64c3814
==== TABLE OF CONTENTS ====
Sharp-integral-fillings-in-CAT(0)-spaces//figures/radial-variation.tex
Sharp-integral-fillings-in-CAT(0)-spaces//glyph-mappings.tex
Sharp-integral-fillings-in-CAT(0)-spaces//macros.tex
Sharp-integral-fillings-in-CAT(0)-spaces//main.tex
Sharp-integral-fillings-in-CAT(0)-spaces//sections/01-introduction.tex
Sharp-integral-fillings-in-CAT(0)-spaces//sections/02-currents.tex
Sharp-integral-fillings-in-CAT(0)-spaces//sections/03-radial.tex
Sharp-integral-fillings-in-CAT(0)-spaces//sections/04-sobolev.tex
Sharp-integral-fillings-in-CAT(0)-spaces//sections/05-minimizer.tex
Sharp-integral-fillings-in-CAT(0)-spaces//sections/06-chords.tex
Sharp-integral-fillings-in-CAT(0)-spaces//sections/07-conclusion.tex
Sharp-integral-fillings-in-CAT(0)-spaces//sections/08-classical.tex
Sharp-integral-fillings-in-CAT(0)-spaces//references.bib
===========================
==== BEGIN Sharp-integral-fillings-in-CAT(0)-spaces//figures/radial-variation.tex ====
\begin{figure}[htbp]
\centering
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\begin{scope}[xshift=0cm]
\node[anchor=west,font=\bfseries\small] at (-.3,3.4) {(a) Slice and sweep};
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\node[above] at (xp) {$\bar x'$};
\node[below] at (p) {$p$};
\node[left] at (pp) {$p'$};
\node[below] at ($(y)!.36!(x)$) {$\lambda r$};
\node[above left] at ($(y)!.27!(xp)$) {$\lambda'r'$};
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\node[anchor=west] at (0,-1.2) {$d(F_t(x),F_t(x'))\le |p-p'|$};
\end{scope}
\end{tikzpicture}
\caption{Radial variation in the whole-current product and its Euclidean
comparison triangle. Panel (a) distinguishes a fixed-time slice from
the swept current; the stack is symbolic and makes no regularity or
injectivity assertion about its image in $Y$. In panel (b), the
barred vertices correspond to $y,x,x'$; $p$ and $p'$ lie at fractions
$\lambda$ and $\lambda'$ on their radial sides. The fractions may
differ even at the same time because $g$ varies with the point.
The comparison distance gives the upper bound
\eqref{eq:radial-two-point}. The exact determinant in
\eqref{eq:radial-total-jacobian} belongs to the quadratic upper form,
rather than an asserted exact differential of the deformed map.}
\label{fig:radial-variation}
\end{figure}
==== END Sharp-integral-fillings-in-CAT(0)-spaces//figures/radial-variation.tex ====
==== BEGIN Sharp-integral-fillings-in-CAT(0)-spaces//glyph-mappings.tex ====
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==== BEGIN Sharp-integral-fillings-in-CAT(0)-spaces//macros.tex ====
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==== BEGIN Sharp-integral-fillings-in-CAT(0)-spaces//main.tex ====
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\documentclass[11pt]{article}
\input{macros}
\title{Sharp integral fillings in CAT(0) spaces}
\author{OpenAI}
\date{September 23, 2026}
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\begin{document}
\maketitle
\begin{abstract}
Every compactly supported integral $n$-cycle, $n\ge2$, in a proper
CAT(0) space bounds a compactly supported integral current with the
sharp Euclidean mass bound. The theorem allows arbitrary integer
multiplicities and unrestricted ambient dimension. In particular, we prove
the Euclidean Cartan--Hadamard isoperimetric conjecture in dimensions
at least three.
\end{abstract}
\input{sections/01-introduction}
\input{sections/02-currents}
\input{sections/03-radial}
\input{sections/04-sobolev}
\input{sections/05-minimizer}
\input{sections/06-chords}
\input{sections/07-conclusion}
\input{sections/08-classical}
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==== END Sharp-integral-fillings-in-CAT(0)-spaces//main.tex ====
==== BEGIN Sharp-integral-fillings-in-CAT(0)-spaces//sections/01-introduction.tex ====
\section{Introduction}\label{sec:introduction}
The Euclidean isoperimetric inequality bounds the volume enclosed by a
boundary of prescribed area. Its current-theoretic form asks for a filling
of a cycle and permits singular supports, integer multiplicities, and
arbitrary codimension. We prove that the sharp Euclidean filling bound
holds in every proper CAT(0) space, in all cycle dimensions at least two.
Write $\I_j(X)$ for the integral $j$-currents of Ambrosio and
Kirchheim~\cite{AK2000}, $\partial$ for current boundary, and $\M$
for mass. An integral $n$-cycle is a current $T\in\I_n(X)$ with
$\partial T=0$; an integral filling is a current $S\in\I_{n+1}(X)$
with $\partial S=T$. A metric space is \emph{proper} if its closed
bounded balls are compact. A CAT(0) space is a geodesic metric space in which the distance
between two points on a geodesic triangle is at most the distance between
the corresponding points of its Euclidean comparison triangle.
Let $\omega_j$ be the volume of the unit ball in $\R^j$ and put
\begin{equation}\label{eq:constants}
s_n=(n+1)\omega_{n+1}=\operatorname{area}(\mathbb S^n),
\qquad c_n=\frac{1}{(n+1)s_n^{1/n}}.
\end{equation}
Here and throughout the filling argument, $n$ denotes the cycle
dimension.
\begin{theorem}[Sharp integral filling]\label{thm:main}
Let $X$ be a proper CAT(0) space and $n\ge2$ an integer.
Every compactly supported $T\in\I_n(X)$ with $\partial T=0$ admits a
compactly supported $S\in\I_{n+1}(X)$ such that
\begin{equation}\label{eq:main}
\partial S=T,\qquad
\M(S)\le c_n\M(T)^{(n+1)/n}
=\frac{\M(T)^{(n+1)/n}}
{(n+1)^{(n+1)/n}\omega_{n+1}^{1/n}}.
\end{equation}
There is no restriction on the dimension of $X$ or on the integer
multiplicities of the currents. The coefficient is optimal.
\end{theorem}
For the first two dimensions, $s_2=4\pi$ and $s_3=2\pi^2$, so
\[
c_2=\frac1{6\sqrt\pi},\qquad
c_3=(128\pi^2)^{-1/3}.
\]
Euclidean spheres attain the coefficient, as verified in
Section~\ref{sec:classical}.
\subsection{The Cartan--Hadamard consequence}
A \emph{Cartan--Hadamard manifold} is a complete, simply connected
Riemannian manifold of nonpositive sectional curvature. The Euclidean
Cartan--Hadamard conjecture asks whether domains in such a manifold
satisfy the same volume--perimeter bound as Euclidean domains.
Theorem~\ref{thm:main}, applied to boundary currents, gives the
following positive answer in dimensions at least three.
\begin{corollary}\label{cor:cartan-hadamard}
Let $M$ be a Cartan--Hadamard manifold of dimension $d\ge3$.
For every bounded domain $\Omega\subset M$ with smooth boundary,
\begin{equation}\label{eq:classical}
\operatorname{area}(\partial\Omega)
\ge d\omega_d^{1/d}\vol(\Omega)^{(d-1)/d}.
\end{equation}
The same inequality holds for relatively compact sets of finite
perimeter, with ambient perimeter in place of boundary area.
\end{corollary}
The reduction uses the uniqueness of a compactly supported
top-dimensional filling of a prescribed boundary. Its proof, and the
classical two-dimensional case, appear in Section~\ref{sec:classical}.
Together they give the Euclidean inequality for bounded smooth domains
in every ambient dimension $d\ge2$.
The smooth domain problem has a long independent history. The surface
inequality goes back to Weil and Beckenbach--Rad\'o
\cite{Weil1926,BeckenbachRado1933}; Kleiner proved the
three-dimensional comparison~\cite{Kleiner1992}, and Croke proved
the four-dimensional Euclidean inequality~\cite{Croke1984}.
Ghomi and Spruck showed that a corresponding total-curvature bound
implies the domain inequality in every dimension
\cite[Theorem~7.1]{GhomiSpruck2022}.
Recent results extend these comparisons in several directions. Chen,
Ghomi and Wang prove the Euclidean comparison in dimensions three
through nine~\cite[Theorem~1.1]{ChenGhomiWang2026HigherDimensions}.
Wheeler proves it in all dimensions under a controlled variation
condition on the negative-curvature scale
\cite[Theorem~1.1 and Definition~1.2]{Wheeler2026}.
Agnoletto, Da Silva, Nardulli and Resende prove small-volume comparison
with the constant-curvature model under a nonpositive sectional-curvature
upper bound and a Ricci lower bound,
and reduce unrestricted-volume comparison to equality rigidity in a
class of Alexandrov spaces
\cite[Theorems~1.1 and~1.3; Assumption~1]{AgnolettoDaSilvaNardulliResende2026}.
Our domain corollary follows from the integral filling theorem and
top-dimensional constancy; it requires no separate smooth comparison
argument or additional curvature hypothesis.
\subsection{The filling problem and its predecessors}
Almgren established the sharp isoperimetric inequality for integral
currents in Euclidean space in arbitrary dimension and
codimension~\cite{Almgren1986}. In a general metric space, a comparable
statement first requires notions of orientation, integer multiplicity,
mass, and boundary that do not depend on an ambient smooth structure.
Ambrosio and Kirchheim supplied this theory, including rectifiable
representation, slicing, and compactness~\cite{AK2000}.
Kirchheim's metric differentiation theorem~\cite{Kirchheim1994}
provides the local normed geometry of its rectifiable charts. In CAT(0)
spaces these chart norms are Euclidean; we give the short comparison
argument and the exact mass normalization in Section~\ref{sec:currents}.
For Lipschitz loops in complete CAT(0) spaces, Reshetnyak's majorization
theorem already gives a Lipschitz disc whose parametrized Hausdorff area
is at most the squared loop length divided by $4\pi$; see
\cite[Section~3.2, Lemma~3.2]{LytchakWenger2018}.
Wenger proved isoperimetric inequalities with the Euclidean exponent
in complete spaces with a convex geodesic bicombing, including CAT(0)
spaces~\cite{Wenger2005Iso}. His constant depends on the cycle
dimension. He also established the relation between weak convergence
and filling convergence used to obtain variational extremizers
here~\cite{Wenger2007Flat}. Theorem~\ref{thm:main} identifies the optimal
coefficient. For two-cycles in smooth Cartan--Hadamard manifolds,
Schulze proved the sharp constant-curvature model comparison in every
codimension, together with equality rigidity
\cite[Theorems~1.2--1.3]{Schulze2020}. The present theorem extends the
Euclidean filling bound to all cycle dimensions $n\ge2$ and singular CAT(0)
ambient spaces.
Recent work identifies geometric hypotheses under which fillings
have smaller growth exponents at large mass. In complete CAT(0)
spaces of finite asymptotic Nagata dimension, Lang, Stadler and Urech
obtain almost-linear bounds for compact integral $k$-cycles, $k\ge2$,
when the asymptotic rank is at most two
\cite[Theorem~1.1]{LangStadlerUrech2026}. With the same Nagata dimension
hypothesis and finite asymptotic rank $k_0$, Peteranderl obtains linear
bounds for integral $k$-cycles with $k\ge\max\{k_0,1\}$, and compact
fillings for compact inputs~\cite[Theorem~1.2]{Peteranderl2026}.
The constants depend on the space. The estimate here fixes the universal
Euclidean coefficient at every mass in an arbitrary proper CAT(0) space.
The problem also belongs to the broader study of filling geometry
developed by Gromov~\cite{Gromov1983}. His sharp shrinking and filling
formulation for CAT(0) targets asks for volume-controlled extensions of
maps on specified domains~\cite[Section~2.3]{Gromov2014}.
Theorem~\ref{thm:main} concerns integral-current fillings: it does not
prescribe the topology or parametrization of the filling. Those
additional extension requirements are not supplied by a mass and
boundary estimate alone.
\subsection{Main ideas and proof route}
The extremizer/curvature-estimate strategy is motivated by Almgren
\cite[Section~0, p.~454]{Almgren1986}, and the two-dimensional
curvature estimate also by Schulze \cite[Theorem~1.4]{Schulze2020}.
The singular CAT(0) radial implementation is established here; these
references are methodological motivation, not imports of a smooth
curvature theorem into the singular setting.
The proof proceeds by induction on $n$, with its two-dimensional case
proved directly. A closed convex ball containing the given compact
support reduces the problem to a compact CAT(0) space $Y$. For an
integral cycle $B$ in $Y$, let $V(B)$ be its least integral filling mass.
If the sharp bound fails, compactness and filling continuity give a
nonzero cycle $A$ maximizing
\[
V(B)-c\M(B)^{(n+1)/n}\qquad(c>c_n).
\]
A constant rescaling makes its mass $m=\M(A)$ smaller than $s_n$.
This is the small-mass inequality that the rest of the proof contradicts.
The first step varies $A$ inward along geodesics from a fixed center
$y$. Write $r(x)=d(y,x)$, and let $Dr$ denote the differential of $r$
along the Euclidean current charts. Triangle comparison bounds the
deformation by a quadratic form in the time and chart variables.
Its determinant retains the factor $\sqrt{1-|Dr|^2}$ in the swept-mass
bound. In a smooth Euclidean model this is the transverse component of
the radial unit direction. The argument needs neither an ambient
normal bundle nor normality of a separate chart restriction.
Section~\ref{sec:radial} proves the resulting radial inequality.
For $n=2$, an inverse-square cutoff and Euclidean lower density
already force $m\ge4\pi=s_2$, establishing the base case.
For $n>2$, the induction hypothesis fills the boundaries of small
restrictions of $A$. Replacing each restriction by such a filling
gives an almost Euclidean small-set perimeter estimate. Intrinsic
coarea and rearrangement turn this into a critical Sobolev inequality
in Section~\ref{sec:sobolev}. The sharp coefficient is the classical
Aubin--Talenti coefficient~\cite{Aubin1976,Talenti1976}; its radial
form is proved here by the mass-transport method of
Cordero-Erausquin, Nazaret and Villani
\cite[Section~2]{CorderoNazaretVillani2004}.
To use this estimate variationally, we close the chart gradient in
$L^2(\mu)$, where $\mu=\|A\|$ is the mass measure. We prove compactness
by first projecting whole weighted currents and then isolating charts
in total variation. With $\beta=1/(n-2)$ and $p=2n/(n-2)$, the
almost sharp inequality and Br\'ezis--Lieb splitting
\cite{BrezisLieb1983} produce a nonnegative minimizer of
\[
\frac{4\beta\int |Du|^2\,d\mu+n\int u^2\,d\mu}
{\bigl(\int |u|^p\,d\mu\bigr)^{2/p}}
\]
on the completion for the norm
$(\|u\|_2^2+\int|Du|^2\,d\mu)^{1/2}$. Section~\ref{sec:minimizer} normalizes this
minimizer $v$ so that $0<W:=\int v^p\,d\mu<s_n$, and establishes
the moment estimates needed for its nonlinear tests.
The final comparison concerns the weighted chord
\[
v(y)^\beta d(y,x)v(x)^\beta
\]
and the probability measure $d\nu=v^p\,d\mu/W$.
Radial variation and the minimizer equation control a scalar transform
of the chord distribution. Euclidean tangent density then bounds its
mean square by $2(W/s_n)^{2/n}<2$ for almost every center $y$.
The actual composite tests remain differentiable at $v=0$, even when
$\beta<1$; no Sobolev regularity of $v^\beta$ is presumed.
Section~\ref{sec:chords} supplies this upper bound.
Section~\ref{sec:conclusion} uses CAT(0) variance to show that the
average of the same squared chords is at least $2$. The contradiction
completes the induction and yields attained, compactly supported
integral fillings.
==== END Sharp-integral-fillings-in-CAT(0)-spaces//sections/01-introduction.tex ====
==== BEGIN Sharp-integral-fillings-in-CAT(0)-spaces//sections/02-currents.tex ====
\section{Compact fillings and Euclidean current charts}
\label{sec:currents}
We first reduce the filling problem to a compact ambient space. Standard
compactness and filling continuity then turn a hypothetical failure of the
sharp bound into a normalized extremizing cycle. To vary this cycle
without losing the sharp coefficient, we derive exact Euclidean chart
mass and density formulas. These local formulas apply to every integral
current. All current dimensions in this section are finite positive
integers; the ambient space has no dimension bound.
\subsection{Reduction to a compact convex ball}
\begin{lemma}[Compact reduction]
\label{lem:compact-reduction}
Suppose that the asserted filling inequality in a fixed cycle dimension
holds in every compact \(\operatorname{CAT}(0)\) space. Then it holds for
every compactly supported integral cycle of that dimension in every proper
\(\operatorname{CAT}(0)\) space, with a compactly supported integral filling.
\end{lemma}
\begin{proof}
Let \(T\) be the cycle in a proper \(\operatorname{CAT}(0)\) space \(X\).
If \(T=0\), take the zero filling. Otherwise choose a closed ball
\(Y=\overline B(o,R)\) whose interior contains \(\spt T\).
Properness makes \(Y\) compact. Triangle comparison implies that balls
are convex, so the induced metric makes \(Y\) a compact
\(\operatorname{CAT}(0)\) space.
For \(x\in X\), let \(P(x)\) be the point of the segment \([o,x]\)
at distance \(\min\{R,d(o,x)\}\) from \(o\).
Geodesics in a \(\operatorname{CAT}(0)\) space are unique. Comparing two
such radial points with their counterparts in the Euclidean comparison
triangle, and then using nonexpansion of Euclidean radial projection onto
a closed ball, gives
\[
d(P(x),P(x'))\le d(x,x').
\]
One can also identify \(P(x)\) as the nearest point of \(Y\): the triangle
inequality bounds the distance from \(x\) to \(Y\) below by
\(\max\{0,d(o,x)-R\}\), and the radial point attains this bound.
Lipschitz pushforward therefore gives an integral cycle \(P_\#T\) in
\(Y\) with \(\M(P_\#T)\le\M(T)\). If
\(\iota:Y\hookrightarrow X\) denotes inclusion, locality gives
\(\iota_\#P_\#T=T\), because \(\iota\circ P\) is the identity on a
neighborhood of \(\spt T\). Apply the compact-space inequality to
\(P_\#T\) and include its filling into \(X\). Its mass does not increase,
its boundary is \(T\), and its support is contained in the compact set
\(Y\).
\end{proof}
We work henceforth in a fixed compact \(\operatorname{CAT}(0)\) space
\(Y\). The elementary comparison facts just used, and the midpoint and
geodesic contraction inequalities used below, follow from the defining
triangle comparison; see also \cite[Chapter~II.2]{BridsonHaefliger1999}.
For an integral \(k\)-cycle \(B\) in \(Y\), define
\begin{equation}
\label{eq:filling-volume-definition}
V(B)=\inf\{\M(S):S\in\I_{k+1}(Y),\ \partial S=B\}.
\end{equation}
The mass measure of a current \(C\) is denoted by \(\|C\|\), and its total
mass by \(\M(C)\). We use the Ambrosio--Kirchheim convention for the action
\(C(b,\pi_1,\ldots,\pi_k)\): the first coefficient is bounded Lipschitz,
and the differentiated scalar coordinates are Lipschitz. The mass bound
extends the first coefficient to bounded Borel functions. In particular,
if a finite Borel measure \(\eta\) satisfies
\begin{equation}
\label{eq:mass-controlling-measure}
|C(b,\pi_1,\ldots,\pi_k)|\le\int |b|\dd\eta
\end{equation}
for every bounded Lipschitz \(b\) and every tuple of \(1\)-Lipschitz
coordinates, then \(\|C\|\le\eta\). This is the minimality property in
the definition of mass.
\subsection{The current-theoretic inputs}
The following theorem records the standard foundations in the precise
form used here. The current representation, compactness, closure and
slicing assertions are theorems of Ambrosio--Kirchheim; the product and
filling assertions use Wenger's results. We include the verifications
specific to the present ambient space.
\begin{theorem}[Current background in a compact \(\operatorname{CAT}(0)\) space]
\label{thm:current-background}
Let \(Y\) be compact and \(\operatorname{CAT}(0)\), and let \(k\ge1\).
The following statements hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Lipschitz pushforward preserves integral currents, commutes with
boundary, and satisfies the Lipschitz mass bound. Restrictions by bounded
Borel coefficients have the usual mass-measure bound; for a Borel set
\(E\subset Y\),
\[
\|C\restr E\|=\|C\|\restr E.
\]
The restricted current need not be normal. If \(u:Y\to\R\)
is Lipschitz and \(C\in\I_k(Y)\), then
\(C\restr\{u>t\}\) is integral for almost every \(t\).
In particular its boundary is an integral \((k-1)\)-cycle.
\item If \(C_j\in\I_k(Y)\) and
\(\sup_j(\M(C_j)+\M(\partial C_j))<\infty\), a subsequence converges
weakly to an integral current. Boundary commutes with weak convergence,
and mass is lower semicontinuous.
\item Every \(C\in\I_k(Y)\) admits a representation by bi-Lipschitz
maps \(\phi_i:E_i\to Y\), where \(E_i\subset\R^k\) are bounded Borel
sets and the images \(Z_i=\phi_i(E_i)\) are pairwise disjoint Borel sets,
and by signed integer multiplicities \(\theta_i\in L^1(E_i)\), such that
\begin{equation}
\label{eq:chart-action}
C(b,\pi_1,\ldots,\pi_k)
=\sum_i\int_{E_i}\theta_i(z)b(\phi_i(z))
\det D\bigl((\pi_1,\ldots,\pi_k)\circ\phi_i\bigr)(z)\dd z.
