Let \((M^n,g)\) be a closed connected Riemannian manifold, where \(n\geq3\)
and \(\Scal_g=-1\), and put \(\Gamma=\pi_1(M)\). Let
-\((\Omega_0,\mu_0)\) be a standard Borel probability space with a
+\((Ω_0,\mu_0)\) be a standard Borel probability space with a
measure-preserving \(\Gamma\)-action; the one-point space is allowed.
Suppose \(\mathcal L_0\) is a countable set with finitely many free
\(\Gamma\)-orbits and
\[
- F^{\mathrm{in}}:\widetilde M\times\Omega_0\longrightarrow
+ F^{in}:\widetilde M\timesΩ_0→
\ell^2(\mathcal L_0),\qquad
- F^{\mathrm{in}}_\ell\geq0,\qquad
- \sum_{\ell\in\mathcal L_0}(F^{\mathrm{in}}_\ell)^2=1
+ F^{in}_\ell\geq0,\qquad
+ \sum_{\ell\in\mathcal L_0}(F^{in}_\ell)^2=1
\]
is equivariant, Borel in the transverse variable, and smooth along
\(\widetilde M\). Assume the following uniformity for this input. In
@@ -2120,32 +2120,32 @@ the supports are locally finite with a uniform bound on their number,
and all leafwise derivatives of the coordinate functions have uniform
bounds. For each orbit representative \(\ell\) there is a compact set
\(K_\ell\subset\widetilde M\) such that
-\(\supp F^{\mathrm{in}}_{\gamma\ell}(\,\cdot\,,\omega)
+\(\supp F^{in}_{\gamma\ell}(\,\cdot\,,\omega)
\subset\gamma K_\ell\) for all \(\gamma\in\Gamma\) and almost every
\(\omega\). The bounds and compact sets in these assumptions may depend
on the input.
For every prescribed elementary function \(\mathcal B\) and every finite
number \(E_{\min}\), there are a standard Borel probability
-\(\Gamma\)-space \((\Omega,\mu)\) extending \((\Omega_0,\mu_0)\), a
+\(\Gamma\)-space \((Ω,\mu)\) extending \((Ω_0,\mu_0)\), a
countable set \(\mathcal L\) with finitely many free \(\Gamma\)-orbits, and
-equivariant leafwise data on \(\widetilde M\times\Omega\): a metric \(h\),
+equivariant leafwise data on \(\widetilde M\timesΩ\): a metric \(h\),
a function \(\phi\), a positive density \(\rho\), and a root probability
map \(F\) with labels \(\mathcal L\). There are also \(E>E_{\min}\), an
-integer \(q\geq n+1\), and a constant \(C=C(n,\mathcal B)<\infty\) such
+integer \(q\geq n+1\), and a constant \(C=C(n,\mathcal B)<∞\) such
that, on an invariant conull set of transverse parameters,
-\begin{equation}\label{eq:entropy-contract}
-\begin{split}
+
+
&\rho\leq dV_g,\qquad
- W:=\log\frac{dV_h}{\rho}\geq0,\\
+ W:=\log(dV_h)/(\rho)\geq0,\\
&\wscal(h,\phi)\geq
- -C+2\left(W-\frac n2\log E\right)+E e_h(F),
+ -C+2(W-\frac n2\log E)+E e_h(F),
\qquad e_h(F)^{1/2}\leq E^{-1/100},\\
&E>\mathcal B(q).
-\end{split}
-\end{equation}
+
+
Here the extension means that there is an equivariant measure-preserving
-map \(\Omega\to\Omega_0\). All output objects are Borel transversely and
+map \(Ω\toΩ_0\). All output objects are Borel transversely and
smooth along leaves. In the same lifted charts, the coefficients of
\(h\), their derivatives of every order, and the coefficients of \(h^{-1}\)
have uniform bounds; the function \(\phi\), the density ratio
@@ -2165,19 +2165,19 @@ Take
q\geq \max\{n+1,\#(\Gamma\backslash\mathcal L),\Delta+1\}
\]
and a Borel equivariant coloring
-\(\chi:\mathcal L\times\Omega\to\{1,\ldots,q\}\) such that labels of
+\(\chi:\mathcal L\timesΩ\to\{1,...,q\}\) such that labels of
graph distance one or two have different colors. The set \(\mathcal L\)
can be taken to consist of finitely many copies of \(\mathcal L_0\).
The constant \(C\) is independent of the manifold, metric, input family,
all the uniform bounds just specified, and \(E_{\min}\).
-\end{theorem}
-Prove the theorem using Proposition~\ref{prop:sweep}. The bootstrap
+
+Prove the theorem using Proposition~prop:sweep. The bootstrap
will fix the density \(\rho\), which then remains fixed through all later
sweeps and conformal changes. For
\(W=\log(dV_h/\rho)\), the balance propagated after the bootstrap is
\[
- \wscal(h,\phi)-\epsilon e_h(F)-2W+n\log\epsilon\geq-C+P,
+ \wscal(h,\phi)-ε e_h(F)-2W+n\logε\geq-C+P,
\qquad W\geq0,\quad P\geq0.
\]
First raise an initially small energy coefficient to a dimensional
@@ -2187,13 +2187,13 @@ that surplus pays for a conformal change which makes the gradient
absolutely small. The small gradient then controls how fast the fixed
graph of label overlaps can grow in the next group of sweeps.
The inverse-metric telescope and the assembly of logarithmic shortening
-also occur in \cite{MaWangXieYuZhu2026};
+also occur in [cite];
the proof below establishes the forcing and entropy estimates needed here.
First supply the colors required for the sweeps. Let \(\mathcal H\)
be any fixed invariant graph of maximum degree
-\(\Delta_{\mathcal H}<\infty\) on a countable label set \(\mathcal L\).
+\(\Delta_{\mathcal H}<∞\) on a countable label set \(\mathcal L\).
Adjoin to the transverse space independent uniform priorities in
\([0,1]^{\mathcal L}\), with the \(\Gamma\)-action permuting coordinates.
This is a standard Borel probability extension. Direct every edge
@@ -2208,7 +2208,7 @@ with probability \(1/(k+1)!\). The probability of arbitrarily long
such paths is therefore zero. Since the label set is countable, on
an invariant conull set every vertex has finite descending depth.
Greedily assign to a vertex the least color in
-\(\{1,\ldots,\Delta_{\mathcal H}+1\}\) unused by its lower-priority
+\(\{1,...,\Delta_{\mathcal H}+1\}\) unused by its lower-priority
neighbors, recursively in that finite depth. This defines a proper
coloring. It is Borel, since each vertex's color is determined in
some finite descending neighborhood, and it is equivariant, since
@@ -2227,46 +2227,46 @@ implies that only finitely many translates of any of these compact
sets can meet another. Joining all such possible pairs, over all
transverse parameters, gives one invariant finite-degree graph for
that step. The color construction just proved supplies the colors
-of Proposition~\ref{prop:sweep}. The graph and color choices are
+of Proposition~prop:sweep. The graph and color choices are
made before the analytic errors for the corresponding sweep are
chosen. This proves availability at each step of the finite
bootstrap, with input-dependent degree. The phase argument below
will control the degree relative to the eventual energy coefficient.
Here and below all the notation of a single sweep is that of
-Proposition~\ref{prop:sweep}. In particular, its input is
+Proposition~prop:sweep. In particular, its input is
\((h_0,\phi_0,F)\), its output is \((h_+,\phi_+,Y)\), and
\[
-\begin{gathered}
- N^2=l\epsilon,\qquad a=\tfrac12+\theta,\qquad
+
+ N^2=lε,\qquad a=\tfrac12+\theta,\qquad
\lambda=\tfrac12-2\theta,\\
r=\sum_i f_i^2>0,\qquad Y=f/\sqrt r,\qquad
W_p=\sum_i\log L_i,\qquad m=\sum_i A_i^2.
-\end{gathered}
+
\]
The weights \(\nu_i=f_i^2/r\) sum to one. Recall the probability
statistics and their centered combinations from
-\eqref{eq:sweep-statistics}:
+(eq:sweep-statistics):
\[
-\begin{gathered}
- u_i=\frac{A_i^2}{\epsilon\nu_i},\qquad
- \sigma_i=\frac{F_i^2}{\nu_i},\qquad
- \bar u=\frac m\epsilon,\\
- M_0=a(1-r)-\frac{\bar u}{2},\qquad
+
+ u_i=(A_i^2)/(ε\nu_i),\qquad
+ \sigma_i=(F_i^2)/(\nu_i),\qquad
+ \bar u=\frac mε,\\
+ M_0=a(1-r)-(\bar u)/(2),\qquad
V_i=a(\sigma_i-1)+\lambda(u_i-\bar u),\qquad
- t_i=\frac{A_i^2}{f_i}.
-\end{gathered}
+ t_i=(A_i^2)/(f_i).
+
\]
Indices of vanishing bumps are omitted. In particular
\(\E_\nu u_i=\bar u\) and \(\E_\nu\sigma_i=1\); the ratios \(\sigma_i\)
need not be at most one. Write \(\E\) and \(\Var\) for
this finite weighted mean and variance in the pointwise calculations.
The graph operations give
-\begin{equation}\label{eq:entropy-sweep-density}
+
\frac{dV_{h_+}}{dV_{h_0}}=e^{W_p},\qquad
\phi_+=\phi_0-W_p,\qquad
N^2 e_{h_+}(Y)\leq \frac mr.
-\end{equation}
+
The added graph tensors are nonnegative. Thus, as quadratic forms,
\(h_0\leq h_{i-1}\leq h_i\leq h_+\); the inverse forms reverse these
inequalities. In particular, for every covector \(\alpha\),
@@ -2275,39 +2275,39 @@ inequalities. In particular, for every covector \(\alpha\),
\leq|\alpha|_{h_{i-1}}\leq|\alpha|_{h_0}.
\]
The positive reserve \(\Pi_p\) is defined in
-\eqref{eq:sweep-positive-reserve}, with coefficient exactly one on
+(eq:sweep-positive-reserve), with coefficient exactly one on
\(T_0\). In a sweep without boost its final term is omitted; in a
boosted sweep its constants \(c_*,C_*\) may depend on the fixed lower
-threshold for \(\epsilon\). The sweep inequality reads
-\begin{equation}\label{eq:entropy-sweep-form}
-\begin{split}
- Q&:=\frac{rN^2}{\epsilon}
- \bigl[\wscal(h_+,\phi_+)-\wscal(h_0,\phi_0)
- +\epsilon e_{h_0}(F)-\epsilon e_{h_+}(Y)\bigr]\\
- &\geq \Pi_p-4\theta^2m-\frac r\epsilon\delta_0 .
-\end{split}
-\end{equation}
+threshold for \(ε\). The sweep inequality reads
+
+
+ Q&:=(rN^2)/(ε)
+ [\wscal(h_+,\phi_+)-\wscal(h_0,\phi_0)
+ +ε e_{h_0}(F)-ε e_{h_+}(Y)]\\
+ &\geq \Pi_p-4\theta^2m-\frac rε\delta_0 .
+
+
The positive errors \(\delta_0,\eta_0\) can be prescribed independently
before each sweep. The construction realizes those errors by choosing
the analytic parameters for each color after the geometry of the
preceding colors has been fixed.
-\begin{lemma}[Conformal bootstrap]\label{lem:entropy-bootstrap}
-There are constants \(\epsilon_*>0\) and \(C_{\mathrm{boot}}<4\),
+[Conformal bootstrap]
+There are constants \(ε_*>0\) and \(C_{boot}<4\),
depending only on \(n\), with the following property. Starting with
-the input of Theorem~\ref{thm:entropy}, finitely many sweeps without
+the input of Theorem~thm:entropy, finitely many sweeps without
boost, followed individually by conformal changes, produce a metric
\(h_*\), a root probability map \(F_*\), and
-\(\epsilon_0\in[\epsilon_*,2\epsilon_*]\) such that
-\begin{equation}\label{eq:entropy-bootstrap-output}
+\(ε_0\in[ε_*,2ε_*]\) such that
+
\rho:=dV_{h_*}\leq dV_g,\qquad
- \Scal_{h_*}\geq-C_{\mathrm{boot}}+\epsilon_0e_{h_*}(F_*).
-\end{equation}
+ \Scal_{h_*}\geq-C_{boot}+ε_0e_{h_*}(F_*).
+
The metric and map have the uniformity and support properties in
-Theorem~\ref{thm:entropy}. The number of sweeps may depend on the input.
-\end{lemma}
+Theorem~thm:entropy. The number of sweeps may depend on the input.
+
-\begin{proof}
+Proof.
Use sweeps with \(\phi_0=0\), no boost, and \(l=\theta\), where \(l>0\)
will be a sufficiently small fixed dimensional number. Then
\(\phi_+=-W_p\). After a sweep set
@@ -2315,215 +2315,215 @@ will be a sufficiently small fixed dimensional number. Then
k=e^{-2W_p/(n-1)}h_+ .
\]
The scalar conformal formula and the definition of \(\wscal\) give
-\begin{equation}\label{eq:entropy-bootstrap-reset}
+
\Scal_k=e^{2W_p/(n-1)}
- \left(\wscal(h_+,\phi_+)+\frac{|dW_p|_{h_+}^2}{n-1}\right).
-\end{equation}
+ (\wscal(h_+,\phi_+)+\frac{|dW_p|_{h_+}^2}{n-1}).
+
The same reset contracts the volume density:
-\begin{equation}\label{eq:entropy-bootstrap-density}
+
dV_k=e^{-nW_p/(n-1)}dV_{h_+}
=e^{-W_p/(n-1)}dV_{h_0}\leq dV_{h_0}.
-\end{equation}
-Claim that there are dimensional \(C_b\) and \(\epsilon_b>0\) such
-that, whenever \(2\leq C_0\leq4\), \(0<\epsilon\leq\epsilon_b\), and
-\(\Scal_{h_0}\geq-C_0+\epsilon e_{h_0}(F)\), one can choose the errors
+
+Claim that there are dimensional \(C_b\) and \(ε_b>0\) such
+that, whenever \(2\leq C_0\leq4\), \(0<ε\leqε_b\), and
+\(\Scal_{h_0}\geq-C_0+ε e_{h_0}(F)\), one can choose the errors
in the sweep so that
-\begin{equation}\label{eq:entropy-bootstrap-step}
- \Scal_k\geq-(C_0+C_b l\epsilon)+(1+l)\epsilon e_k(Y).
-\end{equation}
+
+ \Scal_k\geq-(C_0+C_b lε)+(1+l)ε e_k(Y).
+
First derive the budget needed for this step. The reset gives
\(e_k(Y)=e^{2W_p/(n-1)}e_{h_+}(Y)\). Substituting the definition of
\(Q\) and the input scalar inequality into
-\eqref{eq:entropy-bootstrap-reset}, and dropping its nonnegative
+(eq:entropy-bootstrap-reset), and dropping its nonnegative
gradient term, shows that it suffices to have
\[
- \frac{\epsilon Q}{rN^2}\geq
- l\epsilon e_{h_+}(Y)+C_0(1-e^{-2W_p/(n-1)})
- -C_b l\epsilon e^{-2W_p/(n-1)}.
+ (ε Q)/(rN^2)\geq
+ lε e_{h_+}(Y)+C_0(1-e^{-2W_p/(n-1)})
+ -C_b lε e^{-2W_p/(n-1)}.
\]
Multiply by \(rN^2\), and use
-\(N^2 e_{h_+}(Y)\leq m/r\) and \(N^2=l\epsilon\). A sufficient budget
-for \eqref{eq:entropy-bootstrap-step} is therefore
-\begin{equation}\label{eq:entropy-bootstrap-budget}
- \epsilon Q\geq
- l\epsilon m+C_0l\epsilon r(1-e^{-2W_p/(n-1)})
- -C_b l^2\epsilon^2r e^{-2W_p/(n-1)}.
-\end{equation}
-
-To pay this budget when \(\epsilon\) is small, one needs control of both
+\(N^2 e_{h_+}(Y)\leq m/r\) and \(N^2=lε\). A sufficient budget
+for (eq:entropy-bootstrap-step) is therefore
+
+ ε Q\geq
+ lε m+C_0lε r(1-e^{-2W_p/(n-1)})
+ -C_b l^2ε^2r e^{-2W_p/(n-1)}.
+
+
+To pay this budget when \(ε\) is small, one needs control of both
\(m\) and the deviation of \(r\) from one. The sweep inequality implies
-\begin{equation}\label{eq:entropy-bootstrap-coercivity}
- \epsilon Q\geq
- c_n\bigl(\epsilon^2(1-r)^2+m^2\bigr)
- -C_n\epsilon l^2m-r\delta'_0 ,
-\end{equation}
+
+ ε Q\geq
+ c_n(ε^2(1-r)^2+m^2)
+ -C_nε l^2m-r\delta'_0 ,
+
where \(\delta'_0>0\) can be chosen arbitrarily small. Here is a proof.
Fix a sufficiently small dimensional \(m_b>0\). If \(m\geq m_b\) and
-\(\epsilon_b\) is small enough, then \(\bar u=m/\epsilon\geq4a\). Since
+\(ε_b\) is small enough, then \(\bar u=m/ε\geq4a\). Since
\[
- -M_0=ar+\frac{\bar u}{2}-a,
+ -M_0=ar+(\bar u)/(2)-a,
\]
-the square \(\epsilon^2M_0^2\) controls
-\(\epsilon^2(1-r)^2+m^2\). This is one of the positive terms in
-\(\epsilon\Pi_p\).
+the square \(ε^2M_0^2\) controls
+\(ε^2(1-r)^2+m^2\). This is one of the positive terms in
+\(ε\Pi_p\).
Suppose next that \(m<m_b\). The prescription term \(D_i^0\) of
-Proposition~\ref{prop:sweep} satisfies
+Proposition~prop:sweep satisfies
\[
- ND_i^0+t_i=\frac{\epsilon f_i}{r}
- \left[V_i+M_0+(\lambda+\tfrac12)\bar u+
- (\tfrac32-\lambda)u_i\right].
+ ND_i^0+t_i=(ε f_i)/(r)
+ [V_i+M_0+(\lambda+\tfrac12)\bar u+
+ (\tfrac32-\lambda)u_i].
\]
Both displayed coefficients of \(\bar u\) and \(u_i\) are positive and
bounded below. Moving \(V_i+M_0\) to the other side, taking positive
parts, and summing squares therefore yields
-\begin{equation}\label{eq:entropy-bootstrap-t}
+
\sum_i t_i^2\leq
- \frac{C\epsilon^2}{r}\bigl(\E V_i^2+M_0^2\bigr)
+ (Cε^2)/(r)(\E V_i^2+M_0^2)
+C\sum_i(ND_i^0+t_i)_+^2 .
-\end{equation}
-The prescription error relation in Proposition~\ref{prop:sweep} is
-\(ND_i=ND_i^0-\beta_i f_i+\mathrm{err}_i\), with
-\(\sum_i|\mathrm{err}_i|^2\) arbitrarily small. Moreover, the
+
+The prescription error relation in Proposition~prop:sweep is
+\(ND_i=ND_i^0-\beta_i f_i+err_i\), with
+\(\sum_i|err_i|^2\) arbitrarily small. Moreover, the
longitudinal component of \(\II_i\) is \(-X_i/N\), so
\[
ND_i+t_i=-(X_i-t_i)
- +N\bigl(\tr\II_i^{\mathrm{rem}}-U_i(\phi_{i-1})\bigr).
+ +N(\tr\II_i^{rem}-U_i(\phi_{i-1})).
\]
-The terms \((X_i-t_i)^2\), \(N^2|\II_i^{\mathrm{rem}}|^2\), and
+The terms \((X_i-t_i)^2\), \(N^2|\II_i^{rem}|^2\), and
\(\beta_i^2f_i^2\) occur in \(\mathcal X\).
For the remaining term, choose \(m_b<1/4\). Then
\(L_i\leq C\) and \(|d\log L_i|_{h_i}\leq C A_i|\II_i|\).
Because the metrics increase during the sweep and
\(\phi_{i-1}=-\sum_{j<i}\log L_j\), Cauchy--Schwarz gives
\[
-\begin{split}
+
\sum_iN^2U_i(\phi_{i-1})^2
&\leq C m^2\sum_jN^2|\II_j|^2\\
- &\leq C' m^2\left(\mathcal X+\sum_jt_j^2\right).
-\end{split}
+ &\leq C' m^2(\mathcal X+\sum_jt_j^2).
+
\]
For example, the first line follows by bounding
\(|d\phi_{i-1}|_{h_{i-1}}\) by
\(C\sum_{j<i}A_j|\II_j|\), squaring with
\(\sum_jA_j^2=m\), and then summing the factor \(A_i^2\).
Choosing \(m_b\) small absorbs the last sum of \(t_j^2\) in
-\eqref{eq:entropy-bootstrap-t}. Since
+(eq:entropy-bootstrap-t). Since
\[
- m^2=\left(\sum_i f_i t_i\right)^2\leq r\sum_i t_i^2,
+ m^2=(\sum_i f_i t_i)^2\leq r\sum_i t_i^2,
\qquad
- \epsilon^2(1-r)^2\leq C\bigl(\epsilon^2M_0^2+m^2\bigr),
+ ε^2(1-r)^2\leq C(ε^2M_0^2+m^2),
\]
fractions of the \(M_0^2,\E V_i^2,\mathcal X\) terms in
-\(\epsilon\Pi_p\) control both required squares. The arbitrarily
+\(ε\Pi_p\) control both required squares. The arbitrarily
small errors in the prescription contribute \(r\delta'_0\).
-The term \(-4\theta^2m\) in \eqref{eq:entropy-sweep-form} is
-\(-4l^2m\). This proves \eqref{eq:entropy-bootstrap-coercivity}.
+The term \(-4\theta^2m\) in (eq:entropy-sweep-form) is
+\(-4l^2m\). This proves (eq:entropy-bootstrap-coercivity).
Now compare the right side of
-\eqref{eq:entropy-bootstrap-coercivity} with the required budget.
+(eq:entropy-bootstrap-coercivity) with the required budget.
When \(m\geq m_b\), the first two terms on the right of
-\eqref{eq:entropy-bootstrap-budget} are at most
-\(l\epsilon(m+4r)\). The squares
-\(m^2+\epsilon^2(1-r)^2\) dominate this expression with a fixed
-fraction to spare after \(l\) is chosen small and then \(\epsilon_b\)
+(eq:entropy-bootstrap-budget) are at most
+\(lε(m+4r)\). The squares
+\(m^2+ε^2(1-r)^2\) dominate this expression with a fixed
+fraction to spare after \(l\) is chosen small and then \(ε_b\)
is decreased if needed. One may check this separately for \(r\leq2\),
where \(m\geq m_b\) controls the constant part, and for \(r>2\), where
-Young's inequality controls \(\epsilon r\) by
-\(\epsilon^2(r-1)^2\) and a sufficiently small fixed part of \(m^2\).
-The term \(C_n\epsilon l^2m\) is covered at the same time.
+Young's inequality controls \(ε r\) by
+\(ε^2(r-1)^2\) and a sufficiently small fixed part of \(m^2\).
+The term \(C_nε l^2m\) is covered at the same time.
When \(m<m_b\), the identity
\(L_i=(1-A_i^2)^{-1/2}\) gives \(W_p\leq C m\).
For \(r\notin(1/2,2)\), Young's inequality applied to
-\(l\epsilon m(1+r)\) again gives the required domination by the two
+\(lε m(1+r)\) again gives the required domination by the two
squares, with \(l\) small. For \(1/2<r<2\), the positive terms on the
-right of \eqref{eq:entropy-bootstrap-budget}, together with the term
-\(C_n\epsilon l^2m\) from the coercivity estimate, are at most
-\(C_nl\epsilon m\). Here
+right of (eq:entropy-bootstrap-budget), together with the term
+\(C_nε l^2m\) from the coercivity estimate, are at most
+\(C_nlε m\). Here
\[
- C_nl\epsilon m\leq \frac{c_n}{2}m^2+C'_n l^2\epsilon^2.
+ C_nlε m\leq (c_n)/(2)m^2+C'_n l^2ε^2.
\]
On this region \(r e^{-2W_p/(n-1)}\) has a dimensional positive lower
bound. Therefore choose \(C_b\) dimensionally to cover the
last term. These comparisons can be made strict by increasing \(C_b\)
-and decreasing \(l,\epsilon_b\). They leave a positive multiple of
-\(l^2\epsilon^2r\) in the middle region, and positive square margins
+and decreasing \(l,ε_b\). They leave a positive multiple of
+\(l^2ε^2r\) in the middle region, and positive square margins
outside it; dividing the latter margins by \(r\) gives a positive
-lower bound for fixed \(l,\epsilon\). Consequently choose
+lower bound for fixed \(l,ε\). Consequently choose
\(\delta'_0\) small enough for the sweep. This proves
-\eqref{eq:entropy-bootstrap-step}.
+(eq:entropy-bootstrap-step).
The input map has a finite uniform energy bound in \(g\). Thus for
-some positive, possibly input-dependent \(\epsilon_{\mathrm{in}}\)
+some positive, possibly input-dependent \(ε_{in}\)
small enough,
-\(\Scal_g=-1\geq-2+\epsilon_{\mathrm{in}}e_g(F^{\mathrm{in}})\).
-Choose a dimensional \(\epsilon_*>0\) below \(\epsilon_b\), small
-enough that \(2+2C_b\epsilon_*<4\), and decrease
-\(\epsilon_{\mathrm{in}}\) further so that
-\(0<\epsilon_{\mathrm{in}}<\epsilon_*\). Iterate
-\eqref{eq:entropy-bootstrap-step} until the coefficient first reaches
-\(\epsilon_*\). The output coefficient is at most
-\((1+l)\epsilon_*\leq2\epsilon_*\), and
-\[
- \sum_{\text{bootstrap sweeps}}l\epsilon
- =\epsilon_0-\epsilon_{\mathrm{in}}\leq2\epsilon_*.
-\]
-This bounds the total scalar cost by \(2C_b\epsilon_*\), independently
-of the number of sweeps. By \eqref{eq:entropy-bootstrap-density},
+\(\Scal_g=-1\geq-2+ε_{in}e_g(F^{in})\).
+Choose a dimensional \(ε_*>0\) below \(ε_b\), small
+enough that \(2+2C_bε_*<4\), and decrease
+\(ε_{in}\) further so that
+\(0<ε_{in}<ε_*\). Iterate
+(eq:entropy-bootstrap-step) until the coefficient first reaches
+\(ε_*\). The output coefficient is at most
+\((1+l)ε_*\leq2ε_*\), and
+\[
+ \sum_{bootstrap sweeps}lε
+ =ε_0-ε_{in}\leq2ε_*.
+\]
+This bounds the total scalar cost by \(2C_bε_*\), independently
+of the number of sweeps. By (eq:entropy-bootstrap-density),
every reset contracts the volume density. The final density
\(\rho=dV_{h_*}\) therefore satisfies
-\eqref{eq:entropy-bootstrap-output}. The sweep construction and
+(eq:entropy-bootstrap-output). The sweep construction and
finitely many smooth conformal changes preserve all the stated
input-uniform properties.
-\end{proof}
-Fix \(\epsilon_*\) from Lemma~\ref{lem:entropy-bootstrap}. Now use
-the boost in Proposition~\ref{prop:sweep}, choosing its constant
+
+Fix \(ε_*\) from Lemma~lem:entropy-bootstrap. Now use
+the boost in Proposition~prop:sweep, choosing its constant
\(\kappa>0\) sufficiently small depending on this fixed threshold.
All constants arising from the boost are consequently dimensional.
The increment estimate has a coarse region, where a fixed fraction of
the reserve remains after paying the budget, and a region near
\(m=0,r=1\), where the exact quadratic coefficient will matter.
-\begin{lemma}[Entropy increment]\label{lem:entropy-increment}
+[Entropy increment]
There are dimensional constants \(m_0>0\), \(D_*\geq1\), \(l_0>0\),
-\(\gamma_n>0\), and \(C_n<\infty\) with the following property.
-For \(\epsilon\geq\epsilon_*\), take a boosted sweep with
+\(\gamma_n>0\), and \(C_n<∞\) with the following property.
+For \(ε\geqε_*\), take a boosted sweep with
\[
0<l\leq l_0,\qquad \theta=D_*l\leq\tfrac1{16},
- \qquad N^2=l\epsilon.
+ \qquad N^2=lε.
\]
The errors can be chosen so that
-\begin{equation}\label{eq:entropy-increment}
-\begin{split}
- \wscal(h_+,\phi_+)-(1+l)\epsilon e_{h_+}(Y)
- \geq{}&\wscal(h_0,\phi_0)-\epsilon e_{h_0}(F)\\
+
+
+ \wscal(h_+,\phi_+)-(1+l)ε e_{h_+}(Y)
+ \geq{}&\wscal(h_0,\phi_0)-ε e_{h_0}(F)\\
&+2W_p-n\log(1+l)-C_nl^2+P_p ,
-\end{split}
-\end{equation}
+
+
where the nonnegative, pointwise surplus is
-\begin{equation}\label{eq:entropy-surplus-definition}
- P_p=\gamma_n\frac{\epsilon}{rN^2}\Pi_p\,
+
+ P_p=\gamma_n(ε)/(rN^2)\Pi_p\,
\indic_{\{m\geq m_0\}\,\cup\,\{|1-r|\geq1/4\}}.
-\end{equation}
+
In particular, on the indicated region \(P_p\) retains a fixed fraction
of every positive term in the sweep inequality.
-\end{lemma}
-\begin{proof}
-The output energy bound in \eqref{eq:entropy-sweep-density} shows that,
+
+Proof.
+The output energy bound in (eq:entropy-sweep-density) shows that,
in the units of \(Q\), it suffices to prove
-\begin{equation}\label{eq:entropy-budget}
+
Q\geq lm+2lrW_p-nrl\log(1+l)-C_nr l^3
- +\frac{rN^2}{\epsilon}P_p .
-\end{equation}
-The term \(lm\) pays for the additional \(l\epsilon e_{h_+}(Y)\).
-Choose \(\delta_0\leq\epsilon l^3\) in
-\eqref{eq:entropy-sweep-form}. It remains to compare the other terms
-there with the budget in \eqref{eq:entropy-budget}.
+ +(rN^2)/(ε)P_p .
+
+The term \(lm\) pays for the additional \(lε e_{h_+}(Y)\).
+Choose \(\delta_0\leqε l^3\) in
+(eq:entropy-sweep-form). It remains to compare the other terms
+there with the budget in (eq:entropy-budget).
