64c3814
==== TABLE OF CONTENTS ====
Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//01-introduction.tex
Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//02-maps.tex
Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//03-sobolev.tex
Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//04-profile.tex
Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//04a-bv-calculus.tex
Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//05-comparison.tex
Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//06-dimensions.tex
Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//07-equality-regularity.tex
Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//08-rigidity.tex
Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//09-consequences.tex
Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//confined-boundary.tex
Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//glyph-mappings.tex
Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//main.tex
Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//offset-sphere.tex
Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//references.bib
===========================
==== BEGIN Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//01-introduction.tex ====
\section{Introduction}\label{sec:introduction}
A \emph{Cartan--Hadamard manifold} is a complete simply connected smooth
Riemannian manifold with nonpositive sectional curvature. The generalized
Cartan--Hadamard isoperimetric conjecture asserts that if its sectional
curvature is at most $\kappa\le0$, then every relatively compact smooth
domain has boundary area at least that of the equal-volume ball in the
simply connected space form of curvature $\kappa$. This compares arbitrary
domains, so geodesic-ball volume comparison alone does not suffice.
We prove the conjecture in every dimension, for arbitrary measurable sets
of finite volume and finite perimeter.
\subsection{The model comparison}
Write $\omega_n$ for the volume of the unit ball in $\R^n$, set
$m=n-1$, and put $s_m=|\mathbb S^m|=n\omega_n$. For $b\ge0$, define
\begin{equation}\label{eq:model-functions}
\sn_b(r)=
\begin{cases}r,&b=0,\\ b^{-1}\sinh(br),&b>0,\end{cases}
\qquad \cs_b(r)=\sn_b'(r).
\end{equation}
The area and volume of a radius-$r$ ball in curvature $-b^2$ are
\begin{equation}\label{eq:model-area-volume}
\mathcal A_b(r)=s_m\sn_b(r)^m,
\qquad
\mathcal V_b(r)=s_m\int_0^r\sn_b(t)^m\,dt.
\end{equation}
Both functions increase continuously from zero to infinity. The model
isoperimetric profile is
\[
\mathcal I_b(V)=\mathcal A_b\bigl(\mathcal V_b^{-1}(V)\bigr),
\qquad V\ge0.
\]
In particular, $\mathcal I_b(0)=0$, and $\mathcal I_b$ is continuous
and strictly increasing on $[0,\infty)$.
For a measurable set $E\subset M$, its \emph{ambient perimeter} is
\begin{equation}\label{eq:ambient-perimeter}
P_g(E)=|D\chi_E|_g(M)
=\sup\left\{\int_E\divg_g X\,d\vol_g:
X\in C_c^\infty(TM),\ |X|_g\le1\right\}.
\end{equation}
Here $\chi_E$ is the indicator of $E$, and $|D\chi_E|_g$ is its
variation measure. A set has finite perimeter when this quantity is
finite. Sets are identified up to ambient volume zero. For a relatively
compact smooth domain, perimeter is its boundary area.
\begin{theorem}\label{thm:main}
Let $n\ge2$, let $\kappa\le0$, and let $(M^n,g)$ be a complete simply
connected smooth Riemannian manifold with $\Sec_g\le\kappa$.
Every measurable set $E\subset M$ with $\vol_g(E)<\infty$ and
$P_g(E)<\infty$ satisfies
\[
P_g(E)\ge \mathcal I_{\sqrt{-\kappa}}\bigl(\vol_g(E)\bigr).
\]
\end{theorem}
For $\kappa=0$, the conclusion reads
\[
P_g(E)
\ge n\omega_n^{1/n}\vol_g(E)^{(n-1)/n}.
\]
For $\kappa=-1$, if
$\vol_g(E)=n\omega_n\int_0^r\sinh^{n-1}t\,dt$, it reads
\[
P_g(E)\ge n\omega_n\sinh^{n-1}r.
\]
The comparison concerns full ambient perimeter, including boundary on
any confining sphere used in the proof. It applies to unbounded sets
of finite volume as well as bounded ones. Balls in the model spaces
attain the bounds.