\end{equation}
The chart summands have summable masses. Zero multiplicities may be
discarded. Scalar functions on a Borel chart domain are differentiated
by Lipschitz extension to \(\R^k\); their derivatives are independent
of that extension at almost every density point of the domain.
\item For \(h>0\), the whole-current product \([0,h]\times C\), with
the Euclidean product metric, is an integral \((k+1)\)-current and obeys
\begin{equation}
\label{eq:interval-product-boundary}
\partial([0,h]\times C)
=[h]\times C-[0]\times C-[0,h]\times\partial C.
\end{equation}
Writing \(b_t(x)=b(t,x)\) and \(\pi_{a,t}(x)=\pi_a(t,x)\), its action is
\begin{equation}
\label{eq:interval-product-action}
\begin{split}
&([0,h]\times C)(b,\pi_1,\ldots,\pi_{k+1})\\
&\quad=\sum_{a=1}^{k+1}(-1)^{a+1}\int_0^h
C\bigl(b_t\,\partial_t\pi_a(t,\cdot),
\pi_{1,t},\ldots,\widehat{\pi_{a,t}},\ldots,
\pi_{k+1,t}\bigr)\dd t.
\end{split}
\end{equation}
The hat indicates omission. The time derivative is interpreted almost
everywhere and as a bounded Borel first coefficient in the current action.
\item There are finite constants \(K_k,L_k\), independent of the cycle,
such that every integral \(k\)-cycle \(B\) in \(Y\) satisfies
\begin{equation}
\label{eq:coarse-fillings}
V(B)\le K_k\M(B)^{(k+1)/k},\qquad
V(B)\le L_k\M(B).
\end{equation}
Every such \(B\) has a mass-minimizing integral filling in \(Y\).
If integral \(k\)-cycles \(B_j\) have bounded masses and converge weakly
to an integral cycle \(B\), then
\begin{equation}
\label{eq:filling-continuity}
V(B_j-B)\longrightarrow0,
\qquad V(B_j)\longrightarrow V(B).
\end{equation}
\end{enumerate}
\end{theorem}
\begin{proof}
The pushforward and Borel-coefficient properties are part of the basic
current calculus of \cite{AK2000}. The exact restriction identity also
follows from the restriction mass bound and the decomposition
\(C=C\restr E+C\restr(Y\setminus E)\), using minimality of the mass
measure in both directions. The representation in (iii) is the
integer-rectifiable representation
\cite[Lemma~4.1 and Theorem~4.5]{AK2000}; subdivision makes chart domains
bounded and disjointification makes their images disjoint. The slicing
assertion is \cite[Theorems~5.6--5.7]{AK2000}. The compactness and closure
theorems \cite[Theorems~5.2 and~8.5]{AK2000} give (ii): completeness is
automatic, the bounded normal masses are assumed, and compactness of
\(Y\) supplies uniform tightness. The product statements, including the
action convention, are
\cite[Definition~3.1 and Theorem~3.2]{Wenger2007Flat}. They apply because
an integral current is normal and its support in \(Y\) is bounded.
Here is the geometric verification of the filling hypotheses. Choose a
point \(o\) in the support of a nonzero cycle \(B\), and let
\(H(t,x)\) be the point at fraction \(t\) along \([o,x]\).
Triangle comparison gives
\[
d(H(t,x),H(t,x'))\le t\,d(x,x'),\qquad
d(H(t,x),H(s,x))=|t-s|d(o,x).
\]
Thus \(H\) is jointly Lipschitz on \([0,1]\times Y\).
On the support of \(B\), its time speed is bounded by
\(D=\diam(\spt B)\), and its spatial Lipschitz constant is at most one.
In each of the \(k+1\) terms of \eqref{eq:interval-product-action}, a
\(1\)-Lipschitz scalar test after composition with \(H\) has time
derivative at most \(D\), and the remaining spatial derivatives have
Lipschitz bounds at most one. The mass bound therefore gives
\begin{equation}
\label{eq:cone-mass-bound}
\M\bigl(H_\#([0,1]\times B)\bigr)
\le (k+1)D\M(B).
\end{equation}
The boundary is \(B\): the terminal map is the identity, the initial
map is constant and hence has zero pushforward in positive dimension,
and \(\partial B=0\). The filling is supported on segments from \(o\)
to \(\spt B\), all lying within distance \(D\) of \(o\).
Consequently the space has local cone inequalities in every positive
current dimension. Taking \(D\le\diam Y\) gives the second estimate
in \eqref{eq:coarse-fillings}, for example with
\(L_k=(k+1)\diam Y\).
The complete space \(Y\) has the convex geodesic bicombing given by its
unique geodesics. Wenger's Euclidean-exponent isoperimetric theorem
\cite[Theorem~1.2 and Corollary~1.4]{Wenger2005Iso}, or its cone-inequality
form applied successively in dimension, gives the first estimate in
\eqref{eq:coarse-fillings} for every \(k\ge1\). No sharp value of
\(K_k\) is used here.
For a fixed boundary \(B\), the cone gives at least one filling. A
minimizing sequence has bounded mass and fixed boundary, so (ii) gives
an integral weak limit with that boundary. Lower semicontinuity proves
attainment. Finally, \(Y\) is complete and geodesic, hence quasiconvex,
and it has the local cone inequalities in dimensions \(1,\ldots,k\)
just verified. The cycle assertion in
\cite[Theorem~1.4]{Wenger2007Flat} applies to the weakly convergent
sequence \(B_j\): its normal masses are bounded because its boundary
masses are zero. It gives \(V(B_j-B)\to0\) directly. Finiteness and
subadditivity of filling volume imply
\[
|V(B_j)-V(B)|\le V(B_j-B),
\]
which proves the second limit.
\end{proof}
\subsection{An extremizing cycle and its normalization}
Fix a cycle dimension \(n\ge2\), and put \(q_0=(n+1)/n\).
The following variational device turns a hypothetical violation of the
sharp filling bound into a cycle with a controlled first variation.
\begin{lemma}[Positive extremizer]
\label{lem:extremizer}
Let \(c>0\). If the functional
\[
B\longmapsto V(B)-c\M(B)^{q_0}
\]
is positive on some integral \(n\)-cycle in \(Y\), it attains a positive
maximum on such cycles. A maximizing cycle \(A\), with
\(m=\M(A)>0\), satisfies
\begin{equation}
\label{eq:extremizer}
c\bigl(m^{q_0}-\M(B)^{q_0}\bigr)\le V(A-B)
\qquad\text{for every integral \(n\)-cycle }B\text{ in }Y.
\end{equation}
\end{lemma}
\begin{proof}
The linear bound in \eqref{eq:coarse-fillings} implies
\[
V(B)-c\M(B)^{q_0}\le L_n\M(B)-c\M(B)^{q_0}.
\]
The right side has finite supremum, and positivity implies
\(\M(B)<(L_n/c)^n\). A positive maximizing sequence therefore has
uniformly bounded masses and zero boundaries. By
Theorem~\ref{thm:current-background}, a subsequence converges weakly to an
integral cycle \(A\). Filling continuity and lower semicontinuity of
mass give
\[
V(A)-c\M(A)^{q_0}
\ge\limsup_j\bigl(V(B_j)-c\M(B_j)^{q_0}\bigr).
\]
Thus \(A\) attains the positive supremum, and in particular
\(\M(A)>0\). Maximality and subadditivity now give
\[
c\bigl(\M(A)^{q_0}-\M(B)^{q_0}\bigr)
\le V(A)-V(B)\le V(A-B),
\]
which is \eqref{eq:extremizer}.
\end{proof}
Suppose that the desired sharp inequality fails in dimension \(n\) in
a compact \(\operatorname{CAT}(0)\) space. Attainment in
Theorem~\ref{thm:current-background} makes the failure a strict inequality
\(V(B)>c_n\M(B)^{q_0}\) for some nonzero cycle. Choose
\(c>c_n\) below this ratio, and apply Lemma~\ref{lem:extremizer}.
We write
\(\mu=\|A\|\) and \(m=\M(A)\).
Multiplication of all distances by a positive factor \(a\) multiplies
the mass of a \(j\)-current by exactly \(a^j\). The upper bound follows
from Lipschitz pushforward under the identity map, and the reverse bound
follows by applying the same estimate to its inverse. This identifies
the integral chain groups before and after scaling and gives the same
scaling law for filling volume. Both terms of the maximizing functional
therefore scale by \(a^{n+1}\), so maximality is preserved. Choose the
factor to arrange
\begin{equation}
\label{eq:normalization}
m=((n+1)c)^{-n}<s_n,
\qquad cq_0m^{1/n}=\frac1n.
\end{equation}
The strict inequality follows from
\(c>c_n=1/((n+1)s_n^{1/n})\).
We retain the notation \(Y,A,\mu,m\) for these rescaled objects. The
space is still compact and \(\operatorname{CAT}(0)\), and the
current-theoretic inputs remain applicable.
\subsection{Metric differentiation and Euclidean chart norms}
The contradiction now rests on an extremizing cycle of mass $m<s_n$.
To compare its mass loss under radial variation with the mass swept out,
we need chart formulas with their exact Euclidean coefficients. We prove
these formulas for a general integral current $C$: first identify the
metric differential, then recover the mass by scalar current tests, and
finally use integer multiplicity to obtain a lower density.
Fix \(C\in\I_k(Y)\) and charts as in
Theorem~\ref{thm:current-background}. Kirchheim's metric differentiation
theorem \cite[Theorem~2]{Kirchheim1994}, in the two-moving-point formulation
\cite[Definition~(1.3) and Theorem~1.1]{Duda2007}, gives at almost every
chart point \(z\) a seminorm \(N_z\) such that
\begin{equation}
\label{eq:two-point-metric-differential}
d\bigl(\phi_i(z+w),\phi_i(z+w')\bigr)
=N_z(w-w')+o(|w|+|w'|)
\end{equation}
as the two domain points tend to \(z\) within \(E_i\).
To apply the Euclidean-domain statement to a Borel domain, first embed
\(Y\) isometrically into \(\ell^\infty(Y)\). More explicitly, choose
\(z_*\in E_i\) and use the coordinates
\[
z\longmapsto d(\phi_i(z),y)-d(\phi_i(z_*),y),\qquad y\in Y.
\]
Each coordinate is \(L\)-Lipschitz for \(L=\Lip(\phi_i)\) and vanishes
at \(z_*\). By McShane's scalar extension theorem, extend each separately
to \(\R^k\) with the same Lipschitz constant; see
\cite[Theorem~2.1 and Proposition~2.2]{Rieffel2006} for the scalar and
coordinatewise formulations. These extensions have absolute value at most
\(L|z-z_*|\), uniformly over the coordinates, so together they define
an \(\ell^\infty(Y)\)-valued Lipschitz map. Metric differentiation of
this extension gives \eqref{eq:two-point-metric-differential} on the
original domain. This auxiliary extension is used only for differentiation;
all curvature comparisons below concern original image points in \(Y\).
\begin{lemma}[Euclidean chart metrics]
\label{lem:euclidean-chart-metrics}
For almost every density point \(z\in E_i\), the seminorm in
\eqref{eq:two-point-metric-differential} is a Euclidean norm. Thus there
is a measurable positive definite matrix \(G_i(z)\) with
\begin{equation}
\label{eq:chart-metric}
N_z(w)^2=w^{\mathsf T}G_i(z)w,
\qquad J_i(z)=\sqrt{\det G_i(z)}.
\end{equation}
In particular,
\begin{equation}
\label{eq:centered-metric-differential}
d(\phi_i(z+w),\phi_i(z))=|w|_{G_i(z)}+o(|w|)
\quad (z+w\in E_i).
\end{equation}
\end{lemma}
\begin{proof}
At a density-one point \(z\) of \(E_i\), every fixed finite configuration
in \(\R^k\) can be approximated by points of \((E_i-z)/t\) as
\(t\downarrow0\). Indeed, a point of such a configuration that stayed
a positive distance from \((E_i-z)/t\) along a sequence of scales would
give a missing Euclidean ball of radius comparable to \(t\) at distance
\(O(t)\) from \(z\), contradicting density one. Applying this observation
to a single direction and using the bi-Lipschitz lower bound for
\(\phi_i\) shows that \(N_z(w)\ge\Lip(\phi_i^{-1})^{-1}|w|\).
The upper Lipschitz bound gives \(N_z(w)\le\Lip(\phi_i)|w|\).
Thus \(N_z\) is a norm.
For any four points \(a,b,c,d\) of a \(\operatorname{CAT}(0)\) space,
\begin{equation}
\label{eq:cat0-quadrilateral}
d(a,c)^2+d(b,d)^2
\le d(a,b)^2+d(b,c)^2+d(c,d)^2+d(d,a)^2.
\end{equation}
To verify this, let \(m\) be the midpoint of \([a,c]\). Midpoint
comparison bounds \(d(b,m)^2\) and \(d(d,m)^2\) by the corresponding
average squared distances to \(a,c\), minus \(d(a,c)^2/4\).
Combine these bounds with
\(d(b,d)^2\le2d(b,m)^2+2d(d,m)^2\) to obtain
\eqref{eq:cat0-quadrilateral}.
Approximate the four Euclidean domain points \(0,u,u+v,v\) by scaled
points of \(E_i-z\). Apply \eqref{eq:cat0-quadrilateral} to their images,
divide by \(t^2\), and use the two-point differential
\eqref{eq:two-point-metric-differential}. It follows that
\[
N_z(u+v)^2+N_z(u-v)^2\le2N_z(u)^2+2N_z(v)^2.
\]
Apply the same inequality to \((u+v)/2\) and \((u-v)/2\), and use
homogeneity, to obtain the reverse inequality. The parallelogram law
therefore holds. Polarization produces a positive definite inner product
and the matrix \(G_i(z)\) in \eqref{eq:chart-metric}. Its entries are
measurable, since they are finite linear combinations of squared metric
differentials in fixed rational directions. Setting \(w'=0\) proves
\eqref{eq:centered-metric-differential}.
\end{proof}
The same bounds give, on each chart separately,
\begin{equation}
\label{eq:chart-metric-bounds}
\Lip(\phi_i^{-1})^{-2}I\le G_i(z)\le\Lip(\phi_i)^2I,
\qquad
\Lip(\phi_i^{-1})^{-k}\le J_i(z)\le\Lip(\phi_i)^k.
\end{equation}
These constants may depend on the chart; no uniform bound across the
chart family will be needed.
For a Lipschitz scalar function \(u\) on \(Y\), define its chart covector
\(Du\) by differentiating \(u\circ\phi_i\) and transporting that covector
to \(Z_i\). Norms and inner products of these covectors use the dual
matrix \(G_i^{-1}\). Differentiating the scalar Lipschitz inequality at
density points gives
\begin{equation}
\label{eq:gradient-lipschitz}
|Du|\le\Lip(u).
\end{equation}
Indeed, the derivative in every direction is bounded by
\(\Lip(u)N_z\), which is exactly the dual-norm bound. The chart covectors
obey the linear and product rules and the usual chain rule for continuously
differentiable scalar compositions. A scalar Lipschitz function has zero
chart derivative almost everywhere on any fixed level set: differentiate
at density points of that level set. These statements initially hold
almost everywhere in each Euclidean chart. Proposition~\ref{prop:chart-mass}
below identifies the same exceptional sets as mass-null sets.
\subsection{Quadratic mass bounds and exact chart mass}
The chart differential is Euclidean. The next two statements convert
that infinitesimal geometry into mass estimates with no dimensional
loss: a quadratic upper form controls mass from above, and inverse
chart coordinates recover the matching lower bound.
\begin{lemma}[A quadratic majorant for mass]
\label{lem:quadratic-mass}
Suppose that a finite-mass \(d\)-current \(C\) has a representation by
countably many determinant integrals with Lipschitz maps
\(\psi_i:F_i\to Y\), Borel sets \(F_i\subset\R^d\), and real signed
weights \(\vartheta_i\). Suppose that, at almost every point carrying
these weights, there is a measurable nonnegative quadratic form
\(P_i(z)\) such that
\begin{equation}
\label{eq:quadratic-distance-majorant}
d(\psi_i(z+w),\psi_i(z))^2
\le P_i(z)[w,w]+o(|w|^2),\qquad z+w\in F_i.
\end{equation}
If the measure
\begin{equation}
\label{eq:quadratic-mass-majorant}
\eta=\sum_i(\psi_i)_\#
\bigl(|\vartheta_i|\sqrt{\det P_i}\dd z\bigr)
\end{equation}
is finite, then \(\|C\|\le\eta\). No normality of the separate chart
currents is required.
\end{lemma}
\begin{proof}
Fix a tuple of \(d\) scalar \(1\)-Lipschitz tests. Almost everywhere on
each domain their compositions with \(\psi_i\) are differentiable in
the sense of scalar Lipschitz extensions. If \(\ell\) is one of the
resulting derivative rows, \eqref{eq:quadratic-distance-majorant} gives
\[
|\ell(w)|\le\sqrt{P_i(z)[w,w]}\qquad(w\in\R^d).
\]
To see the bound in an arbitrary direction, approximate that direction
by rescaled domain points at a density-one point, and then use
differentiability and \eqref{eq:quadratic-distance-majorant}.
If \(P_i\) is positive definite, transform it to the identity and apply
Hadamard's inequality to the rows. Their absolute determinant is at most
\(\sqrt{\det P_i}\). If \(P_i\) is singular, all rows vanish on its
kernel; the full determinant and \(\det P_i\) both vanish, giving the
same bound. Substitution in the representation yields
\eqref{eq:mass-controlling-measure} with the measure \(\eta\).
The exceptional set is allowed to depend on the fixed test tuple. The
measure \(\eta\) is independent of that tuple and controls every tuple
separately; this is precisely what the definition of mass requires.
\end{proof}
\begin{proposition}[Exact chart mass]
\label{prop:chart-mass}
For the representation \eqref{eq:chart-action}, let \(G_i,J_i\) be as
in \eqref{eq:chart-metric}. Then
\begin{equation}
\label{eq:chart-mass}
\|C\|=\sum_i(\phi_i)_\#\bigl(|\theta_i|J_i\dd z\bigr).
\end{equation}
Thus, with \(\rho_i=|\theta_i|J_i\), integration against the current
mass is integration against \(\rho_i\dd z\) on the disjoint charts.
\end{proposition}
\begin{proof}
Denote the right side of \eqref{eq:chart-mass} by \(\eta\), initially
allowing \(\eta(Y)=\infty\). We first prove \(\eta\le\|C\|\); this
will also establish finiteness before we apply the preceding lemma.
Fix \(0<\varepsilon<1\). Each chart domain, up to a Lebesgue-null set,
has a countable Borel partition into pieces \(E\) on which there is a
fixed positive definite matrix \(Q\), depending on the piece, satisfying
\begin{equation}
\label{eq:almost-isometric-chart-piece}
(1-\varepsilon)|z-z'|_Q
\le d(\phi_i(z),\phi_i(z'))
\le(1+\varepsilon)|z-z'|_Q,
\qquad z,z'\in E.
\end{equation}
Here is a construction. Approximate the measurable matrix \(G_i(z)\)
by an element of a countable dense family of positive definite matrices,
with a sufficiently small relative error. The centered estimate
\eqref{eq:centered-metric-differential} then gives a radius \(1/l\),
depending on the point, on which the two inequalities with this fixed
matrix and the prescribed \(\varepsilon\) hold against every other
point of the original chart domain. Group points according to the
matrix and \(l\), subdivide into sets of diameter less than \(1/l\),
and disjointify. These sets can be chosen Borel. Indeed, for any fixed
matrix and radius, the non-strict distance inequalities against all
domain points with \(0<|z-z'|<1/l\) can be tested on a fixed countable
dense subset of the domain, using continuity in \(z'\). All metric
differentiability points are covered by this countable construction.
At almost every density point of a piece \(E\), differentiation of
\eqref{eq:almost-isometric-chart-piece} in domain directions gives
\[
|w|_{G_i(z)}\le(1+\varepsilon)|w|_Q,
\qquad
J_i(z)\le(1+\varepsilon)^k\sqrt{\det Q}.
\]
On \(\phi_i(E)\), every scalar coordinate of
\(Q^{1/2}\phi_i^{-1}\) is \((1-\varepsilon)^{-1}\)-Lipschitz by the
lower inequality in \eqref{eq:almost-isometric-chart-piece}.
By McShane's theorem \cite[Theorem~2.1]{Rieffel2006}, extend these
coordinates separately to all of \(Y\), preserving that Lipschitz
constant, and call the resulting tuple \(\pi\).
For any Borel subset \(D\subset E\), choose the bounded Borel first
coefficient supported on \(\phi_i(D)\) and equal there to
\(\sgn(\theta_i)\circ\phi_i^{-1}\). At density points of \(E\), the
extended coordinates have the same derivatives as \(Q^{1/2}z\), since
they agree with them on the piece. The images of all original charts
are disjoint. Hence \eqref{eq:chart-action}, applied to this coefficient
and tuple, gives exactly
\[
\int_D|\theta_i|\sqrt{\det Q}\dd z
= C(b,\pi_1,\ldots,\pi_k)
\le(1-\varepsilon)^{-k}\|C\|(\phi_i(D)).