Use a trace identity for the amount of the new graph energy that
remains in the final metric. Set
@@ -2531,25 +2531,25 @@ remains in the final metric. Set
m_f:=\sum_i A_i^2(1-C_i)=N^2\sum_i|df_i|_{h_+}^2.
\]
As matrices,
-\(\sum_iN^2df_i\otimes df_i=h_+-h_0\). Hence
-\begin{equation}\label{eq:entropy-trace}
+\(\sum_iN^2df_i ⊗ df_i=h_+-h_0\). Hence
+
m_f=n-\tr(h_+^{-1}h_0)
\leq n(1-e^{-2W_p/n}).
-\end{equation}
+
The inequality is arithmetic--geometric mean applied to the positive
eigenvalues of \(h_+^{-1}h_0\), whose product is \(e^{-2W_p}\).
There also holds
-\begin{equation}\label{eq:entropy-volume-angle}
+
L_i=(1-A_i^2)^{-1/2}.
-\end{equation}
+
First choose a small dimensional \(m_0>0\), in particular
-\(m_0\ll\epsilon_*\) and \(m_0<1/4\). Claim that
-\begin{equation}\label{eq:entropy-coarse-coercivity}
- T_0+\epsilon M_0^2\geq c_n(m+rW_p)
- \quad\text{if }m\geq m_0
- \text{ or if }m<m_0,\ |1-r|\geq\tfrac14.
-\end{equation}
+\(m_0\llε_*\) and \(m_0<1/4\). Claim that
+
+ T_0+ε M_0^2\geq c_n(m+rW_p)
+ \quadif m\geq m_0
+ or if m<m_0,\ |1-r|\geq\tfrac14.
+
For \(m\geq2n\), the \(C_i\) terms of \(T_0\) give
\(T_0\geq m-m_f\geq m-n\geq m/2\).
For \(m_0\leq m\leq2n\), first suppose that every \(A_i^2\) is at most
@@ -2557,9 +2557,9 @@ a small number \(\eta=\eta(n,m_0)>0\). Then
\[
2W_p=\sum_i-\log(1-A_i^2)\leq(1+C\eta)m.
\]
-By \eqref{eq:entropy-trace},
+By (eq:entropy-trace),
\[
- m-m_f\geq m-n\bigl(1-e^{-(1+C\eta)m/n}\bigr)\geq c_n m.
+ m-m_f\geq m-n(1-e^{-(1+C\eta)m/n})\geq c_n m.
\]
For the last inequality, the same expression with \(\eta=0\) divided
by \(m\) has a positive minimum on \([m_0,2n]\), and then \(\eta\)
@@ -2567,10 +2567,10 @@ is chosen small. If instead some \(A_i^2>\eta\), the self term of
that index in \(T_0\) is at least \(c_n(r+u_i)\). It gives a
dimensional positive bound if \(r\) is bounded below. If \(r\) is
small, \(M_0=a(1-r)-\bar u/2\) is bounded away from zero unless \(\bar u\)
-lies in a fixed compact subinterval of \((0,\infty)\). In the former
-case \(\epsilon M_0^2\geq c_n\epsilon_*\). In the latter,
-\(\epsilon=m/\bar u\leq C_n\) and
-\(u_i=A_i^2/(\epsilon\nu_i)\geq c_n\), since \(\nu_i\leq1\).
+lies in a fixed compact subinterval of \((0,∞)\). In the former
+case \(ε M_0^2\geq c_nε_*\). In the latter,
+\(ε=m/\bar u\leq C_n\) and
+\(u_i=A_i^2/(ε\nu_i)\geq c_n\), since \(\nu_i\leq1\).
Each bound controls \(m\leq2n\). This proves the \(m\) part for
\(m\geq m_0\).
@@ -2582,34 +2582,34 @@ For all other indices, \(\log L_i\leq C A_i^2\), so their contribution
to \(rW_p\) is at most \(Crm\). If \(r\leq2\), the bound for \(m\)
already covers this. If \(r>2\), then
\(|M_0|\geq c(r+\bar u)\), and
-\(\epsilon M_0^2\geq c\epsilon r\bar u=c\,rm\).
-This proves \eqref{eq:entropy-coarse-coercivity} when \(m\geq m_0\).
-When \(m<m_0\), one has \(\bar u\leq m_0/\epsilon_*\) small and
+\(ε M_0^2\geq cε r\bar u=c\,rm\).
+This proves (eq:entropy-coarse-coercivity) when \(m\geq m_0\).
+When \(m<m_0\), one has \(\bar u\leq m_0/ε_*\) small and
\[
- W_p=\frac m2+O\left(\sum_iA_i^4\right)\leq C m.
+ W_p=\frac m2+O(\sum_iA_i^4)\leq C m.
\]
If \(|1-r|\geq1/4\), choosing \(m_0\) still smaller gives
\(|M_0|\geq c|1-r|\). The quantity
-\(\epsilon(1-r)^2\) controls \(m+rm\): for bounded \(r\) use
-\(\epsilon\geq\epsilon_*\) and \(m\leq m_0\), and for \(r\geq2\)
+\(ε(1-r)^2\) controls \(m+rm\): for bounded \(r\) use
+\(ε\geqε_*\) and \(m\leq m_0\), and for \(r\geq2\)
use \((1-r)^2\geq r^2/4\). This proves the claim in the remaining
case.
-On the region of \eqref{eq:entropy-coarse-coercivity}, the positive
+On the region of (eq:entropy-coarse-coercivity), the positive
reserve \(\Pi_p\) consequently controls \(m+rW_p\). The positive terms
-\(lm+2lrW_p\) in \eqref{eq:entropy-budget} use at most \(C_nl\Pi_p\),
+\(lm+2lrW_p\) in (eq:entropy-budget) use at most \(C_nl\Pi_p\),
and \(4\theta^2m\) uses at most \(C_nD_*^2l^2\Pi_p\). For \(l\)
sufficiently small after \(D_*\), a fixed fraction of \(\Pi_p\)
remains. Choose \(\gamma_n\) below that fraction. The error term
-\(r\delta_0/\epsilon\) is covered by \(C_nr l^3\). This proves
-\eqref{eq:entropy-budget} on the region where the surplus is retained.
+\(r\delta_0/ε\) is covered by \(C_nr l^3\). This proves
+(eq:entropy-budget) on the region where the surplus is retained.
It remains to prove the budget when
-\begin{equation}\label{eq:entropy-small-region}
+
m<m_0,\qquad |1-r|<\tfrac14.
-\end{equation}
+
Here \(T_0\geq c_nm^2\). Indeed, expanding the exponential in
-\eqref{eq:entropy-trace} and using
+(eq:entropy-trace) and using
\(W_p=m/2+O(\sum_iA_i^4)\) gives
\[
\sum_i A_i^2 C_i=m-m_f
@@ -2627,7 +2627,7 @@ below, take \(l\) small enough that \(4D_*^2l^2m\) is also a small
fraction of \(m^2\).
What remains is the critical regime \(m\leq Kl\), still under
-\eqref{eq:entropy-small-region}. Here
+(eq:entropy-small-region). Here
\(W_p=m/2+O(m^2)\) and \(3/4<r<5/4\), so the budget expands as
\[
lm+2lrW_p-nrl\log(1+l)
@@ -2635,8 +2635,8 @@ What remains is the critical regime \(m\leq Kl\), still under
\]
The target quadratic term is covered exactly by
\[
- \frac{m^2}{n}-2lm+nl^2
- =\left(\frac m{\sqrt n}-\sqrt n\,l\right)^2\geq0.
+ (m^2)/(n)-2lm+nl^2
+ =(\frac m{\sqrt n}-\sqrt n\,l)^2\geq0.
\]
It remains to recover \(m^2/n\) from \(T_0\), while paying the
displayed \(r-1\) error and the errors in that recovery. This is where
@@ -2660,7 +2660,7 @@ losses, their exact earlier- and later-color telescopes are
\quad(A_i>0).
\]
The first identity follows from
-\(h_{i-1}=h_0+N^2\sum_{j<i}df_j\otimes df_j\) and
+\(h_{i-1}=h_0+N^2\sum_{j<i}df_j ⊗ df_j\) and
\(|e_i|_{h_{i-1}}=1\). The second follows by telescoping the rank-one
inverse formula
\[
@@ -2672,18 +2672,18 @@ from \(h_i\) to \(h_+\), and using
The common metric comparison and \(L_j=1+O(m)\) show that each summand
in the exact formulas differs from its \(h_0\) expression by
\(O(m)A_j^2\). Their sums therefore give, for \(A_i\ne0\),
-\begin{equation}\label{eq:entropy-BC-expansion}
+
B_i=\sum_{j<i}A_j^2\xi_{ij}+O(m^2),\qquad
C_i=\sum_{j>i}A_j^2\xi_{ij}+O(m^2).
-\end{equation}
-Inserting \eqref{eq:entropy-BC-expansion} into \(T_0\) yields
-\begin{equation}\label{eq:entropy-sharp-T}
+
+Inserting (eq:entropy-BC-expansion) into \(T_0\) yields
+
T_0\geq
- \tr\left[\left(\sum_i A_i^2\bar e_i\bar e_i^{\mathsf T}\right)^2\right]
- -C_n\left(m^3+m^2|r-1|
- +m\sum_iA_i^2(1-\sigma_i)_+\right).
-\end{equation}
+ \tr[(\sum_i A_i^2\bar e_i\bar e_i^{\mathsf T})^2]
+ -C_n(m^3+m^2|r-1|
+ +m\sum_iA_i^2(1-\sigma_i)_+).
+
To see the leading coefficient, the \(B_i\) and \(C_i\) terms each
give one copy of
\(\sum_{i<j}A_i^2A_j^2\xi_{ij}\). In the \(B_i\) terms use
@@ -2691,41 +2691,41 @@ give one copy of
\(Cm\sum_iA_i^2(1-\sigma_i)_+\). The self terms use
\(L_i-1\geq A_i^2/2\) and \(2a\geq1\). They give
\(\sum_iA_i^4\), up to the displayed \(r-1\) and \(\sigma_i-1\) costs.
-The errors in \eqref{eq:entropy-BC-expansion} sum to \(O(m^3)\).
-This proves \eqref{eq:entropy-sharp-T}. The trace in that estimate
+The errors in (eq:entropy-BC-expansion) sum to \(O(m^3)\).
+This proves (eq:entropy-sharp-T). The trace in that estimate
is at least \(m^2/n\), since the matrix is positive semidefinite
with trace \(m\).
-The last error in \eqref{eq:entropy-sharp-T} is paid for by the
+The last error in (eq:entropy-sharp-T) is paid for by the
centered squares. From
\(a(\sigma_i-1)=V_i-\lambda(u_i-\bar u)\), Cauchy--Schwarz and Young's
inequality give, for every fixed \(c'>0\),
-\begin{equation}\label{eq:entropy-v-error}
-\begin{split}
+
+
m\sum_iA_i^2(1-\sigma_i)_+
- &\leq C m\epsilon\,\E\bigl[u_i(|V_i|+|u_i-\bar u|)\bigr]\\
- &\leq c'\epsilon\bigl(\E V_i^2+\theta\Var u\bigr)
- +C_{c'}\frac{m^2}{\theta}\epsilon(\bar u^2+\Var u).
-\end{split}
-\end{equation}
+ &\leq C mε\,\E[u_i(|V_i|+|u_i-\bar u|)]\\
+ &\leq c'ε(\E V_i^2+\theta\Var u)
+ +C_{c'}(m^2)/(\theta)ε(\bar u^2+\Var u).
+
+
Choose \(c'\) to use a small fraction of the squares in \(\Pi_p\).
-The coefficient of \(\epsilon\Var u\) in the last term, relative to
-\(\theta\epsilon\Var u\), is at most \(C_{c'}K^2/D_*^2\), because
+The coefficient of \(ε\Var u\) in the last term, relative to
+\(\thetaε\Var u\), is at most \(C_{c'}K^2/D_*^2\), because
\(m\leq Kl\) and \(\theta=D_*l\). Choose \(D_*\) sufficiently large
after \(K\) so that this term is absorbed as well. The remaining term
is
\[
- C_{c'}\frac{m^2}{\theta}\epsilon \bar u^2
- =C_{c'}\frac{m^4}{\theta\epsilon}=O_n(l^3).
+ C_{c'}(m^2)/(\theta)ε \bar u^2
+ =C_{c'}(m^4)/(\thetaε)=O_n(l^3).
\]
-Here \(\epsilon\geq\epsilon_*\), and \(K,D_*,\epsilon_*\) are already
+Here \(ε\geqε_*\), and \(K,D_*,ε_*\) are already
fixed dimensionally. Also
\[
|r-1|\leq C(|M_0|+\bar u).
\]
Consequently a term \(O_n(l^2)|r-1|\) is bounded by a small fraction
-of \(\epsilon M_0^2\) plus \(O_n(l^3)\): use Young's inequality on
-\(l^2|M_0|\), and use \(\bar u\leq Kl/\epsilon_*\) on the other part.
+of \(ε M_0^2\) plus \(O_n(l^3)\): use Young's inequality on
+\(l^2|M_0|\), and use \(\bar u\leq Kl/ε_*\) on the other part.
The errors \(m^3\), \(m^2|r-1|\), and \(4D_*^2l^2m\) in the present
region are therefore covered with a remaining error \(O_n(l^3)\).
@@ -2734,49 +2734,49 @@ The just established mean-square estimate also covers the
\(m^2/n\) then covers \(2lm-nl^2\) by the displayed square.
Since \(r\) is bounded above and below, the remaining \(O_n(l^3)\)
terms are of the form \(C_nr l^3\). This proves
-\eqref{eq:entropy-budget} in \eqref{eq:entropy-small-region}.
+(eq:entropy-budget) in (eq:entropy-small-region).
The choices just made have a definite order: first \(m_0\) after
-\(\epsilon_*\), then \(K\), then \(D_*\), and finally \(l_0>0\)
+\(ε_*\), then \(K\), then \(D_*\), and finally \(l_0>0\)
small enough for all the comparisons and for \(D_*l_0\leq1/16\).
The errors in each sweep are chosen afterward. Thus every constant
-in \eqref{eq:entropy-increment} is dimensional.
-\end{proof}
-
-\begin{lemma}[Bounds supplied by the surplus]\label{lem:entropy-surplus}
-For every sweep of Lemma~\ref{lem:entropy-increment},
-\begin{equation}\label{eq:entropy-pass-volume}
- W_p\leq C_n\epsilon^{1/4}(\epsilon+P_p)^{3/4}.
-\end{equation}
-There is also a dimensional \(\epsilon_{\mathrm{exit}}\) such that,
-if \(\epsilon\geq\epsilon_{\mathrm{exit}}\) and \(r\leq1/4\), then
-\begin{equation}\label{eq:entropy-last-surplus}
-\begin{split}
- P_p&\geq c_n\frac{\epsilon}{lr}
- \left(1+\frac{m^2}{\epsilon^2}\right),\\
- \sum_i\bigl(D_i^2+|\II_i|^2+|d\log L_i|_{h_i}^2\bigr)
- &\leq C_n\frac{P_p}{\theta}.
-\end{split}
-\end{equation}
+in (eq:entropy-increment) is dimensional.
+
+
+[Bounds supplied by the surplus]
+For every sweep of Lemma~lem:entropy-increment,
+
+ W_p\leq C_nε^{1/4}(ε+P_p)^{3/4}.
+
+There is also a dimensional \(ε_{exit}\) such that,
+if \(ε\geqε_{exit}\) and \(r\leq1/4\), then
+
+
+ P_p&\geq c_n(ε)/(lr)
+ (1+(m^2)/(ε^2)),\\
+ \sum_i(D_i^2+|\II_i|^2+|d\log L_i|_{h_i}^2)
+ &\leq C_n(P_p)/(\theta).
+
+
The errors in the prescription may simultaneously be chosen to have
-\(\sum_i|\mathrm{err}_i|^2\leq1\).
-\end{lemma}
+\(\sum_i|err_i|^2\leq1\).
-\begin{proof}
-If \(\bar u=m/\epsilon\) is larger than a fixed dimensional number, then
+
+Proof.
+If \(\bar u=m/ε\) is larger than a fixed dimensional number, then
\(m\geq m_0\), so the surplus retains the mean square and boost.
Since \(|M_0|\geq c(r+\bar u)\) in this range and \(l\leq1\),
\[
- P_p\geq c_n\epsilon\,
+ P_p\geq c_nε\,
\frac{(r+\bar u)^2+c_*r^6\indic_{\{r\geq C_*\}}}{r}
- \geq c'_n\epsilon \bar u^{5/3}.
+ \geq c'_nε \bar u^{5/3}.
\]
For the last inequality, if \(r\leq \bar u^{1/3}\) use \(\bar u^2/r\);
if \(r\geq \bar u^{1/3}\) use \(r^5\), taking the threshold for \(\bar u\)
large enough that \(\bar u^{1/3}\geq C_*\). It follows in all cases that
-\begin{equation}\label{eq:entropy-m-surplus}
- m\leq C_n\epsilon(1+P_p/\epsilon)^{3/5}.
-\end{equation}
+
+ m\leq C_nε(1+P_p/ε)^{3/5}.
+
If any \(A_i>1/2\), then \(m>1/4>m_0\), and the self terms in \(T_0\)
give
\[
@@ -2784,89 +2784,89 @@ give
\]
Indeed their contribution to \(T_0\) is at least
\(c r\sum_{A_i>1/2}(L_i-1)\), and
-\(\epsilon/(rN^2)=1/(lr)\geq1/r\).
+\(ε/(rN^2)=1/(lr)\geq1/r\).
For \(A_i\leq1/2\), \(\log L_i\leq C A_i^2\).
There are at most \(4m\) indices with \(A_i>1/2\). Concavity of
\(\log(1+s)\) on those indices, together with the last estimate,
therefore gives
-\begin{equation}\label{eq:entropy-log-surplus}
- W_p\leq C_n m\bigl(1+\log(1+P_p/m)\bigr)
-\end{equation}
+
+ W_p\leq C_n m(1+\log(1+P_p/m))
+
when \(m>0\); when \(m=0\), \(W_p=0\).
To justify the bound when the number of those indices is smaller
than \(4m\), use that \(s\log(1+b/s)\) is increasing in \(s>0\).
-Put \(x=m/\epsilon\), \(y=P_p/\epsilon\). The elementary bound
+Put \(x=m/ε\), \(y=P_p/ε\). The elementary bound
\(1+\log(1+t)\leq C(1+t)^{1/4}\) for \(t\geq0\), applied with
\(t=y/x\), makes the right side of
-\eqref{eq:entropy-log-surplus}, divided by \(\epsilon\), at most
+(eq:entropy-log-surplus), divided by \(ε\), at most
\[
C_n x^{3/4}(x+y)^{1/4}
\leq C_n(1+y)^{7/10}
\leq C_n(1+y)^{3/4},
\]
-where \eqref{eq:entropy-m-surplus} was used in the middle inequality.
-This proves \eqref{eq:entropy-pass-volume}.
+where (eq:entropy-m-surplus) was used in the middle inequality.
+This proves (eq:entropy-pass-volume).
Now suppose \(r\leq1/4\). The surplus retains all positive terms
because \(|1-r|\geq3/4\). Outside a fixed compact subinterval of
-\(\bar u\in(0,\infty)\), the expression \(M_0=a(1-r)-\bar u/2\) satisfies
-\(M_0^2\geq c(1+\bar u^2)\). On that compact interval, for \(\epsilon\)
+\(\bar u\in(0,∞)\), the expression \(M_0=a(1-r)-\bar u/2\) satisfies
+\(M_0^2\geq c(1+\bar u^2)\). On that compact interval, for \(ε\)
sufficiently large,
\[
- T_0\geq m-m_f\geq\epsilon \bar u-n
- \geq c_n\epsilon(1+\bar u^2)
+ T_0\geq m-m_f\geqε \bar u-n
+ \geq c_nε(1+\bar u^2)
\]
-by \eqref{eq:entropy-trace}. Multiplying the appropriate retained
+by (eq:entropy-trace). Multiplying the appropriate retained
term by \(\gamma_n/(lr)\) proves the first inequality in
-\eqref{eq:entropy-last-surplus}.
+(eq:entropy-last-surplus).
For the second, the retained derivative terms imply
-\begin{equation}\label{eq:entropy-X-retained}
+
\mathcal X\leq C_n N^2P_p,\qquad
- \frac{\epsilon^2}{rN^2}
- \bigl(\E V_i^2+\theta\Var u\bigr)\leq C_nP_p.
-\end{equation}
+ (ε^2)/(rN^2)
+ (\E V_i^2+\theta\Var u)\leq C_nP_p.
+
Also
\[
- \sum_i t_i^2=\frac{\epsilon^2}{r}(\bar u^2+\Var u).
+ \sum_i t_i^2=(ε^2)/(r)(\bar u^2+\Var u).
\]
-The first inequality of \eqref{eq:entropy-last-surplus} controls
+The first inequality of (eq:entropy-last-surplus) controls
the \(\bar u^2\) part of this sum divided by \(N^2\), and the variance
-term in \eqref{eq:entropy-X-retained} controls the other part with
+term in (eq:entropy-X-retained) controls the other part with
a factor \(1/\theta\). Thus
\(\sum_i t_i^2/N^2\leq C_nP_p/\theta\). Since \(\mathcal X\) contains
-\((X_i-t_i)^2+N^2|\II_i^{\mathrm{rem}}|^2+N^2|d\log L_i|^2\), this proves the
+\((X_i-t_i)^2+N^2|\II_i^{rem}|^2+N^2|d\log L_i|^2\), this proves the
claimed bounds for \(\II_i\) and \(d\log L_i\).
For \(D_i\), use
\[
- ND_i^0=\frac{\epsilon f_i}{r}
- \left(a(\sigma_i-r)+\frac{u_i}{2}\right),\qquad
- \E \sigma_i^2\leq C\bigl(1+\E V_i^2+\Var u\bigr).
+ ND_i^0=(ε f_i)/(r)
+ (a(\sigma_i-r)+(u_i)/(2)),\qquad
+ \E \sigma_i^2\leq C(1+\E V_i^2+\Var u).
\]
Here the second inequality follows by solving the definition of \(V_i\)
for \(\sigma_i-1\). Since \(r\leq1/4\),
\[
\sum_i(D_i^0)^2\leq
- \frac{C\epsilon^2}{rN^2}
- \bigl(1+\bar u^2+\E V_i^2+\Var u\bigr)
+ (Cε^2)/(rN^2)
+ (1+\bar u^2+\E V_i^2+\Var u)
\leq C_nP_p/\theta.
\]
-The relation \(ND_i=ND_i^0-\beta_i f_i+\mathrm{err}_i\) finishes the
-estimate: \(\sum_i\beta_i^2f_i^2\leq\mathcal X\), and one chooses \(\sum_i|\mathrm{err}_i|^2\leq1\). Its contribution \(1/N^2\) is
+The relation \(ND_i=ND_i^0-\beta_i f_i+err_i\) finishes the
+estimate: \(\sum_i\beta_i^2f_i^2\leq\mathcal X\), and one chooses \(\sum_i|err_i|^2\leq1\). Its contribution \(1/N^2\) is
bounded by \(C_nP_p\) using the first inequality of
-\eqref{eq:entropy-last-surplus}, \(\epsilon\geq1\), and \(r\leq1/4\).
-\end{proof}
+(eq:entropy-last-surplus), \(ε\geq1\), and \(r\leq1/4\).
+
The step size has two jobs: the scalar losses \(O_n(l^2)\) must be
summable, and the number of support enlargements before reaching \(E\)
must grow at most cubically in \(\log E\). Since one sweep changes
-\(\log(1+\epsilon)\) by a quantity comparable to \(l\), one takes \begin{equation}\label{eq:entropy-step-schedule}
- l(\epsilon)=\frac{l_*}{(1+\log(1+\epsilon))^2},
-\end{equation}
+\(\log(1+ε)\) by a quantity comparable to \(l\), one takes
+ l(ε)=(l_*)/((1+\log(1+ε))^2),
+
where the dimensional \(l_*>0\) is chosen below \(l_0\) from
-Lemma~\ref{lem:entropy-increment}. Thus all the parameter choices in
+Lemma~lem:entropy-increment. Thus all the parameter choices in
that lemma, including \(\theta=D_*l\), apply.
The fixed exponents in the next two lemmas leave room, respectively,
@@ -2874,35 +2874,35 @@ for the number of sweeps, the conformal scalar cost, the outgoing
volume estimate, and the final gradient target:
\[
0.26=\tfrac14+0.01,\qquad
- 0.26+0.04<\frac{1+0.32}{4},\qquad
+ 0.26+0.04<(1+0.32)/(4),\qquad
0.26+\tfrac34(0.04)<0.32,\qquad
0.02>\tfrac1{100}.
\]
The estimates below show how each margin is used.
-\begin{lemma}[Accumulation of entropy passes]\label{lem:entropy-accumulation}
-Suppose at parameter \(\epsilon_{\mathrm{in}}\geq\epsilon_*\) that
+[Accumulation of entropy passes]
+Suppose at parameter \(ε_{in}\geqε_*\) that
\[
- \wscal(h,\phi)\geq-C+2W-n\log\epsilon_{\mathrm{in}}
- +\epsilon_{\mathrm{in}}e_h(F)+P_{\mathrm{in}},
- \qquad W=\log(dV_h/\rho)\geq0,\quad P_{\mathrm{in}}\geq0.
+ \wscal(h,\phi)\geq-C+2W-n\logε_{in}
+ +ε_{in}e_h(F)+P_{in},
+ \qquad W=\log(dV_h/\rho)\geq0,\quad P_{in}\geq0.
\]
-Successive sweeps of Lemma~\ref{lem:entropy-increment}, with
-\(\epsilon_{\mathrm{new}}=(1+l(\epsilon))\epsilon\), preserve this
+Successive sweeps of Lemma~lem:entropy-increment, with
+\(ε_{new}=(1+l(ε))ε\), preserve this
form of inequality: at the final parameter \(E\) the surplus is
-\(P=P_{\mathrm{in}}+\sum_pP_p\), the volume logarithm is
-\(W_{\mathrm{new}}=W+\sum_pW_p\), and the increase of the scalar
-constant is \(C_n\sum_p l(\epsilon_p)^2\). Along the entire increasing
+\(P=P_{in}+\sum_pP_p\), the volume logarithm is
+\(W_{new}=W+\sum_pW_p\), and the increase of the scalar
+constant is \(C_n\sum_p l(ε_p)^2\). Along the entire increasing
sequence of parameters,
-\begin{equation}\label{eq:entropy-summability}
- \sum_p l(\epsilon_p)^2\leq C_n,\qquad
- \#\{p:\epsilon_p<E\}\leq C_n(1+\log^3(1+E)).
-\end{equation}
+
+ \sum_p l(ε_p)^2\leq C_n,\qquad
+ \#\{p:ε_p<E\}\leq C_n(1+\log^3(1+E)).
+
If the initial volume logarithm is zero, then, for all sufficiently
large dimensional \(E\),
-\begin{equation}\label{eq:entropy-phase-volume}
- W_{\mathrm{new}}\leq E^{0.26}(E+P)^{0.75}.
-\end{equation}
+
+ W_{new}\leq E^{0.26}(E+P)^{0.75}.
+
The same assertion holds if the initial data have parameter \(E_0\),
surplus \(P_0\), and
\[
@@ -2910,57 +2910,57 @@ surplus \(P_0\), and
\]
provided \(E\geq E_0\) is sufficiently large dimensionally and
\(E_0^{0.32}\leq\tfrac12E^{0.26}\).
-\end{lemma}
-\begin{proof}
+
+Proof.
The accumulation assertion follows by adding
-\eqref{eq:entropy-increment}. The energy coefficients telescope
+(eq:entropy-increment). The energy coefficients telescope
multiplicatively, and
-\(\sum_p\log(1+l(\epsilon_p))=\log(E/\epsilon_{\mathrm{in}})\).
-The density identity in \eqref{eq:entropy-sweep-density} gives the
+\(\sum_p\log(1+l(ε_p))=\log(E/ε_{in})\).
+The density identity in (eq:entropy-sweep-density) gives the
sum of the \(W_p\).
-Put \(t_p=\log(1+\epsilon_p)\). Since
-\(\epsilon_p\geq\epsilon_*>0\) and \(l_*\) is fixed small,
+Put \(t_p=\log(1+ε_p)\). Since
+\(ε_p\geqε_*>0\) and \(l_*\) is fixed small,
\[
t_{p+1}-t_p
- =\log\left(1+l(\epsilon_p)\frac{\epsilon_p}{1+\epsilon_p}\right)
- \asymp_n l(\epsilon_p)
- =\frac{l_*}{(1+t_p)^2}.
+ =\log(1+l(ε_p)(ε_p)/(1+ε_p))
+ \asymp_n l(ε_p)
+ =(l_*)/((1+t_p)^2).
\]
Comparison with the integrals of \((1+t)^2\) and \((1+t)^{-2}\)
respectively proves the number bound and
\[
- \sum_p l(\epsilon_p)^2
- \leq C_n l_*\int_{t_0}^{\infty}\frac{dt}{(1+t)^2}\leq C_n.
+ \sum_p l(ε_p)^2
+ \leq C_n l_*\int_{t_0}^{∞}(dt)/((1+t)^2)\leq C_n.
\]
The comparison is valid on each interval \([t_p,t_{p+1}]\), since
its endpoints differ by a uniformly bounded small number.
Let \(k\) be the number of sweeps in the group under consideration.
-By Lemma~\ref{lem:entropy-surplus}, since
-\(\epsilon_p\leq E\) and \(P_p\leq P\),
+By Lemma~lem:entropy-surplus, since
+\(ε_p\leq E\) and \(P_p\leq P\),
\[
\sum_pW_p\leq C_n k E^{1/4}(E+P)^{3/4}.
\]
For all sufficiently large \(E\),
\(C_n k\leq \tfrac12 E^{0.01}\) by
-\eqref{eq:entropy-summability}. This proves
-\eqref{eq:entropy-phase-volume} for zero initial volume logarithm.
+(eq:entropy-summability). This proves
+(eq:entropy-phase-volume) for zero initial volume logarithm.
In the other case \(P\geq P_0\) and \(E\geq E_0\), so the initial
term is at most \(E_0^{0.32}(E+P)^{0.75}\). The stated condition on
\(E_0\) pays for this with the other half of
\(E^{0.26}(E+P)^{0.75}\).