The Euclidean equality case has the following classification.
\begin{theorem}[Euclidean equality rigidity]\label{thm:equality}
Let $(M^n,g)$ be complete, simply connected and smooth, with
$n\ge2$ and $\Sec_g\le0$. Let $E\subset M$ be bounded and measurable,
with $0<\vol_g(E)<\infty$ and $P_g(E)<\infty$. Then
\[
P_g(E)=n\omega_n^{1/n}\vol_g(E)^{(n-1)/n}
\]
if and only if $E$ agrees up to ambient volume zero with an open region
whose induced Riemannian metric is isometric to a round Euclidean ball.
\end{theorem}
Thus rigidity determines the metric on the entire equality region.
It does not require the ambient manifold outside that region to be
flat. Boundedness and positive volume belong to the equality theorem;
Theorem~\ref{thm:main} has the full finite-volume scope stated above.
\subsection{Model-ball spectral and torsional consequences}
For a smooth relatively compact domain $D$ in a Riemannian manifold
$(M,g)$ and $1<p<\infty$, let
$W^{1,p}_0(D)$ be the closure of $C_c^\infty(D)$ in the Sobolev norm, and
define
\[
\begin{aligned}
\lambda_{1,p}(D)&=\inf_{0\ne u\in W^{1,p}_0(D)}
\frac{\int_D|\nabla_g u|_g^p\,d\vol_g}{\int_D|u|^p\,d\vol_g},\\
T_p(D)&=\sup_{0\ne u\in W^{1,p}_0(D)}
\frac{\bigl(\int_D|u|\,d\vol_g\bigr)^p}
{\int_D|\nabla_g u|_g^p\,d\vol_g}.
\end{aligned}
\]
The first is the Dirichlet variational value for
$-\Delta_{p,g}u=\lambda|u|^{p-2}u$, where\newline
$\Delta_{p,g}u=\divg_g(|\nabla_g u|_g^{p-2}\nabla_g u)$ is the
unnormalized $p$-Laplacian. The second uses the quotient normalization
of $p$-torsional rigidity in \cite[(1.1)--(1.6)]{BrianiButtazzoPrinari2022}.
\begin{corollary}[Model-ball Faber--Krahn and Saint--Venant comparisons]
\label{cor:model-ball-spectral}
Let $n\ge2$, $\kappa\le0$ and $1<p<\infty$. Let $(M^n,g)$ be complete,
simply connected, smooth and without boundary, with $\Sec_g\le\kappa$.
For a nonempty smooth domain $D\Subset M$, let $B_\kappa(|D|)$ be a
geodesic ball of volume $|D|=\vol_g(D)$ in the simply connected
$n$-dimensional space form of curvature $\kappa$. Then
\[
\lambda_{1,p}(D)\ge\lambda_{1,p}\bigl(B_\kappa(|D|)\bigr),
\qquad T_p(D)\le T_p\bigl(B_\kappa(|D|)\bigr).
\]
\end{corollary}
Theorem~\ref{thm:main} supplies the isoperimetric input for
equimeasurable radial rearrangement into the model ball
\cite{NobiliViolo2025}. This rearrangement preserves the $L^1$ and
$L^p$ integrals and does not increase the $p$-Dirichlet energy,
giving the two quotient comparisons. The proof appears in
Section~\ref{sec:consequences}.
\subsection{History and related work}
The surface inequality originates in Weil's work and the
subharmonic-function approach of Beckenbach and Rad\'o; Bol developed
the comparison with nonzero model curvature
\cite{Weil1926,BeckenbachRado1933,Bol1941}.
We use the smooth surface formulation of Chavel and Feldman
\cite{ChavelFeldman1980}. Kleiner proved the model comparison in
dimension three, including its equality case
\cite[Theorem~2]{Kleiner1992}. Croke proved the sharp Euclidean
comparison in dimension four \cite{Croke1984}. Kloeckner and Kuperberg
subsequently developed optical and chord-based comparison methods in
dimensions two and four; their four-dimensional negative-curvature
comparison includes an explicit restriction on the domain's chord
length and the radius of its model ball
\cite[Theorem~1.5]{KloecknerKuperberg2019}.