\]
The mass bound is legitimate for this Borel coefficient by its standard
extension from the first slot. Combining the last two estimates gives
\[
\int_D|\theta_i|J_i\dd z
\le\left(\frac{1+\varepsilon}{1-\varepsilon}\right)^k
\|C\|(\phi_i(D)).
\]
For a Borel set \(B\subset Y\), take
\(D=E\cap\phi_i^{-1}(B)\) and sum over all pieces and all charts.
Their images are disjoint, while discarded Lebesgue-null sets make no
contribution to \(\eta\). Thus
\[
\eta(B)\le
\left(\frac{1+\varepsilon}{1-\varepsilon}\right)^k\|C\|(B).
\]
This proves that \(\eta\) is finite; letting
\(\varepsilon\downarrow0\) gives \(\eta\le\|C\|\).
Conversely, \eqref{eq:centered-metric-differential} gives the quadratic
majorant \(P_i=G_i\) for every chart. Its measure
\eqref{eq:quadratic-mass-majorant} is exactly the now finite measure
\(\eta\). Lemma~\ref{lem:quadratic-mass} yields \(\|C\|\le\eta\),
which completes the proof of the equality.
\end{proof}
We have established the exact mass formula, including arbitrary signed
integer multiplicities. Its next consequence is a lower density with
the Euclidean unit-ball coefficient; this is where integrality will
enter the sharp radial contradiction quantitatively.
\begin{lemma}[Euclidean lower density]
\label{lem:euclidean-density}
For an integral \(k\)-current \(C\) in \(Y\),
\begin{equation}
\label{eq:euclidean-lower-density}
\liminf_{\rho\downarrow0}
\rho^{-k}\|C\|(B(y,\rho))\ge\omega_k
\qquad\text{for }\|C\|\text{-almost every }y.
\end{equation}
The balls in this statement are open.
\end{lemma}
\begin{proof}
By Proposition~\ref{prop:chart-mass}, it suffices to take
\(y=\phi_i(z)\), where \(z\) is a density-one point of \(E_i\), the
metric differential exists, \(z\) is a Lebesgue point of
\(\rho_i=|\theta_i|J_i\) extended by zero off \(E_i\), and
\(|\theta_i(z)|\ge1\). These conditions hold at almost every point
carrying chart mass. Fix \(\varepsilon>0\). For all sufficiently small
\(\rho\), the domain points in the ellipsoid
\[
\{z+w:|w|_{G_i(z)}<\rho/(1+\varepsilon)\}
\]
map into \(B(y,\rho)\), by
\eqref{eq:centered-metric-differential}. This ellipsoid has volume
\[
\frac{\omega_k\rho^k}{(1+\varepsilon)^kJ_i(z)}.
\]
Lebesgue differentiation on dilates of this fixed ellipsoid shows that
the contribution of this chart to \(\rho^{-k}\|C\|(B(y,\rho))\) has
lower limit at least
\(\omega_k|\theta_i(z)|/(1+\varepsilon)^k\).
Other charts contribute nonnegative mass. Letting
\(\varepsilon\downarrow0\) and using \(|\theta_i(z)|\ge1\) proves
\eqref{eq:euclidean-lower-density}.
\end{proof}
All subsequent chart covectors and their inner products are understood
with these metrics and the exact mass formula. In particular
\eqref{eq:gradient-lipschitz} holds \(\|C\|\)-almost everywhere.
For a cycle \(C\), the boundary action and the product rule give
\begin{equation}
\label{eq:cycle-product-rule}
C(b,u,\pi_1,\ldots,\pi_{k-1})
=-C(u,b,\pi_1,\ldots,\pi_{k-1})
\end{equation}
for Lipschitz scalar tests. Indeed, the sum of the two sides before
moving a term is \(C(1,bu,\pi)=\partial C(bu,\pi)=0\).
These identities concern the whole cycle. None of the preceding
arguments asserts that the separate restrictions to Borel chart images
have finite boundary mass.
The normalized cycle $A$ has all the chart formulas just proved. In the
next section we apply them to its radial deformation and to the swept
current, and compare the two masses through \eqref{eq:extremizer}.
==== END Sharp-integral-fillings-in-CAT(0)-spaces//sections/02-currents.tex ====
==== BEGIN Sharp-integral-fillings-in-CAT(0)-spaces//sections/03-radial.tex ====
\section{Radial variation and the two-dimensional base case}
\label{sec:radial}
Fix a dimension $n\geq 2$ and suppose that the sharp filling inequality
fails in a compact CAT(0) space. Let $Y$, $A$, $\mu=\|A\|$, $m$, and
$c>c_n$ be the rescaled space, cycle, mass measure, mass, and constant
provided by Lemma~\ref{lem:extremizer}. Thus
\eqref{eq:extremizer} and \eqref{eq:normalization} hold, with
$q_0=(n+1)/n$. The differential notation throughout this section is
that of Proposition~\ref{prop:chart-mass}; in particular, norms and inner
products of spatial covectors use the inverse chart metric.
\begin{proposition}[Radial first variation]
\label{prop:radial}
For every center $y\in Y$, write $r(x)=d(y,x)$. Every nonnegative
Lipschitz function $g:Y\to\R$ satisfies
\begin{equation}
\int_Y \bigl(ng+r\,Dr\cdot Dg\bigr)\dd\mu
\leq n\int_Y rg\sqrt{1-|Dr|^2}\dd\mu.
\label{eq:radial}
\end{equation}
The square root is defined $\mu$-almost everywhere by
$|Dr|\leq 1$.
\end{proposition}
\begin{proof}
Fix $y$ and $g$. Put $R=\diam Y$ and $M_g=\|g\|_\infty$, and choose
$h>0$ small enough that $hM_g\leq 1/2$. For $0\leq t\leq h$, let
\[
\lambda(t,x)=1-tg(x),\qquad
F(t,x)=[y,x]_{\lambda(t,x)},\qquad F_t(x)=F(t,x),
\]
where $[y,x]_a$ denotes the point at fraction $a$ along the geodesic
from $y$ to $x$. In particular, $F_0$ is the identity and
$1/2\leq\lambda\leq1$. Equal-fraction CAT(0) comparison and the
constant-speed parametrization of a geodesic give
\begin{align*}
d(F(t,x),F(t',x'))
&\leq d(x,x')+R\,|tg(x)-t'g(x')|\\
&\leq \bigl(1+hR\Lip(g)\bigr)d(x,x')
+RM_g|t-t'|.
\end{align*}
Thus $F$ is jointly Lipschitz. The currents
\[
B_h=(F_h)_\#A\in\I_n(Y),\qquad
D_h=F_\#([0,h]\times A)\in\I_{n+1}(Y)
\]
are integral by Theorem~\ref{thm:current-background}. Their boundaries
are $\partial B_h=0$ and $\partial D_h=B_h-A$.
\emph{Comparison forms.}
We first establish precise differential upper bounds for these maps.
For $x,x'\in Y$ and two fractions $\lambda,\lambda'\in[0,1]$, Euclidean
comparison for the triangle with vertices $y,x,x'$ gives
\begin{equation}
d([y,x]_\lambda,[y,x']_{\lambda'})^2
\leq (\lambda r-\lambda'r')^2
+\lambda\lambda'\bigl(d(x,x')^2-(r-r')^2\bigr),
\label{eq:radial-two-point}
\end{equation}
where $r=d(y,x)$ and $r'=d(y,x')$. Indeed, the expression on the
right is exactly the squared distance between the corresponding
points in the Euclidean comparison triangle. This remains valid for
degenerate triangles.
Take the disjoint charts $\phi_i:E_i\to Y$ from
Proposition~\ref{prop:chart-mass}, with metric matrices $G_i$,
$J_i=\sqrt{\det G_i}$, and signed multiplicities $\theta_i$.
At almost every density point $z\in E_i$, the metric differential
\eqref{eq:chart-metric} and the derivatives of $r\circ\phi_i$ and
$g\circ\phi_i$ exist. At such a point write
\[
\alpha=D(r\circ\phi_i)(z),\qquad
\gamma=D(g\circ\phi_i)(z),
\]
and abbreviate $r=r(\phi_i(z))$, $g=g(\phi_i(z))$, and
$\lambda=1-tg$. Thus $\alpha$ and $\gamma$ represent $Dr$ and $Dg$
in this chart. Equation~\eqref{eq:gradient-lipschitz} yields
$|\alpha|_{G_i^{-1}}\leq1$ and
$|\gamma|_{G_i^{-1}}\leq\Lip(g)$.
Set $F_i(t,z)=F(t,\phi_i(z))$. The quadratic forms
\begin{align}
P_{\mathrm{tot}}
&=\lambda^2(G_i-\alpha\otimes\alpha)
+(\lambda\alpha-tr\gamma-rg\,dt)
\otimes(\lambda\alpha-tr\gamma-rg\,dt),
\label{eq:radial-total-form}\\
P_{\mathrm{sp}}
&=\lambda^2(G_i-\alpha\otimes\alpha)
+(\lambda\alpha-tr\gamma)\otimes(\lambda\alpha-tr\gamma)
\label{eq:radial-spatial-form}
\end{align}
are nonnegative. In the first expression, the first summand acts
only on spatial vectors. For increments $(s,w)$ for which
$(t+s,z+w)\in[0,h]\times E_i$, they satisfy
\begin{equation}
d(F_i(t+s,z+w),F_i(t,z))^2
\leq P_{\mathrm{tot}}((s,w),(s,w))
+o(|s|^2+|w|^2).
\label{eq:radial-differential-upper}
\end{equation}
To see this, the derivative of $(1-tg)r$ is
$\lambda\alpha-tr\gamma-rg\,dt$, while
\[
d(\phi_i(z+w),\phi_i(z))^2
-(r(\phi_i(z+w))-r(\phi_i(z)))^2
=(G_i-\alpha\otimes\alpha)(w,w)+o(|w|^2).
\]
Substitution into \eqref{eq:radial-two-point} proves
\eqref{eq:radial-differential-upper}; changing the factor
$\lambda\lambda'$ to $\lambda^2$ contributes only a higher-order
error. Taking $s=0$ gives the corresponding upper bound with
$P_{\mathrm{sp}}$ for the fixed-time map. These statements are
needed separately for each fixed $y$ and $g$; no common exceptional
set for all centers or all functions is required.
\emph{Determinants of the upper forms.}
We next compute the determinants. At the point under consideration
choose spatial coordinates orthonormal for $G_i$. In these
coordinates \eqref{eq:radial-spatial-form} becomes
\begin{align*}
P_{\mathrm{sp}}
&=\lambda^2 I-\lambda tr(\alpha\gamma^\top+
\gamma\alpha^\top)
+t^2r^2\gamma\gamma^\top\\
&=I-t\bigl(2gI+r(\alpha\gamma^\top+
\gamma\alpha^\top)\bigr)+O(t^2).
\end{align*}
The derivative at the identity of $H\mapsto\sqrt{\det H}$ is
$H'\mapsto\tfrac12\operatorname{tr}H'$. Returning to the original
chart coordinates therefore gives
\begin{equation}
\frac{\sqrt{\det P_{\mathrm{sp}}}}{J_i}
=1-t\bigl(ng+r\,Dr\cdot Dg\bigr)+O(t^2).
\label{eq:radial-spatial-jacobian}
\end{equation}
After decreasing $h$ if necessary, the remainder is bounded in
absolute value by $Ct^2$, where $C$ depends only on
$n,R,M_g,$ and $\Lip(g)$. In particular it is independent of the
chart and the base point: all the matrix entries in the orthonormal
coordinates are controlled by these bounds.
For the full form, put $H=\lambda^2(I-\alpha\alpha^\top)$ in the
same coordinates and $u=\lambda\alpha-tr\gamma$. Its matrix in
time-first coordinates is
\[
\begin{pmatrix}0&0\\0&H\end{pmatrix}
+\begin{pmatrix}-rg\\u\end{pmatrix}
\begin{pmatrix}-rg&u^\top\end{pmatrix}.
\]
Its determinant is $(rg)^2\det H$. For positive definite $H$ this
follows directly by subtracting suitable multiples of the first
row and column, or by a block determinant calculation; replacing
$H$ by $H+\varepsilon I$ and letting $\varepsilon\downarrow0$
proves the identity for every nonnegative $H$. Since
$\det(I-\alpha\alpha^\top)=1-|\alpha|^2$, we obtain the exact
identity
\begin{equation}
\frac{\sqrt{\det P_{\mathrm{tot}}}}{J_i}
=rg\lambda^n\sqrt{1-|Dr|^2}.
\label{eq:radial-total-jacobian}
\end{equation}
Here $r,g,\lambda$ are nonnegative. In particular, the formula
also covers $rg=0$ and $|Dr|=1$.
Figure~\ref{fig:radial-variation} separates the fixed-time current
from the swept current and illustrates the unequal-fraction comparison.
The forms above are upper bounds for the deformed metric; the exact
identity is the determinant calculation for the comparison form.
\input{figures/radial-variation}
\emph{From whole-current products to mass.}
The spatial determinant will bound the endpoint cycle mass; the full
determinant will bound the swept filling mass. We now pass from the
comparison forms to these two current bounds. The chart formula
\eqref{eq:chart-action} gives, for a bounded Lipschitz coefficient $b$
and Lipschitz functions
$\pi_1,\ldots,\pi_n$ on $Y$,
\begin{equation}
\begin{split}
B_h(b,\pi_1,\ldots,\pi_n)
=\sum_i\int_{E_i}&\theta_i(z)b(F_i(h,z))\\[-2pt]
&{}\cdot\det D_z(\pi_1\circ F_i(h,\cdot),\ldots,
\pi_n\circ F_i(h,\cdot))(z)\dd z.
\end{split}
\label{eq:radial-pushforward-action}
\end{equation}
For the homotopy current, the corresponding formula is
\begin{equation}
\begin{split}
D_h(b,\pi_1,\ldots,\pi_{n+1})
=\sum_i\int_0^h\int_{E_i}
&\theta_i(z)b(F_i(t,z))\\[-2pt]
&{}\cdot\det D_{(t,z)}
(\pi_1\circ F_i,\ldots,\pi_{n+1}\circ F_i)(t,z)
\dd z\dd t.
\end{split}
\label{eq:radial-product-action}
\end{equation}
To obtain this formula, apply the interval product to the whole
current $A$. Write $\psi_a(t,x)=\pi_a(F(t,x))$ and
$\widetilde b(t,x)=b(F(t,x))$. Inserting these functions into
\eqref{eq:interval-product-action} expresses the left side as
\[
\int_0^h\sum_{a=1}^{n+1}(-1)^{a+1}
A\bigl(\widetilde b_t\,\partial_t\psi_a(t,\cdot),
\psi_1(t,\cdot),\ldots,\widehat{\psi_a(t,\cdot)},\ldots,
\psi_{n+1}(t,\cdot)\bigr)\dd t.
\]
The time derivatives admit bounded Borel versions and exist for
$dt\,d\mu$-almost every $(t,x)$. For a fixed test tuple they have
uniform bounds, as do the spatial chart-covector norms of the
$\psi_a(t,\cdot)$. Inserting \eqref{eq:chart-action}, every
cofactor in the resulting expansion is bounded by a constant
times $J_i$, by Euclidean determinant bounds. The sum of the
absolute integrals is consequently bounded by
$C h\sum_i\int_{E_i}|\theta_i|J_i\dd z=C h m$.
Fubini's theorem therefore permits both the insertion and the
interchange of the sum and integrals. Expansion in the time
column gives exactly \eqref{eq:radial-product-action}.
The separate derivatives agree almost everywhere with the joint
derivatives of the Lipschitz scalar compositions on
$[0,h]\times E_i$. This proves the formula without assigning a
boundary to any individual chart piece.
The two action formulas are determinant representations of the
finite-mass currents $B_h$ and $D_h$. Their maps have the respective
quadratic majorants $P_{\mathrm{sp}}(h,z)$ and
$P_{\mathrm{tot}}(t,z)$ from
\eqref{eq:radial-differential-upper}. Consider the measures obtained
by pushing forward
\[
|\theta_i(z)|\sqrt{\det P_{\mathrm{sp}}(h,z)}\dd z
\quad\hbox{and}\quad
|\theta_i(z)|\sqrt{\det P_{\mathrm{tot}}(t,z)}\dd t\dd z
\]
under $F_i(h,\cdot)$ and $F_i$, respectively, and summing over $i$.
They are finite: \eqref{eq:radial-spatial-jacobian} bounds the
spatial density by a uniform multiple of $|\theta_i|J_i$, while
\eqref{eq:radial-total-jacobian} bounds the product density by
$RM_g|\theta_i|J_i$. Their integrals are therefore controlled by
$m$ and $h m$, using \eqref{eq:chart-mass}.
Lemma~\ref{lem:quadratic-mass} now bounds the two current mass
measures by these measures. That lemma includes singular quadratic
forms and permits differentiation null sets to depend on the fixed
test tuple.
Combining these bounds with \eqref{eq:chart-mass},
\eqref{eq:radial-spatial-jacobian}, and
\eqref{eq:radial-total-jacobian}, and using $\lambda^n\leq1$,
gives
\begin{align}
\M(B_h)
&\leq m-h\int_Y(ng+r\,Dr\cdot Dg)\dd\mu+O(h^2),
\label{eq:radial-spatial-mass}\\
\M(D_h)
&\leq h\int_Y rg\sqrt{1-|Dr|^2}\dd\mu.
\label{eq:radial-homotopy-mass}
\end{align}
The constant in the first remainder is finite because the
pointwise remainder was uniform and $\mu(Y)=m<\infty$.
\emph{Using maximality.}
The spatial mass decrease and the swept filling mass are now controlled
by the same radial test. We apply the extremizer inequality to compare
them. Set
\[
I_g=\int_Y(ng+r\,Dr\cdot Dg)\dd\mu,
\qquad H_g=\int_Y rg\sqrt{1-|Dr|^2}\dd\mu.
\]
Choose a constant $C$ so that
$\M(B_h)\leq U_h:=m-hI_g+Ch^2$ for all sufficiently small
$h>0$. Since $m>0$, also $U_h>0$ for such $h$. The current
$-D_h$ fills $A-B_h$, so \eqref{eq:extremizer} and the
monotonicity of $s\mapsto s^{q_0}$ give
\[
c(m^{q_0}-U_h^{q_0})
\leq c(m^{q_0}-\M(B_h)^{q_0})
\leq V(A-B_h)\leq\M(D_h)\leq hH_g.
\]
Dividing by $h$ and letting $h\downarrow0$ yields
$c q_0m^{1/n}I_g\leq H_g$. The normalization
$c q_0m^{1/n}=1/n$ in \eqref{eq:normalization} is precisely
\eqref{eq:radial}.
\end{proof}
The radial inequality already settles the first dimension of the
induction. An inverse-square cutoff isolates the Euclidean density at
one center and recovers exactly the sphere area $4\pi$.
\begin{proposition}[Sharp filling in cycle dimension two]
\label{prop:base}
Let $Y$ be a compact CAT(0) space. Every integral two-cycle
$T\in\I_2(Y)$ admits $S\in\I_3(Y)$ with
\[
\partial S=T,\qquad
\M(S)\leq c_2\M(T)^{3/2}
=\frac{\M(T)^{3/2}}{6\sqrt{\pi}}.
\]
\end{proposition}
\begin{proof}
If this inequality failed, Lemma~\ref{lem:extremizer} in dimension
$n=2$ would produce a normalized nonzero cycle $A$ as above with
\begin{equation}
m<s_2=4\pi.
\label{eq:base-small-mass}
\end{equation}
Choose a center $y$ for which Lemma~\ref{lem:euclidean-density}
gives
\begin{equation}
\liminf_{\varepsilon\downarrow0}
\varepsilon^{-2}\mu(B(y,\varepsilon))\geq\omega_2=\pi.
\label{eq:base-density}
\end{equation}
Such a center exists because the density statement holds
$\mu$-almost everywhere and $m>0$.
Let $r=d(y,\cdot)$. For $0<\varepsilon<1$ such that
$\mu\{r=\varepsilon\}=0$, the function
\[
g_\varepsilon(x)=\max\{r(x),\varepsilon\}^{-2}
\]
is a nonnegative Lipschitz function on $Y$, so it is admissible
in Proposition~\ref{prop:radial}. On $\{r<\varepsilon\}$ its
gradient is zero and
\[
2g_\varepsilon+r\,Dr\cdot Dg_\varepsilon
=2\varepsilon^{-2},\qquad
2rg_\varepsilon\sqrt{1-|Dr|^2}
\leq 2\varepsilon^{-1}.
\]
On $\{r>\varepsilon\}$ the chain rule gives
$Dg_\varepsilon=-2r^{-3}Dr$. Writing
$b=r^{-1}\sqrt{1-|Dr|^2}$ there, the same two integrands are
$2b^2$ and $2b$, respectively. The separating level has zero
$\mu$-measure by our choice of $\varepsilon$. Therefore
\eqref{eq:radial} implies
\begin{align}
(2-2\varepsilon)\varepsilon^{-2}\mu(B(y,\varepsilon))
&\leq\int_{\{r>\varepsilon\}}(2b-2b^2)\dd\mu\\
&\leq \frac{m}{2},
\label{eq:base-cutoff-estimate}
\end{align}
where the last inequality is the elementary bound
$2b-2b^2=\tfrac12-2(b-\tfrac12)^2\leq\tfrac12$.
Only countably many levels of a real function can carry positive
mass under a finite measure, so eligible $\varepsilon$ tend to
zero. Taking the lower limit of
\eqref{eq:base-cutoff-estimate} along such radii and using
\eqref{eq:base-density} yields $2\pi\leq m/2$, or $m\geq4\pi$.