-\end{proof}
-Call a finite group of these sweeps a \emph{phase}. There is no
+
+Call a finite group of these sweeps a phase. There is no
conformal change inside a phase. Each sweep adds the nonnegative
-tensor in \eqref{eq:sweep-output}; hence, if \(h^0\) is its starting
+tensor in (eq:sweep-output); hence, if \(h^0\) is its starting
metric and \(h_+\) its final metric, the successive metrics and their
inverses satisfy
\[
- h^0\leq h^{(1)}\leq\cdots\leq h_+,\qquad
- (h_+)^{-1}\leq\cdots\leq(h^{(1)})^{-1}\leq(h^0)^{-1}.
+ h^0\leq h^{(1)}\leq...\leq h_+,\qquad
+ (h_+)^{-1}\leq...\leq(h^{(1)})^{-1}\leq(h^0)^{-1}.
\]
In particular, the starting metric is an admissible reference metric
for every sweep in the phase, and
@@ -2974,45 +2974,45 @@ the preceding metric comparison. A conformal enlargement confined
to the small-\(r\) region then reduces the remaining gradient there,
and the last sweep's surplus pays its cost.
-\begin{lemma}[Exit from a phase]\label{lem:entropy-phase-exit}
+[Exit from a phase]
Consider a phase starting with root map \(Y^0\) and metric \(h^0\), and
let \(K_0\) be a finite uniform bound for \(e_{h^0}(Y^0)^{1/2}\).
-Suppose its last sweep has input parameter \(\epsilon\), output
-parameter \(E=(1+l(\epsilon))\epsilon\), and output
+Suppose its last sweep has input parameter \(ε\), output
+parameter \(E=(1+l(ε))ε\), and output
\((h_+,\phi_+,Y)\). Suppose, with \(C\geq0\) and \(P\geq P_p\) including the
surplus from that last sweep, that
-\begin{equation}\label{eq:entropy-before-exit}
-\begin{split}
+
+
\wscal(h_+,\phi_+)&\geq-C+2W-n\log E+E e_{h_+}(Y)+P,\\
W&=\log(dV_{h_+}/\rho)\geq0,\qquad
W\leq E^{0.26}(E+P)^{0.75}.
-\end{split}
-\end{equation}
+
+
If \(E\) is sufficiently large in terms of \(n,K_0\), there are a
metric \(\widehat h\), a weight \(\widehat\phi\), and a root map \(Z\)
on two copies of the labels such that
-\begin{equation}\label{eq:entropy-after-exit}
-\begin{split}
+
+
\wscal(\widehat h,\widehat\phi)
&\geq-(C+C_n)+2\widehat W-n\log E
+E e_{\widehat h}(Z)+P',\\
\widehat W&=\log(dV_{\widehat h}/\rho)\geq0,\qquad
\widehat W\leq E^{0.32}(E+P')^{0.75},\\
e_{\widehat h}(Z)^{1/2}&\leq E^{-1/100}.
-\end{split}
-\end{equation}
+
+
Here \(P'\geq0\), and the added scalar cost \(C_n\) is independent
of \(K_0\) and of all other instance constants. The changes preserve
the uniform smoothness and support properties of
-Theorem~\ref{thm:entropy}.
-\end{lemma}
+Theorem~thm:entropy.
-\begin{proof}
+
+Proof.
In this proof all derivatives, unless otherwise indicated, use \(h_+\).
Write \(r=\sum_i f_i^2\) for the last sweep, set \(s=\sqrt r\), and put
\[
c_E=\frac1{\sqrt E(1+K_0)},\qquad
- Z=\frac{(f,c_EY^0)}{\sqrt{r+c_E^2}}.
+ Z=\frac{(f,c_EY^0)}{√(r+c_E^2)}.
\]
The two vectors in the numerator use disjoint copies of the labels.
The map is a smooth root probability map because its denominator is
@@ -3021,7 +3021,7 @@ one calls \(i\) their common parent label at the start of the phase;
the sweeps within the phase do not change the label set. The first
copy is supported on the last bumps and the second on the starting
map. This parent assignment will be used to compare support overlaps
-between phases. By \eqref{eq:entropy-trace},
+between phases. By (eq:entropy-trace),
\[
N^2\sum_i|df_i|_{h_+}^2=m_f\leq n,\qquad
|ds|_{h_+}^2\leq n/N^2.
@@ -3030,49 +3030,49 @@ Monotonicity of the phase metrics also gives \(e_{h_+}(Y^0)\leq K_0^2\).
Since \(Y\) and \(Y^0\) are unit vectors in orthogonal spaces, their
derivatives are perpendicular to their respective vectors. Differentiating
-\(Z=(sY,c_EY^0)/\sqrt{s^2+c_E^2}\) thus gives the exact identity
+\(Z=(sY,c_EY^0)/√(s^2+c_E^2)\) thus gives the exact identity
\[
- e(Z)=\frac r{r+c_E^2}e(Y)+\frac{c_E^2}{r+c_E^2}e(Y^0)
- +\frac{c_E^2}{(r+c_E^2)^2}|ds|^2.
+ e(Z)=\frac r{r+c_E^2}e(Y)+(c_E^2)/(r+c_E^2)e(Y^0)
+ +(c_E^2)/((r+c_E^2)^2)|ds|^2.
\]
For \(r\geq1/4\), it follows that
\[
E(e(Z)-e(Y))\leq
- \frac{Ec_E^2K_0^2}{r+c_E^2}
- +\frac{Ec_E^2n}{N^2(r+c_E^2)^2}
- \leq 4+\frac{16n}{N^2}\leq C_n
+ (Ec_E^2K_0^2)/(r+c_E^2)
+ +(Ec_E^2n)/(N^2(r+c_E^2)^2)
+ \leq 4+(16n)/(N^2)\leq C_n
\]
-when \(E\) is large, since \(N^2=l\epsilon\) tends to infinity under
-\eqref{eq:entropy-step-schedule}. For \(r\leq1/4\), the norm of the
-derivative of normalization is at most \(1/\sqrt{r+c_E^2}\), so
-\begin{equation}\label{eq:entropy-Z-energy}
- e(Z)\leq\frac{n/N^2+c_E^2K_0^2}{r+c_E^2}.
-\end{equation}
-The first inequality of \eqref{eq:entropy-last-surplus} now implies
+when \(E\) is large, since \(N^2=lε\) tends to infinity under
+(eq:entropy-step-schedule). For \(r\leq1/4\), the norm of the
+derivative of normalization is at most \(1/√(r+c_E^2)\), so
+
+ e(Z)\leq(n/N^2+c_E^2K_0^2)/(r+c_E^2).
+
+The first inequality of (eq:entropy-last-surplus) now implies
\[
- E e(Z)\leq \frac{C_n}{l(r+c_E^2)}
- \leq \frac{C_n}{\epsilon}P_p.
+ E e(Z)\leq (C_n)/(l(r+c_E^2))
+ \leq (C_n)/(ε)P_p.
\]
Thus replacing \(Y\) by \(Z\) in
-\eqref{eq:entropy-before-exit} costs a dimensional constant on
+(eq:entropy-before-exit) costs a dimensional constant on
\(\{r\geq1/4\}\) and an arbitrarily small fraction of \(P_p\) on
\(\{r\leq1/4\}\), once \(E\) is dimensionally large. In particular,
no factor \(K_0\) occurs in this scalar cost.
Choose a smooth function \(u=u(r)\geq0\), supported in
\(\{r\leq E^{-0.32}\}\), with
-\begin{equation}\label{eq:entropy-u-choice}
+
e^u\asymp
- \min\bigl(E^{0.02},\max(1,E^{-0.16}/\sqrt r)\bigr),
- \qquad \left|\frac{du}{dr}\right|\leq\frac C r.
-\end{equation}
+ \min(E^{0.02},\max(1,E^{-0.16}/\sqrt r)),
+ \qquad |(du)/(dr)|\leq\frac C r.
+
For example, smooth, with bounded-width transitions on the logarithmic
scale, the function obtained by truncating
\(-\tfrac12\log r-0.16\log E\) to the interval
\([0,0.02\log E]\). The smoothing may vanish whenever the untruncated
function is nonpositive, and changes its value by a bounded amount;
this gives both the support assertion and the comparison in
-\eqref{eq:entropy-u-choice}. Define
+(eq:entropy-u-choice). Define
\[
\widehat h=e^{2u}h_+,\qquad
\widehat\phi=\phi_+-(n-1)u,\qquad
@@ -3081,17 +3081,17 @@ this gives both the support assertion and the comparison in
In particular \(dV_{\widehat h}=e^{nu}dV_{h_+}\), so this is the
claimed volume logarithm. Differentiating \(r\) gives
\[
- |dr|\leq2\sqrt r\left(\sum_i|df_i|^2\right)^{1/2}
- \leq\frac{2\sqrt{nr}}N,\qquad
- |du|\leq\frac{C_n}{N\sqrt r}.
+ |dr|\leq2\sqrt r(\sum_i|df_i|^2)^{1/2}
+ \leq\frac{2√(nr)}N,\qquad
+ |du|\leq(C_n)/(N\sqrt r).
\]
The coefficient \(n-1\) in the new weight cancels the Laplacian of
\(u\) in the weighted conformal formula. With
\(b=\nabla_{h_+}\phi_+\), that formula is
-\begin{equation}\label{eq:entropy-conformal-exit}
+
\wscal(\widehat h,\widehat\phi)
- =e^{-2u}\left[\wscal(h_+,\phi_+)+2b(u)-(n-1)|du|^2\right].
-\end{equation}
+ =e^{-2u}[\wscal(h_+,\phi_+)+2b(u)-(n-1)|du|^2].
+
With the fixed density \(\rho\), a graph sweep changes \(W\) by
\(+W_p\) and \(\phi\) by \(-W_p\), so it preserves \(\phi+W\).
@@ -3116,27 +3116,27 @@ unused later directions. Writing \(b_i\) for the embedded gradient
of \(\phi_i\), orthogonal projection to the new graph gives the exact
recurrence
\[
- b_i=b_{i-1}-\langle b_{i-1},\mathbf n_i\rangle\mathbf n_i
+ b_i=b_{i-1}-< b_{i-1},\mathbf n_i>\mathbf n_i
-\nabla_{h_i}\log L_i,
\]
where the last gradient is embedded in the same space. The normal
coefficient is
-\(\langle b_{i-1},\mathbf n_i\rangle=-U_i(\phi_{i-1})\), and
+\(< b_{i-1},\mathbf n_i>=-U_i(\phi_{i-1})\), and
its magnitude satisfies
\[
|U_i(\phi_{i-1})|\leq |D_i|+\sqrt n\,|\II_i|,
\]
because \(D_i=\tr\II_i-U_i(\phi_{i-1})\).
-By orthonormality and \eqref{eq:entropy-last-surplus}, the norm of
+By orthonormality and (eq:entropy-last-surplus), the norm of
the sum of these normal corrections is at most
-\(C_n\sqrt{P_p/\theta}\).
+\(C_n√(P_p/\theta)\).
For the subtracted tangent gradients, split the indices according as
\(A_i\leq1/2\) or \(A_i>1/2\). In the first set
\(|d\log L_i|\leq C A_i|\II_i|\); in the second there are at most
\(4m\) indices. Cauchy--Schwarz and
-\eqref{eq:entropy-last-surplus} consequently bound the sum of their
-norms by \(C_n\sqrt{mP_p/\theta}\). The final tangent embedding sends
+(eq:entropy-last-surplus) consequently bound the sum of their
+norms by \(C_n√(mP_p/\theta)\). The final tangent embedding sends
a vector \(v\) to \((v,(Ndf_i(v))_i)\). Summing the recurrence, the
initial gradient has no vertical component in this common extension,
while the vertical component of the final gradient is
@@ -3148,40 +3148,40 @@ The normal and tangent bounds therefore prove
It follows that
\[
|b(u)|\leq\frac C r|b(r)|
- \leq C_n\left(\frac{(1+m)P_p}{\theta N^2r}\right)^{1/2}
- \leq \frac{C_nP_p}{\sqrt{\theta E}}.
+ \leq C_n(((1+m)P_p)/(\theta N^2r))^{1/2}
+ \leq \frac{C_nP_p}{√(\theta E)}.
\]
-For the last inequality use \(E\asymp\epsilon\), \(N^2=l\epsilon\),
+For the last inequality use \(E\asympε\), \(N^2=lε\),
and
\[
- P_p\geq\frac{c_n}{lr}
- \left(\epsilon+\frac{m^2}{\epsilon}\right)
- \geq \frac{c'_n(1+m)}{lr}
+ P_p\geq(c_n)/(lr)
+ (ε+(m^2)/(ε))
+ \geq (c'_n(1+m))/(lr)
\]
-for \(\epsilon\geq1\). The same first surplus bound gives
-\begin{equation}\label{eq:entropy-exit-derivatives}
- |du|^2\leq\frac{C_n}{N^2r}\leq\frac{C_nP_p}{E^2}.
-\end{equation}
+for \(ε\geq1\). The same first surplus bound gives
+
+ |du|^2\leq(C_n)/(N^2r)\leq(C_nP_p)/(E^2).
+
All these estimates are only needed where \(u\ne0\), a subset of
\(\{r\leq1/4\}\) for large \(E\). Since
\(\theta E\asymp_n E/(1+\log E)^2\) tends to infinity, the two
-derivative terms in \eqref{eq:entropy-conformal-exit} cost an
+derivative terms in (eq:entropy-conformal-exit) cost an
arbitrarily small fraction of \(P_p\).
There is enough surplus on this region also for the change in the
entropy term. From \(r\leq E^{-0.32}\) and
-\eqref{eq:entropy-last-surplus},
+(eq:entropy-last-surplus),
\[
P\geq P_p\geq c_n E^{1.32}.
\]
-Using \eqref{eq:entropy-before-exit} and \(e^{2u}\leq C E^{0.04}\),
+Using (eq:entropy-before-exit) and \(e^{2u}\leq C E^{0.04}\),
one obtains there
-\begin{equation}\label{eq:entropy-exit-volume-cost}
+
\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.
-\end{equation}
+
In the first bound use was made of \(E+P\leq C_nP\), valid for large \(E\)
on this region. Both quantities tend to zero with dimensional
constants.
@@ -3189,26 +3189,26 @@ constants.
Now assemble these estimates. After replacing \(Y\) by \(Z\), write
the retained surplus as \(\alpha_n P\), where \(\alpha_n>0\) is fixed
dimensionally. Put \(B_u=2|b(u)|+(n-1)|du|^2\). Substituting in
-\eqref{eq:entropy-conformal-exit} and moving the new entropy term
+(eq:entropy-conformal-exit) and moving the new entropy term
to the left gives
-\begin{align*}
+
&\wscal(\widehat h,\widehat\phi)
-E e_{\widehat h}(Z)-2(W+nu)+n\log E\\
&\quad\geq -(C+C_n)
- +e^{-2u}\bigl[\alpha_n P-B_u
- -2(e^{2u}-1)W-2nu e^{2u}\bigr]
+ +e^{-2u}[\alpha_n P-B_u
+ -2(e^{2u}-1)W-2nu e^{2u}]
+(1-e^{-2u})n\log E .
-\end{align*}
+
Use was made of \(e^{-2u}(C+C_n)\leq C+C_n\). The last term is nonnegative
-for \(E\geq1\). Equations \eqref{eq:entropy-exit-derivatives} and
-\eqref{eq:entropy-exit-volume-cost} leave at least a fixed fraction
+for \(E\geq1\). Equations (eq:entropy-exit-derivatives) and
+(eq:entropy-exit-volume-cost) leave at least a fixed fraction
of \(P\) in the bracket on the support of \(u\); off that support the
volume and derivative costs vanish. Thus set
\[
P'=\alpha'_n e^{-2u}P
\]
for a fixed \(\alpha'_n>0\), proving the scalar inequality in
-\eqref{eq:entropy-after-exit}. The only scalar constant charged was
+(eq:entropy-after-exit). The only scalar constant charged was
the dimensional replacement cost.
The volume estimate at exit follows as well. Since
@@ -3220,52 +3220,52 @@ The volume estimate at exit follows as well. Since
\]
for all sufficiently large dimensional \(E\).
-Finally, \eqref{eq:entropy-Z-energy} and the conformal scaling of
+Finally, (eq:entropy-Z-energy) and the conformal scaling of
energy give
\[
e_{\widehat h}(Z)^{1/2}
- \leq e^{-u}\frac{\sqrt n/N+c_EK_0}{\sqrt{r+c_E^2}}.
+ \leq e^{-u}\frac{\sqrt n/N+c_EK_0}{√(r+c_E^2)}.
\]
-The choice \eqref{eq:entropy-u-choice} implies
+The choice (eq:entropy-u-choice) implies
\[
- \frac{e^{-u}}{\sqrt{r+c_E^2}}
- \leq C\left(\frac{E^{-0.02}}{c_E}+E^{0.16}\right).
+ \frac{e^{-u}}{√(r+c_E^2)}
+ \leq C(\frac{E^{-0.02}}{c_E}+E^{0.16}).
\]
Using \(\sqrt E/N\leq C_n(1+\log E)\), \(c_EK_0\leq E^{-1/2}\),
and \(N\leq\sqrt E\), one concludes that
-\begin{equation}\label{eq:entropy-exit-gradient}
+
e_{\widehat h}(Z)^{1/2}
- \leq C_n\left(
- \frac{(1+K_0)(1+\log E)}{E^{0.02}}+\frac{E^{0.16}}{N}\right).
-\end{equation}
+ \leq C_n(
+ \frac{(1+K_0)(1+\log E)}{E^{0.02}}+\frac{E^{0.16}}{N}).
+
The first summand is \(o(E^{-0.01})\) for fixed \(K_0\); the second is
\(O_n((1+\log E)E^{-0.34})\). Taking \(E\) large enough in terms of
\(n,K_0\) proves the final inequality in
-\eqref{eq:entropy-after-exit}. The functions \(u\) and the normalizing
+(eq:entropy-after-exit). The functions \(u\) and the normalizing
denominator are smooth with finite bounds in each fixed construction.
The supports of \(Z\) are those of the last bumps and of \(Y^0\).
Thus the asserted uniformity and compact support properties persist.
-\end{proof}
-\begin{proof}[Proof of Theorem~\ref{thm:entropy}]
+
+Proof. [Proof of Theorem~thm:entropy]
Fix parameters \(H,A,B\) for the prescribed elementary bound and set
\(J=H+2\). This number of phases is fixed before any phase target
is chosen. The first target will be chosen after the bootstrap data
and their uniform constants have been fixed; the estimates below
hold for every sufficiently large choice of that target.
-Apply Lemma~\ref{lem:entropy-bootstrap}. At its output
+Apply Lemma~lem:entropy-bootstrap. At its output
\(W=\log(dV_{h_*}/\rho)=0\), \(\phi=0\), and
-\(\epsilon_0\in[\epsilon_*,2\epsilon_*]\). Increasing a dimensional
-constant once, \eqref{eq:entropy-bootstrap-output} has the form
+\(ε_0\in[ε_*,2ε_*]\). Increasing a dimensional
+constant once, (eq:entropy-bootstrap-output) has the form
\[
- \wscal(h_*,0)\geq-C_n+2W-n\log\epsilon_0
- +\epsilon_0 e_{h_*}(F_*),
+ \wscal(h_*,0)\geq-C_n+2W-n\logε_0
+ +ε_0 e_{h_*}(F_*),
\]
-because \(\epsilon_0\) ranges in a fixed dimensional interval.
-Use the schedule \eqref{eq:entropy-step-schedule} from now on.
-The bound \eqref{eq:entropy-summability} applies to the entire
+because \(ε_0\) ranges in a fixed dimensional interval.
+Use the schedule (eq:entropy-step-schedule) from now on.
+The bound (eq:entropy-summability) applies to the entire
increasing sequence of energy coefficients, even when it is
partitioned into phases.
@@ -3274,7 +3274,7 @@ first reaches it; call the actual output parameter \(E_1\). The
overshoot is at most a factor \(1+l_*\). In every sweep of this phase
use \(h_*\) as the reference metric. The possible support enlargement
per sweep, including its separation buffers, has a dimensional radius
-in this reference metric by Proposition~\ref{prop:sweep}. The total
+in this reference metric by Proposition~prop:sweep. The total
radius is therefore at most \(C_n(1+\log^3(1+E_1))\).
Record a bound for a single fixed graph sufficient for every
@@ -3292,19 +3292,19 @@ elements of word length at most \(L\) is at most exponential in \(L\).
It follows that the graph joining every pair of labels whose possible
enlarged supports in the first phase meet, or come within a required
buffer, has degree at most
-\begin{equation}\label{eq:entropy-first-graph}
+
\Delta_1+1\leq
- \exp\bigl(C_{\mathrm{inst}}(1+\log E_1)^3\bigr).
-\end{equation}
-Here \(C_{\mathrm{inst}}\) is finite and fixed \emph{before} the
+ \exp(C_{inst}(1+\log E_1)^3).
+
+Here \(C_{inst}\) is finite and fixed before the
first target is chosen. It includes the initial number of label
orbits and the compact support bounds. The two copies of labels
used at exit and their possible cooccurrences change this bound only
-by a fixed factor, which is absorbed in \(C_{\mathrm{inst}}\).
+by a fixed factor, which is absorbed in \(C_{inst}\).
Take \(E_1\) sufficiently large for
-Lemma~\ref{lem:entropy-accumulation} and
-Lemma~\ref{lem:entropy-phase-exit}; in the latter the initial \(K_0\)
+Lemma~lem:entropy-accumulation and
+Lemma~lem:entropy-phase-exit; in the latter the initial \(K_0\)
is the finite input-dependent gradient bound of \(F_*\).
Now there holds the scalar inequality, the volume bound with exponent
\(0.32\), and the absolute gradient bound at \(E_1\).
@@ -3339,9 +3339,9 @@ conclusion holds in that one fixed graph.
After an exit at \(E_j\), one chooses a target that both meets this
overlap condition and grows by one exponential level:
-\begin{equation}\label{eq:entropy-phase-targets}
+
T_{j+1}=\exp(E_j^{1/1000}),
-\end{equation}
+
and call the first parameter reaching it \(E_{j+1}\). Thus
\[
T_{j+1}\leq E_{j+1}\leq(1+l_*)T_{j+1}.
@@ -3352,10 +3352,10 @@ will be smaller than the gain from the preceding gradient bound.
For large \(E_j\) this target satisfies
\(E_j^{0.32}\leq\tfrac12 E_{j+1}^{0.26}\). The outgoing bound with
exponent \(0.32\) from phase \(j\) and
-Lemma~\ref{lem:entropy-accumulation} therefore give the bound with
+Lemma~lem:entropy-accumulation therefore give the bound with
exponent \(0.26\) before the next exit. The next initial gradient
bound is \(K_0\leq E_j^{-1/100}\leq1\), so all further thresholds
-for Lemma~\ref{lem:entropy-phase-exit} are dimensional. Increasing
+for Lemma~lem:entropy-phase-exit are dimensional. Increasing
the first target once ensures these conclusions in every one of a
prescribed finite number of phases.
@@ -3368,9 +3368,9 @@ all separation buffers. The number of sweeps gives
\leq C_n E_j^{3/1000}.
\]
Consequently
-\begin{equation}\label{eq:entropy-overlap-radius}
- R_{j+1}E_j^{-1/100}\leq C_n E_j^{-7/1000}\longrightarrow0.
-\end{equation}
+
+ R_{j+1}E_j^{-1/100}\leq C_n E_j^{-7/1000}→0.
+
For large \(E_j\), the overlap condition therefore holds with
\(K=E_j^{-1/100}\). For every relevant pair the preceding estimate
becomes
@@ -3388,24 +3388,24 @@ copies of each parent. Joining two such copies whenever their parents
are equal or at distance at most two gives a fixed graph
\(\mathcal G_{j+1}\) containing all new cooccurrences. If
\(\Delta_j\) is the degree of \(\mathcal G_j\), this construction gives
-\begin{equation}\label{eq:entropy-graph-recurrence}
+
\Delta_{j+1}+1\leq2(\Delta_j+1)^2.
-\end{equation}
+
It also doubles the number of label orbits. The preliminary coloring
construction applies to these graphs before each sweep.
All analytic support and chart bounds in the resulting finite
construction remain finite for the input, even though they need
not satisfy a useful quantitative bound. They do not enter
-\eqref{eq:entropy-graph-recurrence}.
+(eq:entropy-graph-recurrence).
After \(J\) phases, apply the coloring argument to
\(\mathcal G_J^2\), whose maximum degree is at most \(\Delta_J^2\).
Take \(q\) to be the maximum of its degree plus one, the number of
-label orbits, and \(n+1\). Equations \eqref{eq:entropy-first-graph} and
-\eqref{eq:entropy-graph-recurrence} imply
-\begin{equation}\label{eq:entropy-q-growth}
- \log q\leq C_{\mathrm{inst},J,n}(1+\log E_1)^3.
-\end{equation}
+label orbits, and \(n+1\). Equations (eq:entropy-first-graph) and
+(eq:entropy-graph-recurrence) imply
+
+ \log q\leq C_{inst,J,n}(1+\log E_1)^3.
+
For fixed \(J,n\) the coefficient here is fixed before \(E_1\) is
chosen. The graph \(\mathcal G_J\) and this coloring have exactly
the properties asserted in the theorem.
@@ -3413,17 +3413,17 @@ the properties asserted in the theorem.
It remains to make the first target large enough for the already fixed
\(J=H+2\). Let
\(\mathcal B(q)\leq\exp^{[H]}(B(1+q)^A)\) be the prescribed
-elementary bound. Equation \eqref{eq:entropy-q-growth} implies
+elementary bound. Equation (eq:entropy-q-growth) implies
\[
\mathcal B(q)\leq
- \exp^{[H+1]}\bigl(C'_{\mathrm{inst},J,n,A,B}
- (1+\log E_1)^3\bigr)
+ \exp^{[H+1]}(C'_{inst,J,n,A,B}
+ (1+\log E_1)^3)
\]
for large \(E_1\). On the other hand, the recurrence
-\eqref{eq:entropy-phase-targets} gives, for each fixed \(J\geq2\),
-\begin{equation}\label{eq:entropy-tower-growth}
- E_J\geq\exp^{[J-1]}\bigl(c_J E_1^{1/1000}\bigr)
-\end{equation}
+(eq:entropy-phase-targets) gives, for each fixed \(J\geq2\),
+
+ E_J\geq\exp^{[J-1]}(c_J E_1^{1/1000})
+
for some \(c_J>0\) and all sufficiently large \(E_1\).
One proves this by induction: raising a fixed-height exponential
tower with argument \(c t\) to the power \(1/1000\) still dominates
@@ -3438,7 +3438,7 @@ With the already chosen \(J=H+2\), the tower heights on the
two sides are now the same, and
\[
c_J E_1^{1/1000}>
- C'_{\mathrm{inst},J,n,A,B}(1+\log E_1)^3
+ C'_{inst,J,n,A,B}(1+\log E_1)^3
\]
for all sufficiently large \(E_1\). Thus \(E_J>\mathcal B(q)\).
Choose the first target still larger to have
@@ -3448,18 +3448,18 @@ These choices affect only the size of the final parameter and the
finite uniform bounds of the output objects.
Finally one accounts for the scalar constant. The bootstrap charges
-at most \(2C_b\epsilon_*\), independently of its input-dependent
+at most \(2C_bε_*\), independently of its input-dependent
number of sweeps. The initial logarithmic normalization has a
-dimensional cost because \(\epsilon_0\in[\epsilon_*,2\epsilon_*]\).
+dimensional cost because \(ε_0\in[ε_*,2ε_*]\).
All entropy sweep losses together are bounded by
\(C_n\sum l^2\leq C_n\). Each exit charges at most \(C_n\), and their
number \(J=H+2\) depends only on the prescribed elementary function
and \(n\). The input gradient bound \(K_0\), the initial small
-\(\epsilon_{\mathrm{in}}\), the compact support radii, and all metric
+\(ε_{in}\), the compact support radii, and all metric
and elliptic bounds occur only in the choices of the first target,
graphs, and arbitrarily small analytic errors. None contributes to
the scalar constant. Discard the final nonnegative surplus \(P'\);
-the remaining inequality is \eqref{eq:entropy-contract} with
+the remaining inequality is (eq:entropy-contract) with
\(C=C(n,\mathcal B)\).
The density \(\rho\) has been unchanged since the bootstrap, so
@@ -3469,8 +3469,8 @@ exits preserve the asserted smoothness, uniformity, compact supports,
and finite number of free label orbits. The transverse spaces were
enlarged only by the probability products used for the colorings,
and the sweep constructions are Borel on these spaces by
-Proposition~\ref{prop:sweep}. This proves all the conclusions.
-\end{proof}
+Proposition~prop:sweep. This proves all the conclusions.
+
This section bounds the degree of a map from a measured family of
Riemannian manifolds to a cube. Its integrand combines the weighted scalar
@@ -3485,7 +3485,7 @@ Use the following model of a measured space of leaf manifolds. The
underlying space $X$ is standard Borel and has a countable collection of
flow boxes
\[
- \kappa_a:T_a\times Q_a\longrightarrow X,
+ \kappa_a:T_a\times Q_a→ X,
\]
whose images cover $X$ and which are Borel isomorphisms onto their images.
Here $Q_a$ is an open box
@@ -3513,19 +3513,19 @@ 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 $|G|_\infty=\max_i|G_i|$, with the empty maximum interpreted as
+Write $|G|_∞=\max_i|G_i|$, with the empty maximum interpreted as
zero.
-\begin{definition}[Controlled measured band]\label{def:degree-band}
+[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 $G=(G_1,\ldots,G_d):X\to Q_T^d$.
-For every $0<t<T$ the part $\{|G|_\infty\le t\}$ has a countable flow-box
+a function $\phi$, and a map $G=(G_1,...,G_d):X\to Q_T^d$.
+For every $0<t<T$ the part $\{|G|_∞\le t\}$ has a countable flow-box
cover with the following properties.
-\begin{enumerate}
+
\item The boxes have one common positive size and fixed positive buffers:
their smaller concentric boxes cover the indicated part. They can be
-chosen inside $\{|G|_\infty<t'\}$ 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
leafwise coordinate derivatives are bounded by constants common to the
@@ -3535,11 +3535,11 @@ required.
\item The sum of the measures of the transversals of this cover is finite.
There is a common bound on the number of plaques of the cover whose
buffered boxes meet any one plaque of the cover.
-\end{enumerate}
+
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.