A second line of work connects isoperimetry to total curvature.
Ghomi and Spruck showed that, in a fixed Cartan--Hadamard
$n$-manifold, the lower bound $\int_\Gamma|\mathrm{GK}|\ge s_{n-1}$
for every closed convex $C^{1,1}$ hypersurface $\Gamma$ implies the
Euclidean isoperimetric inequality for bounded positive-volume
finite-perimeter sets, with equality only for regions isometric to
Euclidean balls \cite[Theorem~7.1]{GhomiSpruck2022}. Here
$\mathrm{GK}$ is the Gauss--Kronecker curvature, and such a convex
hypersurface bounds a compact convex set with nonempty interior. Their analysis of minimizers
confined to a geodesic ball includes the obstacle regularity that we
use \cite[Lemma~7.2(i),(ii)]{GhomiSpruck2022}; this regularity lemma
does not assume the total-curvature inequality. Ghomi and Stavroulakis
prove an inequality for sufficiently small $C^2$ perturbations of a
Euclidean ball metric with nonpositive curvature
\cite{GhomiStavroulakis2026}. Their theorem concerns the metric on the
ball itself and does not require an extension to a complete
Cartan--Hadamard manifold. Related results also include Ghomi's
inequalities under negative-curvature pinching \cite{Ghomi2026Pinched}.
Recent preprints overlap with several parts of the problem.
Chen, Ghomi and Wang establish Euclidean comparison and rigidity in
dimensions three through nine, report calibration verification through
at least dimension thirteen, and announce generalized comparison in
dimensions four and five
\cite[Theorems~1.1--1.2 and the subsequent discussion]{ChenGhomiWang2026HigherDimensions}.
Their argument uses boundary-pair integrals, Jacobi fields and a
confined profile; its appendix includes exact-arithmetic certificates.
Agnoletto, Da Silva, Nardulli and Resende prove the small-volume
model comparison under a Ricci lower bound, and reduce the
unrestricted-volume comparison to equality rigidity in a
noncollapsing boundaryless Alexandrov class with two-sided curvature
bounds \cite[Theorems~1.1 and 1.3, Assumption~1]{AgnolettoDaSilvaNardulliResende2026}.
Wheeler proves all-dimensional Euclidean comparison under controlled
variation of a variable negative-curvature scale
\cite[Theorem~1.1 and Definition~1.2]{Wheeler2026}.
These dimensional or curvature restrictions, and the Alexandrov
rigidity condition in the reduction, differ from the hypotheses of
Theorems~\ref{thm:main} and~\ref{thm:equality}.
\subsection{Main ideas and proof organization}
The central analytic estimate is a sharp critical Sobolev inequality
on a hypersurface. Normalize the curvature bound to $-b^2$, with
$b\in\{0,1\}$. For a locally $C^{1,1}$ immersed hypersurface
$\Sigma^m$ with $m=n-1\ge3$, let $d\sigma$ be its induced area
measure, $\nabla_\Sigma$ its tangential gradient, and $h$ its signed
normalized scalar mean curvature for a local unit normal. Put
\[
q=\frac{2m}{m-2},\qquad c=\frac4{m-2},\qquad Y_m=m s_m^{2/m}.
\]
We prove
\begin{equation}\label{eq:intro-sobolev}
\int_\Sigma\left(c|\nabla_\Sigma v|^2+m(h^2-b^2)v^2\right)\,d\sigma
\ge Y_m\left(\int_\Sigma|v|^q\,d\sigma\right)^{2/q}
\end{equation}
for Lipschitz tests compactly supported in the interior of $\Sigma$.
The square $h^2$ is independent of the local normal. The hypersurface
need not be complete or connected. The coefficient is the sharp
Euclidean critical Sobolev constant of Aubin and Talenti in this
normalization \cite{Aubin1976,Talenti1976}.