This contradicts \eqref{eq:base-small-mass}. The least filling
mass is therefore at most $c_2\M(T)^{3/2}$, and attainment from
Theorem~\ref{thm:current-background} supplies the required
integral filling. The zero cycle is filled by the zero current.
\end{proof}
The induction is now initialized, with an attained integral filling in
dimension two. In the remaining sections $n>2$, and the only sharp
lower-dimensional input is the theorem in dimension $n-1$, applied to
boundaries of small restrictions of the extremizing $n$-cycle.
==== END Sharp-integral-fillings-in-CAT(0)-spaces//sections/03-radial.tex ====
==== BEGIN Sharp-integral-fillings-in-CAT(0)-spaces//sections/04-sobolev.tex ====
\section{An almost Euclidean critical Sobolev inequality}
\label{sec:sobolev}
Throughout this section, $n>2$, and the sharp filling theorem in dimension
$n-1$ is assumed. We work with the extremizing cycle $A\in\I_n(Y)$ of
Lemma~\ref{lem:extremizer}, normalized as in
\eqref{eq:normalization}. In particular, $Y$ is compact and CAT(0),
$\mu=\|A\|$, $m=\M(A)>0$, and
\[
q_0=\frac{n+1}{n},\qquad cq_0m^{1/n}=\frac1n.
\]
The covectors $Du$ and their norms are those of
Proposition~\ref{prop:chart-mass}. Set
\begin{equation}
\beta=\frac1{n-2},\qquad p=\frac{2n}{n-2},\qquad
S_*=ns_n^{2/n},\qquad
E(u)=\int_Y|Du|^2\dd\mu.
\label{eq:sobolev-notation}
\end{equation}
Unless another measure is specified, function norms in this section are
taken with respect to $\mu$.
Our objective is an almost sharp critical estimate
\[
(S_*-\epsilon)\|u\|_p^2
\le4\beta E(u)+C_\epsilon\|u\|_2^2
\]
uniformly over Lipschitz $u$. We first use the dimension $n-1$ filling
theorem to control the boundary mass of small restrictions of $A$.
Coarea and rearrangement transfer this estimate to functions with small
support; a level truncation then gives the displayed bound for every $u$.
\subsection{Small sets and intrinsic coarea}
\begin{proposition}[Uniform small-set isoperimetry]
\label{prop:small-set}
For every $\eta\in(0,1)$ there is a number $\delta\in(0,m/2)$, independent
of the Lipschitz function $u\colon Y\to\R$, with the following property.
Put
\[
a(t)=\mu\{u>t\},\qquad
U_t=A\restr\{u>t\},\qquad P(t)=\M(\partial U_t),
\]
where $U_t$ and its boundary are integral for almost every $t$.
At almost every level for which $a(t)\le\delta$,
\begin{equation}
P(t)\ge(1-\eta)n\omega_n^{1/n}a(t)^{(n-1)/n}.
\label{eq:small-set}
\end{equation}
\end{proposition}
\begin{proof}
Fix a level at which $U_t$ is integral, and write $a=a(t)$ and $P=P(t)$.
Its boundary is an integral $(n-1)$-cycle. By the induction hypothesis
there is an integral $n$-current $R$ in $Y$ satisfying
\[
\partial R=\partial U_t,\qquad
d:=\M(R)\le
\left(\frac{P}{n\omega_n^{1/n}}\right)^{n/(n-1)}.
\]
Indeed, the coefficient on the right is the sharp filling coefficient
in dimension $n-1$. The currents $B=A-U_t+R$ and $A-B=U_t-R$ are
integral cycles. Restriction decomposes the mass measure exactly, so
\[
\M(B)\le m-a+d,\qquad \M(A-B)\le a+d.
\]
No disjointness involving $R$ is needed for these upper bounds.
Suppose first that $d<a\le m/2$. The extremizer inequality
\eqref{eq:extremizer} and the coarse filling estimate in
Theorem~\ref{thm:current-background} give
\begin{align*}
cq_0(m/2)^{1/n}(a-d)
&\le c\bigl(m^{q_0}-(m-a+d)^{q_0}\bigr)\\
&\le V(A-B)
\le K_n(a+d)^{q_0}
\le K_n(2a)^{q_0}.
\end{align*}
The first inequality follows by integrating the derivative of
$s^{q_0}$ over $[m-a+d,m]\subset[m/2,m]$. Thus
\begin{equation}
d\ge a-Ca^{1+1/n},\qquad
C=\frac{K_n2^{q_0}}{cq_0(m/2)^{1/n}}
=nK_n2^{1+2/n}.
\label{eq:small-set-replacement}
\end{equation}
If $d\ge a$, the same lower bound holds automatically. Choose
$0<\delta<m/2$ so small that
\[
C\delta^{1/n}\le1-(1-\eta)^{n/(n-1)}.
\]
For $0<a\le\delta$, combining \eqref{eq:small-set-replacement} with the
upper bound for $d$ yields
\[
P\ge n\omega_n^{1/n}a^{(n-1)/n}
(1-Ca^{1/n})^{(n-1)/n}
\ge (1-\eta)n\omega_n^{1/n}a^{(n-1)/n}.
\]
The case $a=0$ is immediate. All constants used to choose $\delta$
depend only on the fixed extremizer, the dimension, and $\eta$.
\end{proof}
We now have the Euclidean perimeter coefficient, up to an arbitrarily
small loss, for every sufficiently small superlevel restriction. To turn
this into an energy inequality, coarea must retain the chart derivative
$|Du|$. Its uniform upper bound $\Lip(u)$ would lose the required
Dirichlet energy. The next lemma recovers each slice's total mass from
countably many scalar evaluations, so that the integrated cycle identity
can be used at almost every level.
\begin{lemma}[A countable family of mass tests]
\label{lem:countable-mass-tests}
Let $Z$ be a nonempty compact metric space and let $k\ge1$. There is a
countable collection $\mathcal T_k(Z)$ of finite lists
\[
\bigl((b_1,\pi^1),\ldots,(b_N,\pi^N)\bigr),
\]
where the $b_j$ are Lipschitz, $\sum_j|b_j|\le1$ pointwise, and each
$\pi^j$ is a $k$-tuple of $1$-Lipschitz functions, such that every
$k$-current $C$ of finite mass on $Z$ satisfies
\begin{equation}
\M(C)=\sup_{\mathcal T\in\mathcal T_k(Z)}
\sum_{(b,\pi)\in\mathcal T}C(b,\pi).
\label{eq:countable-mass-tests}
\end{equation}
The collection is independent of $C$.
\end{lemma}
\begin{proof}
Fix a point $z_0\in Z$ and normalize each differentiated test by
$\pi_i(z_0)=0$; adding constants does not change its current evaluation.
The normalized $1$-Lipschitz functions form a compact metric space in
the uniform topology, by the Arzel\`a--Ascoli theorem. Choose a
countable dense family of normalized $k$-tuples
$(\pi^j)_{j\ge1}$.
Write $\lambda=\|C\|$. For each tuple $\pi$, the first-slot extension
to bounded Borel functions gives a finite signed measure
$b\mapsto C(b,\pi)$ with density $\rho_\pi$ relative to $\lambda$,
where $|\rho_\pi|\le1$ almost everywhere. Choose measurable versions
for the countable family and put
\[
\rho(z)=\sup_{j\ge1}|\rho_{\pi^j}(z)|.
\]
This satisfies $0\le\rho\le1$. If a normalized tuple $\pi$ is
approximated uniformly by tuples in the family, current continuity,
with their common Lipschitz bound, shows for every Lipschitz $b$ that
\[
|C(b,\pi)|\le\int_Z|b|\rho\dd\lambda.
\]
Thus $\rho\lambda$ is a mass-controlling measure for all normalized
$1$-Lipschitz tuples and hence, by rescaling the differentiated slots,
for all tests. Minimality of the mass measure gives
$\lambda\le\rho\lambda$. It follows that $\rho=1$ almost everywhere.
For any finite set of the tuples, select at each point the first index
where the largest $|\rho_{\pi^j}|$ is attained and select its sign.
This produces bounded Borel functions $b_j$ with
$\sum_j|b_j|\le1$ and
\[
\sum_j C(b_j,\pi^j)
=\int_Z\max_j|\rho_{\pi^j}|\dd\lambda.
\]
These coefficients can be replaced, with an arbitrarily small error
in the displayed sum, by Lipschitz coefficients satisfying the same
constraint. To see this, regularity of the finite Borel measure
$\lambda$ gives Lipschitz approximations to each $b_j$ in
$L^1(\lambda)$. For example, an indicator is approximated between a
compact set and an open neighborhood by a Lipschitz function obtained
from distances to those sets, and simple functions suffice for the
general case. Apply to the vector of approximations the Lipschitz map
\[
(v_1,\ldots,v_N)\longmapsto
\frac{(v_1,\ldots,v_N)}{\max\{1,\sum_j|v_j|\}}.
\]
This retains $L^1$ convergence and enforces the constraint.
Monotone convergence of the finite maxima to $\rho=1$ now proves that
the supremum over \emph{all} admissible finite Lipschitz lists equals
$\M(C)$; the reverse inequality follows directly from the mass bound.
It remains to make the lists countable without depending on $C$.
For each fixed list length, the admissible lists of coefficients and
normalized tuples form a separable metric space in the uniform
topology: they are a subspace of a finite product of $C(Z)$, which is
separable because $Z$ is compact metric. Choose a countable dense
subset within that admissible space and take the union over list
lengths. An admissible list can be approximated by these lists with
its constraint and differentiated Lipschitz bounds preserved.
The first-slot errors are controlled by their uniform norms times
$\M(C)$, and convergence in the differentiated slots follows from
current continuity. Hence the countable collection has the same
supremum. A common Lipschitz bound on the first coefficients is
neither asserted nor needed.
\end{proof}
\begin{lemma}[Intrinsic coarea inequalities]
\label{lem:intrinsic-coarea}
For a real Lipschitz function $u$ on $Y$, put
\[
a(t)=\mu\{u>t\},\qquad
e(t)=\int_{\{u>t\}}|Du|^2\dd\mu,
\qquad P(t)=\M\bigl(\partial(A\restr\{u>t\})\bigr)
\]
at the almost every level where the restriction is integral. Then
\begin{equation}
P(t)^2\le(-a'(t))(-e'(t)),\qquad
P(t)\le\Lip(u)(-a'(t))
\quad\text{for almost every }t.
\label{eq:intrinsic-coarea}
\end{equation}
The derivatives are the densities of the absolutely continuous parts
of the corresponding distribution measures; neither distribution
function is assumed absolutely continuous.
\end{lemma}
\begin{proof}
For a bounded open interval $I$, define the bounded Lipschitz function
\[
H(s)=\int_I1_{\{t<s\}}\dd t.
\]
Let $b$ be Lipschitz and let $\pi$ be an $(n-1)$-tuple of
$1$-Lipschitz functions. Restriction, the chart representation, and
Fubini give
\begin{equation}
\int_I\partial U_t(b,\pi)\dd t
=A(H(u),b,\pi)=-A(b,H(u),\pi).
\label{eq:integrated-slice}
\end{equation}
For the first equality, the boundary evaluation has the measurable
representative $U_t(1,b,\pi)$, obtained by integrating the chart
determinants with the factor $1_{\{u>t\}}$. It is integrable on bounded
intervals. The second equality is the product rule applied to the
cycle identity $\partial A=0$.
The chain rule in the charts gives
\[
D(H(u))=1_{\{u\in I\}}Du\qquad\mu\text{-almost everywhere}.
\]
There is no ambiguity at the two endpoints of $I$: the differential
of a Lipschitz scalar function vanishes almost everywhere on each
fixed level set. This follows by applying Euclidean differentiation
at density points of that level set in each chart. At points of the
charts the other differentiated covectors have norm at most one, so
the Euclidean determinant bound in
Proposition~\ref{prop:chart-mass} applies.
In particular, for an admissible finite list $\mathcal T$ as in
Lemma~\ref{lem:countable-mass-tests}, \eqref{eq:integrated-slice} yields
\begin{equation}
\left|\int_I\sum_{(b,\pi)\in\mathcal T}
\partial U_t(b,\pi)\dd t\right|
\le\int_{\{u\in I\}}|Du|\dd\mu.
\label{eq:integrated-mass-list}
\end{equation}
Let $\gamma=u_\#(|Du|\mu)$. Lebesgue differentiation of
\eqref{eq:integrated-mass-list}, simultaneously for the countable
collection of lists on $Y$, gives
\[
P(t)\le\frac{d\gamma_{\mathrm{ac}}}{dt}(t)
\qquad\text{for almost every }t.
\]
Here we used \eqref{eq:countable-mass-tests} for each integral slice.
Thus the passage from a signed integrated evaluation to total slice
mass excludes only a countable union of null sets.
Set $\lambda=u_\#\mu$ and $\zeta=u_\#(|Du|^2\mu)$. For every interval
$I$, Cauchy--Schwarz on $u^{-1}(I)$ and the bound
$|Du|\le\Lip(u)$ imply
\[
\gamma(I)^2\le\lambda(I)\zeta(I),\qquad
\gamma(I)\le\Lip(u)\lambda(I).
\]
Divide by the appropriate powers of the interval length and let
symmetric intervals shrink to an almost every point. The densities
of $\lambda_{\mathrm{ac}}$ and $\zeta_{\mathrm{ac}}$ are respectively
$-a'$ and $-e'$. Combining the resulting inequalities with the bound
for $P$ proves \eqref{eq:intrinsic-coarea}.
\end{proof}
\subsection{Rearrangement with an arbitrary distribution function}
The small-set perimeter bound and intrinsic coarea now have their
Euclidean coefficients up to the prescribed loss. We construct a radial
Euclidean function with the same distribution of values and controlled
energy. The distribution may have atoms or a singular continuous part;
the generalized inverse below includes both.
\begin{lemma}[Rearranged energy]
\label{lem:rearranged-energy}
Fix $\eta\in(0,1)$ and the corresponding $\delta$ from
Proposition~\ref{prop:small-set}. If $u\ge0$ is Lipschitz and
$\mu\{u>0\}\le\delta$, there is a nonnegative, radial nonincreasing,
compactly supported Lipschitz function $u^*$ on $\R^n$ such that
\begin{equation}
\int_{\R^n}(u^*)^q\dd x=\int_Yu^q\dd\mu\quad(q>0),
\qquad
\int_{\R^n}|\nabla u^*|^2\dd x
\le(1-\eta)^{-2}E(u).
\label{eq:rearranged-energy-bound}
\end{equation}
\end{lemma}
\begin{proof}
If $u=0$ almost everywhere, take $u^*=0$. Otherwise let
$L=\Lip(u)$ and $U=\esssup_\mu u$. Then $U>0$ and $L>0$: a function
with Lipschitz constant zero is constant, and a positive constant
would have positivity set of mass $m>\delta$.
For $0\le t\le U$, define
\[
a(t)=\mu\{u>t\},\qquad
R(t)=\left(\frac{a(t)}{\omega_n}\right)^{1/n}.
\]
The function $R$ is nonincreasing and right-continuous,
$R(t)>0$ for $t<U$, and $R(U)=0$.
For almost every $t\in(0,U)$, Proposition~\ref{prop:small-set} and
Lemma~\ref{lem:intrinsic-coarea} give
\begin{equation}
-R'(t)=\frac{-a'(t)}{n\omega_n R(t)^{n-1}}
\ge c_0,\qquad c_0=\frac{1-\eta}{L}>0.
\label{eq:radius-distribution-slope}
\end{equation}
This estimate also controls finite differences, even if $R$ has a
singular part. Indeed, on every compact subinterval of $(0,U)$,
the nonnegative measure $-dR$ has absolutely continuous density
$-R'\ge c_0$ and a nonnegative singular part. Therefore
\begin{equation}
R(s)-R(t)\ge c_0(t-s)
\qquad(0<s<t<U).
\label{eq:radius-distribution-difference}
\end{equation}
Define the generalized inverse for $\rho\ge0$ by
\begin{equation}
h(\rho)=\sup\{0\le t<U:\rho<R(t)\},
\qquad \sup\varnothing:=0.
\label{eq:generalized-inverse}
\end{equation}
It is nonincreasing, $h(0)=U$, and $h(\rho)=0$ for
$\rho\ge R(0)$. It is also $c_0^{-1}$-Lipschitz. To check the latter
claim, let $\rho_1<\rho_2$ with $h(\rho_1)>h(\rho_2)$ and choose
\[
h(\rho_2)<s<t<h(\rho_1).
\]
The definition and monotonicity imply
$R(s)\le\rho_2$ and $R(t)>\rho_1$, whence
$c_0(t-s)\le R(s)-R(t)\le\rho_2-\rho_1$.
Letting $s$ and $t$ approach the two inverse values proves the claim.
For $0<t<U$, right continuity of $R$ implies
\begin{equation}
h(\rho)>t\quad\Longleftrightarrow\quad \rho<R(t).
\label{eq:inverse-superlevels}
\end{equation}
In the reverse implication, if $\rho<R(t)$, right continuity supplies
some $s>t$ with $\rho<R(s)$; the other implication follows from
monotonicity. Consequently $u^*(x)=h(|x|)$ is Lipschitz and compactly
supported, and its superlevel sets have Euclidean volume $a(t)$.
The layer-cake formula proves the asserted equality of all positive
power integrals.
We give the exact energy substitution, including possible atoms and
singular continuous parts of $a$. Strict decrease in
\eqref{eq:radius-distribution-difference} implies
\begin{equation}
h(R(t))=t\qquad(0<t<U).
\label{eq:inverse-identity}
\end{equation}
Indeed, all levels $s<t$ satisfy $R(s)>R(t)$, while no $s\ge t$ is
eligible in \eqref{eq:generalized-inverse} with $\rho=R(t)$.
If $R$ is continuous at $t$, \eqref{eq:inverse-superlevels} and this
continuity show that the entire fiber $h^{-1}(\{t\})$ is the singleton
$\{R(t)\}$.
The monotone function $R$ is continuous except at countably many
points and differentiable with finite derivative almost everywhere.
Its derivative, wherever it is relevant to
\eqref{eq:radius-distribution-slope}, is nonzero.
Let $N$ be the null set where the Lipschitz function $h$ is not
differentiable. The set $h(N)$ is null because Lipschitz functions
map Lebesgue null sets to null sets. By \eqref{eq:inverse-identity},
every level $t$ with $R(t)\in N$ lies in $h(N)$. It follows that for
almost every $t\in(0,U)$ both derivatives needed below exist, and
differentiating the inverse identity gives
\begin{equation}
h'(R(t))=\frac1{R'(t)}.
\label{eq:inverse-derivative}
\end{equation}
For example, this follows directly by letting $s\to t$ in
$h(R(s))-h(R(t))=s-t$ at a continuity and differentiability point
of $R$.
Since $h$ is absolutely continuous and nonincreasing on $[0,R(0)]$,
the measure $|h'(\rho)|\dd\rho$ pushes forward under $h$ to Lebesgue
measure on $(0,U)$. One may verify this first on an interval
$(a,b)\subset(0,U)$: integration of $-h'$ over its inverse image
equals $b-a$ by the fundamental theorem of calculus; constant
portions have zero derivative. Intervals determine the measure.
At almost every target level the fiber consists of the single point
$R(t)$, so this substitution and \eqref{eq:inverse-derivative} give
\begin{align}
\int_{\R^n}|\nabla u^*|^2\dd x
&=n\omega_n\int_0^{R(0)}\rho^{n-1}|h'(\rho)|^2\dd\rho \notag\\
&=\int_0^U\frac{n\omega_n R(t)^{n-1}}{-R'(t)}\dd t \notag\\
&=\int_0^U
\frac{\bigl(n\omega_n R(t)^{n-1}\bigr)^2}{-a'(t)}\dd t.
\label{eq:rearranged-energy-identity}
\end{align}
The denominators are positive and finite almost everywhere by
\eqref{eq:radius-distribution-slope}. The preceding substitution
also justifies the equality for a nonnegative integrand before its
finiteness is known.
For clarity, a jump of $R$ produces an interval on which $h$ is
constant. In particular, an atom at $U$ produces a central constant
part of $u^*$. Such intervals contribute no energy. A singular
continuous decrease causes no additional term either: the exact
substitution uses the absolutely continuous measure
$|h'|\dd\rho$, and the inverse image of any null set of levels has
zero measure for this measure. Thus no absolute-continuity
assumption on $a$ has entered
\eqref{eq:rearranged-energy-identity}.
Finally, \eqref{eq:small-set} gives
\[
\bigl(n\omega_n R(t)^{n-1}\bigr)^2
\le(1-\eta)^{-2}P(t)^2.
\]
Using \eqref{eq:intrinsic-coarea} in
\eqref{eq:rearranged-energy-identity} therefore yields
\[
\int_{\R^n}|\nabla u^*|^2\dd x
\le(1-\eta)^{-2}\int_0^U(-e'(t))\dd t
\le(1-\eta)^{-2}E(u).
\]
The last inequality only uses monotonicity of $e$ and remains true
if its distributional derivative has a singular part.
\end{proof}
\subsection{The sharp radial Euclidean inequality}
Rearrangement reduces the small-support estimate to a radial function
on $\R^n$. The next lemma identifies its sharp energy coefficient;
afterwards only the truncation to arbitrary functions remains.