-\end{definition}
+
For $d\ge1$, 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
@@ -3550,9 +3550,9 @@ metric and inverse-metric bounds give uniform bounds for these derivatives
in geometric norms. Consequently, for a leafwise smooth $(d-1)$-form
$\eta$ supported over a compact subcube, with $\eta$ and $d\eta$ uniformly
bounded there, chartwise Stokes and Fubini give
-\begin{equation}\label{eq:degree-stokes}
+
\int_X d\eta=0.
-\end{equation}
+
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
@@ -3565,49 +3565,49 @@ 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
-\begin{equation}\label{eq:degree-definition}
+
\mathcal D_X(G)=\int_X G^*\omega.
-\end{equation}
+
The integral is absolute: its support lies in an inner band, the
derivatives of $G$ there are bounded, and that band has finite measured
volume. For $d\ge1$, a compactly supported top form on a cube of integral
-zero has a compactly supported primitive. Applying \eqref{eq:degree-stokes} to the
-pullback of this primitive shows that \eqref{eq:degree-definition} is
-independent of $\omega$. This is the \emph{measured degree}, with the
+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 $\mathcal D$
when the band and map are understood.
-\begin{theorem}[Measured degree estimate]\label{thm:degree}
+[Measured degree estimate]
There are absolute constants $C_0,C_1>0$ with the following property.
Let $q\ge2$, let $d$ be an integer with $0\le d\le q$, and let $\tau>0$.
For $0\le r\le d$ set
-\begin{equation}\label{eq:degree-widths}
- T_r=2\tau+\frac{r\tau}{q},\qquad
- b_r=\tau\left(1+\frac1{2q}+\frac r{4q^2}\right).
-\end{equation}
+
+ T_r=2\tau+(r\tau)/(q),\qquad
+ b_r=\tau(1+\frac1{2q}+\frac r{4q^2}).
+
Suppose $(X,h,\phi,G)$ is a controlled measured $d$-band over $Q_{T_d}^d$
-in the sense of Definition~\ref{def:degree-band}. Let $B\ge1$ and
-$L_1,\ldots,L_d>0$. Suppose throughout $X$ that
-\begin{equation}\label{eq:degree-horizon-bound}
+in the sense of Definition~def:degree-band. Let $B\ge1$ and
+$L_1,...,L_d>0$. Suppose throughout $X$ that
+
L_i|dG_i|_h\le B-\frac dq\qquad(1\le i\le d),
-\end{equation}
+
and suppose that
-\begin{equation}\label{eq:degree-horizon}
- \frac{C_0(1+B)^{2/3}}{L_i}<\frac{\tau}{10q}
+
+ \frac{C_0(1+B)^{2/3}}{L_i}<(\tau)/(10q)
\qquad(1\le i\le d).
-\end{equation}
+
Define $H_i(G)=2(L_i|dG_i|_h-1)_+$. Then
-\begin{equation}\label{eq:degree-estimate}
+
|\mathcal D_X(G)|(2\tau)^d
\le \frac{\exp(C_1d/q)}{\prod_{i=1}^dL_i}
- \int_{\{|G|_\infty<b_d\}}
- \exp\left(\frac{-\wscal(h,\phi)+\sum_{i=1}^dH_i(G)}2\right)dV_h.
-\end{equation}
+ \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
unsigned count.
-\end{theorem}
+
The proof removes one coordinate at a time. For $r\ge1$, the gaps
$T_r-T_{r-1}=\tau/q$ and $b_r-b_{r-1}=\tau/(4q^2)$ supply, respectively,
@@ -3621,54 +3621,54 @@ of the same degree; a scalar gain will pay exactly for the volume of the
graphs used to construct it. The following formulation separates that
construction from the final induction.
-\begin{lemma}[One step of measured descent]\label{lem:degree-descent}
-Assume the hypotheses of Theorem~\ref{thm:degree} with $d\ge1$, put
+[One step of measured descent]
+Assume the hypotheses of Theorem~thm:degree with $d\ge1$, put
$p=d-1$, and put
\[
- \Theta=\exp\left(\frac{-\wscal(h,\phi)+\sum_{i=1}^dH_i(G)}2\right).
+ Θ=\exp(\frac{-\wscal(h,\phi)+\sum_{i=1}^dH_i(G)}2).
\]
There is a Borel family of oriented regular hypersurface portions
-$\Sigma_s$ in the leaves, $-\tau<s<\tau$, and, for almost every $s$, a
-positive function $v_s$ on $\Sigma_s$, 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 $s$.
-\begin{enumerate}
+
\item For every nonnegative Borel function $F$ supported in
-$\{|G|_\infty<b_d\}$,
-\begin{equation}\label{eq:degree-coarea}
- \int_{-\tau}^{\tau}\!ds\int_{\Sigma_s}v_sF\,dA_h
+$\{|G|_∞<b_d\}$,
+
+ \int_{-\tau}^{\tau}\!ds\int_{Σ_s}v_sF\,dA_h
\le\int_XF\,dV_h.
-\end{equation}
+
\item For each such $s$ there is a controlled measured $p$-band
$(X_s^+,h_s^+,\phi_s^+,J_s^+)$ over $Q_{T_p}^p$ of degree
-$\mathcal D_X(G)$. Its coordinates are indexed by $j=2,\ldots,d$ and
+$\mathcal D_X(G)$. 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
-\begin{equation}\label{eq:degree-child-integral}
- \begin{split}
- &\int_{\{|J_s^+|_\infty<b_p\}}
- \exp\left(\frac{-\wscal(h_s^+,\phi_s^+)+\sum_{j=2}^dH_j^+}{2}\right)
+
+
+ &\int_{\{|J_s^+|_∞<b_p\}}
+ \exp(\frac{-\wscal(h_s^+,\phi_s^+)+\sum_{j=2}^dH_j^+}{2})
dV_{h_s^+}\\
&\hspace{25mm}\le \frac{e^{C_*/q}}{L_1}
- \int_{\Sigma_s\cap\{|G|_\infty<b_d\}}v_s\Theta\,dA_h.
- \end{split}
-\end{equation}
+ \int_{Σ_s\cap\{|G|_∞<b_d\}}v_sΘ\,dA_h.
+
+
In dimension $p=0$ the left side is the measured unsigned count.
-\end{enumerate}
+
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.
-\end{lemma}
-Prove Lemma~\ref{lem:degree-descent} in this and the next two
+
+Prove Lemma~lem:degree-descent in this and the next two
subsections. Fix intermediate widths
-\begin{equation}\label{eq:degree-intermediate-widths}
+
T_p<t_1<t_2<t_3<t_4<T_d.
-\end{equation}
+
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
@@ -3679,86 +3679,86 @@ Choose once and for all a smooth even function $\gamma$ with
$0\le\gamma(y)\le y^2$ and with $\gamma(y)$ bounded above and below by
positive absolute multiples of $|y|^{3/2}$ for $|y|\ge1$. Let $P_0$ be the
solution about zero of
-\begin{equation}\label{eq:degree-pressure}
+
P_0(0)=0,\qquad P_0'=L_1\psi(P_0),\qquad
- \psi(y)=1+\frac{\gamma(y)}{8(1+B)}.
-\end{equation}
+ \psi(y)=1+(\gamma(y))/(8(1+B)).
+
It is odd and increasing and has poles at $\pm b$, where
\[
- b=\frac1{L_1}\int_0^\infty\frac{dy}{\psi(y)}
+ b=\frac1{L_1}\int_0^∞(dy)/(\psi(y))
\le\frac{C(1+B)^{2/3}}{L_1}.
\]
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
-\eqref{eq:degree-horizon} at least this $C$. Thus
-\begin{equation}\label{eq:degree-pole-width}
- b<\frac{\tau}{10q}.
-\end{equation}
+(eq:degree-horizon) at least this $C$. Thus
+
+ b<(\tau)/(10q).
-\paragraph{The local slice calculation.}
-First record the analytic purpose of the pressure law. Let $\Sigma$
+
+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
\[
- \begin{gathered}
- D=H+\partial_\nu\phi,\qquad
- H=\operatorname{tr}\mathrm{II},\qquad
- \mathrm{II}(Y,Z)=\langle\nabla_Y\nu,Z\rangle,\\
- \mathcal P=|\mathrm{II}|^2+
- (\operatorname{Ric}_h-\nabla^2\phi)(\nu,\nu).
- \end{gathered}
+
+ D=H+∂_\nu\phi,\qquad
+ H=trII,\qquad
+ II(Y,Z)=<\nabla_Y\nu,Z>,\\
+ \mathcal P=|II|^2+
+ (Ric_h-\nabla^2\phi)(\nu,\nu).
+
\]
Put $\mathfrak a=L_1\psi(D)>0$. When $D=P_0(s-G_1)$, the derivatives
-of the raw pressure are $\partial_sP_0(s-G_1)=\mathfrak a$ and
+of the raw pressure are $∂_sP_0(s-G_1)=\mathfrak a$ and
$d_xP_0(s-G_1)=-\mathfrak a\,dG_1$. The following calculation uses
only a positive smooth function $v$ satisfying the local equation
-\begin{equation}\label{eq:degree-local-lapse}
- -\bigl(\Delta_{\Sigma,\phi}+\mathcal P
- -\mathfrak a\,dG_1(\nu)\bigr)v=\mathfrak a.
-\end{equation}
-Here $\Delta_{\Sigma,\phi}$ is weighted by
-$e^{\phi|_\Sigma}dA_h$, and slice norms use the induced metric. The
+
+ -(\Delta_{Σ,\phi}+\mathcal P
+ -\mathfrak a\,dG_1(\nu))v=\mathfrak a.
+
+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 $\phi_\Sigma=\phi|_\Sigma+\ln v$ and
-$S_\Sigma=\wscal(h|_{T\Sigma},\phi_\Sigma)$. Gauss' equation and the
+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
\[
- \wscal(h|_{T\Sigma},\phi|_\Sigma)
- =\wscal(h,\phi)+D^2-2\mathcal P+|\mathrm{II}|^2.
+ \wscal(h|_{TΣ},\phi|_Σ)
+ =\wscal(h,\phi)+D^2-2\mathcal P+|II|^2.
\]
Adding $\ln v$ to the weight changes this by
-$-2\Delta_{\Sigma,\phi}v/v+|d\ln v|^2$. Substitution of
-\eqref{eq:degree-local-lapse} yields the exact identity
-\begin{equation}\label{eq:degree-slice-identity}
- S_\Sigma=\wscal(h,\phi)+D^2-2\mathfrak a\,dG_1(\nu)
- +|\mathrm{II}|^2+\frac{2\mathfrak a}{v}+|d\ln v|^2.
-\end{equation}
+$-2\Delta_{Σ,\phi}v/v+|d\ln v|^2$. Substitution of
+(eq:degree-local-lapse) yields the exact identity
+
+ S_Σ=\wscal(h,\phi)+D^2-2\mathfrak a\,dG_1(\nu)
+ +|II|^2+(2\mathfrak a)/(v)+|d\ln v|^2.
+
This is also the hypersurface calculation behind
-Lemma~\ref{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 $t=L_1|dG_1|_h\le B$ and $x=v/L_1$. Since $\gamma(D)\le D^2$,
\[
-2\mathfrak a\,dG_1(\nu)
- \ge-2t\left(1+\frac{\gamma(D)}{8(1+B)}\right),
- \qquad \frac{t\gamma(D)}{4(1+B)}\le\frac{D^2}{4}.
+ \ge-2t(1+(\gamma(D))/(8(1+B))),
+ \qquad (t\gamma(D))/(4(1+B))\le(D^2)/(4).
\]
Discard the remaining nonnegative part of $D^2$, and use
$2\mathfrak a/v\ge2/x$. Finally,
$t\le1+(t-1)_+$ and $x^{-1}+\ln x\ge1$ for $x>0$. Thus
-\eqref{eq:degree-slice-identity} implies
-\begin{equation}\label{eq:degree-slice-bound}
- S_\Sigma\ge\wscal(h,\phi)-H_1(G)-2\ln(v/L_1)+|d\ln v|^2.
-\end{equation}
+(eq:degree-slice-identity) implies
+
+ S_Σ\ge\wscal(h,\phi)-H_1(G)-2\ln(v/L_1)+|d\ln v|^2.
+
The square term will be used later when the weight is changed near
-singular points. For $j=2,\ldots,d$, put
-$H_j^\Sigma=2(L_j|d_\Sigma G_j|-1)_+$. Tangential restriction gives
-$H_j^\Sigma\le H_j(G)$, so the full pointwise density comparison is
-\begin{equation}\label{eq:degree-raw-integrand}
- \exp\left(\frac{-S_\Sigma+\sum_{j=2}^dH_j^\Sigma}{2}\right)
- \le\frac v{L_1}e^{-|d\ln v|^2/2}\Theta
- \le\frac v{L_1}\Theta.
-\end{equation}
+singular points. For $j=2,...,d$, put
+$H_j^Σ=2(L_j|d_Σ G_j|-1)_+$. Tangential restriction gives
+$H_j^Σ\le H_j(G)$, so the full pointwise density comparison is
+
+ \exp(\frac{-S_Σ+\sum_{j=2}^dH_j^Σ}{2})
+ \le\frac v{L_1}e^{-|d\ln v|^2/2}Θ
+ \le\frac v{L_1}Θ.
+
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.
@@ -3772,15 +3772,15 @@ $\widehat P$ and set
\]
Choose $\eta$ sufficiently small for the given band and then keep
it fixed. On every boundary portion used below, $\widehat P$ will agree
-with $P_0$ and hence obey \eqref{eq:degree-pressure}.
+with $P_0$ and hence obey (eq:degree-pressure).
-\paragraph{Least minimizers, including on noncompact leaves.}
-Write $P_\phi(E;U)=\int_{\partial^*E\cap U}e^\phi\,dA_h$ for the weighted
+Least minimizers, including on noncompact leaves.:
+Write $P_\phi(E;U)=\int_{∂^*E\cap U}e^\phi\,dA_h$ for the weighted
perimeter of a set of locally finite perimeter. Use local differences
of the functional
-\begin{equation}\label{eq:degree-functional}
+
P_\phi(E)-\int_E\mathfrak h_s e^\phi\,dV_h.
-\end{equation}
+
This is a weighted $\mu$-bubble functional. For soap bubbles in Gromov's
terminology, see \cite[Section~5$\frac56$]{Gromov1996}; for the weighted
functional and its local first- and second-variation formulas, see also
@@ -3789,7 +3789,7 @@ Neither term is required to be finite on the whole leaf. In particular,
this construction uses no global weighted-volume bound.
-\par\noindent{\footnotesize\url{https://doi.org/10.1007/978-1-4612-4098-3_1}}\par
+\par\noindent{\footnotesizehttps://doi.org/10.1007/978-1-4612-4098-3_1}\par
\par\noindent{\footnotesize\ttfamily src-chodoshli2024.pdf}\par
The output will be a canonical nested family $E_s$ on each open leaf,
@@ -3800,8 +3800,8 @@ 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.
-\emph{Compact domains and leastness.}
-For a compact subset $K$ of a leaf, minimize \eqref{eq:degree-functional}
+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
smooth and positive and the forcing is bounded. A bound on the functional
@@ -3820,9 +3820,9 @@ up to null sets, in every other one. It is consequently unique as an
$L^1$ set; call it the least minimizer in $K$.
-\par\noindent{\footnotesize\url{https://doi.org/10.1017/CBO9781139108133}}\par
+\par\noindent{\footnotesizehttps://doi.org/10.1017/CBO9781139108133}\par
-\emph{Order and exhaustion.}
+Order and exhaustion.
The same inequality proves both monotonicities one needs. If $K\subset K'$
and two forcings satisfy $\mathfrak h\le\mathfrak h'$, a minimizer for
$(K,\mathfrak h)$ and one for $(K',\mathfrak h')$ obey
@@ -3830,19 +3830,19 @@ $(K,\mathfrak h)$ and one for $(K',\mathfrak h')$ obey
\mathcal F_{\mathfrak h}(E\cap F)+\mathcal F_{\mathfrak h'}(E\cup F)
\le \mathcal F_{\mathfrak h}(E)+\mathcal F_{\mathfrak h'}(F),
\]
-where $\mathcal F$ denotes \eqref{eq:degree-functional}; the inequality for
-the volume terms follows on $E\setminus F$ from
+where $\mathcal F$ denotes (eq:degree-functional); the inequality for
+the volume terms follows on $E∖ F$ from
$\mathfrak h\le\mathfrak h'$. Minimality in
the two domains makes both inequalities equalities, and leastness in $K$
implies $E\subset F$. Exhaust a leaf by increasing finite unions of
compact coordinate subboxes whose interiors exhaust it. The least
minimizers increase with this exhaustion. Their union, interpreted in
-$L^1_{\mathrm{loc}}$, 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 $E_s\subset E_t$
for $s<t$.
-\emph{Local comparison.}
+Local comparison.
The limit must still minimize for compact variations, since all later
regularity and compactness arguments use that property.
Ball replacement in a relatively compact chart gives uniform local
@@ -3866,7 +3866,7 @@ convergence of the metric, density, and forcing coefficients. Below, apply it w
the local perimeter measures. Multiplying a density by a positive
constant has no effect on any of these comparisons.
-\emph{Borel dependence.}
+Borel dependence.
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
@@ -3922,28 +3922,28 @@ on the measured space. Use the density-one representative of
$E_s$; the support of its boundary measure is also Borel, by testing its
mass on a countable basis of coordinate balls.
-\paragraph{Confinement.}
+Confinement.:
For sufficiently small $\eta$, uniformly for $-\tau<s<\tau$, the boundary
-support in $\{|G|_\infty\le t_4\}$ satisfies
-\begin{equation}\label{eq:degree-confinement}
-\operatorname{supp}\partial[E_s]\cap\{|G|_\infty\le t_4\}
+support in $\{|G|_∞\le t_4\}$ satisfies
+
+supp∂[E_s]\cap\{|G|_∞\le t_4\}
\subset\{|s-G_1|<b-\eta\}.
-\end{equation}
+
In this inner band the side $G_1\le s-b+\eta$ is full and the side
$G_1\ge s+b-\eta$ is empty, up to null sets. To prove this uniform claim,
the ODE at the positive pole gives
\[
- b-x=\frac1{L_1}\int_{P_0(x)}^\infty\frac{dy}{\psi(y)}
- \asymp_{\mathrm{inst}}P_0(x)^{-1/2}
- \quad(x\uparrow b).
+ b-x=\frac1{L_1}\int_{P_0(x)}^∞(dy)/(\psi(y))
+ \asymp_{inst}P_0(x)^{-1/2}
+ \quad(x↑ b).
\]
Thus at distance comparable to $\eta$ from a pole the absolute forcing is
-at least $c_{\mathrm{inst}}\eta^{-2}$. The prolongation is monotone, so
+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
-$r=c'_{\mathrm{inst}}\eta$ centered at a point with $s-G_1\ge b-\eta$
-has forcing at least $H=c''_{\mathrm{inst}}\eta^{-2}$ throughout it, if
-$c'_{\mathrm{inst}}$ is small. The balls are taken in a buffered atlas
+$r=c'_{inst}\eta$ centered at a point with $s-G_1\ge b-\eta$
+has forcing at least $H=c''_{inst}\eta^{-2}$ throughout it, if
+$c'_{inst}$ 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
@@ -3953,35 +3953,35 @@ 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 $t<r$
\[
- P_{\mathrm{norm}}(E_s;B_t)+H V(t)\le C V'(t).
+ P_{norm}(E_s;B_t)+H V(t)\le C V'(t).
\]
The chart isoperimetric inequality applied to that complementary phase,
extended by zero outside $B_t$, gives, when $V(t)>0$,
\[
- V(t)^{(d-1)/d}\le C\bigl(P_{\mathrm{norm}}(E_s;B_t)+V'(t)\bigr).
+ V(t)^{(d-1)/d}\le C(P_{norm}(E_s;B_t)+V'(t)).
\]
If the complementary phase occurs in $B_{r/4}$, the two inequalities first
give $V'\ge cV^{(d-1)/d}$ and hence $V(r/2)\ge cr^d$. They also give
$V'\ge cHV$ on $[r/2,r]$. It follows that
$V(r)\ge cr^d\exp(cHr)$, contrary to $V(r)\le Cr^d$ when $Hr$ is large.
This argument includes $d=1$, where the first differential inequality
-has exponent zero. Since $Hr\asymp_{\mathrm{inst}}\eta^{-1}$, small
+has exponent zero. Since $Hr\asymp_{inst}\eta^{-1}$, 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
-the forcing estimate, proves \eqref{eq:degree-confinement}. In particular
-the unprolonged Equation~\eqref{eq:degree-pressure} holds on all the
+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.
-\paragraph{The local regularity and compactness inputs.}
+The local regularity and compactness inputs.:
Record precisely the local geometric measure theory used from now on.
-Set $\Sigma_s^{\mathrm{all}}=\operatorname{supp}\partial[E_s]$. For
+Set $Σ_s^{all}=supp∂[E_s]$. For
$d\ge2$, its regular portion is a smooth multiplicity-one hypersurface.
-The remaining set is closed in $\Sigma_s^{\mathrm{all}}$ and has Hausdorff
+The remaining set is closed in $Σ_s^{all}$ and has Hausdorff
dimension at most
-\begin{equation}\label{eq:degree-singular-dimension}
+
p-7.
-\end{equation}
+
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 $\phi_0$
@@ -3995,7 +3995,7 @@ 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
-\eqref{eq:degree-singular-dimension}; see
+(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
@@ -4009,19 +4009,19 @@ The set comparison therefore implies that formulation as well.
\par\noindent{\footnotesize\ttfamily src-eichmair2009.pdf}\par
-\emph{Local compactness claim.}
+Local compactness claim.
Suppose $d\ge2$ and $E_j$ minimize the absorbed-perimeter functionals
for all compact variations in a common buffered normalized outer chart.
Assume that their absorbed metrics and converted volume forcings
converge smoothly on smaller boxes and that $E_j\to E$ in
-$L^1_{\mathrm{loc}}$. Then $E$ is locally minimizing, the normalized
+$L^1_{loc}$. 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.
Near every regular point of the limiting boundary, the boundaries are
-single smooth graphs converging in $C^\infty$ 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
-$\partial U$ is zero, the perimeter masses on $U$ converge. For the
+$∂ U$ 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
@@ -4031,37 +4031,37 @@ Here is the proof, including the local perimeter assertion. In the
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 $C^\infty$ 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 $\widehat h_j$ converge smoothly to a
-positive metric $\widehat h_\infty$. Ball comparison gives the local
+positive metric $\widehat h_∞$. Ball comparison gives the local
perimeter bounds used above, so BV compactness gives a further
-subsequence converging in $L^1_{\mathrm{loc}}$ 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.
Write $\mu_j=P_{\widehat h_j}(E_j;\,\cdot\,)$ and
-$\mu_\infty=P_{\widehat h_\infty}(E;\,\cdot\,)$. To prove convergence
+$\mu_∞=P_{\widehat h_∞}(E;\,\cdot\,)$. To prove convergence
of these measures, begin with a relatively compact coordinate domain
$V$ from a convergence-determining family, chosen so that
-$\mu_\infty(\partial V)=0$. Surround it by a thin outer belt with level
-domains $V_t$ contained in an outer enlargement $V_\epsilon$. Coarea
+$\mu_∞(∂ V)=0$. Surround it by a thin outer belt with level
+domains $V_t$ contained in an outer enlargement $V_ε$. Coarea
chooses a level $t_j$ at which the trace mismatch between $E_j$ and $E$
tends to zero. Splice the unchanged $E$ inside $V_{t_j}$ to $E_j$
outside. The common coefficient bounds make the extra perimeter at the
splice tend to zero, and the bounded converted forcing times the
$L^1$ mismatch tends to zero as well. Minimality of $E_j$ gives
\[
- \limsup_{j\to\infty}\mu_j(V)\le\mu_\infty(V_\epsilon).
+ limsup_{j\to∞}\mu_j(V)\le\mu_∞(V_ε).
\]
Here the perimeter of the fixed set $E$ in the varying metric converges
by uniform convergence of the metric integrands on unit conormals.
-Shrinking the belt makes the right side tend to $\mu_\infty(V)$, because
-$\mu_\infty(\partial V)=0$. 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 $\mu_\infty$, and their
+Consequently $\mu_j$ converge weakly locally to $\mu_∞$, and their
masses converge on every relatively compact continuity subdomain $U$.
For precision in passing to varifolds, restrict to a relatively compact
@@ -4070,20 +4070,20 @@ continuity subdomains can exhaust the smaller box. To check weak
convergence of the restrictions of $D\mathbf1_{E_j}$ as vector measures
on $\overline U$, extend a continuous vector test there to the outer
chart and multiply it by a cutoff supported in $U$ and equal to one away
-from a $\delta$-collar of $\partial U$. Local weak BV convergence
+from a $\delta$-collar of $∂ U$. Local weak BV convergence
handles this cutoff test. On the compact chart uniform ellipticity
gives $|D\mathbf1_{E_j}|\le C\mu_j$. For a compact closed
$\delta$-collar $F_\delta$, weak perimeter convergence and the closed-set
inequality give
-$\limsup_j\mu_j(F_\delta)\le\mu_\infty(F_\delta)$, which tends to zero
-as $\delta\downarrow0$ because $\mu_\infty(\partial U)=0$. The same
+$limsup_j\mu_j(F_\delta)\le\mu_∞(F_\delta)$, which tends to zero
+as $\delta↓0$ because $\mu_∞(∂ U)=0$. The same
bound for the limiting vector measure lets the collar shrink and proves
the restriction convergence. The perimeter masses on $U$ also converge.
In coordinates the perimeter
integrand for a metric $k$ is
-$\mathfrak p_k(x,z)=\sqrt{\det k(x)}\,|z|_{k^{-1}(x)}$.
+$\mathfrak p_k(x,z)=√(\det k(x))\,|z|_{k^{-1}(x)}$.
Uniform convergence of this integrand on unit conormals and the local
-mass bound replace $\widehat h_j$ by $\widehat h_\infty$ in that mass
+mass bound replace $\widehat h_j$ by $\widehat h_∞$ 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
@@ -4091,13 +4091,13 @@ 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
-$\mathfrak p_{\widehat h_\infty}(x,z)a(x,\ker z)$ for $z\ne0$ and zero
+$\mathfrak p_{\widehat h_∞}(x,z)a(x,\ker z)$ for $z\ne0$ and zero
at $z=0$. This gives precisely the local varifold convergence; the
uniform convergence of the metric integrands gives it also for the
original varying metrics.
-\par\noindent{\footnotesize\url{https://doi.org/10.1007/BF02196453}}\par
+\par\noindent{\footnotesizehttps://doi.org/10.1007/BF02196453}\par
To prove the support and graph assertions, the first variation in
the metric $\widehat h$ is, by BV Gauss--Green, integration of the bounded
@@ -4111,21 +4111,21 @@ support has positive limit mass, so weak convergence forces nearby
support points. Near a fixed smooth multiplicity-one sheet of the
limit, Allard's interior regularity theorem gives single graphs with a
common $C^{1,\alpha_{\mathrm A}}$ bound for some
-$\alpha_{\mathrm A}\in(0,1)$ on a smaller cylinder \cite{Allard1972}.
+$\alpha_{\mathrm A}\in(0,1)$ on a smaller cylinder [cite].
Compactness and identification of the already known varifold limit give
$C^{1,\beta}$ graph convergence for every $0<\beta<\alpha_{\mathrm A}$.
The prescribed equation, with the smoothly converging coefficients,
-then gives $C^\infty$ 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.
-\par\noindent{\footnotesize\url{https://doi.org/10.2307/1970868}}\par
+\par\noindent{\footnotesizehttps://doi.org/10.2307/1970868}\par
The varying-metric use of these local theorems reduces to one smooth
ambient metric as follows. Pass further in a smaller compact chart to a subsequence converging so
-rapidly in $C^\infty$ 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
disjoint smooth interval cutoffs. The resulting product block-diagonal
@@ -4139,7 +4139,7 @@ argument to a sequence on which local convergence failed proves the
local support and graphical convergence asserted above.
-\par\noindent{\footnotesize\url{https://doi.org/10.2307/1969989}}\par
+\par\noindent{\footnotesizehttps://doi.org/10.2307/1969989}\par
For $d=1$ the needed facts are elementary. A locally minimizing set has
locally finitely many jumps by the perimeter bound. Two consecutive
@@ -4156,21 +4156,21 @@ singular points in this case.
Now extract a smooth normal speed from the ordered family. Write
-$\Sigma_s$ for the regular part of $\Sigma_s^{\mathrm{all}}$ in
-$\{|G|_\infty<t_3\}$. The boundary has its exterior coorientation and
+$Σ_s$ 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.
-\paragraph{Differentiating the ordered family.}
-On each leaf the sets $E_t$ converge to $E_s$ in $L^1_{\mathrm{loc}}$ as
+Differentiating the ordered family.:
+On each leaf the sets $E_t$ converge to $E_s$ in $L^1_{loc}$ as
$t\to s$ except at countably many $s$. In fact, on a compact chart set
$K$ the function $t\mapsto\int_K\mathbf1_{E_t}$ is monotone, and nestedness
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
-$\Sigma_t$ converge smoothly and graphically near each regular point of
-$\Sigma_s$ as $t\to s$.
+$Σ_t$ converge smoothly and graphically near each regular point of
+$Σ_s$ as $t\to s$.
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
@@ -4180,10 +4180,10 @@ their tangent planes agree at contact. The comparison of the graph mean
curvatures at that minimum of $r(t,\cdot)-r(s,\cdot)$ gives
$D_t\le D_s$, whereas their equations give
$D_t=\mathfrak h_t>\mathfrak h_s=D_s$ at the same point. This is
-impossible. Use the conventions for $D$ and $\mathrm{II}$ from the
+impossible. Use the conventions for $D$ and $II$ from the
local slice calculation, with $\nu$ the exterior unit normal. The
equation $D=\mathfrak h_s$ is the first variation of
-\eqref{eq:degree-functional} on a regular portion.
+(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
@@ -4191,13 +4191,13 @@ 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
\[
- t\longmapsto\int_{z_-}^{z_+}\mathbf1_{E_t}(y,z)\,dz,
+ t⟼\int_{z_-}^{z_+}\mathbf1_{E_t}(y,z)\,dz,
\]
which equals $r(t,y)-z_-$ in the single-graph situation. Consequently,
for almost every continuity parameter $s$, the nonnegative difference
quotients
\[
- w_t(y)=\frac{r(t,y)-r(s,y)}{t-s},\qquad t\ne s,
+ w_t(y)=(r(t,y)-r(s,y))/(t-s),\qquad t\ne s,
\]
converge to the finite column-derivative values as $t\to s$ at all these
dense test coordinates. In particular they are bounded there.