Sections~\ref{sec:maps} and~\ref{sec:sobolev} establish this estimate.
Radial maps into a unit sphere come with positive weights whose
logarithmic Hessians and map differentials satisfy compatible
curvature-comparison bounds. The center and scale can be chosen so
that every spherical coordinate has zero mean for a prescribed
compactly supported nonatomic probability measure. The bounds combine
into a completed square in the hypersurface energy.
If the critical quotient were below the Euclidean threshold, local
compactness and Br\'ezis--Lieb splitting would give a Dirichlet
minimizer \cite{BrezisLieb1983}. Balancing its critical mass and using
the spherical coordinates as variations forces a first-order identity.
A weighted zero extension then has zero gradient and nonzero mass,
contradicting the Dirichlet condition. The centering and coordinate
variations are related to Li and Yau's conformal-volume argument
\cite[proof of Theorem~1]{LiYau1982}; the maps and weighted centering
needed here are constructed directly.
To compare domains, the difficulty is that the mean curvature of an
arbitrary boundary is uncontrolled. Sections~\ref{sec:profile}
and~\ref{sec:comparison} instead minimize full ambient perimeter at
fixed volume inside a large geodesic ball. This is the confined-profile
strategy of Kleiner, with the regularity formulation of Ghomi and Spruck
\cite{Kleiner1992,GhomiSpruck2022}. Free variations identify the
boundary curvature with the profile derivative; inward variations
control it at contact with the confining sphere. A deletion-and-volume-repair argument first proves area growth near the singular set.
Cutoffs with vanishing Dirichlet energy then make the constant test
legitimate in \eqref{eq:intro-sobolev}. This gives one differential
inequality whose integral is exactly the model comparison.
Section~\ref{sec:dimensions} supplies the classical two- and
three-dimensional inputs, their strict BV approximation, a common
finite-volume cutoff argument, and metric scaling to arbitrary
$\kappa<0$.
For equality, Section~\ref{sec:equality-regularity} first uses the
proved Euclidean comparison to make a bounded equality set a local
perimeter almost-minimizer. Local regularity
\cite[Corollary~1.6]{AntonelliPasqualettoPozzetta2022}, followed by a
difference-quotient argument, gives a locally $C^{1,1}$ regular
boundary with the same signed constant mean curvature on every
component. Singular cutoffs extend first variation and the Sobolev
inequality to ambient tests on the whole perimeter support.
Section~\ref{sec:rigidity} locates that support and then recovers the
region. In dimensions at least four, the constant equality test in
\eqref{eq:intro-sobolev} gives a sharp spectral bound. Logarithm
coordinates centered at a squared-distance barycenter use this bound
to give an upper estimate for the radial second moment. A radial flux
identity gives the matching lower estimate. Equality forces the
outward normal to be radial with constant radius, so the perimeter
support lies on a sphere. Periodic Wirtinger replaces the spectral
argument in dimension two; a punctured flux identity replaces it in
dimension three and produces a sphere in a tangent space whose center
may be offset from the logarithm origin. In all dimensions, BV phase
constancy and boundedness identify the occupied interior. Exponential
metric domination and equality of volume then force every metric
eigenvalue to be Euclidean throughout that ball. This last step proves
whole-region rigidity, beyond the preceding containment of its boundary.
\subsection{Conventions}
All hypersurface metrics and area measures are induced. For a local
unit normal $N$, we use the signed trace curvature
$H=\divg_\Sigma N$ and the signed normalized scalar $h=H/m$.
For a smooth ambient function $u$, its restriction satisfies
\[
\Delta_\Sigma u=\tr_{T\Sigma}\Hess_M u-HNu.
\]
Thus a Euclidean sphere of radius $r$ has outward $h=1/r$, and a
curvature-$-1$ sphere has $h=\coth r$. Reversing $N$ reverses $h$;
only normal-independent expressions are used when no global normal is
chosen. For boundaries of sets, the occupied phase fixes $N$.