This is the radial case of the sharp Sobolev inequality of Aubin and
Talenti~\cite{Aubin1976,Talenti1976}. Its proof is an explicit radial
version of the mass-transport argument of Cordero-Erausquin, Nazaret and
Villani~\cite[Section~2]{CorderoNazaretVillani2004}: cumulative masses
determine the transport, and determinant comparison followed by
integration by parts recovers the sharp coefficient.
\begin{lemma}[Sharp radial Sobolev inequality]
\label{lem:radial-sobolev}
For every nonnegative, radial nonincreasing, compactly supported
Lipschitz function $f$ on $\R^n$, where $n>2$,
\begin{equation}
ns_n^{2/n}
\left(\int_{\R^n}f^p\dd x\right)^{2/p}
\le \frac4{n-2}\int_{\R^n}|\nabla f|^2\dd x.
\label{eq:radial-sobolev}
\end{equation}
\end{lemma}
\begin{proof}
The assertion is immediate for $f=0$. By homogeneity, assume
$\int f^p\dd x=1$. Write $f=f(r)$, $r=|x|$. Continuity and radial
monotonicity give a finite $r_0>0$ such that $f>0$ on $[0,r_0)$ and
$f=0$ on $[r_0,\infty)$.
For $D>0$ define
\[
g_D(r)=A_D(1+r^2)^{-(n-2)/2}\quad(0\le r<D),
\qquad g_D(r)=0\quad(r\ge D),
\]
where $A_D>0$ is chosen so that $\int g_D^p\dd x=1$.
For $0<r<r_0$ match the cumulative radial masses by
\begin{equation}
n\omega_n\int_0^r s^{n-1}f(s)^p\dd s
=n\omega_n\int_0^{t(r)}s^{n-1}g_D(s)^p\dd s.
\label{eq:radial-transport-cdf}
\end{equation}
Both cumulative functions are $C^1$ with strictly positive derivative
on their open radial intervals. Their inverse composition $t$ is
therefore $C^1$ on $(0,r_0)$, strictly increasing, and satisfies
$t(0+)=0$ and $t(r_0-)=D$. Differentiation gives
\begin{equation}
d_T:=t'(r)\left(\frac{t(r)}r\right)^{n-1}
=\frac{f(r)^p}{g_D(t(r))^p}.
\label{eq:radial-transport-jacobian}
\end{equation}
The radial map $T(x)=t(|x|)x/|x|$ has positive eigenvalues
$t'$ and $n-1$ copies of $t/r$ and transports $f^p\dd x$ to
$g_D^p\dd x$. This follows either from
\eqref{eq:radial-transport-cdf} by radial integration, or from the
change of variables and \eqref{eq:radial-transport-jacobian}.
The jump of $g_D$ at $D$ is harmless: the calculation uses its
positive continuous interior profile and never differentiates the
cutoff.
Let
\[
\ell=p\left(1-\frac1n\right)=\frac{2(n-1)}{n-2},
\qquad \ell-1=\frac p2.
\]
Change of variables, the arithmetic--geometric mean inequality for
the positive eigenvalues of $DT$, and radial integration by parts
give
\begin{align}
n\int_{\R^n}g_D^\ell\dd x
&=n\int_{|x|<r_0}f^\ell d_T^{1/n}\dd x \notag\\
&\le\int_{|x|<r_0}f^\ell
\left(t'+(n-1)\frac tr\right)\dd x \notag\\
&=-\ell\int_{|x|<r_0}f^{p/2}f't\dd x \notag\\
&\le\ell\left(\int_{\R^n}|\nabla f|^2\dd x\right)^{1/2}
\left(\int_{\R^n}|x|^2g_D^p\dd x\right)^{1/2}.
\label{eq:radial-transport-estimate}
\end{align}
The last equality needed for Cauchy--Schwarz is the transport
identity $\int f^pt^2\dd x=\int|x|^2g_D^p\dd x$.
To justify the integration by parts, first integrate over
$\varepsilon<r<R<r_0$. The boundary term is
$n\omega_n[r^{n-1}tf^\ell]_{\varepsilon}^R$.
It tends to zero at $r_0$ because $f(r_0)=0$ and $t\le D$, and at zero
because $n>1$ and $t,f$ are bounded. The resulting derivative
integral is absolutely integrable: $f'$ is essentially bounded,
$t$ is bounded, and the source ball is bounded. The source profile
is Lipschitz, so its radial integration by parts is justified by
absolute continuity.
Now put
\[
I_0=\int_{\R^n}(1+|x|^2)^{-n}\dd x,
\qquad A_\infty=I_0^{-1/p},\qquad
g(r)=A_\infty(1+r^2)^{-(n-2)/2}.
\]
The normalizing constants $A_D$ converge to $A_\infty$ as
$D\to\infty$. Moreover,
\[
\int g_D^\ell\dd x\longrightarrow\int g^\ell\dd x,
\qquad
\int|x|^2g_D^p\dd x\longrightarrow\int|x|^2g^p\dd x.
\]
For both integrals the unnormalized radial integrand is of order
$r^{1-n}$ at infinity. Thus both are finite for every $n>2$, and
the convergence follows by monotone convergence for the
unnormalized integrals and convergence of $A_D$.
In particular, no limit of the transport maps is required.
The ratio on the limiting left side of
\eqref{eq:radial-transport-estimate} can be evaluated from $g$ itself.
Direct differentiation shows that
\[
-g'(r)=Krg(r)^{p/2},\qquad
K=(n-2)A_\infty^{-2/(n-2)}>0.
\]
If $J=\int g^\ell\dd x$ and $M_2=\int|x|^2g^p\dd x$, radial
integration by parts with the vector field $x$ gives
\[
nJ=-\ell\int g^{p/2}g'r\dd x=\ell K M_2,
\qquad \int|\nabla g|^2\dd x=K^2M_2.
\]
Its boundary term $r^ng(r)^\ell$ vanishes at zero and is
$O(r^{2-n})$ at infinity. Therefore
\begin{equation}
\frac{nJ}{\ell M_2^{1/2}}
=\left(\int|\nabla g|^2\dd x\right)^{1/2}.
\label{eq:bubble-transport-ratio}
\end{equation}
Taking $D\to\infty$ in \eqref{eq:radial-transport-estimate} proves
$\int|\nabla f|^2\dd x\ge\int|\nabla g|^2\dd x$.
The transport argument has reduced the lower energy bound to the
single comparison profile $g$. It remains to compute this energy and
thus fix the threshold used in the current's critical minimization.
The inverse stereographic map
\[
x\longmapsto\frac{(2x,|x|^2-1)}{1+|x|^2}\in\mathbb S^n
\]
pulls the sphere metric back to $4(1+|x|^2)^{-2}$ times the Euclidean
metric. Its volume Jacobian is $2^n(1+|x|^2)^{-n}$, so
\begin{equation}
I_0=2^{-n}s_n.
\label{eq:stereographic-integral}
\end{equation}
For $w(r)=(1+r^2)^{-(n-2)/2}$, direct differentiation gives
\[
-\Delta w=n(n-2)(1+r^2)^{-(n+2)/2}.
\]
Integration by parts consequently yields
\begin{align*}
\int_{\R^n}|\nabla g|^2\dd x
&=n(n-2)A_\infty^2I_0\\
&=n(n-2)I_0^{1-2/p}
=\frac{n(n-2)}4s_n^{2/n}.
\end{align*}
Here the boundary term $r^{n-1}g(r)g'(r)$ vanishes at zero and is
$O(r^{2-n})$ at infinity; also $1-2/p=2/n$.
Multiplying the energy lower bound by $4/(n-2)$ proves
\eqref{eq:radial-sobolev} for the normalized $f$, and homogeneity
proves the general statement. All moments and boundary limits
used above remain valid when $n=3$ or $n=4$; no $L^2$ integrability
of the full profile $g$ is needed.
The coefficient is also optimal within the class in the statement.
Indeed, choose a nonnegative nonincreasing Lipschitz cutoff $\chi$
equal to one on $[0,1]$ and zero on $[2,\infty)$, and set
$g_R(x)=g(x)\chi(|x|/R)$. Then $g_R\to g$ in $L^p$, and
$\nabla g_R\to\nabla g$ in $L^2$: the tail of $\nabla g$ tends to
zero, while the extra cutoff energy is bounded by
\[
\frac{\Lip(\chi)^2}{R^2}
\int_{R<|x|<2R}g(x)^2\dd x=O(R^{2-n})\longrightarrow0.
\]
Each $g_R$ is admissible, and its Sobolev quotient tends to the
computed value for $g$.
\end{proof}
\subsection{Globalization}
The preceding two lemmas give the sharp coefficient, with an
arbitrarily small loss, for functions supported on a small amount of
current mass. Truncation at a level controlled by $\|u\|_2$ yields the
section's asserted inequality for every Lipschitz function.
\begin{proposition}[Almost Euclidean critical Sobolev bound]
\label{prop:almost-sobolev}
For every $\epsilon\in(0,S_*)$ there is a finite constant
$C_\epsilon\ge0$ such that every real Lipschitz function $u$ on $Y$
satisfies
\begin{equation}
(S_*-\epsilon)\|u\|_p^2
\le4\beta E(u)+C_\epsilon\|u\|_2^2.
\label{eq:almost-sobolev}
\end{equation}
The constant is independent of $u$.
\end{proposition}
\begin{proof}
Fix temporarily $\eta\in(0,1)$ and its corresponding $\delta$.
Combining Lemmas~\ref{lem:rearranged-energy} and
\ref{lem:radial-sobolev} gives
\begin{equation}
S_*(1-\eta)^2\|z\|_p^2\le4\beta E(z)
\quad\text{if }z\ge0\text{ is Lipschitz and }\mu\{z>0\}\le\delta.
\label{eq:small-support-sobolev}
\end{equation}
This also holds when $z=0$ almost everywhere.
For a general Lipschitz $u$ with $\|u\|_2>0$, set
\[
\tau=\frac{\|u\|_2}{\sqrt\delta},\qquad z=(|u|-\tau)_+.
\]
Chebyshev's inequality gives $\mu\{z>0\}\le\delta$.
Scalar Lipschitz composition in the charts gives $E(z)\le E(u)$;
at the finitely many corner levels the gradient vanishes almost
everywhere on the corresponding level sets, so the usual
almost-everywhere chain rule applies. In addition,
$0\le|u|-z\le\tau$, whence
\[
\||u|-z\|_p\le m^{1/p}\tau.
\]
For every $\xi>0$, the triangle inequality followed by
$(b+d)^2\le(1+\xi)b^2+(1+\xi^{-1})d^2$ now yields
\begin{align*}
\frac{S_*(1-\eta)^2}{1+\xi}\|u\|_p^2
&\le S_*(1-\eta)^2\|z\|_p^2
+\frac{S_*(1-\eta)^2}{\xi}m^{2/p}\tau^2\\
&\le4\beta E(u)
+\frac{S_*(1-\eta)^2m^{2/p}}{\xi\delta}\|u\|_2^2.
\end{align*}
Choose $\eta>0$ and $\xi>0$ sufficiently small that
$S_*(1-\eta)^2/(1+\xi)\ge S_*-\epsilon$, and then fix the associated
$\delta$. The last display proves the assertion with, for example,
\[
C_\epsilon=\frac{S_*(1-\eta)^2m^{2/p}}{\xi\delta}.
\]
If $\|u\|_2=0$, then $\|u\|_p=0$ and the asserted inequality is
immediate.
\end{proof}
==== END Sharp-integral-fillings-in-CAT(0)-spaces//sections/04-sobolev.tex ====
==== BEGIN Sharp-integral-fillings-in-CAT(0)-spaces//sections/05-minimizer.tex ====
\section{A subthreshold minimizer on the cycle}
\label{sec:minimizer}
Throughout this section, $n>2$ and $A$ is the extremizing cycle constructed
in Lemma~\ref{lem:extremizer}, with the normalization
\eqref{eq:normalization}. Thus $\mu=\|A\|$ is a finite measure on the
compact space $Y$, and
\[
0<m:=\mu(Y)<s_n.
\]
We use the charts, multiplicities, and covector norms of
Proposition~\ref{prop:chart-mass}. In particular, the chart images $Z_i$
are pairwise disjoint, their multiplicities
$\theta_i$ are nonzero signed integers, and
\[
\mu=\sum_i(\phi_i)_\#\bigl(|\theta_i|J_i\,dz\bigr),
\qquad J_i=\sqrt{\det G_i}.
\]
Set, as in Section~\ref{sec:sobolev},
\[
\beta=\frac1{n-2},\qquad p=\frac{2n}{n-2},\qquad
S_*=ns_n^{2/n},\qquad E(u)=\int_Y|Du|^2\,d\mu.
\]
All function norms in this section are taken with respect to $\mu$.
We use the almost sharp estimate \eqref{eq:almost-sobolev}; in particular,
its constant $C_\epsilon$ is independent of the Lipschitz function to
which it is applied.
We seek a nonnegative minimizer of
\[
\frac{4\beta E(u)+n\|u\|_2^2}{\|u\|_p^2},
\]
initially defined for Lipschitz functions with $\|u\|_p>0$.
To pass to a minimizing limit, we first close the chart gradient and
prove compactness in $L^2$. The almost sharp inequality from
Section~\ref{sec:sobolev} then prevents loss of critical $L^p$ mass
below $S_*$.
\subsection{The closed gradient and calculus}
\begin{proposition}[The Sobolev space and its calculus]
\label{prop:sobolev-space}
Identify real Lipschitz functions on $Y$ that agree $\mu$-almost
everywhere. Their completion in the norm
\[
\|u\|_{\Hcal}^2=\|u\|_2^2+E(u)
\]
is a Hilbert space $\Hcal$ of functions in $L^2(\mu)$. Its gradient is
a well-defined closed operator with values in the Hilbert space of
square-integrable chart covectors. The inclusion
$\Hcal\longrightarrow L^p(\mu)$ is continuous, and
\eqref{eq:almost-sobolev} holds for all $u\in\Hcal$.
The following calculus properties hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item If $u_1,\ldots,u_k\in\Hcal$ and
$\Phi\in C^1(\R^k)$ has bounded first derivatives, then
$\Phi(u_1,\ldots,u_k)\in\Hcal$ and
\[
D\Phi(u_1,\ldots,u_k)
=\sum_{a=1}^k\partial_a\Phi(u_1,\ldots,u_k)Du_a.
\]
\item For every $u\in\Hcal$ and every fixed $t\in\R$,
$Du=0$ almost everywhere on $\{u=t\}$. Consequently, the scalar
chain rule also holds for Lipschitz, piecewise $C^1$ functions with
finitely many corners, with any derivative values assigned at the
corners.
\item The map $u\mapsto u_+$ is continuous from $\Hcal$ to itself.
Every nonnegative element of $\Hcal$ has nonnegative Lipschitz
approximants converging in $\Hcal$.
\item For each fixed center $y\in Y$, the radial inequality
\eqref{eq:radial} holds for every nonnegative $g\in\Hcal$, with
$r=d(y,\cdot)$ and its chart gradient $Dr$.
\end{enumerate}
\end{proposition}
\begin{proof}
Let $\mathcal L^2$ denote the Hilbert space of chart covector fields
$L$ with $\int|L|^2\,d\mu<\infty$. The Lipschitz level-set property
from Section~\ref{sec:currents}, applied to the difference of two
representatives, shows that their gradients agree whenever the functions
agree $\mu$-almost everywhere. We first prove that the graph of
the Lipschitz gradient in $L^2(\mu)\times\mathcal L^2$ is closable.
Suppose that $u_j$ are Lipschitz,
\[
u_j\longrightarrow0\quad\hbox{in }L^2(\mu),\qquad
Du_j\longrightarrow L\quad\hbox{in }\mathcal L^2.
\]
For a Lipschitz scalar function $b$ and a Lipschitz
$(n-1)$-tuple $\pi$, the identity $\partial A=0$ gives
\begin{equation}
A(b,u_j,\pi)=-A(u_j,b,\pi).
\label{eq:cycle-gradient-identity}
\end{equation}
The right-hand side tends to zero by the mass bound and
$\|u_j\|_1\le m^{1/2}\|u_j\|_2$. For the fixed tuple $\pi$, the limit
of the left-hand side is integration of $b$ against the signed measure
\[
\nu_\pi(B)=\sum_i
\int_{\phi_i^{-1}(B\cap Z_i)}
\theta_i\det\bigl(L_i,D(\pi\circ\phi_i)\bigr)\,dz,
\qquad B\subset Y\ \hbox{Borel}.
\]
Here $L_i$ is the row of coordinate components of $L$ in the $i$th
chart. The determinant estimate of
Proposition~\ref{prop:chart-mass} shows that this is a finite signed
measure and that
\[
\|\nu_\pi\|_{\TV}
\le \left(\prod_{a=1}^{n-1}\Lip(\pi_a)\right)
\int|L|\,d\mu.
\]
The same estimate applied to $Du_j-L$ justifies the asserted limit.
Thus $\int b\,d\nu_\pi=0$ for every Lipschitz $b$ on $Y$.
Lipschitz functions are uniformly dense in $C(Y)$, so they determine
finite signed measures on the compact metric space $Y$. It follows
that $\nu_\pi=0$ as a measure.
Fix a chart $i$. Extend each coordinate of $\phi_i^{-1}:Z_i\to\R^n$
to a Lipschitz scalar function $F^a$ on $Y$. At almost every density
point of $E_i$,
\[
D(F^a\circ\phi_i)=e_a^*.
\]
Take for $\pi$ each of the $n$ tuples obtained by omitting one member
of $(F^1,\ldots,F^n)$. Restricting the corresponding zero measures
$\nu_\pi$ to $Z_i$ shows, component by component, that
$\theta_i L_i=0$ almost everywhere on $E_i$. The multiplicity is
nonzero, so $L_i=0$. There are only countably many charts and finitely
many tuples per chart; hence $L=0$ in $\mathcal L^2$.
Notice the order of this argument: we first obtained zero signed
measures by testing on the whole cycle, and then localized those
measures. No boundary or normality property of a chart restriction
is used.
The closure of the graph is therefore the graph of an operator $D$,
and, with its graph norm, it is the asserted Hilbert space $\Hcal$.
In particular, functions that agree almost everywhere have the same
gradient. Applying \eqref{eq:almost-sobolev} to differences of
Lipschitz approximants shows that a graph-norm Cauchy sequence is
Cauchy in $L^p$. Its $L^p$ limit agrees with its $L^2$ limit, and
passing to the limit proves both the continuous inclusion and the
extension of \eqref{eq:almost-sobolev}.
To prove the first calculus assertion, choose Lipschitz approximants
$u_{a,j}\to u_a$ in $\Hcal$, and pass to a common subsequence converging
almost everywhere. Write $U_j=(u_{1,j},\ldots,u_{k,j})$ and
$U=(u_1,\ldots,u_k)$. Bounded first derivatives give
$\Phi(U_j)\to\Phi(U)$ in $L^2$. The constant $\Phi(0)$ is harmless
because $\mu$ is finite. The chart chain rule gives
\[
D\Phi(U_j)=\sum_a\partial_a\Phi(U_j)Du_{a,j}.
\]
For each summand, subtract the proposed limit by writing
\[
\partial_a\Phi(U_j)(Du_{a,j}-Du_a)
+\bigl(\partial_a\Phi(U_j)-\partial_a\Phi(U)\bigr)Du_a.
\]
The first term tends to zero in $\mathcal L^2$ by the derivative
bound; the second does so by dominated convergence. Closedness
proves assertion~\textup{(i)}.
For assertion~\textup{(ii)}, fix a level $t$ and choose
$\eta\in C_c^\infty((-1,1))$ with $0\le\eta\le1$ and $\eta(0)=1$.
The functions
\[
\Phi_\epsilon(s)=\int_{-\infty}^s
\eta\bigl((a-t)/\epsilon\bigr)\,da
\]
converge uniformly to zero, their derivatives are bounded by one,
and $\Phi_\epsilon'(s)\to\mathbf1_{\{t\}}(s)$. Assertion~\textup{(i)}
and dominated convergence give
\[
\Phi_\epsilon(u)\longrightarrow0\quad\hbox{in }L^2,
\qquad
D\Phi_\epsilon(u)\longrightarrow\mathbf1_{\{u=t\}}Du
\quad\hbox{in }\mathcal L^2.
\]
Closedness forces the latter limit to be zero. A Lipschitz,
piecewise $C^1$ scalar function with finitely many corners can now be
smoothed by convolution. The smoothed functions converge uniformly,
their derivatives have a common bound, and the derivatives converge
at every point other than the corners. The level-set conclusion
removes the contribution of those corners, so dominated convergence
and closedness give the claimed chain rule.
In particular, $D(u_+)=\mathbf1_{\{u>0\}}Du$. If $u_j\to u$ in
$\Hcal$, the positive parts converge in $L^2$, and their gradient
difference is
\[
\mathbf1_{\{u_j>0\}}(Du_j-Du)
+\bigl(\mathbf1_{\{u_j>0\}}-\mathbf1_{\{u>0\}}\bigr)Du.
\]
Along any subsequence with almost-everywhere convergence, the second
term tends to zero in $\mathcal L^2$ by the level-set conclusion at
zero and dominated convergence. The first term is bounded in norm
by $\|Du_j-Du\|_2$. Applying this observation to subsequences proves
continuity for the original sequence. Taking positive parts of
Lipschitz approximants proves assertion~\textup{(iii)}.
Finally fix $y\in Y$. The distance $r=d(y,\cdot)$ is bounded,
$|Dr|\le1$ almost everywhere, and $\mu$ is finite. Each term of
\eqref{eq:radial} is therefore continuous as a linear functional of
$g$ in the graph norm. Approximate a nonnegative $g\in\Hcal$ by the
nonnegative Lipschitz functions just constructed and pass to the
limit. This proves assertion~\textup{(iv)}.