@@ -4222,19 +4222,19 @@ derivative of a monotone height.
Differentiation of $D=\mathfrak h_s$ now identifies this normal speed.
Use $\mathcal P$ from the local slice calculation and put
-$\mathfrak a=\partial_s\mathfrak h_s=L_1\psi(D)>0$; the last equality
+$\mathfrak a=∂_s\mathfrak h_s=L_1\psi(D)>0$; the last equality
uses confinement, where the forcing agrees with the raw pressure.
The normal variation formula, with the tangential derivative of
$D-\mathfrak h_s=0$ cancelling, is
-\begin{equation}\label{eq:degree-lapse}
- -\bigl(\Delta_{\Sigma,\phi}+\mathcal P+\partial_\nu\mathfrak h_s\bigr)v
+
+ -(\Delta_{Σ,\phi}+\mathcal P+∂_\nu\mathfrak h_s)v
=\mathfrak a.
-\end{equation}
-The convention for $\Delta_{\Sigma,\phi}$ is weighted by
-$e^{\phi|_\Sigma}dA_h$. The variation of $\partial_\nu\phi$ adds
-$v\nabla^2\phi(\nu,\nu)-\langle d\phi,dv\rangle_\Sigma$ to the ordinary
+
+The convention for $\Delta_{Σ,\phi}$ is weighted by
+$e^{\phi|_Σ}dA_h$. The variation of $∂_\nu\phi$ adds
+$v\nabla^2\phi(\nu,\nu)-< d\phi,dv>_Σ$ to the ordinary
mean-curvature variation, giving the signs in
-\eqref{eq:degree-lapse}. A nonnegative $v$ cannot vanish at an interior
+(eq:degree-lapse). A nonnegative $v$ cannot vanish at an interior
point: there the left side is nonpositive, contrary to $\mathfrak a>0$.
Thus $v>0$. In graph dimension zero the differential terms vanish and
the same equation rules out $v=0$.
@@ -4258,7 +4258,7 @@ integral over the whole parameter interval.
In a graph cylinder, normal speed times area has the expression
\[
- v\,dA_h=J_h(y,r(s,y))\,\partial_s r(s,y)\,dy,
+ v\,dA_h=J_h(y,r(s,y))\,∂_s r(s,y)\,dy,
\]
where $J_h(y,z)\,dy\,dz$ is ambient Riemannian volume in the coordinates.
The one-dimensional change-of-variables inequality for a monotone
@@ -4272,77 +4272,77 @@ ambient point at two parameters, by the noncontact argument above.
Therefore no ambient point is counted more than once on the right.
Equivalently, the counted images can be taken Borel by the Borel
one-to-one image theorem. Transverse invariance on overlaps gives
-\begin{equation}\label{eq:degree-coarea-proof}
- \int_{-\tau}^{\tau}\!ds\int_{\Sigma_s}vF\,dA_h
+
+ \int_{-\tau}^{\tau}\!ds\int_{Σ_s}vF\,dA_h
\le\int_XF\,dV_h
-\end{equation}
-for every nonnegative Borel $F$ supported in $\{|G|_\infty<t_3\}$.
-Since $b_d<t_3$, this proves \eqref{eq:degree-coarea}.
-Require one more consequence of \eqref{eq:degree-lapse}. On a
+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
-patches. Indeed the coefficients in \eqref{eq:degree-lapse} and the
+patches. Indeed the coefficients in (eq:degree-lapse) and the
derivatives of $\mathfrak a$ are uniformly bounded there, and
$\mathfrak a$ has a uniform positive lower bound. Confinement provides
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
patch, and elliptic compactness would give a nonnegative limit with an
-interior zero satisfying \eqref{eq:degree-lapse}, an impossibility.
+interior zero satisfying (eq:degree-lapse), an impossibility.
Thus $v$ has a positive local lower bound. Harnack then bounds its local
supremum by a constant times its value at an interior point, since the
bounded source can be absorbed using that lower bound. Interior
estimates give $|\nabla^r v|\le C_r v$ on still smaller patches, and
differentiating $\ln v$ gives the claim. This does not bound the value
-of $\ln v$, which is not required by Definition~\ref{def:degree-band}.
+of $\ln v$, which is not required by Definition~def:degree-band.
For $p=0$ the assertion about positive-order derivatives is empty.
The local slice calculation now applies to the actual boundaries.
Confinement gives $D=\mathfrak h_s=P_0(s-G_1)$ and hence
-$\partial_\nu\mathfrak h_s=-\mathfrak a\,dG_1(\nu)$ on every retained
-regular portion. Equation~\eqref{eq:degree-lapse} therefore becomes
-exactly \eqref{eq:degree-local-lapse}. With
-$\phi_\Sigma=\phi|_\Sigma+\ln v$, the scalar bound
-\eqref{eq:degree-slice-bound} and the pointwise comparison
-\eqref{eq:degree-raw-integrand} hold on these portions. The derivative
+$∂_\nu\mathfrak h_s=-\mathfrak a\,dG_1(\nu)$ 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
+(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.
-\paragraph{Degree on the boundary.}
-Let $J=(G_2,\ldots,G_d)$. Fix a unit-integral one-form
+Degree on the boundary.:
+Let $J=(G_2,...,G_d)$. Fix a unit-integral one-form
$a_-(z)\,dz$ supported in $(-1.9\tau,-1.6\tau)$ and one
$a_+(z)\,dz$ supported in $(1.6\tau,1.9\tau)$. Let $\beta$ be a
unit-integral smooth $p$-form supported in a sufficiently small cube
about zero in the residual target; for $p=0$ let $\beta=1$. Set
\[
- \chi_{\mathrm{deg}}(z)=\int_{-\infty}^{z}(a_-(u)-a_+(u))\,du.
+ \chi_{deg}(z)=\int_{-∞}^{z}(a_-(u)-a_+(u))\,du.
\]
This function is compactly supported in $(-1.9\tau,1.9\tau)$ and is one
-between the two supports. By \eqref{eq:degree-pole-width} and
-\eqref{eq:degree-confinement}, the low support is in $E_s$, the high
-support is outside $E_s$, and $\chi_{\mathrm{deg}}(G_1)=1$ on the confined
+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
boundary where the residual form is supported. Measured current Stokes therefore gives
-\begin{equation}\label{eq:degree-boundary-degree}
- \begin{split}
- \int_{\partial[E_s]} \chi_{\mathrm{deg}}(G_1)J^*\beta
+
+
+ \int_{∂[E_s]} \chi_{deg}(G_1)J^*\beta
&=\int_{E_s}(a_-(G_1)-a_+(G_1))\,dG_1\wedge J^*\beta\\
&=\int_Xa_-(G_1)\,dG_1\wedge J^*\beta
=\mathcal D_X(G).
- \end{split}
-\end{equation}
+
+
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
-\eqref{eq:degree-stokes}. The singular set contributes no area, and
-the counted part of the boundary is in $\{|G|_\infty<t_3\}$.
-Consequently \eqref{eq:degree-boundary-degree} computes the degree by
+(eq:degree-stokes). The singular set contributes no area, and
+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.
If $2\le d\le7$, 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
-$\{|G|_\infty<t_3\}$, 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
@@ -4353,26 +4353,26 @@ 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
-$\phi_\Sigma$.
+$\phi_Σ$.
Restrict this branch to
\[
- |G|_\infty<t_1,\qquad |J|_\infty<T_p.
+ |G|_∞<t_1,\qquad |J|_∞<T_p.
\]
The $G_1$ sides of the first restriction cannot be reached by confinement,
since $|G_1|<\tau+b<T_p$. Near any other side some residual coordinate
has absolute value near $t_1>T_p$. Hence every inner residual subcube
has buffered graph charts in this restriction. Its degree equals
-$\mathcal D_X(G)$ by \eqref{eq:degree-boundary-degree}, and its coordinate gradients only
+$\mathcal D_X(G)$ by (eq:degree-boundary-degree), and its coordinate gradients only
decrease on restriction to the tangent space. Take $J^+=J$ and use
-\eqref{eq:degree-slice-bound}.
+(eq:degree-slice-bound).
If $d=1$, so $p=0$, let $X_s^+$ be the confined regular points in
-$\Sigma_s\cap\{|G_1|<t_1\}$, each with its boundary orientation, and let
+$Σ_s\cap\{|G_1|<t_1\}$, 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 $\phi_\Sigma$.
+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 \eqref{eq:degree-boundary-degree}.
+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
finer than that separation in the normalized plaque. Each mesh cell
@@ -4383,12 +4383,12 @@ number of point charts, with transversals restricted from the parent and
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~\ref{def:degree-band} are
+weight derivative conditions in Definition~def:degree-band are
vacuous in dimension zero, so this is a controlled $0$-band. Its measured
signed count is $\mathcal D_X(G)$ by
-\eqref{eq:degree-boundary-degree}; its integral is the measured unsigned
-count. At each point \eqref{eq:degree-raw-integrand} reads
-$1\le(v/L_1)\Theta$, and confinement places every counted point in
+(eq:degree-boundary-degree); its integral is the measured unsigned
+count. At each point (eq:degree-raw-integrand) reads
+$1\le(v/L_1)Θ$, and confinement places every counted point in
$|G_1|<b_d$. Return to the common integral estimate after treating
the possible singularities when $p\ge7$.
@@ -4403,28 +4403,28 @@ boundary current.
Construct a controlled child carried by an open regular subset
of this slice, with the inherited orientation and transverse measure,
-and with degree $\mathcal D_X(G)$. For each residual row one requires $|dJ_j^+|_{h^+}\le |dG_j|_h+\varepsilon_{\mathrm{grad}}$ and the
+and with degree $\mathcal D_X(G)$. For each residual row one requires $|dJ_j^+|_{h^+}\le |dG_j|_h+ε_{grad}$ and the
one-sided value bound
-$|J_j^+|\ge |G_j|-\varepsilon_{\mathrm{val}}$, with
-\begin{equation}\label{eq:degree-tolerances}
- \sum_{j=2}^d L_j\varepsilon_{\mathrm{grad}}<\frac1{10q},
- \qquad \varepsilon_{\mathrm{val}}<b_d-b_p=\frac{\tau}{4q^2}.
-\end{equation}
+$|J_j^+|\ge |G_j|-ε_{val}$, with
+
+ \sum_{j=2}^d L_jε_{grad}<\frac1{10q},
+ \qquad ε_{val}<b_d-b_p=(\tau)/(4q^2).
+
The gradient bound controls the residual penalties and the child's
coordinate bound.
The value bound, together with confinement of $G_1$, puts the child's
$b_p$ integration region over the parent's $b_d$ region. Write
$\mathcal W=dV_{h^+}/dA_h$ for its ordinary leafwise volume density
relative to the original induced metric. The scalar and volume target is
-\begin{equation}\label{eq:degree-child-scalar}
- \begin{gathered}
+
+
dV_{h^+}=\mathcal W\,dA_h,\\
\wscal(h^+,\phi^+)\ge\wscal(h,\phi)-H_1(G)-2\ln(v/L_1)
+2\ln\mathcal W-\frac1q.
- \end{gathered}
-\end{equation}
+
+
The coefficient $2$ makes the volume factor cancel after exponentiation,
-leaving the same lapse factor as in \eqref{eq:degree-raw-integrand}.
+leaving the same lapse factor as in (eq:degree-raw-integrand).
The density $\mathcal W$ includes neither $v$ nor the transverse measure.
Every constant and error chosen inside this subsection may depend on this
@@ -4436,17 +4436,17 @@ 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
-\eqref{eq:degree-child-scalar} is at most $1/q$, so chart and elliptic
-constants will not enter the absolute loss in Theorem~\ref{thm:degree}.
+(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 $\{|G|_\infty\le t_3\}$ still
-inside $\{|G|_\infty<t_4\}$. 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
use this atlas.
-\paragraph{Uniform regularity tests and avoidance of the remaining points.}
+Uniform regularity tests and avoidance of the remaining points.:
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
@@ -4469,14 +4469,14 @@ convergent graphical boundary. The graph coding already described shows
that recognition is Borel.
Let $D_k$ be the part of the boundary support in
-$\{|G|_\infty\le t_3\}$ left after removing the open inner portions
+$\{|G|_∞\le t_3\}$ left after removing the open inner portions
recognized through level $k$ in any parent plaque. These are decreasing
Borel sets, closed along each leaf. Every regular point is eventually
removed. More precisely, in any converging sequence of controlled
charts with levels tending to infinity, the $D_k$ can accumulate on a
compact subbox only at singular points of the limiting boundary.
Outside the limit support this follows from support convergence. If
-points of $D_{k_j}$ with $k_j\to\infty$ converged instead to a regular
+points of $D_{k_j}$ with $k_j\to∞$ converged instead to a regular
limit point, one test with a fixed index $\ell$ in the common list would
recognize that point with strict room. Smooth graphical convergence
makes the same test recognize the nearby points for large $j$, and then
@@ -4484,13 +4484,13 @@ $k_j\ge\ell$, contradicting their membership in $D_{k_j}$. This is the
compactness fact used in both constructions that follow.
First make an arbitrarily small smooth perturbation $\bar J$ of $J$ in
-the \emph{parent} band, small in value and first derivative, with all
+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 $k$,
-\begin{equation}\label{eq:degree-gap}
- |\bar J|_\infty>4a_0
+
+ |\bar J|_∞>4a_0
\quad\hbox{on a uniform neighborhood of }D_k.
-\end{equation}
+
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.
@@ -4502,7 +4502,7 @@ the marked set in the box. Otherwise take a sequence of failing plaques,
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 \eqref{eq:degree-singular-dimension} this
+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
given open budget can therefore avoid it with a positive distance.
@@ -4530,20 +4530,20 @@ colors. The budgets can also make the total value and first-derivative
change as small as prescribed; the cutoffs give uniform bounds for all
higher derivatives. Taking $a_0$ smaller than the final gaps and then
using the derivative bound of $\bar J$ extends the gap to a uniform
-neighborhood. This proves \eqref{eq:degree-gap}.
+neighborhood. This proves (eq:degree-gap).
Second, set
-\begin{equation}\label{eq:degree-alpha}
- \alpha=p-\frac{13}{2}>0.
-\end{equation}
+
+ \alpha=p-(13)/(2)>0.
+
The next cover will support a function large on the remaining marked
points while keeping the total kernel coefficients as small as needed.
For any prescribed positive budget and upper radius, a sufficiently
large $k'\ge k$ permits a cover of $D_{k'}$ in each inner plaque box by
finitely many small balls, of radii $r_i$, such that
-\begin{equation}\label{eq:degree-cover-budget}
- \sum_i r_i^\alpha<\text{the prescribed budget}.
-\end{equation}
+
+ \sum_i r_i^\alpha<the prescribed budget.
+
The radii can also have a common positive lower bound, depending on the
budget and the controlled family. To verify the uniform claim, suppose
it failed for levels tending to infinity and for requested lower radii
@@ -4561,52 +4561,52 @@ change of centers and the reserved enlargement. The centers then also
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 \eqref{eq:degree-cover-budget}
+test. The positive lower radius and (eq:degree-cover-budget)
bound the number of balls in any plaque.
-The two uses of \eqref{eq:degree-alpha} are distinct: its strict excess
+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.
-\paragraph{Potentials near the marked set.}
+Potentials near the marked set.:
Choose a small parent normal-coordinate radius $R>0$ so that a ball of
that radius about any cover center stays in the gap region
-$|\bar J|_\infty>3a_0$ and in the buffered atlas. It can also be chosen
+$|\bar J|_∞>3a_0$ and in the buffered atlas. It can also be chosen
small with respect to the local metric, weight, and forcing bounds.
This radius is fixed before the final cover is chosen; require its ball
radii to be much smaller than $R$. For a center $c_i$ and radius $r_i$
let
\[
- K_i(x)=(r_i^2+\operatorname{dist}_h(x,c_i)^2)^{-1/2}
+ K_i(x)=(r_i^2+dist_h(x,c_i)^2)^{-1/2}
\]
on the smaller normal ball, and cut it off within radius $R$. Choose a
radial cutoff which is one on the smaller ball, is positive before the
outer boundary, and near that boundary has the tail
$\exp(-1/(1-t))$ for $t<1$ and zero for $t\ge1$, where
-$t=\operatorname{dist}_h(x,c_i)^2/R^2$. Join this tail smoothly and
+$t=dist_h(x,c_i)^2/R^2$. Join this tail smoothly and
positively to the inner plateau. Every positive power of the cutoff
then has a smooth exponential flat tail, so each power of $K_i$ used
below is smooth. Denote its support by $V_i$.
On the regular boundary in $V_i$, for $j=\alpha,\alpha+4$,
-\begin{equation}\label{eq:degree-kernel-bounds}
- \begin{split}
- -\Delta_{\Sigma,\phi}K_i^j&\ge cK_i^{j+2}-C\mathbf1_{V_i},\\
- |dK_i^j|&\le C\bigl(K_i^{j+1}+\mathbf1_{V_i}\bigr).
- \end{split}
-\end{equation}
+
+
+ -\Delta_{Σ,\phi}K_i^j&\ge cK_i^{j+2}-C\mathbf1_{V_i},\\
+ |dK_i^j|&\le C(K_i^{j+1}+\mathbf1_{V_i}).
+
+
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 $u=\operatorname{dist}_h(x,c_i)^2$.
+first estimate before cutoff, put $u=dist_h(x,c_i)^2$.
The normal-coordinate Hessian bound, the bounded mean curvature
-$H=\mathfrak h_s-\partial_\nu\phi$, and the bounded weight gradient give
+$H=\mathfrak h_s-∂_\nu\phi$, and the bounded weight gradient give
\[
- \Delta_{\Sigma,\phi}u\ge2p-C\sqrt u,\qquad |du|_\Sigma^2\le4u.
+ \Delta_{Σ,\phi}u\ge2p-C\sqrt u,\qquad |du|_Σ^2\le4u.
\]
Differentiating $(r_i^2+u)^{-j/2}$ therefore gives
\[
- -\Delta_{\Sigma,\phi}K_i^j
- \ge j\left(p-j-2-C\sqrt u\right)K_i^{j+2}.
+ -\Delta_{Σ,\phi}K_i^j
+ \ge j(p-j-2-C\sqrt u)K_i^{j+2}.
\]
Both chosen exponents are less than $p-2$, so small $R$ makes the
coefficient positive. The gradient estimate follows by one
@@ -4616,57 +4616,57 @@ 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
-\begin{equation}\label{eq:degree-potentials}
- u_j=\varepsilon_{\mathrm{base}}^j+\sum_i M_*r_i^\alpha K_i^j,
- \qquad \Lambda=\left(\frac{u_{\alpha+4}}{u_\alpha}\right)^{1/4},
+
+ u_j=ε_{base}^j+\sum_i M_*r_i^\alpha K_i^j,
+ \qquad Λ=(\frac{u_{\alpha+4}}{u_\alpha})^{1/4},
\qquad
\mathcal L=|d\ln u_\alpha|^2+|d\ln u_{\alpha+4}|^2.
-\end{equation}
+
These quantities have separate roles. Make $u_\alpha$ large on
$D_{k'}$ so that its logarithm locates the region to be excluded.
-The term $\Lambda^2$ will supply the positive scalar contribution that
+The term $Λ^2$ will supply the positive scalar contribution that
pays for the later coordinate graphs, while $\mathcal L$ controls the
logarithmic derivatives introduced by the weight and the first graph.
-Here $0<\varepsilon_{\mathrm{base}}<1$ 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
-\eqref{eq:degree-cover-budget} allow the total active coefficient
+(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
-$\varepsilon_{\mathrm{base}}$ small and then the cover budget small enough
+$ε_{base}$ small and then the cover budget small enough
gives, for $j=\alpha,\alpha+4$,
-\begin{equation}\label{eq:degree-potential-laplacian}
- -\frac{\Delta_{\Sigma,\phi}u_j}{u_j}
+
+ -\frac{\Delta_{Σ,\phi}u_j}{u_j}
\ge c\frac{u_{j+2}}{u_j}-\delta.
-\end{equation}
-Indeed \eqref{eq:degree-kernel-bounds} leaves the errors
-$c\varepsilon_{\mathrm{base}}^{j+2}+C\sum_{i:x\in V_i}M_*r_i^\alpha$
-before division by $u_j\ge\varepsilon_{\mathrm{base}}^j$. Choose
-$\varepsilon_{\mathrm{base}}^2$ and then the active sum divided by
-$\varepsilon_{\mathrm{base}}^{\alpha+4}$ small enough for the displayed
+
+Indeed (eq:degree-kernel-bounds) leaves the errors
+$cε_{base}^{j+2}+C\sum_{i:x\in V_i}M_*r_i^\alpha$
+before division by $u_j\geε_{base}^j$. Choose
+$ε_{base}^2$ 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
-$\varepsilon_{\mathrm{base}}$ of mass one, and atoms at $K_i$ of masses
+$ε_{base}$ 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
-\begin{equation}\label{eq:degree-moments}
- \frac{u_{\alpha+6}}{u_{\alpha+4}}\ge\Lambda^2
+
+ \frac{u_{\alpha+6}}{u_{\alpha+4}}\geΛ^2
\ge\frac{u_{\alpha+2}}{u_\alpha},\qquad
- |d\ln\Lambda|^2\le\frac{\mathcal L}{8},\qquad
- |d\ln u_\alpha|\le C(\Lambda+1),\qquad
- \Lambda\ge c' u_\alpha^{1/\alpha}.
-\end{equation}
+ |d\lnΛ|^2\le(\mathcal L)/(8),\qquad
+ |d\ln u_\alpha|\le C(Λ+1),\qquad
+ Λ\ge 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
atom position, then
$\mathbb E Z\le(\mathbb E Z^2)^{1/2}$ and
$\mathbb E Z^3\ge(\mathbb E Z^2)^{3/2}$ give the two inequalities. The
second estimate follows directly by differentiating
-$\ln\Lambda=(\ln u_{\alpha+4}-\ln u_\alpha)/4$. For the third, the
-gradient bound in \eqref{eq:degree-kernel-bounds} bounds
+$\lnΛ=(\ln u_{\alpha+4}-\ln u_\alpha)/4$. For the third, the
+gradient bound in (eq:degree-kernel-bounds) bounds
$|du_\alpha|$ by $C\sum_iM_*r_i^\alpha(K_i^{\alpha+1}+1)$.
-The first part divided by $u_\alpha$ is at most $\Lambda$ by H\"older's
+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
@@ -4674,40 +4674,40 @@ $u_{\alpha+4}/u_\alpha\ge(u_\alpha/2)^{4/\alpha}$, 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
-\begin{equation}\label{eq:degree-potential-on-bad}
+
u_\alpha\ge c''M_*\quad\hbox{on }D_{k'}.
-\end{equation}
+
The positive lower bound on all radii and the bounded number of balls per
plaque make all $u_j$ smooth in the parent with uniform derivative bounds
for the final slice. These bounds are allowed to depend on that radius
lower bound.
-Add $\eta_0=(\ln u_\alpha+\ln u_{\alpha+4})/4$ to $\phi_\Sigma$ on the
+Add $\eta_0=(\ln u_\alpha+\ln u_{\alpha+4})/4$ to $\phi_Σ$ on the
regular boundary. Put $Q_1=d\eta_0$. The weight-change formula used in
-\eqref{eq:degree-slice-identity}, now with
-$\Delta_{\Sigma,\phi_\Sigma}=\Delta_{\Sigma,\phi}
-+\langle d\ln v,d\,\cdot\,\rangle$, yields from
-\eqref{eq:degree-potential-laplacian} the scalar gains
-$c\Lambda^2+\mathcal L/2-2\langle d\ln v,Q_1\rangle-|Q_1|^2-\delta$.
-Here the high moment in \eqref{eq:degree-moments} supplies $\Lambda^2$.
+(eq:degree-slice-identity), now with
+$\Delta_{Σ,\phi_Σ}=\Delta_{Σ,\phi}
++< d\ln v,d\,\cdot\,>$, yields from
+(eq:degree-potential-laplacian) the scalar gains
+$cΛ^2+\mathcal L/2-2< d\ln v,Q_1>-|Q_1|^2-\delta$.
+Here the high moment in (eq:degree-moments) supplies $Λ^2$.
Since $|Q_1|^2\le\mathcal L/8$,
\[
- \frac{\mathcal L}{2}+|d\ln v|^2
- -2\langle d\ln v,Q_1\rangle-|Q_1|^2
- \ge\frac{\mathcal L}{2}-2|Q_1|^2\ge\frac{\mathcal L}{4}.
+ (\mathcal L)/(2)+|d\ln v|^2
+ -2< d\ln v,Q_1>-|Q_1|^2
+ \ge(\mathcal L)/(2)-2|Q_1|^2\ge(\mathcal L)/(4).
\]
-Combining with \eqref{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
-\begin{equation}\label{eq:degree-weight-gain}
- \wscal(h|_{T\Sigma},\phi_\Sigma+\eta_0)
+
+ \wscal(h|_{TΣ},\phi_Σ+\eta_0)
\ge\wscal(h,\phi)-H_1(G)-2\ln(v/L_1)
- +c\Lambda^2+\frac{\mathcal L}{4}-\delta.
-\end{equation}
+ +cΛ^2+(\mathcal L)/(4)-\delta.
+
A large scalar allowance has been obtained near the marked set. The next
construction uses it to move the residual map out of the target band
there, while paying for the graph volume created in doing so.
-\paragraph{Two graph prescriptions and their costs.}
+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 $\ln u_\alpha$,
@@ -4720,92 +4720,92 @@ Their scalar gains will pay the exact graph volume factors.
For a current metric and
weight $(k,\varphi)$, a length-one graph of a function $f$ has
\[
- L_f=(1+|df|_k^2)^{1/2},\quad U_f=\frac{\nabla_k f}{L_f},\quad
- k_f=k+df\otimes df,\quad \varphi_f=\varphi-\ln L_f,
- \quad D_f=-\operatorname{div}_{k,\varphi}U_f.
+ L_f=(1+|df|_k^2)^{1/2},\quad U_f=(\nabla_k f)/(L_f),\quad
+ k_f=k+df ⊗ df,\quad \varphi_f=\varphi-\ln L_f,
+ \quad D_f=-div_{k,\varphi}U_f.
\]
-Lemma~\ref{lem:graph-identity} says
-\begin{equation}\label{eq:degree-graph-identity}
+Lemma~lem:graph-identity says
+
\wscal(k_f,\varphi_f)-\wscal(k,\varphi)
- =D_f^2-2U_f(D_f)+|\mathrm{II}_f|^2+|d\ln L_f|_{k_f}^2.
-\end{equation}
+ =D_f^2-2U_f(D_f)+|II_f|^2+|d\ln L_f|_{k_f}^2.
+
Also
-\begin{equation}\label{eq:degree-graph-elementary}
- \begin{gathered}
+
+
|U_f|_k\le1,\qquad U_f(f)=L_f-L_f^{-1},\qquad
|df|_{k_f}\le1,\\
dV_{k_f}=L_f\,dV_k,\qquad L_f-L_f^{-1}\ge\ln L_f.
- \end{gathered}
-\end{equation}
+
+
The last inequality holds for every $L_f\ge1$ by differentiation. These
relations will make the scalar cost and the volume change use the same
factor $L_f$.
For the first graph put
\[
- f_0=\sigma\ln u_\alpha,\qquad C_f=B_0+\Lambda,
+ f_0=\sigma\ln u_\alpha,\qquad C_f=B_0+Λ,
\qquad D_f=C_f\tan(f_0-f),
\]
with the argument between its two poles. Here $B_0\ge2$ is fixed and
$\sigma>0$ will be small. Pole avoidance gives $|f-f_0|<\pi/2<2$.
Differentiating the prescribed $D_f$ gives the exact terms
-\begin{equation}\label{eq:degree-first-graph-differentiation}
- \begin{split}
+
+
D_f^2-2U_f(D_f)
={}&D_f^2-2D_fU_f(\ln C_f)
-2(C_f+D_f^2/C_f)U_f(f_0)\\
&+2(C_f+D_f^2/C_f)(L_f-L_f^{-1}).
- \end{split}
-\end{equation}
+
+
The final term is at least $2\ln L_f$ by
-\eqref{eq:degree-graph-elementary}. For the derivative of $C_f$, Young's
-inequality and \eqref{eq:degree-moments} give
+(eq:degree-graph-elementary). For the derivative of $C_f$, Young's
+inequality and (eq:degree-moments) give
\[
2|D_fU_f(\ln C_f)|
\le0.8D_f^2+1.25|d\ln C_f|^2
- \le0.8D_f^2+\frac{1.25}{8}\mathcal L.
+ \le0.8D_f^2+(1.25)/(8)\mathcal L.
\]
The $f_0$ term has absolute value at most
\[
- 2\sigma C(C_f+D_f^2/C_f)(\Lambda+1).
+ 2\sigma C(C_f+D_f^2/C_f)(Λ+1).
\]
-Since $(\Lambda+1)/C_f\le1$, a sufficiently small $\sigma$ spends less
+Since $(Λ+1)/C_f\le1$, a sufficiently small $\sigma$ spends less
than the remaining $0.2D_f^2$ on its square term. Moreover
-$C_f(\Lambda+1)\le C_{B_0}(1+\Lambda^2)$, so a further small choice of
-$\sigma$ spends less than half the $c\Lambda^2$ in
-\eqref{eq:degree-weight-gain} and makes the remaining constant loss as
+$C_f(Λ+1)\le C_{B_0}(1+Λ^2)$, so a further small choice of
+$\sigma$ spends less than half the $cΛ^2$ in
+(eq:degree-weight-gain) and makes the remaining constant loss as
small as prescribed. The available $\mathcal L/4$ covers
-$1.25\mathcal L/8$. Combining \eqref{eq:degree-weight-gain} and
-\eqref{eq:degree-graph-identity}, and discarding nonnegative terms, thus
+$1.25\mathcal L/8$. Combining (eq:degree-weight-gain) and
+(eq:degree-graph-identity), and discarding nonnegative terms, thus
gives
-\begin{equation}\label{eq:degree-first-graph-gain}
+
\wscal(k_f,\varphi_f)\ge\wscal(h,\phi)-H_1(G)-2\ln(v/L_1)
- +c_1\Lambda^2+2\ln L_f-\delta_1,
-\end{equation}
+ +c_1Λ^2+2\ln L_f-\delta_1,
+
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 \eqref{eq:degree-weight-gain} was proved.
+induced metric for which (eq:degree-weight-gain) was proved.