All functions are real-valued. We often write $|E|=\vol_g(E)$ and
$P(E)=P_g(E)$, and use $P(E;D)=|D\chi_E|(D)$ for the perimeter
measure on a Borel set $D$. For $p\in M$ we write
$\log_p=\exp_p^{-1}$, $r_p(x)=d(p,x)$ and $D_p=r_p^2/2$;
Section~\ref{sec:maps} proves the comparisons used for these functions.
==== END Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//01-introduction.tex ====
==== BEGIN Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//02-maps.tex ====
\section{Comparison maps into a sphere}
\label{sec:maps}
We seek a sphere-valued map with a controlled differential and a
compatible Hessian bound for its weight. These two estimates will combine
in the hypersurface Sobolev energy. A second step chooses the map's
parameters to balance a given measure.
Fix $b\in\{0,1\}$ and a complete simply connected manifold $(M^n,g)$
with $\Sec_g\le-b^2$. We use the model functions $\sn_b$ and $\cs_b$
from Section~\ref{sec:introduction}; thus
\[
\sn_b'=\cs_b,\qquad \cs_b'=b^2\sn_b,
\qquad \cs_b^2-b^2\sn_b^2=1.
\]
The Cartan--Hadamard theorem makes $\exp_p:T_pM\to M$ a diffeomorphism
for every $p$; see \cite[Theorem~6.2.2]{PetersenManuscript}.
For $x\ne p$, put $r=d(p,x)$ and $\theta=\exp_p^{-1}(x)/r$.
The sphere $\mathbb S(T_pM\oplus\R)$ has its round metric induced by
$g_p$ and the usual metric on $\R$.
Write $g_{\mathbb S_p^{n-1}}$ for the round metric on the unit sphere
in $T_pM$.
In the radial coordinate $\sn_b(r/2)/\cs_b(r/2)$, the model polar
metric $dr^2+\sn_b(r)^2g_{\mathbb S_p^{n-1}}$ is conformal to the
Euclidean polar metric. We dilate this coordinate by $a>0$ and apply
inverse stereographic projection, obtaining the map below and its model
conformal factor $f_{p,a}$:
\begin{equation}\label{eq:map-definition}
\begin{gathered}
z=a\frac{\sn_b(r/2)}{\cs_b(r/2)},\qquad
\Phi_{p,a}(x)=\left(\frac{2z}{1+z^2}\theta,
\frac{1-z^2}{1+z^2}\right),\\
f_{p,a}(x)=e^{u_{p,a}(x)}
=\frac{a}{\cs_b(r/2)^2+a^2\sn_b(r/2)^2}.
\end{gathered}
\end{equation}
At the pole set $\Phi_{p,a}(p)=(0,1)$ and $f_{p,a}(p)=a$.
In particular the zero-curvature formulas are explicitly
$z=ar/2$ and $f_{p,a}=a/(1+a^2r^2/4)$.
\begin{lemma}[Comparison maps]\label{lem:comparison-maps}
The map $\Phi_{p,a}$ and the positive function $f_{p,a}$ are smooth on
$M$. For every $X\in T_xM$,
\begin{equation}\label{eq:map-comparison}
\begin{aligned}
|d\Phi_{p,a}(X)|&\le f_{p,a}|X|,\\
\Hess u_{p,a}&\le du_{p,a}\otimes du_{p,a}
-\frac12\bigl(b^2+f_{p,a}^2+|\nabla u_{p,a}|^2\bigr)g.
\end{aligned}
\end{equation}
The second inequality is one of quadratic forms. The functions depend
smoothly on $(p,a,x)$. For fixed $o\in M$, parallel transport of the
first component of $\Phi_{p,a}$ from $p$ to $o$ along the unique geodesic
also depends smoothly on $(p,a,x)$.
\end{lemma}
\begin{proof}
The two components of $\Phi_{p,a}$ have squared norms summing to one.