\end{proof}
\subsection{Compactness}
The gradient is now defined on the completed space, and its critical
embedding is continuous. We still need strong $L^2$ convergence for
bounded sequences. The proof obtains compactness for projections of
whole weighted currents, then isolates charts in total variation.
\begin{proposition}[Compact inclusion]
\label{prop:compact-embedding}
The inclusion $\Hcal\longrightarrow L^2(\mu)$ is compact.
\end{proposition}
\begin{proof}
\emph{Projecting the whole weighted current.}
First consider Lipschitz functions $u_j$ with
$\sup_j\|u_j\|_{\Hcal}\le M_0<\infty$. Fix a chart $i$, extend its
inverse coordinates to a Lipschitz map $F=(F^1,\ldots,F^n):Y\to\R^n$,
and fix a Lipschitz function $\chi$ on $Y$. Define finite signed
measures $\lambda_{j,\chi}$ on $\R^n$ by
\begin{equation}
\int h\,d\lambda_{j,\chi}
=A\bigl((h\circ F)\chi u_j,F^1,\ldots,F^n\bigr)
\label{eq:projected-function-measure}
\end{equation}
for bounded Borel $h$. The chart representation defines these
measures directly. They are supported in the fixed compact set
$F(Y)$, and the current mass bound gives
\[
\sup_j\|\lambda_{j,\chi}\|_{\TV}
\le \left(\prod_{a=1}^n\Lip(F^a)\right)
\|\chi\|_\infty m^{1/2}M_0.
\]
For $h\in C_c^\infty(\R^n)$, expanding the row
$D(h\circ F)=\sum_a(\partial_a h\circ F)DF^a$ in a determinant shows
that
\[
\int\partial_l h\,d\lambda_{j,\chi}
=A\bigl(\chi u_j,F^1,\ldots,F^{l-1},h\circ F,
F^{l+1},\ldots,F^n\bigr).
\]
All terms except $a=l$ in the row expansion vanish because they have
a repeated row. Apply the cycle identity
\eqref{eq:cycle-gradient-identity}, with a permutation of rows if
necessary, to move $h\circ F$ to the first coefficient. The intrinsic
determinant bound then gives
\begin{equation}
\left|\int\partial_lh\,d\lambda_{j,\chi}\right|
\le C_F\|h\|_\infty\int|D(\chi u_j)|\,d\mu
\le C_\chi\|h\|_\infty,
\label{eq:projected-measure-derivative}
\end{equation}
uniformly in $j$ and $l$. Indeed,
\[
\int|D(\chi u_j)|\,d\mu
\le m^{1/2}\bigl(\|\chi\|_\infty\|Du_j\|_2
+\Lip(\chi)\|u_j\|_2\bigr).
\]
Constants here may depend on the fixed chart, $\chi$, and $M_0$.
\emph{Translation compactness.}
We give the compactness consequence of
\eqref{eq:projected-measure-derivative} in detail. This is the usual
translation and mollification proof of Euclidean BV compactness
\cite[Chapter~2, Theorem~2.6]{Simon2014}, expressed directly for the
projected signed measures. Let
$\tau_w(x)=x+w$. For a smooth compactly supported test $h$, the
fundamental theorem of calculus along $x+tw$ and
\eqref{eq:projected-measure-derivative} imply
\[
\left|\int h\,d\bigl((\tau_w)_\#\lambda_{j,\chi}
-\lambda_{j,\chi}\bigr)\right|
\le C_\chi\|h\|_\infty\sum_{l=1}^n|w_l|.
\]
Smooth compactly supported tests recover the total variation of
finite signed Radon measures, by regularity and approximation of
continuous tests. After changing the constant, we obtain
\begin{equation}
\| (\tau_w)_\#\lambda_{j,\chi}-\lambda_{j,\chi}\|_{\TV}
\le C_\chi|w|.
\label{eq:measure-translation-bound}
\end{equation}
Choose a nonnegative smooth convolution kernel $\rho_\epsilon$ of
integral one supported in the ball of radius $\epsilon$. Averaging
\eqref{eq:measure-translation-bound} gives
\[
\|\lambda_{j,\chi}-(\rho_\epsilon*\lambda_{j,\chi})\,dx\|_{\TV}
\le C_\chi\epsilon.
\]
For fixed $\epsilon$, the densities
$\rho_\epsilon*\lambda_{j,\chi}$ have a common compact support and
uniform bounds on their values and first derivatives. The
Arzel\`a--Ascoli theorem makes them precompact in the uniform norm
on that support and hence in $L^1(\R^n)$. Approximation by these
families, first fixing $\epsilon$ and then letting it decrease to
zero, proves that $\{\lambda_{j,\chi}:j\ge1\}$ is totally bounded
in total variation. The space of finite signed measures is complete
in that norm, so this family is precompact.
\emph{Isolating a chart in total variation.}
We now isolate the chart. Formula
\eqref{eq:projected-function-measure} also defines
$\lambda_{j,\mathbf1_{Z_i}}$, using the bounded Borel coefficient
$\mathbf1_{Z_i}$. By regularity of $\mu$ and distance cutoffs on the
compact metric space $Y$, choose Lipschitz $\chi_k$ with
\[
\|\chi_k-\mathbf1_{Z_i}\|_2\longrightarrow0.
\]
The mass bound and Cauchy--Schwarz give, uniformly in $j$,
\begin{align}
\|\lambda_{j,\chi_k}-\lambda_{j,\mathbf1_{Z_i}}\|_{\TV}
&\le \left(\prod_{a=1}^n\Lip(F^a)\right)
\int|\chi_k-\mathbf1_{Z_i}|\,|u_j|\,d\mu \notag\\
&\le C_FM_0\|\chi_k-\mathbf1_{Z_i}\|_2.
\label{eq:chart-cutoff-approximation}
\end{align}
For each fixed $k$ the family on the left with coefficient $\chi_k$
is precompact by the preceding argument. Uniform approximation
\eqref{eq:chart-cutoff-approximation} therefore makes
$\{\lambda_{j,\mathbf1_{Z_i}}:j\ge1\}$ precompact as well. There is
no need for the Lipschitz constants of $\chi_k$ to remain bounded.
Since $F\circ\phi_i$ is the identity on $E_i$, the density of the
isolated measure is exactly
\[
d\lambda_{j,\mathbf1_{Z_i}}(z)
=\mathbf1_{E_i}(z)\theta_i(z)(u_j\circ\phi_i)(z)\,dz.
\]
Consequently, for any $j,k$,
\begin{align*}
\|u_j-u_k\|_{L^1(Z_i,\mu)}
&=\int_{E_i}|\theta_i|J_i
|(u_j-u_k)\circ\phi_i|\,dz\\
&\le\Lip(\phi_i)^n
\|\lambda_{j,\mathbf1_{Z_i}}
-\lambda_{k,\mathbf1_{Z_i}}\|_{\TV}.
\end{align*}
Here $J_i\le\Lip(\phi_i)^n$. This calculation allows arbitrary
signed, unbounded integer multiplicities: the fixed $\theta_i$
appears as $|\theta_i|$ in the total variation of a difference.
Likewise, \eqref{eq:chart-cutoff-approximation} already uses the full
mass measure $|\theta_i|J_i\,dz$, so it requires no bound or regularity
of the multiplicities. All integrations by parts above concerned
the whole cycle $A$ with a Lipschitz coefficient. None concerned a
restriction to one chart.
\emph{Assembling the charts.}
We may now take a diagonal subsequence that is Cauchy in
$L^1(Z_i,\mu)$ for every $i$. The omitted charts have uniformly
small contributions, because
\[
\int_{\bigcup_{i>N}Z_i}|u_j|\,d\mu
\le M_0\,\mu\left(\bigcup_{i>N}Z_i\right)^{1/2}
\longrightarrow0
\]
uniformly in $j$. The chart images cover $\mu$ up to a null set,
and $\mu$ is finite. The subsequence is thus Cauchy in $L^1(\mu)$.
Proposition~\ref{prop:sobolev-space} supplies a uniform $L^p$ bound.
Since $p>2$, interpolation with
$\alpha=(p-2)/(2(p-1))>0$ gives
\[
\|u_j-u_k\|_2
\le\|u_j-u_k\|_1^\alpha\|u_j-u_k\|_p^{1-\alpha}
\longrightarrow0.
\]
Finally, approximate the $j$th member of an arbitrary bounded
sequence in $\Hcal$ by a Lipschitz function with graph-norm error
at most $1/j$. The result just proved for those approximants proves
compactness for the original sequence.
\end{proof}
\subsection{A minimizer and its powers}
We now have weak compactness in $\Hcal$ and strong compactness in
$L^2$. These do not alone give strong convergence at the critical
exponent $p$. The strict inequality $m<s_n$ places the quotient
infimum below $S_*$; the almost sharp estimate will exclude any missing
$L^p$ mass.
Define the quadratic functional and its critical constrained infimum by
\begin{equation}
Q(u)=4\beta E(u)+n\|u\|_2^2,
\qquad
\Lambda=\inf\{Q(u):u\in\Hcal,\ \|u\|_p=1\}.
\label{eq:critical-variational-problem}
\end{equation}
\begin{proposition}[Existence and normalization of the minimizer]
\label{prop:minimizer}
One has $0<\Lambda<S_*$. There exists a nonzero nonnegative
$v\in\Hcal$ minimizing $Q(u)/\|u\|_p^2$ over nonzero $u\in\Hcal$
and satisfying
\begin{equation}
\begin{split}
0<W:=\int v^p\,d\mu
=\left(\frac\Lambda n\right)^{n/2}<s_n,
\qquad Q(v)=nW.
\end{split}
\label{eq:minimizer-normalization}
\end{equation}
For every $\psi\in\Hcal$ it satisfies the weak equation
\begin{equation}
4\beta\int Dv\cdot D\psi\,d\mu+n\int v\psi\,d\mu
=n\int v^{p-1}\psi\,d\mu.
\label{eq:euler}
\end{equation}
\end{proposition}
\begin{proof}
Taking $\epsilon=S_*/2$ in \eqref{eq:almost-sobolev} shows that
\[
\frac{S_*}{2}\|u\|_p^2
\le4\beta E(u)+C_{S_*/2}\|u\|_2^2
\le\max\{1,C_{S_*/2}/n\}\,Q(u).
\]
Thus $\Lambda>0$. The constant function $m^{-1/p}$ has $L^p$ norm
one, so
\begin{equation}
\Lambda\le n m^{1-2/p}=n m^{2/n}<ns_n^{2/n}=S_*.
\label{eq:strict-subthreshold}
\end{equation}
Let $u_j$ be an $L^p$-normalized minimizing sequence. It is bounded
in $\Hcal$, since $Q$ is a positive definite quadratic form equivalent
to the squared graph norm. By weak compactness in a Hilbert space
and Proposition~\ref{prop:compact-embedding}, after taking a
subsequence we have
\[
u_j\rightharpoonup u\quad\hbox{in }\Hcal,
\qquad u_j\to u\quad\hbox{in }L^2(\mu)
\quad\hbox{and almost everywhere}.
\]
Put $z_j=u_j-u$ and $t=\int|u|^p\,d\mu$. Fatou's lemma gives
$0\le t\le1$. We give the Br\'ezis--Lieb
splitting~\cite[Theorem~1]{BrezisLieb1983} in the form needed here:
\begin{equation}
\int|z_j|^p\,d\mu\longrightarrow1-t.
\label{eq:critical-mass-splitting}
\end{equation}
For every $\epsilon>0$, the mean-value theorem and Young's
inequality give a finite $C_\epsilon$ such that, for real $a,b$,
\[
\bigl||a+b|^p-|a|^p\bigr|
\le\epsilon|a|^p+C_\epsilon|b|^p.
\]
Apply this with $a=z_j$ and $b=u$, and set
$R_j=|u_j|^p-|z_j|^p-|u|^p$. The functions
\[
\bigl(|R_j|-\epsilon|z_j|^p\bigr)_+
\]
converge to zero almost everywhere and are bounded by
$(C_\epsilon+1)|u|^p$. Dominated convergence, followed by
$\epsilon\downarrow0$, shows that $R_j\to0$ in $L^1$, since the
$L^p$ norms of $z_j$ are bounded. This proves
\eqref{eq:critical-mass-splitting}.
Weak convergence in $\Hcal$ also gives the quadratic splitting
\[
Q(u_j)=Q(u)+Q(z_j)+o(1).
\]
The definition of $\Lambda$ gives $Q(u)\ge\Lambda t^{2/p}$, with
the same conclusion if $u=0$. For fixed $\epsilon>0$, apply
\eqref{eq:almost-sobolev} to $z_j$ and use $\|z_j\|_2\to0$ to obtain
\[
\liminf_j Q(z_j)\ge(S_*-\epsilon)(1-t)^{2/p}.
\]
Letting $\epsilon\downarrow0$ therefore yields
\begin{equation}
\Lambda\ge\Lambda t^{2/p}+S_*(1-t)^{2/p}.
\label{eq:no-critical-mass-loss}
\end{equation}
If $t<1$, the inequalities $a^{2/p}\ge a$ for $0\le a\le1$ and
$S_*>\Lambda$ make the right-hand side strictly larger than
$\Lambda$: indeed it is at least
$\Lambda t+S_*(1-t)>\Lambda$. Thus $t=1$.
Weak lower semicontinuity now gives $Q(u)\le\Lambda$, so $u$ is a
minimizer with $\|u\|_p=1$. Replacing it by $|u|$ preserves both
its $L^p$ norm and its energy, by
Proposition~\ref{prop:sobolev-space}. We take $u\ge0$.
For each $\psi\in\Hcal$, differentiate the quotient
$Q(u+s\psi)/\|u+s\psi\|_p^2$ at $s=0$. Its denominator is nonzero
for sufficiently small $s$, and H\"older's inequality justifies the
derivative of the $L^p$ integral: for $|s|\le1$ the derivative
integrand is bounded by
$p(|u|+|\psi|)^{p-1}|\psi|\in L^1$. Since $u$ is a minimizer and
$\|u\|_p=1$, this gives
\[
4\beta\int Du\cdot D\psi\,d\mu+n\int u\psi\,d\mu
=\Lambda\int u^{p-1}\psi\,d\mu.
\]
Set
\[
v=\left(\frac\Lambda n\right)^{1/(p-2)}u.
\]
Homogeneity preserves the quotient minimum, and
$\Lambda(\Lambda/n)^{-1}=n$ gives \eqref{eq:euler}.
Since $p/(p-2)=n/2$, its $p$th moment is
$W=(\Lambda/n)^{n/2}<s_n$ by
\eqref{eq:strict-subthreshold}. Testing \eqref{eq:euler} with
$\psi=v$ gives $Q(v)=nW$, completing
\eqref{eq:minimizer-normalization}.
\end{proof}
The normalized minimizer has the strict weighted mass bound $W<s_n$
needed for the contradiction. The remaining task is to justify nonlinear
tests in its weak equation and radial first variation. The following
moment and weighted-energy estimates do so without assuming that $v$
is positive everywhere, smooth, or essentially bounded.
The proof adapts the critical-potential absorption method of
Br\'ezis and Kato~\cite{BrezisKato1979} to the closed chart gradient.
For the truncated-power formulation, see also \cite[Lemma~6, p.~178]{Brezis1986}.
The adaptation is given in full, without invoking a Euclidean regularity
theorem in the present Sobolev space.
\begin{lemma}[Powers of the minimizer]
\label{lem:powers}
For every real $\gamma\ge1$,
\[
v^\gamma\in\Hcal\cap L^p(\mu),\qquad
D(v^\gamma)=\gamma v^{\gamma-1}Dv.
\]
Moreover, $\int v^q\,d\mu<\infty$ for every finite $q>0$, and
\begin{equation}
\int v^q|Dv|^2\,d\mu<\infty
\qquad\hbox{for every finite }q\ge0.
\label{eq:polynomial-weighted-energy}
\end{equation}
\end{lemma}
\begin{proof}
We first prove the power assertion under the additional assumption
$v\in L^{2\gamma}$. It is immediate for $\gamma=1$, so suppose
$\gamma>1$. For $N\ge1$ define
\[
w_N=v\min(v,N)^{\gamma-1},\qquad
\psi_N=v\min(v,N)^{2\gamma-2},\qquad
d_\gamma=\frac{2\gamma-1}{\gamma^2}.
\]
The scalar functions in these definitions, extended by zero for
negative arguments, are Lipschitz and piecewise $C^1$. Thus both
compositions belong to $\Hcal$. Their derivatives show that
\[
Dv\cdot D\psi_N\ge d_\gamma|Dw_N|^2.
\]
Indeed there is equality below $N$, while above $N$ the ratio of
the coefficients is one and $0<d_\gamma\le1$. The gradient vanishes
on the level set $\{v=N\}$, so no value at the corner matters.
Testing \eqref{eq:euler} with $\psi_N$ is legitimate; for example,
$v^{p-1}\psi_N\le N^{2\gamma-2}v^p$ is integrable. Dropping the
nonnegative term $n\int v\psi_N\,d\mu$ gives
\begin{equation}
4\beta d_\gamma E(w_N)
\le n\int v^{p-2}w_N^2\,d\mu.
\label{eq:truncated-power-energy}
\end{equation}
Apply \eqref{eq:almost-sobolev} to $w_N$ with $\epsilon=S_*/2$ and
multiply by $d_\gamma$. By
\eqref{eq:truncated-power-energy},
\[
\frac{d_\gamma S_*}{2}\|w_N\|_p^2
\le n\int v^{p-2}w_N^2\,d\mu
+d_\gamma C_{S_*/2}\|w_N\|_2^2.
\]
H\"older's inequality with conjugate exponents $p/(p-2)$ and $p/2$
gives
\[
n\int_{\{v>K\}}v^{p-2}w_N^2\,d\mu
\le n\left(\int_{\{v>K\}}v^p\,d\mu\right)^{(p-2)/p}
\|w_N\|_p^2.
\]
Since $v\in L^p$, choose $K\ge1$, depending on $\gamma$ but not $N$,
so that the coefficient on the right is at most
$d_\gamma S_*/4$. The contribution of $\{v\le K\}$ is at most
$nK^{p-2}\|w_N\|_2^2$. Absorbing the tail yields
\[
\frac{d_\gamma S_*}{4}\|w_N\|_p^2
\le\bigl(nK^{p-2}+d_\gamma C_{S_*/2}\bigr)\|w_N\|_2^2.
\]
The assumed $L^{2\gamma}$ moment bounds the right-hand side uniformly
in $N$, because $w_N\le v^\gamma$. Thus $\|w_N\|_p$ is uniformly
bounded. Equation~\eqref{eq:truncated-power-energy}, with another
application of H\"older, then uniformly bounds $E(w_N)$.
On $\{v<N\}$ the gradient of $w_N$ is
$\gamma v^{\gamma-1}Dv$. Monotone convergence in these sets shows
that $\gamma v^{\gamma-1}Dv$ belongs to $\mathcal L^2$. At every
truncation level the formulas and the level-set property give
\[
|w_N|\le v^\gamma,\qquad
|Dw_N|\le\gamma v^{\gamma-1}|Dv|.
\]
The values and gradients converge almost everywhere to $v^\gamma$
and $\gamma v^{\gamma-1}Dv$, respectively. The displayed bounds
give strong $L^2$ convergence of both by dominated convergence.
Closedness proves $v^\gamma\in\Hcal$ with the asserted gradient,
and the continuous inclusion into $L^p$ gives $v^\gamma\in L^p$.
Starting with $v\in L^p$, set $\ell_j=p(p/2)^j$ for $j\ge0$.
The conditional assertion just proved, with $\gamma=\ell_j/2$, shows
\[
v\in L^{\ell_j}\quad\Longrightarrow\quad v\in L^{\ell_{j+1}}.
\]
Since $p/2>1$, these exponents tend to infinity. Finiteness of $\mu$ then gives
every finite positive moment of $v$. The conditional assertion
therefore applies to every $\gamma\ge1$. Finally, for $q\ge0$
take $\gamma=1+q/2$ to obtain
\[
\int v^q|Dv|^2\,d\mu=\gamma^{-2}E(v^\gamma)<\infty,
\]
which is \eqref{eq:polynomial-weighted-energy}.
\end{proof}
All positive moments and polynomially weighted energies are now
finite. We next turn those integrability statements into an admissible
chain rule for the actual two-variable tests used in the chord
comparison.
\begin{lemma}[Compositions with polynomial growth]
\label{lem:composites}
Let $r\in\Hcal$ satisfy $0\le r\le R<\infty$ almost everywhere and
$|Dr|\in L^\infty(\mu)$. Suppose that $F$ is $C^1$ on an open
neighborhood of $[0,R]\times[0,\infty)$ and that, on this strip,
\[
|F(a,t)|+|\partial_1F(a,t)|+|\partial_2F(a,t)|
\le C(1+t)^M
\]
for some finite $C$ and nonnegative integer $M$. Then
$F(r,v)\in\Hcal$ and
\begin{equation}
D(F(r,v))=\partial_1F(r,v)Dr+\partial_2F(r,v)Dv.
\label{eq:polynomial-composite-chain-rule}
\end{equation}
In particular, this applies with $r=d(y,\cdot)$ for each fixed
$y\in Y$.