For the remaining graphs, let $\zeta(f)$ be a smooth function valued in
$[0,1]$, zero for $f\le K$ and one for $f\ge K+1$, with derivative bound
independent of the threshold $K$. Write
-$(k^{(1)},\varphi^{(1)})=(k_f,\varphi_f)$. For row $j=2,\ldots,d$, its
+$(k^{(1)},\varphi^{(1)})=(k_f,\varphi_f)$. For row $j=2,...,d$, its
graph quantities use the metric and weight after all preceding rows:
\[
- \begin{gathered}
+
L'_j=(1+|df'_j|_{k^{(j-1)}}^2)^{1/2},\qquad
U'_j=\frac{\nabla_{k^{(j-1)}}f'_j}{L'_j},\\
- D'_j=-\operatorname{div}_{k^{(j-1)},\varphi^{(j-1)}}U'_j,\\
- k^{(j)}=k^{(j-1)}+df'_j\otimes df'_j,\qquad
+ D'_j=-div_{k^{(j-1)},\varphi^{(j-1)}}U'_j,\\
+ k^{(j)}=k^{(j-1)}+df'_j ⊗ df'_j,\qquad
\varphi^{(j)}=\varphi^{(j-1)}-\ln L'_j.
- \end{gathered}
+
\]
Prescribe successively
-\begin{equation}\label{eq:degree-row-graph}
+
\ell_j=A\zeta(f)\bar J_j,\qquad
- D'_j=B'\tan\left(\frac{\ell_j-e f'_j}{w}\right),\qquad
+ D'_j=B'\tan((\ell_j-e f'_j)/(w)),\qquad
J_j^+=\bar J_j+e f'_j.
-\end{equation}
+
The constants $A,w,e,B'$ are positive; their order of choice is given
below. Put $h^+=k^{(d)}$, $\phi^+=\varphi^{(d)}$, and
\[
@@ -4814,106 +4814,106 @@ below. Put $h^+=k^{(d)}$, $\phi^+=\varphi^{(d)}$, and
The successive volume identities give $dV_{h^+}=\mathcal W\,dV_k$
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 \eqref{eq:degree-child-scalar}. In the metric after the first graph, $|df|\le1$ by
-\eqref{eq:degree-graph-elementary}, and later graph metrics only
+density in (eq:degree-child-scalar). In the metric after the first graph, $|df|\le1$ by
+(eq:degree-graph-elementary), and later graph metrics only
increase. The values and first derivatives of $\bar J$ are uniformly
bounded in the region in question. It follows that in each current
metric
-\begin{equation}\label{eq:degree-ell-bound}
+
|d\ell_j|\le C_\ell\mathbf1_{\{f\ge K\}},
-\end{equation}
+
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
-\eqref{eq:degree-first-graph-differentiation} is
+(eq:degree-first-graph-differentiation) is
\[
(D'_j)^2-2U'_j(D'_j)
- =(D'_j)^2-\frac2w\left(B'+\frac{(D'_j)^2}{B'}\right)U'_j(\ell_j)
- +\frac{2e}{w}\left(B'+\frac{(D'_j)^2}{B'}\right)
+ =(D'_j)^2-\frac2w(B'+((D'_j)^2)/(B'))U'_j(\ell_j)
+ +(2e)/(w)(B'+((D'_j)^2)/(B'))
(L'_j-(L'_j)^{-1}).
\]
Choose $B'$ after $w,e$ so that
-\begin{equation}\label{eq:degree-row-B-choice}
- \frac{B'e}{w}\ge2,\qquad
- \frac{2C_\ell}{wB'}\le\frac12.
-\end{equation}
+
+ (B'e)/(w)\ge2,\qquad
+ (2C_\ell)/(wB')\le\frac12.
+
Then the differentiated $\ell_j$ spends at most half of $(D'_j)^2$ and
the fixed amount $2B'C_\ell/w$ on $\{f\ge K\}$, while the last term pays
$2\ln L'_j$ everywhere. The other terms of
-\eqref{eq:degree-graph-identity} are nonnegative. After all $p$ rows,
+(eq:degree-graph-identity) are nonnegative. After all $p$ rows,
the sum of fixed errors is therefore at most
-$C_{\mathrm{rows}}\mathbf1_{\{f\ge K\}}$, with
-$C_{\mathrm{rows}}$ independent of $K,M_*,k'$ and the cover budget.
+$C_{rows}\mathbf1_{\{f\ge K\}}$, with
+$C_{rows}$ independent of $K,M_*,k'$ and the cover budget.
On this support, $f<f_0+2$ implies
\[
u_\alpha>\exp((K-2)/\sigma),\qquad
- \Lambda\ge c'\exp((K-2)/(\sigma\alpha))
+ Λ\ge c'\exp((K-2)/(\sigma\alpha))
\]
-by \eqref{eq:degree-moments}. Choose $K$ large enough that the remaining
-$c_1\Lambda^2$ in \eqref{eq:degree-first-graph-gain} covers
-$C_{\mathrm{rows}}$ 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 \eqref{eq:degree-row-graph} also gives the output
+Pole avoidance in (eq:degree-row-graph) also gives the output
estimates that will be used for the band sides and its gradients:
-\begin{equation}\label{eq:degree-output-estimates}
- \begin{split}
+
+
|J_j^+-(1+A\zeta(f))\bar J_j|&<2w,\\
- |dJ_j^+|_{h^+}&\le |dG_j|_h+\varepsilon_{\mathrm{grad}},\\
- |J_j^+|&\ge |G_j|-\varepsilon_{\mathrm{val}}.
- \end{split}
-\end{equation}
+ |dJ_j^+|_{h^+}&\le |dG_j|_h+ε_{grad},\\
+ |J_j^+|&\ge |G_j|-ε_{val}.
+
+
For the gradient inequality,
$|df'_j|\le1$ in its own graph metric and hence in $h^+$, and all these
metrics dominate the original induced metric and the parent metric on
tangent vectors. Thus
$|dJ_j^+|_{h^+}\le|d\bar J_j|_h+e$. For the value inequality use
$1+A\zeta\ge1$ in the first line. The quantities
-$\varepsilon_{\mathrm{grad}}$ and $\varepsilon_{\mathrm{val}}$ can be as
+$ε_{grad}$ 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
-target budgets \eqref{eq:degree-tolerances} and also to be smaller than
+target budgets (eq:degree-tolerances) and also to be smaller than
all the fixed side margins used below.
The full dependency order is
\[
- \begin{aligned}
- &\text{perturbation budgets}\ \longrightarrow\ (\bar J,a_0,k)
- \ \longrightarrow\ R\\
- &\qquad\longrightarrow\ (\text{scalar error},\varepsilon_{\mathrm{base}})
- \ \longrightarrow\ B_0\ \longrightarrow\ \sigma\\
- &\qquad\longrightarrow\ A\ \longrightarrow\ (w,e)
- \ \longrightarrow\ B'\ \longrightarrow\ K\ \longrightarrow\ M_*\\
- &\qquad\longrightarrow\ (\text{cover budget},k',\text{radius lower bound}).
- \end{aligned}
+
+ &perturbation budgets\ →\ (\bar J,a_0,k)
+ \ →\ R\\
+ &\qquad→\ (scalar error,ε_{base})
+ \ →\ B_0\ →\ \sigma\\
+ &\qquad→\ A\ →\ (w,e)
+ \ →\ B'\ →\ K\ →\ M_*\\
+ &\qquad→\ (cover budget,k',radius lower bound).
+
\]
Each later choice may use the preceding ones. The intended scalar error
-is much smaller than $1/q$, and $\varepsilon_{\mathrm{base}}$ is small
-enough for \eqref{eq:degree-potential-laplacian}. The constants in
-\eqref{eq:degree-moments} and the positive constant in
-\eqref{eq:degree-weight-gain} are independent of the future cover, so
+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
-\eqref{eq:degree-first-graph-gain} at this stage. Choose $A$ large compared with
+(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 \eqref{eq:degree-tolerances}. Choose $B'$ by
-\eqref{eq:degree-row-B-choice}, and only then $K$ to cover the row errors.
+and $e$ small for (eq:degree-tolerances). Choose $B'$ by
+(eq:degree-row-B-choice), and only then $K$ to cover the row errors.
Also make $K$ larger than the baseline value
-$\sigma\ln(\varepsilon_{\mathrm{base}}^\alpha)$ by a fixed margin.
+$\sigma\ln(ε_{base}^\alpha)$ by a fixed margin.
Choose $M_*$ so large that
-\eqref{eq:degree-potential-on-bad} implies
-\begin{equation}\label{eq:degree-high-on-bad}
+(eq:degree-potential-on-bad) implies
+
f_0>K+100\quad\hbox{on }D_{k'}.
-\end{equation}
+
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
lower radius bound. This final selection is possible for every fixed
-$M_*$ by \eqref{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 $M_*$
was allowed to depend on the final level or cover.
-\paragraph{Solving on a complete regular region.}
+Solving on a complete regular region.:
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
@@ -4922,36 +4922,36 @@ 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 $f_0$.
-$\Sigma_s$ need not be complete. Use the operations
+$Σ_s$ need not be complete. Use the operations
only on the open region
-\begin{equation}\label{eq:degree-auxiliary-region}
- \Omega=\Sigma_s\cap\{|G|_\infty<t_2,\ f_0<K+12\}.
-\end{equation}
-By \eqref{eq:degree-high-on-bad} it avoids $D_{k'}$ with a large margin in
+
+ Ω=Σ_s\cap\{|G|_∞<t_2,\ f_0<K+12\}.
+
+By (eq:degree-high-on-bad) it avoids $D_{k'}$ with a large margin in
$f_0$; by $t_2<t_3$ it also has a margin within the region of the tests.
The uniform derivative bounds on $f_0$ turn these into uniform spatial
margins. Thus the graph tests through level $k'$ give regular graph
charts with uniform buffers and all estimates on a neighborhood of this
region. The estimates for derivatives of $\ln v$ apply there. The
-functions $u_j,\Lambda,f_0,\bar J$ have uniform derivatives because of
+functions $u_j,Λ,f_0,\bar J$ have uniform derivatives because of
the radius lower bound. The weight
-$\phi_\Sigma+(\ln u_\alpha+\ln u_{\alpha+4})/4$ therefore has the
+$\phi_Σ+(\ln u_\alpha+\ln u_{\alpha+4})/4$ therefore has the
positive-order derivative bounds required for a graph solve.
-Choose a smooth function $0<\xi\le1$ on $\Omega$ which tends to zero
-at the sides in \eqref{eq:degree-auxiliary-region} and is one, with
-buffers, through the smaller bounds $|G|_\infty\le t_1$ and
+Choose a smooth function $0<\xi\le1$ on $Ω$ which tends to zero
+at the sides in (eq:degree-auxiliary-region) and is one, with
+buffers, through the smaller bounds $|G|_∞\le t_1$ and
$f_0\le K+7$. It can be made from smooth functions of
$t_2^2-G_i^2$ 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 $\Omega$ by
+Replace the induced metric on $Ω$ by
\[
- k_{\mathrm{aux}}=\xi^{-2}h|_{T\Sigma}.
+ k_{aux}=\xi^{-2}h|_{TΣ}.
\]
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 $\Omega$, since $\xi(x)$ is bounded
+fixed multiple of $\xi(x)$ remains in $Ω$, since $\xi(x)$ is bounded
by a constant times each scalar distance to a side. The bound on
$d\xi$ makes $\xi$ comparable to $\xi(x)$ on that ball. Rescale the
coordinate ball by $\xi(x)$. The coefficients of $\xi^{-2}h$ and their
@@ -4964,16 +4964,16 @@ completeness. A curve of finite remaining length, or a Cauchy sequence,
eventually stays within the outer part of one such buffered box and
converges there; it cannot escape a leaf through a side or an uncharted
end. Equivalently, approach to a side already has infinite length
-because $|d\ln\xi|_{k_{\mathrm{aux}}}$ is bounded. This argument used
-only the controlled charts near \eqref{eq:degree-auxiliary-region},
+because $|d\ln\xi|_{k_{aux}}$ is bounded. This argument used
+only the controlled charts near (eq:degree-auxiliary-region),
not completeness of a parent leaf.
The restricted chart descriptions are Borel and accessible charts still
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~\ref{lem:graph-prescription}
+the graph existence theorem. Apply Lemma~lem:graph-prescription
on these complete leaves. For the first prescription its positive
-coefficient is $B_0+\Lambda$, its height target is $f_0$, and its
+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
@@ -4984,34 +4984,34 @@ countably enumerated charts. It applies successively: a solved graph
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
-\eqref{eq:degree-weight-gain}--\eqref{eq:degree-first-graph-gain}, use
+(eq:degree-weight-gain)--(eq:degree-first-graph-gain), use
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
-\begin{equation}\label{eq:degree-child-domain}
- X_s^+=\{x\in\Omega: |G|_\infty<t_1,\ f_0<K+7,
- \ |J^+|_\infty<T_p\},
-\end{equation}
+
+ X_s^+=\{x\inΩ: |G|_∞<t_1,\ f_0<K+7,
+ \ |J^+|_∞<T_p\},
+
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
-$\varepsilon_{\mathrm{val}}<(t_1-T_p)/4$; then
-\eqref{eq:degree-output-estimates} puts $|J_j^+|$ beyond $T_p$ already
+$ε_{val}<(t_1-T_p)/4$; then
+(eq:degree-output-estimates) puts $|J_j^+|$ beyond $T_p$ already
on a uniform neighborhood of that side. For the $f_0=K+7$ side, if
$f_0\ge K+4$ some kernel is active, since the baseline value of $f_0$
-is less than $K$. Its support lies in $|\bar J|_\infty>3a_0$.
+is less than $K$. Its support lies in $|\bar J|_∞>3a_0$.
Moreover $f>f_0-2\ge K+2$, so $\zeta(f)=1$. Choose $A$ so that
$(1+A)3a_0>T_p+4a_0$ and take $w<a_0/2$. The first line of
-\eqref{eq:degree-output-estimates} then puts some $|J_j^+|$ beyond
+(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~\ref{def:degree-band} for the child. To be explicit about its
+Definition~def:degree-band for the child. To be explicit about its
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
@@ -5024,17 +5024,17 @@ measure and the plaque overlap bound persist. The transverse measure
is precisely the restricted parent measure. It has no lapse factor and
no graph-volume factor; those belong to the leafwise integrations.
-\paragraph{Preserving the degree through the modifications.}
-Use in \eqref{eq:degree-boundary-degree} a residual form $\beta$ supported
-in $|z|_\infty<a_0$, reducing $a_0$ if necessary. First homotope $J$ to
+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
$\bar J$ on the whole parent boundary current. The small value
perturbation ensures that the preimages of this support throughout the
homotopy remain in a fixed old inner band. One can justify invariance
directly by the homotopy formula. The difference of the two pullbacks
of $\beta$ is $d\mathcal H\beta$, where $\mathcal H\beta$ is its usual
transgression along the homotopy. Multiply by the cutoff
-$\chi_{\mathrm{deg}}(G_1)$ used
-in \eqref{eq:degree-boundary-degree}. The current $\partial[E_s]$ 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
@@ -5042,125 +5042,125 @@ of the difference zero. This uses the boundary current across its
singular set and so does not assume regularity of the homotopy there.
The resulting integral for $\bar J$ is entirely in the regular region
-used for the solves. Indeed $|\bar J|_\infty<a_0$ avoids a neighborhood
-of $D_k$ by \eqref{eq:degree-gap} and avoids every kernel support, which
-lies in $|\bar J|_\infty>3a_0$. Thus $f_0$ has its baseline value there.
+used for the solves. Indeed $|\bar J|_∞<a_0$ avoids a neighborhood
+of $D_k$ by (eq:degree-gap) and avoids every kernel support, which
+lies in $|\bar J|_∞>3a_0$. Thus $f_0$ has its baseline value there.
The small perturbation also keeps $|G|$ in the smaller inner band.
Now homotope on this regular region first from $\bar J$ to
$(1+A\zeta(f))\bar J$ and then to $J^+$. Along the first homotopy the
multiplier is at least one, so a preimage of the small target support
-has $|\bar J|_\infty<a_0$. Along the second, the first line of
-\eqref{eq:degree-output-estimates} shows that it has
-$|\bar J|_\infty<a_0+2w<2a_0$. Both supports thus avoid the marked
+has $|\bar J|_∞<a_0$. Along the second, the first line of
+(eq:degree-output-estimates) shows that it has
+$|\bar J|_∞<a_0+2w<2a_0$. Both supports thus avoid the marked
region and the kernel supports, and retain the same spatial and
$G_1$ margins. Ordinary measured Stokes on the regular buffered charts
applies to these homotopies. The final support lies in
-\eqref{eq:degree-child-domain}, so its integral is the measured degree
+(eq:degree-child-domain), so its integral is the measured degree
of the child. It equals $\mathcal D_X(G)$.
On the child the initial auxiliary metric is the induced metric, so
-\eqref{eq:degree-graph-elementary} verifies the volume identity in
-\eqref{eq:degree-child-scalar} for the product $\mathcal W$ defined above.
+(eq:degree-graph-elementary) verifies the volume identity in
+(eq:degree-child-scalar) for the product $\mathcal W$ defined above.
Choose all the adjustable scalar errors above with sum at most $1/q$.
The absorption of the row errors and
-\eqref{eq:degree-first-graph-gain} give the scalar inequality in
-\eqref{eq:degree-child-scalar} on the child.
+(eq:degree-first-graph-gain) give the scalar inequality in
+(eq:degree-child-scalar) on the child.
This completes the construction in high dimensions. In the case
$d\le7$ treated above, use $\mathcal W=1$. The volume identity is then
-immediate, and \eqref{eq:degree-slice-bound} gives the scalar inequality
-in \eqref{eq:degree-child-scalar}. In dimension $p=0$ the same statements
+immediate, and (eq:degree-slice-bound) gives the scalar inequality
+in (eq:degree-child-scalar). In dimension $p=0$ the same statements
hold on the counted points with the zero-dimensional scalar convention.
-\begin{proof}[Completion of the proof of Lemma~\ref{lem:degree-descent}]
-The coarea assertion is \eqref{eq:degree-coarea-proof}. The constructions
+Proof. [Completion of the proof of Lemma~lem:degree-descent]
+The coarea assertion is (eq:degree-coarea-proof). The constructions
above give a controlled child of degree $\mathcal D_X(G)$ for each typical
$s$. Its coordinate bound follows from
-\eqref{eq:degree-output-estimates} and \eqref{eq:degree-tolerances}:
+(eq:degree-output-estimates) and (eq:degree-tolerances):
\[
- L_j|dJ_j^+|_{h^+}\le B-\frac dq+L_j\varepsilon_{\mathrm{grad}}
+ L_j|dJ_j^+|_{h^+}\le B-\frac dq+L_jε_{grad}
<B-\frac pq.
\]
For a low-dimensional child the same statement holds with zero error.
-The horizon condition \eqref{eq:degree-horizon} for its rows is unchanged,
-since $q,B,\tau,L_2,\ldots,L_d$ are unchanged.
+The horizon condition (eq:degree-horizon) for its rows is unchanged,
+since $q,B,\tau,L_2,...,L_d$ are unchanged.
-Every point with $|J^+|_\infty<b_p$ comes from
-$|G|_\infty<b_d$. For residual rows this follows from the last line of
-\eqref{eq:degree-output-estimates} and
-$\varepsilon_{\mathrm{val}}<b_d-b_p$; in low dimensions it is immediate.
+Every point with $|J^+|_∞<b_p$ comes from
+$|G|_∞<b_d$. For residual rows this follows from the last line of
+(eq:degree-output-estimates) and
+$ε_{val}<b_d-b_p$; in low dimensions it is immediate.
For the first row confinement gives
\[
- |G_1|<\tau+b<\tau+\frac{\tau}{10q}<b_d.
+ |G_1|<\tau+b<\tau+(\tau)/(10q)<b_d.
\]
The same observation covers the counted case $p=0$.
The function $x\mapsto(x-1)_+$ is one-Lipschitz, so
-\eqref{eq:degree-output-estimates} gives
+(eq:degree-output-estimates) gives
\[
\sum_{j=2}^dH_j^+\le\sum_{j=2}^dH_j(G)
- +2\sum_{j=2}^dL_j\varepsilon_{\mathrm{grad}}.
+ +2\sum_{j=2}^dL_jε_{grad}.
\]
Use this and the scalar and volume relations in
-\eqref{eq:degree-child-scalar} to compare the integrands on the child:
-\begin{equation}\label{eq:degree-integrand-cancellation}
- \begin{split}
- &\exp\left(\frac{-\wscal(h^+,\phi^+)+\sum_{j=2}^dH_j^+}{2}\right)
+(eq:degree-child-scalar) to compare the integrands on the child:
+
+
+ &\exp(\frac{-\wscal(h^+,\phi^+)+\sum_{j=2}^dH_j^+}{2})
\mathcal W\\
&\quad\le
- \exp\left(\frac1{2q}+\sum_{j=2}^dL_j\varepsilon_{\mathrm{grad}}\right)
+ \exp(\frac1{2q}+\sum_{j=2}^dL_jε_{grad})
\frac v{L_1}
- \exp\left(\frac{-\wscal(h,\phi)+\sum_{i=1}^dH_i(G)}2\right)\\
- &\quad\le e^{3/(5q)}\frac v{L_1}\Theta.
- \end{split}
-\end{equation}
+ \exp(\frac{-\wscal(h,\phi)+\sum_{i=1}^dH_i(G)}2)\\
+ &\quad\le e^{3/(5q)}\frac v{L_1}Θ.
+
+
The factor $\mathcal W$ cancels exactly against the term
$2\ln\mathcal W$ in the scalar bound. The factor $v/L_1$ is the lapse
factor in coarea. This calculation makes no bound on either factor.
-For a low-dimensional child use \eqref{eq:degree-slice-bound}, with no
+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
-counted point. Integrate \eqref{eq:degree-integrand-cancellation} over
+counted point. Integrate (eq:degree-integrand-cancellation) over
the child and use its containment in the parent $b_d$ region. This is
-\eqref{eq:degree-child-integral}, for example with $C_*=1$.
-\end{proof}
+(eq:degree-child-integral), for example with $C_*=1$.
-\begin{proof}[Proof of Theorem~\ref{thm:degree}]
+
+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 residual lengths. Its hypotheses hold for the child of
-Lemma~\ref{lem:degree-descent}, including the unchanged strict horizon
+Lemma~lem:degree-descent, including the unchanged strict horizon
condition. For almost every $s$ the induction hypothesis and
-\eqref{eq:degree-child-integral} give
+(eq:degree-child-integral) give
\[
|\mathcal D_X(G)|(2\tau)^p
\le \frac{e^{C_1p/q+C_*/q}}{\prod_{i=1}^dL_i}
- \int_{\Sigma_s\cap\{|G|_\infty<b_d\}}v_s\Theta\,dA_h.
+ \int_{Σ_s\cap\{|G|_∞<b_d\}}v_sΘ\,dA_h.
\]
For $p=0$ this is the same use of the unsigned-count bound. Take
$C_1\ge C_*$. Integrate this scalar inequality over $-\tau<s<\tau$
-and apply \eqref{eq:degree-coarea} with
-$F=\mathbf1_{\{|G|_\infty<b_d\}}\Theta$. The factor $2\tau$ on the
+and apply (eq:degree-coarea) with
+$F=\mathbf1_{\{|G|_∞<b_d\}}Θ$. The factor $2\tau$ on the
left changes its power from $p$ to $d$, and the right is bounded by
-the integral in \eqref{eq:degree-estimate}. This proves the induction.
+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
against the same parent integral. That parent integral is a measurable
function of $s$ by the lapse construction and is governed by
-\eqref{eq:degree-coarea}; it is the only function being integrated.
-Its integral is finite because $\Theta$ is bounded on the controlled
+(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
arbitrarily small slice errors were chosen. All geometry-dependent
constants were used only to make those choices.
-\end{proof}
+
The entropy construction supplies a root probability map \(F\) whose
energy \(e_h(F)\) has a large coefficient in the weighted scalar-curvature
inequality. Its coordinatewise square takes values in the probability
simplex. Now turn its pairing with the bounded cocycle into comparison
-degrees to which Theorem~\ref{thm:degree} applies. A common integer
+degrees to which Theorem~thm:degree applies. A common integer
clears the rounded cubical coefficients, giving integer multiplicities;
the resulting measured degree may still be real. Its expectation will
retain the original cocycle pairing exactly. The different lengths
@@ -5168,58 +5168,58 @@ assigned to the tangent and normal directions of the realization are
what remove the auxiliary dimension from the final estimate.
The probability-map and auxiliary-degree strategy has antecedents in
-\cite{MaWangXieYuZhu2026}. Prove the
+[cite]. Prove the
quantitative measured transfer and its estimates here.
-\begin{theorem}\label{thm:scalar-volume}
+
For every integer \(n\geq 3\), there is a finite constant \(C_n>0\) such
that every closed connected oriented smooth \(n\)-manifold \(M\) and every
smooth Riemannian metric \(g\) with \(\Scal_g\equiv-1\) satisfy
\[
\|M\|\leq C_n\Vol_g(M).
\]
-\end{theorem}
+
Prepare the proof for a manifold with \(\|M\|>0\). Choose the
homogeneous alternating bounded cocycle \(c\), the initial probability
map, and the number \(I\geq\|M\|/2\) provided by
-Proposition~\ref{prop:detector}; in particular \(\|c\|_\infty\leq1\).
+Proposition~prop:detector; in particular \(\|c\|_∞\leq1\).
First consider conditional data having the properties asserted by
-Theorem~\ref{thm:entropy}, without yet invoking their existence.
+Theorem~thm:entropy, without yet invoking their existence.
Denote the scalar constant in this contract by \(C_{\rm ent}\).
All functions called elementary in this section have a fixed height of
exponentiation on a polynomial, with parameters depending only on \(n\).
-Write \(\mathcal M=(\widetilde M\times\Omega)/\Gamma\) for such a measured
+Write \(\mathcal M=(\widetilde M\timesΩ)/\Gamma\) for such a measured
family. Denote its metric, weight, density, and
root probability map by \(h,\phi,\rho,F\), and put
\[
\mathfrak e=e_h(F),\qquad W=\log(dV_h/\rho).
\]
Its estimates are
-\begin{equation}\label{eq:transfer-entropy}
+
\rho\leq dV_g,\qquad W\geq0,\qquad
\wscal(h,\phi)\geq-C_{\rm ent}+2W-n\log E+E\mathfrak e,
\qquad \mathfrak e^{1/2}\leq E^{-1/100}.
-\end{equation}
+
There is a fixed invariant graph on the countable label set, with finitely
many free \(\Gamma\)-orbits, containing every cooccurrence of nonzero
coordinates of \(F\). The integer \(q\geq n+1\) bounds the number of
orbits and the maximum graph degree plus one. The equivariant coloring
by \(q\) colors separates any two labels at graph distance at most two.
-Its dependence on \(\omega\in\Omega\) is Borel. Set
+Its dependence on \(\omega\inΩ\) is Borel. Set
\(m=q-n>0\). Integrals over \(\mathcal M\) and over the measured spaces
constructed below include their transverse measures.
-\medskip\noindent\emph{Conditional transfer target.}
+\medskip\noindentConditional transfer target.
For every \(n\geq3\), there are a constant \(K_n>0\) and an elementary
-function \(\Psi_n:[1,\infty)\to(0,\infty)\), chosen from \(n\) alone,
+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
-\begin{equation}\label{eq:transfer-conditional-target}
- E\geq\Psi_n(q)\quad\Longrightarrow\quad
+
+ E\geq\Psi_n(q)\quad⇒\quad
I\leq K_n e^{C_{\rm ent}/2}\int_{\mathcal M}\rho.
-\end{equation}
+
Thus \(K_n\) and \(\Psi_n\) are fixed before the manifold, cocycle,
instance geometry, and \(C_{\rm ent}\) are chosen. Carry out the
estimates for each integer \(q\geq n+1\), uniformly over conditional
@@ -5228,7 +5228,7 @@ The grid, the common rounding, and an upper density bound for the
resolution come first. That density bound permits a small collar;
the ensuing curvature and rotation bounds then determine one comparison
width and the required energy threshold. Only after the conditional
-target is closed will Theorem~\ref{thm:entropy} supply an actual pair
+target is closed will Theorem~thm:entropy supply an actual pair
\((q,E)\) and make \(C_{\rm ent}\) dimension-only.
@@ -5242,7 +5242,7 @@ the flat neighborhood below keeps distinct label configurations with the
same color image on separate sheets for comparison, while retaining
Euclidean coordinates for the grid averages.
-\begin{lemma}[Flat sheet neighborhood]\label{lem:transfer-flat}
+[Flat sheet neighborhood]
Put \(\delta=(1000\sqrt q)^{-1}\). There is an oriented flat smooth
\(q\)-manifold \(\mathcal U_\omega\) with a local isometry
\(z:\mathcal U_\omega\to\mathbb R^q\), and an inclusion of \(K_\omega\)
@@ -5258,15 +5258,15 @@ There is a closed smooth \(n\)-form, again denoted \(\eta\), obtained
in these flat charts by pulling back the detector form through the cutoff
probabilities constructed below. It is Borel and equivariant in the
family, and satisfies
-\begin{equation}\label{eq:transfer-form}
+
\int_{\mathcal M}F^*\eta=I,\qquad |\eta|_{\HS}\leq C_n.
-\end{equation}
+
Here the exterior Hilbert norm uses the Euclidean metric on the charts.
For every fixed derivative order its derivatives admit an elementary
bound in \(q\).
-\end{lemma}
-\begin{proof}
+
+Proof.
Take one copy of \(B(z_y,10\delta)\subset\mathbb R^q\) for each witness
\(y\in K\). At the same point \(z\) in two such balls, identify the two
copies if their witnesses share a label in their nonzero supports. Check that these identifications give ball charts, rather than a branched
@@ -5320,21 +5320,21 @@ coordinate at color \(j\), because its distance from \(z\) is less
than \(10\delta\). The label for every active coordinate is therefore
unambiguous on chart overlaps. Evaluate
\[
- \eta=n!\,c(p,dp,\ldots,dp)
+ \eta=n!\,c(p,dp,...,dp)
\]
using those labels. The cocycle law makes this form closed, as in
-Proposition~\ref{prop:detector}, and a straight probability homotopy
+Proposition~prop:detector, and a straight probability homotopy
from the old probabilities to these new ones preserves the measured
pairing. Its local finiteness and smooth bounds are uniform for the
instance, so the measured Stokes argument of that proposition applies.
-This gives the first identity in \eqref{eq:transfer-form}.
+This gives the first identity in (eq:transfer-form).
For the dimension-only norm bound, first regard the form on nonnegative
unit root vectors \(w\). In the Euclidean coordinate directions
-\(i_1,\ldots,i_n\), its component is
+\(i_1,...,i_n\), its component is
\[
- n!2^n\Bigl(\prod_{j=1}^n w_{i_j}\Bigr)
- \sum_a w_a^2 c(a,i_1,\ldots,i_n).
+ n!2^n(\prod_{j=1}^n w_{i_j})
+ \sum_a w_a^2 c(a,i_1,...,i_n).
\]
Its absolute value is at most \(n!2^n\prod_j w_{i_j}\). The sum of the
squares of these products over all increasing \(n\)-tuples is at most
@@ -5345,10 +5345,10 @@ is bounded below, normalization is an orthogonal projection divided by
that denominator, and \(\chi'\) is absolutely bounded. Pullback
multiplies the exterior Hilbert norm by at most the \(n\)-th power of
this operator bound. This proves the second assertion in
-\eqref{eq:transfer-form}. Differentiating the same formulas a fixed
+(eq:transfer-form). Differentiating the same formulas a fixed
number of times gives elementary bounds (polynomial bounds in \(q\)
already suffice), since \(\delta^{-1}=1000\sqrt q\).
-\end{proof}
+
Choose a rotation \(V\in SO(q)\) with Haar probability, independently
of the entropy data, and put \(\widehat z=Vz\). In these coordinates
@@ -5357,10 +5357,10 @@ uniform in \([0,r)^q\). Fix \(r=r(q)>0\) so that
\(r\sqrt q\ll\delta\) and so that every \(n\)-component of \(\eta\)
in every rotated frame changes by at most \(q^{-n}\) over a ball of
radius \(10r\sqrt q\) in the inner sheet charts. The derivative bounds
-of Lemma~\ref{lem:transfer-flat} permit such a choice with an elementary
+of Lemma~lem:transfer-flat permit such a choice with an elementary
bound on \(r^{-1}\), uniformly in \(V,s,\omega\).
-A primal \(n\)-cell has an axis set \(\mathcal I\subset\{1,\ldots,q\}\)
+A primal \(n\)-cell has an axis set \(\mathcal I\subset\{1,...,q\}\)
of size \(n\), varies along those grid intervals, and has the remaining
coordinates at grid vertices. Orient it in increasing axis order.
Include each lift \(C\) whose center is at distance less than \(4\delta\)
@@ -5370,9 +5370,9 @@ the complementary axes by \([-r/2,r/2]\), and has the other coordinates
fixed. It too fits in the chart. Orient it so that the primal oriented
basis, followed by the negatives of its dual oriented basis, is a
positive basis of \(\mathbb R^q\). Give the primal cell the coefficient
-\begin{equation}\label{eq:transfer-cell-value}
+
u_C=\int_C\eta.
-\end{equation}
+
These coefficients satisfy the cubical cocycle equations where they
are needed. If a primal \((n+1)\)-cell has its center within
@@ -5390,18 +5390,18 @@ facet points by at most \(r\sqrt q\), so they have the stipulated
\(2\delta\) witnesses. Projected grid cells on other sheets give no
incidence: all these statements use lifts in the same flat chart.
-\begin{lemma}[Finite-type unbiased rounding]\label{lem:transfer-rounding}
+[Finite-type unbiased rounding]
There is an integer \(D_0\geq1\), bounded by an elementary function of
\(q\) and constant in \(V,s,\omega\), and a Borel equivariant random
choice of coefficients \(u'_C\in D_0^{-1}\mathbb Z\) such that
-\begin{equation}\label{eq:transfer-rounding}
+
\E_{\rm rd}u'_C=u_C,\qquad |u'_C-u_C|\leq r^nq^{-n}.
-\end{equation}
+
The expectation is conditional on the grid and entropy data. Every
cocycle equation just described is satisfied exactly by the \(u'_C\).
-\end{lemma}
-\begin{proof}
+
+Proof.
The closure equations couple faces whose anchors can differ. Encode all local coefficients in one ordered finite vector and round within the
joint kernel of its incidence equations, so every cell reads from the
same rounded vector. First establish that the vector has elementary
@@ -5414,7 +5414,7 @@ At a selected center choose, breaking ties by color order, its largest
unrotated color coordinate. This coordinate is at least
\(q^{-1/2}-4\delta\), and its label occurs in every witness for the
center. It determines the lift uniquely, by the proof of
-Lemma~\ref{lem:transfer-flat}; call it the anchor label.
+Lemma~lem:transfer-flat; call it the anchor label.
The closed graph one-neighborhood of an anchor has at most \(q\)
labels. Every center-test witness, meaning a witness at distance less
@@ -5465,7 +5465,7 @@ different valid witness cliques have the same value: the
throughout the primal cell.
Impose on this list every integer row \(b\) with
-\(\|b\|_\infty\leq2(n+1)\) for which \(b\cdot v=0\). There are
+\(\|b\|_∞\leq2(n+1)\) for which \(b\cdot v=0\). There are
finitely many candidate rows, so the resulting matrix depends Borelly
on \(v\). In particular it includes the difference row for any two
duplicate valid slots. It also includes every needed cubical equation,
@@ -5474,11 +5474,11 @@ all their values are entries of this same vector \(v\), and Stokes gives
the signed zero sum of those entries. After repeated slots in its
\(2(n+1)\) faces have been combined, each coefficient has the stated
bound. The kernel of this
-integer matrix has a real basis of integer vectors \(b_1,\ldots,b_k\)
+integer matrix has a real basis of integer vectors \(b_1,...,b_k\)
with
\[
k\leq N_0,\qquad
- \max_j\|b_j\|_\infty\leq K_0:=(C_nN_0)^{C N_0}.
+ \max_j\|b_j\|_∞\leq K_0:=(C_nN_0)^{C N_0}.
\]
To see this, choose a nonsingular maximal minor and solve for its pivot
coordinates in terms of the free coordinates; multiplying each basis
@@ -5495,20 +5495,20 @@ expectation is \(t_j\). Choose a single integer
\(D_0\geq N_0K_0r^{-n}q^n\), using uniform elementary upper bounds
for \(N_0,K_0\). The resulting vector \(v'\) lies in the same
kernel, has entries in \(D_0^{-1}\mathbb Z\), has expectation \(v\),
-and satisfies \(\|v'-v\|_\infty\leq N_0K_0/D_0\leq r^nq^{-n}\).
+and satisfies \(\|v'-v\|_∞\leq N_0K_0/D_0\leq r^nq^{-n}\).
All operations, including taking floors and using independent uniform
parameters for the adjacent choices, are Borel. The fixed slot order
is used for every pivot choice. Thus one parameter \(\xi\), independent
-of \(\Omega\), gives one rounded vector \(v'(V,s,\xi)\) for all cells.
+of \(Ω\), gives one rounded vector \(v'(V,s,\xi)\) for all cells.
Set \(u'_C=v'_{t(C,\omega)}\). The invariance of the type under
translation proves equivariance, and the common zero rows prove all
the asserted incidence equations.
-\end{proof}
+
The rounded coefficients define the original oriented dual current
-\begin{equation}\label{eq:transfer-rounded-current}
+
Z_{\rm rd}:=\sum_C D_0u'_C[C^\#].
-\end{equation}
+
Here \([C^\#]\) is integration over the oriented cube, and the
coefficients are integers. Verify the local finiteness implicit
in this notation. The form bound and rounding give
@@ -5526,10 +5526,10 @@ coefficients, proving neighborhood local finiteness.
Let \(\mathcal T_\delta\subset\mathcal U_\omega\) be the strict tube
of points at distance less than \(\delta\) from a witness in its ball
chart. The exact incidence equations in
-Lemma~\ref{lem:transfer-rounding} give
-\begin{equation}\label{eq:transfer-current-closed}
- \partial Z_{\rm rd}\big|_{\mathcal T_\delta}=0.
-\end{equation}
+Lemma~lem:transfer-rounding give
+
+ ∂ Z_{\rm rd}\big|_{\mathcal T_\delta}=0.
+
This is a local assertion: the boundary current vanishes on test forms
compactly supported in \(\mathcal T_\delta\).
@@ -5555,9 +5555,9 @@ identifications preserve that point and its sheet, so even after
doubling they relate only this bounded collection of incident chambers.
Take barycentric subdivision of the doubled cubes. Each top simplex
-\(\sigma\) has vertices colored \(0,\ldots,m\) by the dimensions of
+\(\sigma\) has vertices colored \(0,...,m\) by the dimensions of
its barycentric faces, and every facet pairing respects these colors.
-Write \(\varepsilon_\sigma\in\{1,-1\}\) for its sign relative to that
+Write \(ε_\sigma\in\{1,-1\}\) for its sign relative to that
color order, and call this doubled signed simplicial diagram
\(\mathcal P\). It maps facewise to \(\mathcal U_\omega\), with
elementary subdivision and incidence bounds. The original current
@@ -5583,7 +5583,7 @@ metric bounds.
\par\noindent{\footnotesize\ttfamily src-gaifullin2008.pdf}\par
-\begin{lemma}[Measured resolution]\label{lem:transfer-resolution}
+[Measured resolution]
For the diagram \(\mathcal P\) there is a standard Borel measured
family \(\mathcal A\) of oriented smooth \(m\)-manifolds and a smooth
map \(X:\mathcal A\to\mathcal U_\omega\), equivariant and Borel in
@@ -5594,17 +5594,17 @@ data. The points over \(\mathcal T_\delta\) represented by
original chamber tiles form a Borel leaf-open subset
\(\mathcal A_{\rm orig}\subset X^{-1}(\mathcal T_\delta)\). For every
compactly supported smooth \(m\)-form \(\alpha\) on \(\mathcal T_\delta\),
-\begin{equation}\label{eq:transfer-local-push}
+
\int_{\mathcal A_{\rm orig}}X^*\alpha=Z_{\rm rd}(\alpha).
-\end{equation}
+
For the full doubled diagram, measured oriented integration also gives
-\begin{equation}\label{eq:transfer-push}
+
X_*[\mathcal A]=[\mathcal P].
-\end{equation}
+
For a parameter \(0<\lambda<1/2\) and a choice
-\begin{equation}\label{eq:transfer-gamma}
+
0<2\gamma_0<(m+1)^{-(m+1)}\lambda^{m(m+1)},
-\end{equation}
+
there are a smooth metric \(k\) and a closed seam set in each tile such
that \(\|dX\|_k\leq1\), and \(k=X^*k_{\rm flat}\) off the seam.
There are buffered smooth charts with elementary overlap, metric,
@@ -5623,12 +5623,12 @@ seam in that tile, equal to the identity near all tile facets, and
homotopic to the identity. Its fixed-order chart derivatives and
\(\|d\mathcal R\|_k\) are elementarily bounded under the same
hypotheses.
-\end{lemma}
-\begin{proof}
-\emph{Invariant reflection states.}
+
+Proof.
+Invariant reflection states.
For each nonempty proper subset
-\(S\subset\{0,\ldots,m\}\) and each face with exactly those vertex
+\(S\subset\{0,...,m\}\) and each face with exactly those vertex
colors, pair the incident positive chambers with the incident negative
chambers. The two counts agree. Indeed choose a color omitted by
\(S\); the given codimension-one matching in \(\mathcal P\) across
@@ -5646,14 +5646,14 @@ so the product laws are Borel probability kernels.
Define a move \(s_S\) by
\[
- \sigma\longmapsto p_S\sigma,\qquad t_S\longmapsto1-t_S,
- \qquad p_R\longmapsto p_Sp_Rp_S\quad(R\subsetneq S),
+ \sigma⟼ p_S\sigma,\qquad t_S⟼1-t_S,
+ \qquad p_R⟼ p_Sp_Rp_S\quad(R⊊ S),
\]
leaving the other data unchanged. The conjugated \(p_R\) is again
a pairing at each \(R\)-face: \(p_S\) preserves every such face and
flips the chamber sign, so conjugation preserves its face and its
sign-reversing property. Each \(s_S\) is an involution. If
-\(R\subsetneq S\), the two moves commute. On chambers, for example,
+\(R⊊ S\), the two moves commute. On chambers, for example,
their two orders both give \(p_Sp_R\sigma\), since after \(s_S\) the
\(R\)-pairing is \(p_Sp_Rp_S\). On a pairing indexed by a subset of
\(R\), both orders conjugate by \(p_Sp_R\); on the other indices the
@@ -5670,19 +5670,19 @@ finite set of possible pairings and preserves its uniform law. It
also preserves the independent product of those laws. All moves are
Borel and commute with translating the original label data.
-\medskip\noindent\emph{Smooth leaves and the measured atlas.}
+\medskip\noindentSmooth leaves and the measured atlas.
Attach the following tile to each state. Start with the positive
-simplex \(x_0,\ldots,x_m>0\), \(\sum_i x_i=1\), and take the closure
+simplex \(x_0,...,x_m>0\), \(\sum_i x_i=1\), and take the closure
of its collection of pair ratios \((x_i/x_j)_{i\ne j}\) in the
-product of copies of \([0,\infty]\). For each permutation
-\(i_0,\ldots,i_m\) use the coordinate chart
-\begin{equation}\label{eq:transfer-ratios}
+product of copies of \([0,∞]\). For each permutation
+\(i_0,...,i_m\) use the coordinate chart
+
u_j=x_{i_j}/x_{i_{j-1}}\in[0,4),\qquad 1\leq j\leq m.
-\end{equation}
+
All pair ratios in the chart are products of these or their reciprocals,
so the \(u_j\) are independent coordinates. Sorted orders have
\(u_j\leq1\) and cover the closure. The facet \(u_j=0\) has label
-\(S_j=\{i_0,\ldots,i_{j-1}\}\); the labels of a corner are nested.
+\(S_j=\{i_0,...,i_{j-1}\}\); the labels of a corner are nested.
Glue each facet to the identical resolved coordinates in the state
obtained by \(s_{S_j}\). The commutation just proved makes the gluing
consistent at corners.
@@ -5693,16 +5693,16 @@ tile obtained by \(s_{S_j}\); use the product of these commuting moves
when several coordinates are negative. At zero use the starting tile
as a convention for representing the same glued point.
-Figure~\ref{fig:resolution} illustrates the tile and the four signed
+Figure~fig:resolution illustrates the tile and the four signed
quadrants at a two-facet corner when \(m=2\).
\par\medskip
-\noindent\begin{minipage}{\textwidth}
+\noindent{\textwidth}
\makeatletter\def\@captype{figure}\makeatother
\centering
-\begin{minipage}[t]{0.53\textwidth}
+[t]{0.53\textwidth}
\centering
-\begin{tikzpicture}[scale=1.15, line cap=round, line join=round]
+[scale=1.15, line cap=round, line join=round]
\coordinate (v0) at (-2,1.05);
\coordinate (v1) at (-2.95,-0.65);
\coordinate (v2) at (-1.05,-0.65);
@@ -5715,7 +5715,7 @@ quadrants at a two-facet corner when \(m=2\).
{one barycentric\\chamber \(\sigma\simeq\Delta^2\)};
\draw[-{Stealth[length=2mm]},thick] (-0.25,0.12)--(-0.90,0.12)
node[midway,above] {\(X\)};
- \begin{scope}[shift={(1,0)}]
+ [shift={(1,0)}]
\path[fill=gray!22,draw=black,thick]
(30:1.40)--(90:1.40)--(150:1.40)--(210:1.40)--
(270:1.40)--(330:1.40)--cycle;
@@ -5729,15 +5729,15 @@ quadrants at a two-facet corner when \(m=2\).
\node[font=\small] at (300:0.97) {\(\{2\}\)};
\node[font=\small] at (0:0.87) {\(\{0,2\}\)};
\node[font=\small] at (0,0) {tile};
- \end{scope}
+
\node[font=\small,align=center] at (-0.55,-2.15)
{(a) A chamber and its resolved tile};
-\end{tikzpicture}
-\end{minipage}
+
+
\hfill
-\begin{minipage}[t]{0.43\textwidth}
+[t]{0.43\textwidth}
\centering
-\begin{tikzpicture}[scale=1.15, line cap=round, line join=round]
+[scale=1.15, line cap=round, line join=round]
\fill[blue!12] (0,0) rectangle (1.25,1.25);
\fill[orange!15] (-1.25,0) rectangle (0,1.25);
\fill[orange!15] (0,-1.25) rectangle (1.25,0);
@@ -5759,10 +5759,10 @@ quadrants at a two-facet corner when \(m=2\).
\node[font=\small] at (-1.03,-1.04) {\(+\)};
\node[font=\small,align=center] at (0,-2.15)
{(b) \(u_S=Z_S^2,\quad u_T=Z_T^2\)};
-\end{tikzpicture}
-\end{minipage}
+
+
\caption{The reflected resolution in dimension two, after Gaifullin's
-permutohedral construction \cite{Gaifullin2008}. (a) A barycentric
+permutohedral construction [cite]. (a) A barycentric
chamber \(\sigma\simeq\Delta^2\) and its resolved tile. The six facets
correspond to the nonempty proper subsets of \(\{0,1,2\}\), nested when
the labels are adjacent, and \(X\) maps the tile interior
@@ -5774,8 +5774,8 @@ four quadrants one smooth chart. Signs record whether the chamber
orientation agrees with that of \(\zeta\) (blue) or opposes it (orange).
The collar and strips indicate the seam schematically, with illustrative
widths, and \(X\) is not affine.}
-\label{fig:resolution}
-\end{minipage}
+
+
\par\medskip
Verify that these are ordinary smooth charts. On an overlap of two permutations,
@@ -5823,7 +5823,7 @@ of transversal are the measure-preserving moves on countably many
Borel overlap pieces. A plaque meets only elementarily many plaques
in this atlas: for each second flag only the states in the two finite
flag reflection groups can occur. These observations give precisely
-the measured flow-box properties of Theorem~\ref{thm:degree}.
+the measured flow-box properties of Theorem~thm:degree.
The chamber signs orient the leaves. In a flag chart the Jacobian
sign from positive ratio coordinates to oriented barycentric
@@ -5836,15 +5836,15 @@ through this orientation in integrals of forms. The value of
\(\omega\) is constant along each tile leaf, and translation by
\(\Gamma\) reindexes the states and preserves these measures.
-\medskip\noindent\emph{The map and the local current.}
-Recover the raw \(x_i\) from \eqref{eq:transfer-ratios} by normalizing
-the successive products \(1,u_1,u_1u_2,\ldots,u_1\cdots u_m\).
+\medskip\noindentThe map and the local current.
+Recover the raw \(x_i\) from (eq:transfer-ratios) by normalizing
+the successive products \(1,u_1,u_1u_2,...,u_1... u_m\).
If \(v_{\sigma,i}\) are the vertices of the indicated chamber in its
flat chart, map its tile by
-\begin{equation}\label{eq:transfer-tile-map}
+
X=\sum_{i=0}^m b_i v_{\sigma,i},\qquad
- b_i=\frac{\exp(-1/x_i)}{\sum_l\exp(-1/x_l)},
-\end{equation}
+ b_i=(\exp(-1/x_i))/(\sum_l\exp(-1/x_l)),
+
where \(\exp(-1/x_i)\) is extended by zero at \(x_i=0\). On a facet
with label \(S\), all coefficients outside \(S\) vanish to infinite
order in its signed coordinate. The vertices in \(S\) are the same
@@ -5866,15 +5866,15 @@ preserves orientation. Indeed it extends to the closed simplex
preserving every face, and the linear homotopy to the identity has
the same property, so its relative topological degree is \(+1\).
It follows that the tile interior maps diffeomorphically, with sign
-\(\varepsilon_\sigma\), onto the interior of its chamber. Each
+\(ε_\sigma\), onto the interior of its chamber. Each
chamber's state probability integrates to one and the tile boundaries
have leafwise measure zero. Fubini and change of variables therefore
give, for every compactly supported smooth \(m\)-form \(\alpha\),
\[
\int_{\mathcal A}X^*\alpha
- =\sum_\sigma\varepsilon_\sigma\int_\sigma\alpha.
+ =\sum_\sigmaε_\sigma\int_\sigma\alpha.
\]
-This proves \eqref{eq:transfer-push}. To obtain the local identity,
+This proves (eq:transfer-push). To obtain the local identity,
define \(\mathcal A_{\rm orig}\) as the image in
\(X^{-1}(\mathcal T_\delta)\) of tile points belonging to original
chambers. At an interior point this subset is leaf-open. At a lower
@@ -5886,10 +5886,10 @@ leaf-openness and makes the definition unambiguous at glued points;
the finite indexed charts also make it Borel. Applying the same
change-of-variables calculation only to original tile interiors, for
a form supported in \(\mathcal T_\delta\), gives
-\eqref{eq:transfer-local-push}: each original cube is counted with
+(eq:transfer-local-push): each original cube is counted with
its coefficient \(D_0u'_C\), and each chamber's state fiber has mass one.
-\medskip\noindent\emph{The metric and upper density.}
+\medskip\noindentThe metric and upper density.
The signed charts admit a background metric \(k_0\) with elementary
bounds for its coefficients, inverse, and every fixed number of
derivatives in buffered charts. To construct it, take smooth product
@@ -5903,22 +5903,22 @@ is at most \(4^{O(m)}\). Their fixed-order derivatives consequently
have elementary bounds. These facts give the claimed bounds for the
average and its inverse, and make the construction Borel and
equivariant. The same transition and product estimates, applied to
-\eqref{eq:transfer-tile-map}, give an elementary upper bound for
+(eq:transfer-tile-map), give an elementary upper bound for
\(dX\) and its fixed-order derivatives before any choice of
\(\lambda\). The affine simplexes have diameter at most \(r\sqrt m\).
-Put \(\Theta=\prod_i x_i\). Choose a smooth \(j=j(\Theta)\) equal
-to one for \(\Theta\leq\gamma_0\), zero for
-\(\Theta\geq2\gamma_0\), and between zero and one elsewhere, with
+Put \(Θ=\prod_i x_i\). Choose a smooth \(j=j(Θ)\) equal
+to one for \(Θ\leq\gamma_0\), zero for
+\(Θ\geq2\gamma_0\), and between zero and one elsewhere, with
the rescaled smooth cutoff estimates. It is smooth in the signed
charts and the same on both sides of every facet. Set
-\begin{equation}\label{eq:transfer-resolution-metric}
+
k=X^*k_{\rm flat}+j^2 k_0,
- \qquad \text{seam}=\{\Theta\leq2\gamma_0\}.
-\end{equation}
+ \qquad seam=\{Θ\leq2\gamma_0\}.
+
Call the complement of this closed seam the straight part.
The metric is positive definite near the tile boundary because \(j=1\) there. Where
-\(j<1\), one has \(\Theta>\gamma_0\), hence \(x_i\geq\gamma_0\) for
+\(j<1\), one has \(Θ>\gamma_0\), hence \(x_i\geq\gamma_0\) for
every \(i\), and \(X\) is an immersion. The formula proves
\(\|dX\|_k\leq1\) and equality of metrics off the seam.
@@ -5929,11 +5929,11 @@ Call a common elementary bound \(H_0(q)\). In a sorted box
\(|Z_j|\leq1\). If all \(|Z_j|>\lambda\), normalization of the
successive products gives
\[
- \Theta\geq(m+1)^{-(m+1)}
- \lambda^{2(1+2+\cdots+m)}
+ Θ\geq(m+1)^{-(m+1)}
+ \lambda^{2(1+2+...+m)}
=(m+1)^{-(m+1)}\lambda^{m(m+1)}.
\]
-Condition \eqref{eq:transfer-gamma} shows that the seam lies in the
+Condition (eq:transfer-gamma) shows that the seam lies in the
stated union of slabs. Its coordinate volume in such a box is at
most an elementary factor times \(\lambda\). The bound \(H_0(q)\)
was fixed before this factor was made small.
@@ -5942,12 +5942,12 @@ The remaining metric estimates may depend on \(\gamma_0\). On the part
with \(j<1\), all positive ratios are bounded below by powers of
\(\gamma_0\), so the product formulas control the inverse from raw
barycentric to signed coordinates. The normalized weights in
-\eqref{eq:transfer-tile-map} obey
+(eq:transfer-tile-map) obey
\(b_i\geq e^{-1/\gamma_0}/(m+1)\). On a tangent vector
\(w=(w_i)\), \(\sum_i w_i=0\), their derivative is
\[
- db_i(w)=b_i\left(\frac{w_i}{x_i^2}
- -\sum_l b_l\frac{w_l}{x_l^2}\right).
+ db_i(w)=b_i((w_i)/(x_i^2)
+ -\sum_l b_l(w_l)/(x_l^2)).
\]
These differences control \(w\): if their common subtracted mean is
\(a\), the relation \(\sum_iw_i=0\) expresses \(a\) as a weighted
@@ -5963,7 +5963,7 @@ Differentiating the formula for \(k\) a fixed number of times now gives
elementary bounds, including the curvature bound, when
\(\gamma_0^{-1}\) is elementary.
-\medskip\noindent\emph{Frame interpolation.}
+\medskip\noindentFrame interpolation.
The comparison needs a constant frame ordering tangent and normal axes
on each straight tile, a frame that glues smoothly near facets, and a homotopy to the
identity for the degree calculation. Choose a Borel skew-symmetric logarithm
@@ -5982,7 +5982,7 @@ where \(j=0\), and replacing \(1-j\) by \(t(1-j)\), \(0\leq t\leq1\),
gives the asserted homotopy. The cutoff and metric inverse estimates
give the derivative bounds. The choices can be made on the indexed
chamber data, so they are Borel and equivariant as asserted.
-\end{proof}
+
For the present purpose, choose \(O_\sigma\) to take \(z\)-increments to rotated
grid increments, putting the \(m\) axes tangent to the dual cube first
@@ -5994,19 +5994,19 @@ not affect any row norm or any cubical test.
The parameter selection below fixes \(\lambda(q)\) and \(\gamma_0(q)\)
deterministically. For those choices the bounds in
-Lemma~\ref{lem:transfer-resolution} give fixed elementary numerical
+Lemma~lem:transfer-resolution give fixed elementary numerical
envelopes \(K_R(q)\) and \(K_{\rm scal}(q)\) such that
\[
- \|d\mathcal R\|_{\infty,k}\leq K_R(q),\qquad
+ \|d\mathcal R\|_{∞,k}\leq K_R(q),\qquad
(-\Scal_k)_+\leq K_{\rm scal}(q)
\]
for every allowed diagram and every \(V,s,\omega\), rounding parameter,
and state. These envelopes come from the uniform chart derivative
bounds and do not use \(C_{\rm ent}\). Fix the single number
-\begin{equation}\label{eq:transfer-tau}
+
\tau=\tau(q)=
\frac{1}{300\sqrt q\max\{r^{-1},\delta^{-1},1+K_R(q)\}}.
-\end{equation}
+
It has an elementary reciprocal and is common to all those parameters.
In particular
\(3\tau\sqrt q\leq\min\{r/100,\delta/100,(1+K_R(q))^{-1}\}\).
@@ -6017,42 +6017,42 @@ their shared \(\omega\), and divide by the diagonal \(\Gamma\)-action.
Orient the product with the \(\widetilde M\) directions first and the
\(\mathcal A\) directions second. The resulting measured leaves have
dimension \(n+m=q\). Using the leaf-open set
-\(\mathcal A_{\rm orig}\) from Lemma~\ref{lem:transfer-resolution},
+\(\mathcal A_{\rm orig}\) from Lemma~lem:transfer-resolution,
work only on the open set of pairs \((x,a)\) with
\(a\in\mathcal A_{\rm orig}\) for which \(X(a)\) is in the lift of the ball chart of
\(F(x)\), at distance less than \(\delta/4\). Define there
-\begin{equation}\label{eq:transfer-comparison}
+
G(x,a)=\mathcal R(a)(z_F(x)-z_{X(a)}).
-\end{equation}
+
Here \(z_{X(a)}=z\circ X(a)\). The requirement on the lifts is part of
the domain; coincident color coordinates on incompatible sheets are
never compared. The map is invariant under the diagonal action.
-Use the band \(\{|G|_\infty<3\tau\}\) in that open domain, and put
-\(b_q=\tau(1+3/(4q))\), the inner width in Theorem~\ref{thm:degree}
+Use the band \(\{|G|_∞<3\tau\}\) in that open domain, and put
+\(b_q=\tau(1+3/(4q))\), the inner width in Theorem~thm:degree
when \(d=q\). Let \(\mathcal D\) be its measured degree.
-\begin{lemma}[Comparison degree]\label{lem:transfer-degree}
-The band in \eqref{eq:transfer-comparison} satisfies the chart and
-integration control assumptions of Theorem~\ref{thm:degree}, for each
+[Comparison degree]
+The band in (eq:transfer-comparison) satisfies the chart and
+integration control assumptions of Theorem~thm:degree, for each
fixed instance, the product metric \(h+L^2k\) with any fixed \(L>0\),
and the pullback of the entropy weight \(\phi\). Its degree is Borel
in the auxiliary parameters and satisfies
-\begin{equation}\label{eq:transfer-degree}
+
\E_{\rm rd}\mathcal D=D_0I
-\end{equation}
+
for every fixed \(V,s\).
-\end{lemma}
-\begin{proof}
-The local pushforward \eqref{eq:transfer-local-push} will express the
+
+Proof.
+The local pushforward (eq:transfer-local-push) will express the
degree using a smoothed form of \(Z_{\rm rd}\). Its primal-cell periods
are the integer coefficients \(D_0u'_C\). Their expectations match the periods of
\(D_0\eta\); a compatible piecewise cubical primitive and mollification
then give the same measured pairing with \(F\). First verify the
band control that makes these integrations valid.
-\emph{The controlled comparison band.}
+The controlled comparison band.
Before taking the quotient, the product flow boxes use the entropy
transversal \(\omega\) and the
state transversals of \(\mathcal A\). Variation along the tile keeps
@@ -6066,7 +6066,7 @@ transverse measures and orientations.
For an inner closed comparison cube there is a uniform margin from
the boundary of the allowed sheet domain. Indeed \(\mathcal R\) is
-orthogonal and \eqref{eq:transfer-tau} gives
+orthogonal and (eq:transfer-tau) gives
\(|z_F-z_X|_2\leq3\tau\sqrt q\ll\delta/4\). The large witness
color at \(F\) remains present with a uniform small buffer. In a
flat ball chart a lifted path whose projected coordinates stay in
@@ -6079,7 +6079,7 @@ coordinate charts for all the data in these regions.