\end{lemma}
\begin{proof}
Let $t_N=\min(v,N)$. The scalar calculus in
Proposition~\ref{prop:sobolev-space} gives $t_N\in\Hcal$ and
$Dt_N=\mathbf1_{\{v<N\}}Dv$, where the level-set property handles
$v=N$. For each $N$, choose a smooth, compactly supported cutoff in the given
open neighborhood and equal to one on a neighborhood of the compact
rectangle $[0,R]\times[0,N]$. Multiply $F$ by this cutoff and extend
by zero to obtain a global $C^1$ function with bounded first
derivatives. It agrees with $F$ and its first derivatives on the
rectangle. The bounded-derivative calculus consequently gives
$F(r,t_N)\in\Hcal$ and
\[
D(F(r,t_N))
=\partial_1F(r,t_N)Dr
+\partial_2F(r,t_N)\mathbf1_{\{v<N\}}Dv.
\]
Writing $L=\|\,|Dr|\,\|_\infty$, the growth assumption gives bounds
independent of $N$,
\[
|F(r,t_N)|\le C(1+v)^M,\qquad
|D(F(r,t_N))|\le C(1+v)^M(L+|Dv|).
\]
The squares of these bounds are integrable by
Lemma~\ref{lem:powers}: they are bounded by a constant times
$(1+v^{2M})(1+|Dv|^2)$. Since $v$ is finite almost everywhere,
the values and the displayed gradient formulas converge almost
everywhere to the values and right-hand side in
\eqref{eq:polynomial-composite-chain-rule}. Dominated convergence
gives strong convergence in the two $L^2$ spaces, and closedness
proves the result. A distance function has bounded values and
$|Dr|\le1$, so it satisfies the hypotheses.
\end{proof}
==== END Sharp-integral-fillings-in-CAT(0)-spaces//sections/05-minimizer.tex ====
==== BEGIN Sharp-integral-fillings-in-CAT(0)-spaces//sections/06-chords.tex ====
\section{Conformally weighted chords}\label{sec:chords}
Throughout this section, $n>2$ and the normalized extremizing cycle
$A$ is as in Section~\ref{sec:radial}. We assume the sharp filling
inequality in dimension $n-1$, so that Sections~\ref{sec:sobolev}
and~\ref{sec:minimizer} apply. Keep the notation
\[
\beta=\frac1{n-2},\qquad p=\frac{2n}{n-2},\qquad
Q(u)=4\beta E(u)+n\norm{u}_2^2.
\]
Let $v\ge0$ be the quotient minimizer from
Proposition~\ref{prop:minimizer}. In particular,
\begin{equation}\label{chord:normalization}
0<W:=\int_Y v^p\dd\mu<s_n,\qquad Q(v)=nW,
\qquad \dd\nu=\frac{v^p}{W}\dd\mu.
\end{equation}
The last expression defines a probability measure. By
Lemma~\ref{lem:powers}, $v$ has all finite positive moments and
\begin{equation}\label{chord:weighted-energy}
\int_Y v^q\abs{Dv}^2\dd\mu<\infty\qquad(q\ge0).
\end{equation}
Choose a finite nonnegative Borel representative of $v$; changing
it on a $\mu$-null set has no effect on these assertions.
For a fixed center $y\in Y$ with $v(y)>0$, set
\begin{equation}\label{chord:data}
\kappa=v(y)^\beta,\qquad r(x)=d(y,x),\qquad
s(x)=\kappa r(x)v(x)^\beta,
\qquad L_y=\int_Y s^2\dd\nu.
\end{equation}
These are finite Borel functions or numbers. In particular, with
$D=\diam Y$,
\[
L_y\le\frac{\kappa^2D^2}{W}
\int_Y v^{p+2\beta}\dd\mu<\infty.
\]
We study the chord distribution through the transform
\begin{equation}\label{chord:transform-definition}
J_y(a)=\int_Y(1+as^2)^{-n}\dd\nu\qquad(a\ge0).
\end{equation}
Our first comparison will give
$J_y(a)\le(1+2aL_y)^{-n/2}$ for every center with $v(y)>0$.
Euclidean tangent density will then give a lower bound for
$\liminf_{a\to\infty}a^{n/2}J_y(a)$ at almost every center, forcing
\[
L_y\le2(W/s_n)^{2/n}<2
\quad\text{for $\nu$-almost every center $y$}.
\]
Section~\ref{sec:conclusion} will show that the average of these same
moments is at least $2$.
Every chart derivative in this section is taken in the variable
$x$, at this fixed center $y$. We shall not require $s\in\Hcal$.
Instead, define the square-integrable chart covector field
\begin{equation}\label{chord:xi}
\xi=\kappa\bigl(v^{1+\beta}Dr
+\beta r v^\beta Dv\bigr).
\end{equation}
Its square integrability follows from $\abs{Dr}\le1$, boundedness
of $r$, the moments of $v$, and~\eqref{chord:weighted-energy} with
$q=2\beta$. On $\{v>0\}$ it is the formal expression $vDs$;
the explicit formula~\eqref{chord:xi} defines it also on $\{v=0\}$.
\subsection{Admissible chord tests}
For $a>0$, the profile $w(t)=(1+at^2)^{-(n-2)/2}$ is the rescaled
Euclidean Sobolev profile used in Lemma~\ref{lem:radial-sobolev}.
Its exponent is suited to the transform because
$w(s)^p=(1+as^2)^{-n}$. We will compare the quotient of $vw(s)$
with that of $v$, using $vw(s)^2$ in the minimizer equation.
The following lemma first justifies these tests and the radial tests
needed to control their energy, including at points where $v=0$.
\begin{lemma}[Admissibility of the chord tests]\label{lem:chord-tests}
Fix $a>0$ and put
\[
w(t)=(1+at^2)^{-(n-2)/2}\qquad(t\in\R).
\]
Then $vw(s)$ and $vw(s)^2$ belong to $\Hcal$, and
\begin{equation}\label{chord:product-rules}
\begin{split}
D(vw(s))&=w(s)Dv+w'(s)\xi,\\
D(vw(s)^2)&=w(s)^2Dv+2w(s)w'(s)\xi.
\end{split}
\end{equation}
If $f:\R\to[0,\infty)$ is smooth with bounded value and bounded
first derivative, then
\begin{equation}\label{chord:radial-tests}
g=\kappa^2v^{2+2\beta}f(s),\qquad
\psi=\beta\kappa^2r^2v^{1+2\beta}f(s)
\end{equation}
also belong to $\Hcal$. In particular, $g$ is an admissible
nonnegative test in the extended radial inequality, and $\psi$
is an admissible test in~\eqref{eq:euler}.
\end{lemma}
\begin{proof}
We verify the hypotheses of Lemma~\ref{lem:composites} for the
actual functions of $(r,v)$ that occur here. This is necessary
when $\beta<1$, since the scalar function $v^\beta$ itself is
not $C^1$ at zero.
For $j=1,2$ and $z>0$, define
\[
F_j(r,z)=z\,w(\kappa r z^\beta)^j,
\qquad t=\kappa r z^\beta.
\]
On this half-plane,
\[
\partial_zF_j=w(t)^j+j\beta t\,w(t)^{j-1}w'(t),\qquad
\partial_rF_j=j\kappa z^{1+\beta}w(t)^{j-1}w'(t).
\]
As $z\downarrow0$, uniformly for $r$ in bounded intervals,
$w(t)=1+O(t^2)$ and $w'(t)=O(t)$. Thus
\[
F_j(r,z)=z+O(z^{1+2\beta}),\qquad
\partial_zF_j(r,z)\longrightarrow1,
\qquad \partial_rF_j(r,z)\longrightarrow0.
\]
Extension by $F_j(r,z)=z$ on $z\le0$ is therefore jointly $C^1$.
The functions and their first derivatives have polynomial growth
in $z\ge0$ on bounded $r$ intervals.
For the other tests, write
\[
F_g(r,z)=\kappa^2z^{2+2\beta}f(\kappa r z^\beta),\qquad
F_\psi(r,z)=\beta\kappa^2r^2z^{1+2\beta}
f(\kappa r z^\beta)
\quad(z>0).
\]
Their partial derivatives are
\begin{align*}
\partial_zF_g
&=\kappa^2z^{1+2\beta}
\bigl((2+2\beta)f(t)+\beta t f'(t)\bigr),\\
\partial_rF_g
&=\kappa^3z^{2+3\beta}f'(t),\\
\partial_zF_\psi
&=\beta\kappa^2r^2
\bigl((1+2\beta)z^{2\beta}f(t)
+\beta\kappa r z^{3\beta}f'(t)\bigr),\\
\partial_rF_\psi
&=\beta\kappa^2
\bigl(2r z^{1+2\beta}f(t)
+\kappa r^2z^{1+3\beta}f'(t)\bigr).
\end{align*}
Their values and all these derivatives tend to zero as
$z\downarrow0$, uniformly for bounded $r$. The smallest power
of $z$ in the derivative formulas is $2\beta>0$. Consequently,
extension by zero on $z\le0$ makes both functions jointly $C^1$,
even if $f(0)$ or $f'(0)$ is nonzero. Their values and first
derivatives again have polynomial growth on the range of $r$.
These $C^1$ extensions and polynomial bounds satisfy all the
hypotheses of Lemma~\ref{lem:composites}. That lemma gives membership
in $\Hcal$ and the chain rules without a boundedness assumption on $v$.
On $\{v>0\}$, differentiation gives~\eqref{chord:product-rules}.
On $\{v=0\}$, Proposition~\ref{prop:sobolev-space} gives $Dv=0$
almost everywhere, while~\eqref{chord:xi} gives $\xi=0$.
Thus these formulas hold $\mu$-almost everywhere on all of $Y$.
\end{proof}
\begin{lemma}[A consequence of the radial inequality]
\label{lem:completed-radial}
For every center $y\in Y$, with $r=d(y,\cdot)$, and every
nonnegative $g\in\Hcal$,
\begin{equation}\label{chord:completed-radial}
\int_Y\bigl(ng\abs{Dr}^2+r\ip{Dr}{Dg}\bigr)\dd\mu
\le\frac n4\int_Y r^2g\dd\mu.
\end{equation}
\end{lemma}
\begin{proof}
The extension of~\eqref{eq:radial} to these tests is part of
Proposition~\ref{prop:sobolev-space}. Pointwise, with
$b=\sqrt{1-\abs{Dr}^2}$,
\[
1-rb-\abs{Dr}^2+\frac{r^2}{4}
=\left(b-\frac r2\right)^2\ge0.
\]
Multiply by $ng\ge0$ and combine this inequality
with~\eqref{eq:radial}. All terms are integrable by boundedness
of $r$, $\abs{Dr}\le1$, finite $\mu$, and $g\in\Hcal$.
\end{proof}
\subsection{Distribution comparison}
All required composite tests are now admissible, including on the zero
set of $v$. We combine quotient minimality with radial variation to
control the chord transform. Once the integral estimate is obtained,
the remaining comparison is a scalar differential inequality.
\begin{proposition}[Distribution comparison]\label{prop:chord-transform}
For every center $y$ with $v(y)>0$, and every $a\ge0$,
\begin{equation}\label{eq:chord-transform}
J_y(a)=\int_Y(1+as^2)^{-n}\dd\nu
\le(1+2aL_y)^{-n/2},
\end{equation}
where $s$ and $L_y$ are defined in~\eqref{chord:data}.
\end{proposition}
\begin{proof}
Fix the center and suppress its subscript on $J$. For $a>0$ let
$w(t)=(1+at^2)^{-(n-2)/2}$ as in
Lemma~\ref{lem:chord-tests}. Since $p(n-2)/2=n$,
\[
\int_Y w(s)^p\dd\nu=J(a).
\]
Testing~\eqref{eq:euler} with $vw(s)^2$ and using
the two product formulas~\eqref{chord:product-rules} gives
\begin{equation}\label{chord:ground-state}
Q(vw(s))
=n\int_Y v^p w(s)^2\dd\mu
+4\beta\int_Y w'(s)^2\abs{\xi}^2\dd\mu.
\end{equation}
Indeed the terms $w^2\abs{Dv}^2+2ww'\ip{Dv}{\xi}$ in
$\abs{D(vw)}^2$ equal $\ip{Dv}{D(vw^2)}$. The quotient
minimality of $v$ gives, on the other hand,
\begin{equation}\label{chord:quotient}
Q(vw(s))\ge
\frac{Q(v)}{\norm{v}_p^2}\norm{vw(s)}_p^2
=nW J(a)^{(n-2)/n}.
\end{equation}
The test $vw(s)$ is nonzero because $w$ is strictly positive
and $W>0$.
We next control the last integral
in~\eqref{chord:ground-state}. Let $f$ be smooth, bounded and
nonnegative with bounded derivative, and use $g,\psi$ from
\eqref{chord:radial-tests}. Add
\eqref{chord:completed-radial} to~\eqref{eq:euler} divided by
$4\beta$, with test $\psi$. The identity
$v\psi=\beta r^2g$ cancels the two undifferentiated terms and
yields
\begin{equation}\label{chord:added-tests}
\int_Y\bigl(ng\abs{Dr}^2+r\ip{Dr}{Dg}
+\ip{Dv}{D\psi}\bigr)\dd\mu
\le\frac n4\int_Y s^2f(s)v^p\dd\mu.
\end{equation}
Here we used
$v^{p-1}\psi=\beta s^2f(s)v^p$.
For completeness, after removing the common factor $\kappa^2$,
the coefficients of $\abs{Dr}^2$, $\ip{Dr}{Dv}$, and
$\abs{Dv}^2$ in the left integrand are respectively
\begin{align*}
&v^{2+2\beta}\bigl(nf+sf'\bigr),\\
&r v^{1+2\beta}\bigl((2+4\beta)f+2\beta sf'\bigr),\\
&\beta r^2v^{2\beta}\bigl((1+2\beta)f+\beta sf'\bigr).
\end{align*}
The scalar functions $f,f'$ in this display are evaluated at
$s$. Since
\begin{equation}\label{chord:exponent-identities}
1+2\beta=n\beta,\qquad 2+4\beta=2n\beta=p,
\end{equation}
these are precisely the coefficients of
$(nf(s)+sf'(s))\abs{\xi}^2/\kappa^2$. Thus
\begin{equation}\label{chord:energy}
\int_Y\bigl(nf(s)+sf'(s)\bigr)\abs{\xi}^2\dd\mu
\le\frac n4\int_Ys^2f(s)v^p\dd\mu.
\end{equation}
To bound the energy term in~\eqref{chord:ground-state}, we want
$nf(t)+tf'(t)=w'(t)^2$ in~\eqref{chord:energy}. Integrating this
first-order equation leads to the choice
\begin{equation}\label{chord:f}
f(t)=\int_0^1\tau^{n-1}w'(\tau t)^2\dd\tau
\qquad(t\in\R).
\end{equation}
This is smooth and nonnegative. Both $w'$ and $w''$ are bounded
on $\R$, so $f$ and
$f'(t)=2\int_0^1\tau^nw'(\tau t)w''(\tau t)\dd\tau$
are bounded. Integrating the derivative of
$\tau^nw'(\tau t)^2$ gives
\begin{equation}\label{chord:f-identity}
nf(t)+tf'(t)=w'(t)^2.
\end{equation}
The boundary term at $\tau=0$ is zero. Combining
\eqref{chord:ground-state}, \eqref{chord:quotient},
\eqref{chord:energy}, and~\eqref{chord:f-identity}, and dividing
by $nW$, we obtain
\begin{equation}\label{chord:scalar-start}
J(a)^{(n-2)/n}
\le\int_Y\bigl(w(s)^2+\beta s^2f(s)\bigr)\dd\nu.
\end{equation}
The current and variational estimates have now reduced the comparison
to \eqref{chord:scalar-start}. We evaluate its scalar right side and
solve the resulting differential inequality for the transform. Put
$b=as^2\ge0$ and $q=(n-2)/2>0$. Since
\[
w'(t)^2=(n-2)^2a^2t^2(1+at^2)^{-n},
\]
the substitution $u=\tau^2$ in~\eqref{chord:f} gives
\[
w(s)^2+\beta s^2f(s)
=(1+b)^{-(n-2)}
+q b^2\int_0^1\frac{u^{n/2}}{(1+bu)^n}\dd u.
\]
The exact derivative identity
\[
\frac{\dd}{\dd u}\left(\frac{u^q}{(1+bu)^{n-2}}\right)
=q\frac{u^{q-1}(1-b^2u^2)}{(1+bu)^n}
\]
has an integrable right side on $(0,1)$; this includes $n=3$,
when $q-1=-1/2$. Integrating it shows that
\begin{equation}\label{chord:scalar-identity}
w(s)^2+\beta s^2f(s)
=q\int_0^1\frac{u^{q-1}}{(1+as^2u)^n}\dd u.
\end{equation}
In particular this expression lies in $(0,1]$. Tonelli's theorem
therefore turns~\eqref{chord:scalar-start} into
\begin{equation}\label{chord:J-G}
J(a)^{(n-2)/n}\le G(a),\qquad
G(a)=q\int_0^1J(au)u^{q-1}\dd u.
\end{equation}
We have $G(0)=1$ and $G(a)>0$. For $a>0$, changing variable
$t=au$ gives
\[
G(a)=q a^{-q}\int_0^a J(t)t^{q-1}\dd t,
\qquad aG'(a)=q\bigl(J(a)-G(a)\bigr).
\]
Continuity of $J$ suffices for this differentiation. The finite
second moment in~\eqref{chord:data} also allows differentiation
at zero from the right, because
\[
\left|\frac{\partial}{\partial a}(1+as^2)^{-n}\right|
\le ns^2\qquad(a\ge0).
\]
Consequently
\begin{equation}\label{chord:initial-derivatives}
J'(0+)=-nL_y,
\qquad G'(0+)=\frac{q}{q+1}J'(0+)=-(n-2)L_y.
\end{equation}
Set $h=G^{-1/q}=G^{-2/(n-2)}$. For $a>0$,
\[
ah'=h-JG^{-n/(n-2)}\ge h-1,
\]
where the last step follows from~\eqref{chord:J-G}. Thus
\[
\left(\frac{h(a)-1}{a}\right)'
=\frac{ah'(a)-h(a)+1}{a^2}\ge0.
\]
By~\eqref{chord:initial-derivatives}, the quotient on the left
has limit $2L_y$ at zero. It follows that
$h(a)\ge1+2aL_y$. Finally~\eqref{chord:J-G} gives
\[
J(a)\le G(a)^{n/(n-2)}=h(a)^{-n/2}
\le(1+2aL_y)^{-n/2}.
\]
At $a=0$ both sides equal one. If $L_y=0$, then $s=0$
$\nu$-almost everywhere, so $J=G=h=1$ and the same conclusion
holds. No higher chord moment is needed in the scalar comparison.
\end{proof}
\subsection{Tangent density}
The transform estimate bounds $J_y$ in terms of its second moment
$L_y$, but does not yet bound that moment. At almost every center,
integer Euclidean tangent density supplies a lower coefficient for
$J_y$ at large parameter. Combining the two estimates gives the
required strict bound on $L_y$.
\begin{lemma}[Tangent density of the chord transform]\label{lem:chord-density}
For $\nu$-almost every center $y$,
\begin{equation}\label{chord:transform-density}
\liminf_{a\to\infty}a^{n/2}J_y(a)
\ge\frac{s_n}{2^nW}.
\end{equation}
Consequently,
\begin{equation}\label{eq:chord-upper}
L_y\le2\left(\frac{W}{s_n}\right)^{2/n}<2
\qquad\text{for $\nu$-almost every }y.
\end{equation}
\end{lemma}
\begin{proof}
Use the disjoint chart representation of
Proposition~\ref{prop:chart-mass}. For one chart
$\phi_i:E_i\to Z_i\subset Y$, write
\[
u_i=v\circ\phi_i,
\qquad \rho_i=\abs{\theta_i}J_i,
\qquad J_i=\sqrt{\det G_i},
\]
and extend $u_i$ and $\rho_i$ by zero to $\R^n$. The
bi-Lipschitz bounds give a positive lower bound for $J_i$ on
this chart almost everywhere. Since $\abs{\theta_i}\ge1$,
$v\in L^p(\mu)$ and $\mu(Y)<\infty$, $u_i$ is Lebesgue
integrable. The function $\rho_i$ is integrable as well, by
the mass representation. Thus ordinary Lebesgue differentiation
applies to both functions and to $1_{E_i}$.
For $\nu$-almost every $y=\phi_i(z)$, we may therefore require
all of the following: $z$ is a density point of $E_i$; the centered
metric differential is the Euclidean norm $\abs{\cdot}_{G_0}$,
with $G_0=G_i(z)>0$; and $u_i,\rho_i$ have Lebesgue values
\begin{equation}\label{chord:lebesgue-values}
v_0=v(y)>0,
\qquad \rho_0=\abs{\theta_i(z)}\sqrt{\det G_0}>0.
\end{equation}
Indeed the exceptional coordinate sets are Lebesgue-null and
therefore have zero chart mass, while $\{v=0\}$ has zero
$\nu$-mass. There are only countably many charts. The representative
of $v$ agrees with its Lebesgue value at almost every such point.
Fix a center with these properties.
Let $\ell=\liminf_{a\to\infty}a^{n/2}J_y(a)$. If
$\ell=\infty$, the desired lower bound is automatic. Otherwise,
choose $a_j\to\infty$ realizing this lower limit and put
$\varepsilon_j=a_j^{-1/2}$. On every bounded set of
$h\in\R^n$, the functions
\[
u_i(z+\varepsilon_jh),\qquad
\rho_i(z+\varepsilon_jh),\qquad
1_{E_i}(z+\varepsilon_jh)
\]
converge in measure to $v_0,\rho_0,1$, respectively. For example,
this follows by changing variables in the defining mean convergence
at the Lebesgue point $z$. A diagonal subsequence over bounded
balls makes all three convergences hold almost everywhere in
$\R^n$. The subsequence still realizes $\ell$.