To check the finiteness condition on those charts, choose lifts of a
finite buffered atlas on \(M\). On their compact closures \(F\) uses
only uniformly finitely many possible labels, by the lifted support
-bounds in Theorem~\ref{thm:entropy}. Any nearby cell lift has its
+bounds in Theorem~thm:entropy. Any nearby cell lift has its
anchor in a closed one-neighborhood of one of these labels. The
numbers of projected grid indices, chamber subdivisions, and copies
there are bounded as above. Thus only finitely many chamber indices
@@ -6090,17 +6090,17 @@ overlaps are bounded by the tile overlap bound and the bounded
overlap of the base atlas. In the quotient only finitely many deck
changes meet two of these buffered base charts. Subdividing their
inner boxes and, when necessary, restricting to Borel transverse sets
-gives the flow boxes stipulated in Theorem~\ref{thm:degree}. All
+gives the flow boxes stipulated in Theorem~thm:degree. All
metric and data bounds in this verification may depend on the
instance. It also justifies Fubini and measured Stokes for the
integrations that follow.
-\medskip\noindent\emph{The local convolution form.}
+\medskip\noindentThe local convolution form.
Choose a smooth unit-integral top form \(\beta\) on \(\mathbb R^q\)
supported in a Euclidean ball about zero of radius less than
\(\min\{\tau,r/10\}\). In calculating the degree using \(\beta\),
homotope \(\mathcal R\) to the identity as in
-Lemma~\ref{lem:transfer-resolution}. All maps in this homotopy are
+Lemma~lem:transfer-resolution. All maps in this homotopy are
orthogonal on the difference variable. Their pullbacks of \(\beta\)
are thus supported where the difference has this same small norm,
strictly inside both the sheet domain and the original band. Measured
@@ -6114,11 +6114,11 @@ More explicitly, in a flat chart put \(\mathsf d(z,w)=z-w\). Integrate
\(\mathsf d^*\beta\) along each compatible \(C^\#\) in the second
variable, multiply by \(D_0u'_C\), and sum. The sum is locally finite
and uses the original side. By the local identity
-\eqref{eq:transfer-local-push}, Fubini, and change of variables on tile
+(eq:transfer-local-push), Fubini, and change of variables on tile
interiors, the degree is
-\begin{equation}\label{eq:transfer-convolution-degree}
+
\mathcal D=\int_{\mathcal M}F^*\alpha_{\rm rd}.
-\end{equation}
+
Boundary cancellation of the dual cubes shows that
\(d\alpha_{\rm rd}=0\) on, for example, the tube of radius
\(\delta/8\) around \(K\). To verify this directly, apply fiber
@@ -6128,9 +6128,9 @@ tube and cancel with their paired incidence signs.
For every primal \(n\)-cell sufficiently near \(K\) in the same
sheet,
-\begin{equation}\label{eq:transfer-convolution-cells}
+
\int_C\alpha_{\rm rd}=D_0u'_C.
-\end{equation}
+
Indeed a primal \(n\)-cell and a complementary dual \(m\)-cell of
this grid meet only when their indices match, and then at their common
center. If their axis sets differ, a coordinate fixed for both is a
@@ -6147,14 +6147,14 @@ orientation. The differential of \(\mathsf d(z,w)=z-w\) sends this
ordered frame to the primal basis followed by the negative dual basis,
so its orientation is positive by the choice of the dual orientation.
Its integral is therefore
-one, proving \eqref{eq:transfer-convolution-cells}. The small support
+one, proving (eq:transfer-convolution-cells). The small support
and the chart buffers ensure that only the indicated lifts contribute.
-\medskip\noindent\emph{The detector pairing.}
+\medskip\noindentThe detector pairing.
Set \(\alpha=\E_{\rm rd}\alpha_{\rm rd}-D_0\eta\). It is a smooth
-closed form on that tube, and \eqref{eq:transfer-rounding},
-\eqref{eq:transfer-cell-value}, and
-\eqref{eq:transfer-convolution-cells} give \(\int_C\alpha=0\) on all
+closed form on that tube, and (eq:transfer-rounding),
+(eq:transfer-cell-value), and
+(eq:transfer-convolution-cells) give \(\int_C\alpha=0\) on all
the primal \(n\)-cells there. Explain why its measured pairing
with \(F\) vanishes, including when \(F\) meets grid faces.
@@ -6168,10 +6168,10 @@ project a one-form \(f(t)dt\) to the constant one-form with the same
integral. The homotopy for a one-form is the function
\[
\mathsf h(f(t)dt)(t)=\int_a^t
- \left(f(v)-r^{-1}\int_a^{a+r}f(s)\,ds\right)dv,
+ (f(v)-r^{-1}\int_a^{a+r}f(s)\,ds)dv,
\]
which vanishes at both endpoints; the homotopy on functions is zero.
-Direct differentiation gives \(\mathrm{id}-\mathsf P
+Direct differentiation gives \(id-\mathsf P
=d\mathsf h+\mathsf h d\). Apply these operations one coordinate
at a time, with the ordinary exterior differential signs. The
tensor-product homotopy still commutes with tangential restriction
@@ -6198,10 +6198,10 @@ zero; convergence and the uniform chart bounds give
\(\int_{\mathcal M}F^*\alpha=0\). All expectations and integrals
above are measurable by their locally finite chart formulas, and
their absolute bounds for the instance justify their interchange.
-Using \eqref{eq:transfer-form} in
-\eqref{eq:transfer-convolution-degree} now proves
-\eqref{eq:transfer-degree}.
-\end{proof}
+Using (eq:transfer-form) in
+(eq:transfer-convolution-degree) now proves
+(eq:transfer-degree).
+
Finish by estimating these degrees. It will be seen separately how
@@ -6209,22 +6209,22 @@ the straight part uses the dimension-only Hilbert norm of \(\eta\)
and how the small coordinate volume of the seam pays for its rougher
bound.
-\begin{proof}[Proof of Theorem~\ref{thm:scalar-volume}]
+Proof. [Proof of Theorem~thm:scalar-volume]
The case \(\|M\|=0\) is immediate. Otherwise, first prove the
conditional target for the data above, choosing \(\Psi_n\) from the
requirements on \(E\) derived below, and then apply it to the entropy
output. Choose a positive constant \(b_*\), depending only on \(n\),
whose smallness will be fixed in the estimate below, and put
-\begin{equation}\label{eq:transfer-lengths}
- L_T=b_*\sqrt E,\qquad L_N=b_*\sqrt{qE},\qquad
+
+ L_T=b_*\sqrt E,\qquad L_N=b_*√(qE),\qquad
h_{\rm tot}=h+L_T^2k.
-\end{equation}
+
Once the degree hypotheses are verified, its length denominator, the
product volume, and the entropy contribution will combine as
-\begin{equation}\label{eq:transfer-prefactor}
+
\frac{L_T^mE^{n/2}}{L_T^mL_N^n}\,e^{-W}dV_h
=b_*^{-n}q^{-n/2}\rho.
-\end{equation}
+
This is the cancellation of the tangent volume and all powers of
\(E\). The estimates below justify the degree application and bound
the straight angular sum by a dimensional multiple of \(q^{n/2}\),
@@ -6235,10 +6235,10 @@ product scalar formula and the definition of \(\wscal\) give
\(\wscal(h_{\rm tot},\phi)=\wscal(h,\phi)+L_T^{-2}\Scal_k\).
By the definition of \(K_{\rm scal}(q)\), the lower bound
\(E\geq b_*^{-2}K_{\rm scal}(q)\) ensures
-\(L_T^{-2}\Scal_k\geq-1\). By \eqref{eq:transfer-entropy},
-\begin{equation}\label{eq:transfer-total-scalar}
+\(L_T^{-2}\Scal_k\geq-1\). By (eq:transfer-entropy),
+
\wscal(h_{\rm tot},\phi)\geq-C_{\rm ent}-1+2W-n\log E+E\mathfrak e.
-\end{equation}
+
Assign \(L_i=L_T\) to the first \(m\) comparison rows and
\(L_i=L_N\) to the last \(n\) rows. On the band, the derivative in
@@ -6246,39 +6246,39 @@ tile directions of each row of \(G\), measured with \(k\), has norm
at most two. The \(dX\) term has norm at most one, and the
\(d\mathcal R\) term has norm at most
\(3\tau\sqrt q\|d\mathcal R\|_k\leq1\) by
-\eqref{eq:transfer-tau}. The horizontal derivative is a row of a
+(eq:transfer-tau). The horizontal derivative is a row of a
rotation of \(dz_F\). It follows that
\[
L_i|dG_i|_{h_{\rm tot}}
- \leq C_n\bigl(\sqrt q+\sqrt{qE}\,E^{-1/100}\bigr).
+ \leq C_n(\sqrt q+√(qE)\,E^{-1/100}).
\]
Take \(B\) to be this upper bound increased by two. Then
\(L_i|dG_i|\leq B-1=B-q/q\). The horizon condition of
-Theorem~\ref{thm:degree} holds after an elementary increase of \(E\):
+Theorem~thm:degree holds after an elementary increase of \(E\):
it suffices to test the smallest length \(L_T\), and, for \(E\geq1\),
\[
\frac{C(1+B)^{2/3}}{L_T}
\leq c_{{\rm hor},n}q^{1/3}E^{-13/75}
- <\frac{\tau}{10q}.
+ <(\tau)/(10q).
\]
Here \(c_{{\rm hor},n}\) depends only on the absolute horizon constant
and the dimensional choice of \(b_*\), and
\(13/75=1/2-(2/3)(1/2-1/100)>0\). Thus the displayed strict horizon
inequality follows if
\(E\geq(20c_{{\rm hor},n}q^{4/3}/\tau)^{75/13}\).
-Therefore apply Theorem~\ref{thm:degree}
+Therefore apply Theorem~thm:degree
with \(d=q\).
Write \(H_i=2(L_i|dG_i|-1)_+\), as in that theorem. At a straight
point in a tile, let \(\mathcal I\) be the \(n\)-set of grid axes
normal to its dual cube. Put \(A_i=(Vdz_F)_i\), with covector norms
measured by \(h\), and define
-\begin{equation}\label{eq:transfer-U}
+
U_{\mathcal I}^2=
\frac q{\mathfrak e}\sum_{i\in\mathcal I}|A_i|_h^2
- \quad\text{if }\mathfrak e>0,
- \qquad U_{\mathcal I}=0\quad\text{if }\mathfrak e=0.
-\end{equation}
+ \quadif \mathfrak e>0,
+ \qquad U_{\mathcal I}=0\quadif \mathfrak e=0.
+
Here \(\sum_i|A_i|^2=\mathfrak e\). At a straight point
\(k=X^*k_{\rm flat}\) and \(\mathcal R=Q_\sigma V\) is constant.
For a tangent comparison row corresponding to axis \(i\), the squared
@@ -6287,32 +6287,32 @@ row norm is exactly \(L_T^{-2}+|A_i|^2\). Thus
\sum_{\rm tangent}H_i
\leq L_T^2\mathfrak e=b_*^2E\mathfrak e,
\]
-by \(2(\sqrt{1+t}-1)\leq t\). Normal rows have no tile derivative,
+by \(2(√(1+t)-1)\leq t\). Normal rows have no tile derivative,
so Cauchy--Schwarz gives
\[
\sum_{\rm normal}H_i
\leq2L_N\sum_{i\in\mathcal I}|A_i|
- \leq2b_*\sqrt n\sqrt{E\mathfrak e}\,U_{\mathcal I}.
+ \leq2b_*\sqrt n√(E\mathfrak e)\,U_{\mathcal I}.
\]
-Completing the square in \(\sqrt{E\mathfrak e}\), for \(b_*<1\),
+Completing the square in \(√(E\mathfrak e)\), for \(b_*<1\),
shows that
-\begin{equation}\label{eq:transfer-straight-penalty}
- \frac{\sum_iH_i-E\mathfrak e}{2}
+
+ (\sum_iH_i-E\mathfrak e)/(2)
\leq c_n U_{\mathcal I}^2,
- \qquad c_n=\frac{n b_*^2}{2(1-b_*^2)}.
-\end{equation}
+ \qquad c_n=(n b_*^2)/(2(1-b_*^2)).
+
Choose \(b_*\) sufficiently small that
-\begin{equation}\label{eq:transfer-sphere}
+
\E_V\exp(2c_nU_{\mathcal I}^2)\leq C_n
-\end{equation}
+
for every fixed horizontal differential and every fixed
\(\mathcal I\). Here is a direct verification of this choice. If
\(u\) is one coordinate of a uniform point on \(S^{q-1}\), its sphere
moments satisfy
\[
\E(q u^2)^j
- =\frac{q^j(1\cdot3\cdots(2j-1))}{q(q+2)\cdots(q+2j-2)}
+ =(q^j(1\cdot3...(2j-1)))/(q(q+2)...(q+2j-2))
\leq2^j j!.
\]
Consequently \(\E e^{tqu^2}\leq(1-2t)^{-1}\) for \(0\leq t<1/2\).
@@ -6321,9 +6321,9 @@ The singular-value decomposition of \(dz_F\) expresses each
variables, with its singular-value weights. Jensen's inequality
gives the same exponential bound for that convex combination.
Another application of Jensen to the sum of the \(n\) rows in
-\eqref{eq:transfer-U} bounds its exponential moment by
+(eq:transfer-U) bounds its exponential moment by
\((1-2nt)^{-1}\). Choosing, for instance, \(2nc_n<1/4\) proves
-\eqref{eq:transfer-sphere}. No independence of the rows of \(V\)
+(eq:transfer-sphere). No independence of the rows of \(V\)
is used.
At a seam point the uniform tile row bound gives a rougher estimate
@@ -6331,42 +6331,42 @@ which is still independent of the instance. By the triangle inequality
for each row, \((a-1)_+\leq a\), and Cauchy--Schwarz on the horizontal
rows,
\[
- \sum_iH_i\leq 2L_N\sqrt{q\mathfrak e}
+ \sum_iH_i\leq 2L_N√(q\mathfrak e)
+4q L_N/L_T
- \leq2b_*q\sqrt{E\mathfrak e}+4q^{3/2}.
+ \leq2b_*q√(E\mathfrak e)+4q^{3/2}.
\]
Completing the square once more gives
-\begin{equation}\label{eq:transfer-seam-penalty}
- \frac{\sum_iH_i-E\mathfrak e}{2}\leq c_{{\rm seam},n}q^2,
- \qquad c_{{\rm seam},n}=2+\frac{b_*^2}{2}.
-\end{equation}
+
+ (\sum_iH_i-E\mathfrak e)/(2)\leq c_{{\rm seam},n}q^2,
+ \qquad c_{{\rm seam},n}=2+(b_*^2)/(2).
+
Apply the degree inequality, then use
-\eqref{eq:transfer-total-scalar}. Its loss is \(\exp(Cq/q)\), an
+(eq:transfer-total-scalar). Its loss is \(\exp(Cq/q)\), an
absolute constant. The product volume form is
\(dV_{h_{\rm tot}}=L_T^m dV_h\,dV_k\), so the measure factor is
-exactly \eqref{eq:transfer-prefactor}. Together with
-\eqref{eq:transfer-degree} and \(|\E\mathcal D|\leq\E|\mathcal D|\),
+exactly (eq:transfer-prefactor). Together with
+(eq:transfer-degree) and \(|\E\mathcal D|\leq\E|\mathcal D|\),
this gives the following inequality, with
\(C_{\rm out}=\exp(C_{\rm deg}+(C_{\rm ent}+1)/2)b_*^{-n}\), where
-\(C_{\rm deg}\) is the absolute exponent in Theorem~\ref{thm:degree}:
-\begin{equation}\label{eq:transfer-average}
- \begin{split}
+\(C_{\rm deg}\) is the absolute exponent in Theorem~thm:degree:
+
+
D_0I(2\tau)^q\leq C_{\rm out}q^{-n/2}
\int_{\mathcal M}\rho\,
\E_{V,s,\rm rd}
- \int_{\{a:\ |G(x,a)|_\infty<b_q\}}
+ \int_{\{a:\ |G(x,a)|_∞<b_q\}}
\mathcal B(x,a)\,dV_k(a),
- \end{split}
-\end{equation}
+
+
where the fiber integral includes its state transversals and only the
compatible original-side band, and
\[
\mathcal B(x,a)=
- \begin{cases}
- \exp(c_nU_{\mathcal I}^2),&a\text{ straight over axes }\mathcal I,\\
- \exp(c_{{\rm seam},n}q^2),&a\text{ in the seam}.
- \end{cases}
+
+ \exp(c_nU_{\mathcal I}^2),&a straight over axes \mathcal I,\\
+ \exp(c_{{\rm seam},n}q^2),&a in the seam.
+
\]
The entropy constant enters only through this final prefactor, not the
geometric bounds used below. The length and volume prefactor has no
@@ -6374,33 +6374,33 @@ exponential loss in \(m\); the seam exponential will be absorbed by
the choice of \(\lambda\).
Fix \(x,\omega\), and put \(y=F(x)\). At a coupled pair in
-\eqref{eq:transfer-average}, a fixed projected grid index has at most
+(eq:transfer-average), a fixed projected grid index has at most
one possible cell lift in the ball chart of \(y\). Its full primal
-and dual cells fit in that chart by \eqref{eq:transfer-tau} and the
+and dual cells fit in that chart by (eq:transfer-tau) and the
buffers, and the lift there is unique. Moreover each point of its
primal cell is within \(10r\sqrt q\) of \(y\): the dual point is
within \(b_q\sqrt q\) of \(y\), and the dual and primal extents are
at most \(r\sqrt m/2\) and \(r\sqrt n/2\). The variation choice
for \(r\) and the rounding error now give the copy-count estimate
-\begin{equation}\label{eq:transfer-copies}
+
D_0|u'_C|\leq D_0r^n
- \bigl(|\eta_{\mathcal I}(y)|+2q^{-n}\bigr),
-\end{equation}
+ (|\eta_{\mathcal I}(y)|+2q^{-n}),
+
where the component is in the \(V\)-rotated frame.
On a straight tile the \(k\)-volume is exactly the flat volume of
its image in the chamber. The signed permutation in \(G\) does not
change its cube test, which is
-\(|\widehat z_y-\widehat z_X|_\infty<b_q\). Summing the chamber
+\(|\widehat z_y-\widehat z_X|_∞<b_q\). Summing the chamber
pieces in a copy is thus bounded by the \(m\)-area of its whole dual
cube inside that test; the state probabilities for each chamber
integrate to one. For a fixed axis set \(\mathcal I\), summing over
all projected centers and averaging the shift gives exactly
-\begin{equation}\label{eq:transfer-shift}
+
r^{-q}\int_{\mathbb R^q}\int_{[-r/2,r/2]^m}
- \indic_{\{|\widehat z_y-c-t|_\infty<b_q\}}\,dt\,dc
+ \indic_{\{|\widehat z_y-c-t|_∞<b_q\}}\,dt\,dc
=r^{-q}r^m(2b_q)^q.
-\end{equation}
+
Here \(t\) is placed in the complementary axes. This identity is
just the density \(r^{-q}\) of centers under uniform shift; extending
the sum to the full grid is an upper bound for the selected lifted
@@ -6408,34 +6408,34 @@ cells.
The form bound holds in every rotated frame. Using Cauchy--Schwarz
jointly in \(V\) and the \(\binom qn\) axis sets, and then
-\eqref{eq:transfer-sphere}, one obtains \begin{equation}\label{eq:transfer-form-average}
- \begin{split}
+(eq:transfer-sphere), one obtains
+
\E_V\sum_{|\mathcal I|=n}
(|\eta_{\mathcal I}(y)|+2q^{-n})e^{c_nU_{\mathcal I}^2}
&\leq
- \left(\E_V\sum_{\mathcal I}|\eta_{\mathcal I}(y)|^2\right)^{1/2}
- \left(\E_V\sum_{\mathcal I}e^{2c_nU_{\mathcal I}^2}\right)^{1/2}
+ (\E_V\sum_{\mathcal I}|\eta_{\mathcal I}(y)|^2)^{1/2}
+ (\E_V\sum_{\mathcal I}e^{2c_nU_{\mathcal I}^2})^{1/2}
+C_nq^{-n}\binom qn\\
&\leq C_nq^{n/2}.
- \end{split}
-\end{equation}
+
+
This use of Cauchy--Schwarz allows the form components and row norms
to depend on the same rotation. Combining
-\eqref{eq:transfer-copies}--\eqref{eq:transfer-form-average}, the
+(eq:transfer-copies)--(eq:transfer-form-average), the
straight part of the inner expectation in
-\eqref{eq:transfer-average} is at most
+(eq:transfer-average) is at most
\(C_nD_0q^{n/2}(2b_q)^q\), because \(r^{n+m-q}=1\). Multiplying
by its prefactor \(q^{-n/2}\) bounds that part, for a fixed
dimension-only constant \(C_{{\rm str},n}\), by
-\begin{equation}\label{eq:transfer-straight-average}
+
C_{{\rm str},n}D_0(2\tau)^q,
-\end{equation}
+
since \((b_q/\tau)^q=(1+3/(4q))^q\leq e^{3/4}\).
For the seam use only the consequence
\(|z_X-z_y|_2\leq b_q\sqrt q\) of the cube test. Discard the tile
boundaries, which have leafwise measure zero. The remaining set is
-the disjoint union of tile interiors indexed by their \emph{actual}
+the disjoint union of tile interiors indexed by their actual
chamber \(\sigma\) and its state arrays. Partition each such interior
into positive sorted-coordinate pieces, for example by taking the
first sorted permutation in a fixed order. On these pieces
@@ -6446,7 +6446,7 @@ before \(\lambda\).
Retain the compatible-sheet restriction at this point. For a fixed
projected grid index the uniqueness proved before
-\eqref{eq:transfer-copies} allows at most one compatible geometric
+(eq:transfer-copies) allows at most one compatible geometric
lift. Apply the rough copy bound
\(D_0|u'_C|\leq C_{\eta,n}D_0r^n\) from that equation, where
\(C_{\eta,n}\) depends only on the form norm, and use the uniform
@@ -6455,10 +6455,10 @@ on available sheets, roundings, and chambers and sum over all projected
centers and chamber shapes.
For a fixed actual chamber shape, normal axis set \(\mathcal I\), and
-positive sorted flag, \eqref{eq:transfer-tile-map} has rotated position
+positive sorted flag, (eq:transfer-tile-map) has rotated position
\[
- \widehat z_X=s+r\left(k+\tfrac12\indic_{\mathcal I}\right)
- +d_{\mathcal I,\mathrm{shape},\mathrm{flag}}(Z).
+ \widehat z_X=s+r(k+\tfrac12\indic_{\mathcal I})
+ +d_{\mathcal I,shape,flag}(Z).
\]
Here \(\indic_{\mathcal I}\) is the coordinate indicator vector of
\(\mathcal I\).
@@ -6467,13 +6467,13 @@ of \(\lambda,\gamma_0\), and of the shift; it is determined by the
fixed chamber shape and the barycentric formula. For each fixed \(Z\)
the shift integral of the rough test is consequently
\[
- \begin{split}
+
\frac1{r^q}\int_{[0,r)^q}\sum_{k\in\mathbb Z^q}
\indic_{\{|s+r(k+\frac12\indic_{\mathcal I})+d(Z)-\widehat z_y|_2
\leq b_q\sqrt q\}}\,ds
&=r^{-q}\Vol_{\mathbb R^q}(B_{b_q\sqrt q})\\
&\leq r^{-q}(4\tau\sqrt q)^q.
- \end{split}
+
\]
This calculation is valid despite the shift dependence of the actual
diagram, because the copy, lift, and density bounds were imposed before
@@ -6481,18 +6481,18 @@ the enlargement to all centers. Tonelli's theorem applies to these
nonnegative bounds.
In each positive sorted piece the seam lies in
-\(\bigcup_j\{Z_j\leq\lambda\}\), of coordinate volume at most
+\(∪_j\{Z_j\leq\lambda\}\), of coordinate volume at most
\(m\lambda\). A cube has \(2^m m!\) barycentric top chambers and
at most \((m+1)!\) sorted flags per chamber. Thus the seam part of
the inner expectation is bounded by
-\begin{equation}\label{eq:transfer-seam-average}
+
C_*(q)\lambda D_0(2\tau)^q,
-\end{equation}
+
where one may take the elementary envelope
\[
- C_*(q)=\max\left\{1,\,
+ C_*(q)=\max{1,\,
C_{\eta,n}H_0(q)\binom qn(2^m m!)(m+1)!\,m\,
- r^{n-q}(2\sqrt q)^q\exp(c_{{\rm seam},n}q^2)\right\}.
+ r^{n-q}(2\sqrt q)^q\exp(c_{{\rm seam},n}q^2)}.
\]
Every factor here was bounded before choosing \(\lambda\). Thus
\(C_*\) is independent of \(\lambda,\gamma_0,\tau\), of the bounds
@@ -6500,15 +6500,15 @@ Every factor here was bounded before choosing \(\lambda\). Thus
count have already canceled those in \((2\tau)^q\).
The two fiber estimates now have the same normalization. Insert
-\eqref{eq:transfer-straight-average} and
-\eqref{eq:transfer-seam-average} into \eqref{eq:transfer-average},
+(eq:transfer-straight-average) and
+(eq:transfer-seam-average) into (eq:transfer-average),
using \(q^{-n/2}\leq1\) for the seam, and cancel the positive factor
\(D_0(2\tau)^q\). For the conditional data and parameter bounds used
so far, this gives
-\begin{equation}\label{eq:transfer-before-choices}
- I\leq C_{\rm out}\bigl(C_{{\rm str},n}+C_*(q)\lambda\bigr)
+
+ I\leq C_{\rm out}(C_{{\rm str},n}+C_*(q)\lambda)
\int_{\mathcal M}\rho.
-\end{equation}
+
Now fix these bounds before invoking the entropy theorem. For each
integer \(q\geq n+1\), the form derivative bound first gives a
@@ -6516,7 +6516,7 @@ deterministic \(r(q)\) with elementary reciprocal. The size bound for
the universal type list, \(D_0\), incidence counts, and the bounds for
\(k_0,dX\), including \(H_0(q)\), have elementary envelopes uniform
over all permitted graphs and cocycles. Fix \(b_*\) using
-\eqref{eq:transfer-sphere}; this determines the dimensional constants
+(eq:transfer-sphere); this determines the dimensional constants
\(c_{{\rm seam},n}\) and \(c_{{\rm hor},n}\), independently of
\(C_{\rm ent}\). The displayed formula for \(C_*(q)\) is consequently
available at this stage. Set
@@ -6525,18 +6525,18 @@ available at this stage. Set
\gamma_0(q)=\frac14(m+1)^{-(m+1)}\lambda(q)^{m(m+1)}.
\]
These deterministic choices have elementary reciprocals and satisfy
-\eqref{eq:transfer-gamma}. The resolution estimates now give common
+(eq:transfer-gamma). The resolution estimates now give common
elementary envelopes for its fixed-order chart derivatives, hence
\(K_R(q),K_{\rm scal}(q)\), and then the common \(\tau(q)\) in
-\eqref{eq:transfer-tau}. None of these choices uses the entropy constant.
+(eq:transfer-tau). None of these choices uses the entropy constant.
For integer \(j\geq n+1\), define
\[
- \Psi_n^\circ(j)=\max\left\{1,\ b_*^{-2}K_{\rm scal}(j),
- \left(\frac{20c_{{\rm hor},n}j^{4/3}}{\tau(j)}\right)^{75/13}
- \right\},
+ \Psi_n^\circ(j)=\max{1,\ b_*^{-2}K_{\rm scal}(j),
+ (\frac{20c_{{\rm hor},n}j^{4/3}}{\tau(j)})^{75/13}
+ },
\]
-and define a majorant on all \(t\in[1,\infty)\) by
+and define a majorant on all \(t\in[1,∞)\) by
\[
\Psi_n(t)=1+\max_{\substack{j\in\mathbb Z\\
n+1\leq j\leq\max\{n+1,\lceil t\rceil\}}}
@@ -6548,23 +6548,23 @@ maximum is bounded by an increasing elementary envelope at
majorant fixed solely from \(n\), and it dominates both the curvature
and strict horizon requirements. Hence every conditional datum with
\(E\geq\Psi_n(q)\) satisfies all hypotheses used in
-\eqref{eq:transfer-before-choices}. Our choice
+(eq:transfer-before-choices). Our choice
\(C_*(q)\lambda(q)=1/4\) and the definition of \(C_{\rm out}\) prove
-\eqref{eq:transfer-conditional-target} with
+(eq:transfer-conditional-target) with
\[
K_n=e^{C_{\rm deg}+1/2}b_*^{-n}
- \bigl(C_{{\rm str},n}+1/4\bigr).
+ (C_{{\rm str},n}+1/4).
\]
This closes the conditional target for all the specified data before
any entropy family is chosen.
-Now apply Theorem~\ref{thm:entropy} with \(\mathcal B=\Psi_n\),
-\(E_{\min}=1\), the detector root map as \(F^{\mathrm{in}}\), and the
-one-point transverse space \(\Omega_0\). The detector construction uses
+Now apply Theorem~thm:entropy with \(\mathcal B=\Psi_n\),
+\(E_{\min}=1\), the detector root map as \(F^{in}\), and the
+one-point transverse space \(Ω_0\). The detector construction uses
finitely many compactly supported bump representatives and their deck
translates, giving the required uniform support and derivative bounds in
a lifted atlas and the equivariant compact support sets. After pulling
-the detector map to \(\Omega\) and embedding its labels equivariantly
+the detector map to \(Ω\) and embedding its labels equivariantly
among the output's finite copies, the measured affine probability homotopy
of Section~2 preserves \(I\); the output therefore supplies the conditional
data with \(E>\Psi_n(q)\). Its constant
@@ -6577,169 +6577,169 @@ probability measure,
\]
Finally \(I\geq\|M\|/2\), so the constant
\(C_n=2K_ne^{C_{\rm ent}/2}\) proves the theorem.
-\end{proof}
-\begin{proof}[Proof of Theorem~\ref{thm:main}]
+
+Proof. [Proof of Theorem~thm:main]
Fix \(n\geq3\), and let \(C_n>0\) be the constant for the normalized
-scalar-curvature estimate in Theorem~\ref{thm:scalar-volume}.
-Proposition~\ref{prop:yamabe-reduction} gives
+scalar-curvature estimate in Theorem~thm:scalar-volume.
+Proposition~prop:yamabe-reduction gives
\[
\|M\|\leq C_n\int_M(\Scal_g^-)^{n/2}\,dV_g
\]
for every metric in the theorem. Taking \(a_n=C_n^{-1}>0\) proves
the stated integral bound.
-\end{proof}
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