At almost every $h$, the points $z+\varepsilon_jh$ consequently
belong to $E_i$ for all sufficiently large $j$. The centered
metric differential then gives
\begin{equation}\label{chord:rescaled-distance}
R_j(h):=\varepsilon_j^{-1}
d\bigl(y,\phi_i(z+\varepsilon_jh)\bigr)
\longrightarrow\abs{h}_{G_0}.
\end{equation}
When the chart point is absent, define the integrand below to
be zero. Restricting the nonnegative integral defining $J_y$
to this chart and changing variables gives, for every fixed
bounded ball $D_R\subset\R^n$ centered at zero,
\[
W a_j^{n/2}J_y(a_j)
\ge\int_{D_R}
\frac{1_{E_i}(z+\varepsilon_jh)
u_i(z+\varepsilon_jh)^p\rho_i(z+\varepsilon_jh)}
{\bigl(1+\kappa^2R_j(h)^2
u_i(z+\varepsilon_jh)^{2\beta}\bigr)^n}\dd h.
\]
The factor $a_j^{n/2}$ has canceled the Jacobian
$\varepsilon_j^n$. Fatou's lemma,
\eqref{chord:lebesgue-values}, and~\eqref{chord:rescaled-distance}
now imply
\[
W\ell\ge v_0^p\rho_0
\int_{D_R}\bigl(1+v_0^{4\beta}\abs{h}_{G_0}^2\bigr)^{-n}
\dd h.
\]
This step uses no uniform integrability of the rescaled densities.
Letting $R\to\infty$ and applying monotone convergence gives the
same bound with $D_R$ replaced by $\R^n$.
The linear substitution $H=v_0^{2\beta}G_0^{1/2}h$ has Jacobian
$v_0^{2n\beta}\sqrt{\det G_0}$. Since $2n\beta=p$, we obtain
\begin{align*}
W\ell
&\ge \abs{\theta_i(z)}
\int_{\R^n}(1+\abs{H}^2)^{-n}\dd H\\
&=\abs{\theta_i(z)}\,2^{-n}s_n
\ge2^{-n}s_n.
\end{align*}
The middle equality is the stereographic integral
\eqref{eq:stereographic-integral}. The final inequality uses the
nonzero integer multiplicity $\abs{\theta_i(z)}\ge1$. This proves
\eqref{chord:transform-density} without a bound on the multiplicity.
If $L_y>0$, Proposition~\ref{prop:chord-transform} and
\eqref{chord:transform-density} give
\[
\frac{s_n}{2^nW}\le(2L_y)^{-n/2}.
\]
Rearranging yields~\eqref{eq:chord-upper}. If $L_y=0$, that
upper bound is immediate. Its strict inequality follows from
$W<s_n$.
\end{proof}
\begin{remark}[The round sphere]\label{rem:chord-sphere}
The constants can be checked simultaneously in every dimension
on the unit round $S^n\subset\R^{n+1}$ with $v=1$. Here
$W=s_n$, $\nu$ is normalized area, and $L_y=2$ by
rotational symmetry. In stereographic coordinates chosen so that
$\abs{x-y}^2=4\abs{z}^2/(1+\abs{z}^2)$,
\[
J_y(a)=\frac{2^n}{s_n}
\int_{\R^n}\bigl(1+(1+4a)\abs{z}^2\bigr)^{-n}\dd z
=(1+4a)^{-n/2}.
\]
Thus the transform estimate is an equality, and its limiting
coefficient is $2^{-n}$, exactly the value in
\eqref{chord:transform-density} when $W=s_n$.
\end{remark}
Only the scalar values $v(y),v(x)$ and the distance $d(y,x)$
occur in~\eqref{eq:chord-upper}. The kernel
\[
(y,x)\longmapsto
v(y)^{2\beta}d(y,x)^2v(x)^{2\beta}
\]
is Borel and is integrable against $\nu\otimes\nu$, since $Y$
has finite diameter and $v^{p+2\beta}\in L^1(\mu)$. Accordingly,
the almost-everywhere upper bound can be averaged without any
jointly measurable choice of the covectors $D_xd(y,x)$.
==== END Sharp-integral-fillings-in-CAT(0)-spaces//sections/06-chords.tex ====
==== BEGIN Sharp-integral-fillings-in-CAT(0)-spaces//sections/07-conclusion.tex ====
\section{Averaging and completion of the proof}
\label{sec:conclusion}
We first derive the lower bound that contradicts the chord estimate of
Section~\ref{sec:chords}. Throughout this argument, \(n>2\), and we retain
the normalized extremizing cycle \(A\), its mass measure \(\mu\), and the
nonnegative minimizer \(v\) of Proposition~\ref{prop:minimizer}. Thus
\[
\beta=\frac1{n-2},\qquad p=\frac{2n}{n-2}=2+4\beta,
\qquad W=\int_Y v^p\dd\mu>0,
\qquad \dd\nu=\frac{v^p}{W}\dd\mu.
\]
We use the same finite, nonnegative Borel representative of \(v\) as in
Section~\ref{sec:chords}.
\begin{lemma}[Opposite averaged chord bound]
\label{lem:opposite-average}
Define
\begin{equation}
\label{finish:weighted-moment}
B_*:=\int_Y v^{p+2\beta}\dd\mu,
\qquad
I(y):=\int_Y d(y,x)^2v(x)^{p+2\beta}\dd\mu(x).
\end{equation}
Then \(0<B_*<\infty\), the function \(I\) is finite and continuous on
\(Y\), and
\begin{equation}
\label{finish:all-centers}
W^2\le B_*I(y)\qquad\text{for every }y\in Y.
\end{equation}
In particular, for the chord moments
\[
L_y=\int_Y
\bigl[v(y)^\beta d(y,x)v(x)^\beta\bigr]^2\dd\nu(x),
\]
one has
\begin{equation}
\label{finish:opposite-average}
\int_Y L_y\dd\nu(y)\ge2.
\end{equation}
\end{lemma}
\begin{proof}
Lemma~\ref{lem:powers} gives all finite moments of \(v\), so \(B_*\) is
finite. It is positive because \(W>0\). Write \(D_Y=\diam Y\). Then
\(0\le I(y)\le D_Y^2B_*\), and the triangle inequality gives
\[
|I(y)-I(z)|\le2D_YB_*d(y,z)\qquad(y,z\in Y).
\]
This proves the stated continuity.
Fix any \(y\in Y\) and put \(r=d(y,\cdot)\). By
Lemma~\ref{lem:powers}, the nonnegative function \(g=v^p\) belongs to
\(\Hcal\), and \(Dg=pv^{p-1}Dv\). The extension of the radial inequality
\eqref{eq:radial} furnished by Proposition~\ref{prop:sobolev-space}
therefore applies to this \(g\). Dividing by \(n\) and using
\(p/n=2\beta\) gives
\begin{equation}
\label{finish:radial-test}
W\le\int_Y r\left(v^p\sqrt{1-|Dr|^2}
-2\beta v^{p-1}\ip{Dr}{Dv}\right)\dd\mu.
\end{equation}
At almost every point of each current chart, take the direct sum of its
cotangent inner-product space with \(\R\). The vector
\[
\left(\sqrt{1-|Dr|^2},Dr\right)
\]
has norm one. Moreover, the identities
\[
\frac p2+\beta+1+\beta=p,
\qquad
\frac p2+\beta+\beta=p-1
\]
give the factorization
\begin{align*}
&v^p\sqrt{1-|Dr|^2}-2\beta v^{p-1}\ip{Dr}{Dv}\\
&\quad=
v^{p/2+\beta}
\left\langle
\left(\sqrt{1-|Dr|^2},Dr\right),
\left(v^{1+\beta},-2\beta v^\beta Dv\right)
\right\rangle\\
&\quad\le
v^{p/2+\beta}
\left(v^{2+2\beta}+4\beta^2v^{2\beta}|Dv|^2\right)^{1/2}.
\end{align*}
All powers in this formula are nonnegative powers of \(v\), and the
identity also holds on \(\{v=0\}\). Applying the integral
Cauchy--Schwarz inequality to \eqref{finish:radial-test} now yields
\begin{equation}
\label{finish:cauchy}
W^2\le I(y)
\int_Y\left(v^{2+2\beta}
+4\beta^2v^{2\beta}|Dv|^2\right)\dd\mu.
\end{equation}
Both factors are finite: the first was bounded above, and the second is
finite by the moment estimates and the Sobolev power
\(v^{1+\beta}\) in Lemma~\ref{lem:powers}.
To identify the last integral, use
\(\psi=v^{1+2\beta}\in\Hcal\) in the Euler equation
\eqref{eq:euler}. Its gradient is
\(D\psi=(1+2\beta)v^{2\beta}Dv\). Thus
\[
4\beta(1+2\beta)\int_Yv^{2\beta}|Dv|^2\dd\mu
+n\int_Yv^{2+2\beta}\dd\mu
=n\int_Yv^{p+2\beta}\dd\mu.
\]
Since \(1+2\beta=n\beta\), division by \(n\) proves
\begin{equation}
\label{finish:euler-moment}
\int_Y\left(v^{2+2\beta}
+4\beta^2v^{2\beta}|Dv|^2\right)\dd\mu=B_*.
\end{equation}
Equations~\eqref{finish:cauchy} and \eqref{finish:euler-moment}
prove \eqref{finish:all-centers}. The argument fixed an arbitrary center
before taking chart derivatives. Hence its conclusion holds for every
center, although the null set on which \(Dr\) is undefined may depend
on that center.
We have proved the all-center estimate \(W^2\le B_*I(y)\).
It remains to combine it with the CAT(0) variance inequality
\cite[Proposition~4.4]{Sturm2003}, which doubles this lower contribution
when we average over centers. We recall its proof for the finite measure
$v^{p+2\beta}\mu$.
Compactness of \(Y\) and continuity of \(I\) provide a minimizer
\(o\in Y\). This is a barycenter of the finite measure
\(v^{p+2\beta}\mu\). For \(y\in Y\), let \(o_t\) be the point at
fraction \(t\in(0,1)\) on the segment from \(o\) to \(y\). The
CAT(0) squared-distance inequality, integrated against
\(v(x)^{p+2\beta}\dd\mu(x)\), gives
\[
I(o)\le I(o_t)
\le(1-t)I(o)+tI(y)-t(1-t)B_*d(o,y)^2.
\]
Subtracting \((1-t)I(o)\), dividing by \(t\), and letting
\(t\downarrow0\) proves the variance inequality
\begin{equation}
\label{finish:variance}
I(y)\ge I(o)+B_*d(o,y)^2\qquad(y\in Y).
\end{equation}
This uses only the segment from \(o\) to \(y\).
The chord kernel is a nonnegative Borel function on \(Y\times Y\).
Its integral is finite, since its expression below is bounded above by
\(D_Y^2B_*^2/W^2\). Tonelli's theorem and
\eqref{finish:weighted-moment} give
\begin{equation}
\label{finish:double-integral}
\begin{aligned}
\int_YL_y\dd\nu(y)
&=\iint_{Y\times Y}
\bigl[v(y)^\beta d(y,x)v(x)^\beta\bigr]^2
\dd\nu(x)\dd\nu(y)\\
&=\frac1{W^2}\int_Y I(y)v(y)^{p+2\beta}\dd\mu(y)\\
&\ge\frac1{W^2}\left(
B_*I(o)+B_*\int_Y d(o,y)^2v(y)^{p+2\beta}\dd\mu(y)
\right)\\
&=\frac{2B_*I(o)}{W^2}\ge2.
\end{aligned}
\end{equation}
Here the first inequality is \eqref{finish:variance}, and the last is
\eqref{finish:all-centers} at \(y=o\). The averaged expressions contain
only distances and values of the Borel representative of \(v\); the
proof requires no joint choice of the derivatives of distance
functions. This proves \eqref{finish:opposite-average}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
We first prove the statement in every compact CAT(0) space by induction
on the cycle dimension \(n\ge2\). For such a space \(Y\), let
\(V(T)\) denote the infimum of the masses of integral fillings of an
integral \(n\)-cycle \(T\) in \(Y\). Theorem~\ref{thm:current-background}
ensures that this infimum is finite and attained. It therefore suffices
in each dimension to prove
\begin{equation}
\label{finish:compact-bound}
V(T)\le c_n\M(T)^{(n+1)/n}
\qquad\text{for every integral }n\text{-cycle }T\text{ in }Y.
\end{equation}
The zero cycle has the zero current as a filling.
In dimension \(n=2\), a violation of
\eqref{finish:compact-bound} would, by Lemma~\ref{lem:extremizer},
produce a nonzero extremizing cycle that can be rescaled to satisfy
\eqref{eq:normalization}. Proposition~\ref{prop:base} excludes this
cycle: its radial inequality and Euclidean lower density force
\(\M(A)\ge s_2\), whereas the normalization requires
\(\M(A)<s_2\). Thus \eqref{finish:compact-bound} holds in
dimension two, and attainment gives an integral filling with that mass
bound.
Now let \(n>2\), and assume the asserted existence of sharp integral
fillings in dimension \(n-1\) in every compact CAT(0) space. Suppose
that \eqref{finish:compact-bound} fails for a cycle \(T\) in a compact
CAT(0) space \(Y\). Since \(T\ne0\), we may choose
\[
c_n<c<\frac{V(T)}{\M(T)^{(n+1)/n}}.
\]
Lemma~\ref{lem:extremizer} produces a nonzero cycle \(A\) maximizing
\(V(B)-c\M(B)^{(n+1)/n}\) over integral \(n\)-cycles. After a constant
rescaling of the metric it satisfies \eqref{eq:extremizer} and
\eqref{eq:normalization}, in particular
\[
m:=\M(A)=((n+1)c)^{-n}<s_n.
\]
The rescaled space remains compact and CAT(0), so the induction
hypothesis continues to apply there.
Proposition~\ref{prop:radial} gives \eqref{eq:radial} for this cycle.
The sharp filling statement in dimension \(n-1\) is exactly the
induction input to Proposition~\ref{prop:small-set}. Consequently
Section~\ref{sec:sobolev}, and in particular the almost sharp Sobolev
inequality of Proposition~\ref{prop:almost-sobolev}, applies.
Propositions~\ref{prop:sobolev-space} and
\ref{prop:compact-embedding} supply the completed calculus and
compactness used in Proposition~\ref{prop:minimizer}. We therefore
obtain a nonnegative minimizer \(v\) satisfying \eqref{eq:euler} and
\eqref{eq:minimizer-normalization}, with
\[
0<W=\int_Yv^p\dd\mu<s_n.
\]
Lemmas~\ref{lem:powers} and \ref{lem:composites} justify the powers
and nonlinear tests used in Section~\ref{sec:chords} and in
Lemma~\ref{lem:opposite-average}.
For the probability measure \(\nu=v^p\mu/W\), the chord upper bound
\eqref{eq:chord-upper} holds for \(\nu\)-almost every center. Integrating
that bound and using Lemma~\ref{lem:opposite-average} gives
\[
2\le\int_YL_y\dd\nu(y)
\le2\left(\frac W{s_n}\right)^{2/n}<2,
\]
a contradiction. Hence no violating cycle \(T\) exists, and
\eqref{finish:compact-bound} holds in dimension \(n\). By attainment
in Theorem~\ref{thm:current-background}, every integral \(n\)-cycle
has an integral filling realizing \(V(T)\), with the required mass
bound. Its support is compact because it is a closed subset of \(Y\).
This completes the induction for every \(n\ge2\).
Lemma~\ref{lem:compact-reduction} now transfers this bound to every
compactly supported integral cycle \(T\in\I_n(X)\) in a proper
CAT(0) space. It supplies a compactly supported integral filling
\(S\) with
\[
\partial S=T,\qquad
\M(S)\le c_n\M(T)^{(n+1)/n}
=\frac{\M(T)^{(n+1)/n}}
{(n+1)^{(n+1)/n}\omega_{n+1}^{1/n}}.
\]
The zero cycle has the zero filling. This proves the theorem.
\end{proof}
==== END Sharp-integral-fillings-in-CAT(0)-spaces//sections/07-conclusion.tex ====
==== BEGIN Sharp-integral-fillings-in-CAT(0)-spaces//sections/08-classical.tex ====
\section{The classical domain inequality and sharpness}
\label{sec:classical}
We prove Corollary~\ref{cor:cartan-hadamard} and verify the optimality
of the filling coefficient. Both conclusions use the absence of
compactly supported top-dimensional cycles in a connected noncompact
oriented manifold.
\begin{lemma}\label{lem:top-dimensional-constancy}
Let $M$ be a connected, noncompact, oriented Riemannian manifold of
dimension $d$. If $C\in\I_d(M)$ has compact support and
$\partial C=0$, then $C=0$.
\end{lemma}
\begin{proof}
In a relatively compact oriented coordinate ball, the
top-dimensional representation of an integral current is integration
against an integer-valued locally integrable coefficient $\theta$.
Testing $\partial C=0$ with compactly supported smooth
$(d-1)$-forms shows that every distributional partial derivative of
$\theta$ is zero. Thus $\theta$ is constant almost everywhere on that
ball. For completeness, convolution with a smooth kernel gives
functions with zero ordinary gradient on each smaller ball; passing
to the local $L^1$ limit proves the assertion. Compatibility on
overlaps and connectedness give one constant on $M$. Some nonempty
open set lies outside the compact support, so this constant is zero.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:cartan-hadamard}]
By the Cartan--Hadamard theorem and Hopf--Rinow, $M$ is a proper
CAT(0) space and is diffeomorphic to $\R^d$; in particular it is
oriented and noncompact. See~\cite[Sections~I.3, II.1A and II.4]{BridsonHaefliger1999}
for the metric comparison and completeness statements.
Let $[[\Omega]]$ be its integration
current. Its mass is $\vol(\Omega)$, and the mass of
$T=\partial[[\Omega]]$ is $\operatorname{area}(\partial\Omega)$.
Theorem~\ref{thm:main}, with $n=d-1$, gives a compactly supported
$S\in\I_d(M)$ with $\partial S=T$ and
\[
\M(S)\le
\frac{\operatorname{area}(\partial\Omega)^{d/(d-1)}}
{d^{d/(d-1)}\omega_d^{1/(d-1)}}.
\]
Lemma~\ref{lem:top-dimensional-constancy} applies to
$S-[[\Omega]]$, so $S=[[\Omega]]$. Rearranging its mass bound
proves~\eqref{eq:classical}.
For a relatively compact set $\Omega$ of finite perimeter, the
standard representation of its integration current gives
$[[\Omega]]\in\I_d(M)$, with mass $\vol(\Omega)$ and boundary
mass equal to its perimeter; see~\cite[Chapter~3, Section~4, Theorem~4.3;
Chapter~6, Section~3, Remark~3.15]{Simon2014}. The same
argument applies verbatim.
\end{proof}
For $d=2$, the classical disk inequality gives
$L(\partial D)^2\ge4\pi\operatorname{area}(D)$ for a smooth
Riemannian disk of Gaussian curvature at most zero; see
\cite[Section~1.2, Theorem~3]{Izmestiev2014} for this formulation.
Filling the holes in a bounded smooth domain on a Cartan--Hadamard
surface increases area and decreases boundary length. Applying the disk
inequality to each resulting component, and then summing, proves the
same bound for the domain because $(\sum_j L_j)^2\ge\sum_jL_j^2$.
Together with Corollary~\ref{cor:cartan-hadamard}, this establishes the
Euclidean Cartan--Hadamard domain inequality in every ambient dimension.
To see that the coefficient in Theorem~\ref{thm:main} is optimal,
take $X=\R^{n+1}$ and let $T$ be the oriented boundary of the
Euclidean ball $B_R$. Its mass is $s_nR^n$, while
\[
\M([[B_R]])=\omega_{n+1}R^{n+1}
=c_n(s_nR^n)^{(n+1)/n}.
\]
Any compactly supported integral filling of $T$ equals $[[B_R]]$
by Lemma~\ref{lem:top-dimensional-constancy}. Hence equality occurs
and no smaller universal coefficient is possible.
==== END Sharp-integral-fillings-in-CAT(0)-spaces//sections/08-classical.tex ====
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title = {A proof of the {Cartan--Hadamard} conjecture for small volumes under a {Ricci} curvature lower bound},
year = {2026},
eprint = {2609.17093},
archivePrefix = {arXiv},
primaryClass = {math.DG},
note = {arXiv:2609.17093v2, revised 21 September 2026; \url{https://arxiv.org/abs/2609.17093v2}},
url = {https://arxiv.org/abs/2609.17093v2}
}
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year = {2026},
eprint = {2609.14505},
archivePrefix = {arXiv},
primaryClass = {math.DG},
note = {arXiv:2609.14505v1, 13 September 2026; \url{https://arxiv.org/abs/2609.14505v1}},
url = {https://arxiv.org/abs/2609.14505v1}
}
@misc{Peteranderl2026,
author = {Peteranderl, Jonas W.},
title = {Linear isoperimetric filling inequalities in {Hadamard} spaces at and above the asymptotic rank},
year = {2026},
eprint = {2608.26059},
archivePrefix = {arXiv},
primaryClass = {math.MG},
note = {arXiv:2608.26059v1, 26 August 2026; \url{https://arxiv.org/abs/2608.26059v1}},
url = {https://arxiv.org/abs/2608.26059v1}
}
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}
==== END Sharp-integral-fillings-in-CAT(0)-spaces//references.bib ====