In normal coordinates $y=\exp_p^{-1}(x)$, its first component is
$2z\,y/(r(1+z^2))$. The coefficient, the last component, and $f_{p,a}$
are smooth functions of $r^2$ at zero. Their denominators are positive.
This proves smoothness at the pole, including that of $u=\log f$.
The map $(p,y)\mapsto(p,\exp_p y)$ from $TM$ to $M\times M$ is a
bijective local diffeomorphism, hence a diffeomorphism. Its inverse
and the same even radial expressions prove joint smoothness.
The paths $t\mapsto\exp_o(t\exp_o^{-1}p)$ vary smoothly with $p$;
smooth dependence for the parallel-transport equation proves the
last assertion.
We next establish the one-sided distance comparison needed below:
\begin{equation}\label{eq:distance-hessian-comparison}
\mathcal H:=\Hess r-\frac{\cs_b(r)}{\sn_b(r)}
(g-dr\otimes dr)\ge0\qquad(r>0).
\end{equation}
Indeed, for a vector $X$ perpendicular to a unit radial geodesic of
length $r$, the Jacobi field with endpoint values $0,X$ gives
\[
\Hess r(X,X)
=\int_0^r\bigl(|D_tJ|^2-\langle R(J,\dot\gamma)\dot\gamma,J\rangle\bigr)\,dt
\ge\int_0^r(|D_tJ|^2+b^2|J|^2)\,dt.
\]
In a parallel trivialization the last energy is minimized by
$J_0(t)=\sn_b(t)X/\sn_b(r)$ and equals
$\cs_b(r)|X|^2/\sn_b(r)$. To verify the minimum, write $J=J_0+V$;
integration by parts makes the cross term zero because $V$ vanishes
at the endpoints and $J_0''=b^2J_0$. The remaining energy of $V$ is
nonnegative. The radial and mixed components of $\Hess r$ vanish,
proving \eqref{eq:distance-hessian-comparison} without a lower
curvature bound.
For an angular variation field $J$ along a radial geodesic, Gauss's
lemma and commutation of the variation fields give
\[
\frac d{dr}|J|^2=2\Hess r(J,J)
\ge2\frac{\cs_b(r)}{\sn_b(r)}|J|^2.
\]
Thus $|J|^2/\sn_b(r)^2$ is nondecreasing, with limit equal to the
initial angular squared norm at zero. Applying this to every angular
vector proves the polar metric comparison
\begin{equation}\label{eq:polar-comparison}
g\ge dr^2+\sn_b(r)^2g_{\mathbb S_p^{n-1}}.
\end{equation}
The round stereographic metric is
$4(dz^2+z^2g_{\mathbb S_p^{n-1}})/(1+z^2)^2$.
The identities
\[
\frac{2z'}{1+z^2}=f,\qquad
\frac{2z}{1+z^2}=f\sn_b(r)
\]
therefore give
\begin{equation}\label{eq:map-pullback}
\Phi_{p,a}^*g_{\mathbb S(T_pM\oplus\R)}
=f^2\bigl(dr^2+\sn_b(r)^2g_{\mathbb S_p^{n-1}}\bigr)\le f^2g.
\end{equation}
This proves the differential estimate off the pole.
For the Hessian calculation write
\[
A=a^2+b^2,\qquad
D=\cs_b(r/2)^2+a^2\sn_b(r/2)^2=1+A\sn_b(r/2)^2.
\]
Then $f=a/D$, $D'=A\sn_b(r)/2$, and $D''=A\cs_b(r)/2$.
The double-angle identities give
\[
b^2D^2+a^2+\frac{A^2}{4}\sn_b(r)^2=A\cs_b(r)D.
\]
For example, substituting $t=\sn_b(r/2)^2$ makes both sides
$A+(A^2+2b^2A)t+2b^2A^2t^2$. Consequently
\begin{equation}\label{eq:radial-identities}
\begin{gathered}
u'=-\frac{A\sn_b(r)}{2D}\le0,\qquad
u''=(u')^2+u'\frac{\cs_b(r)}{\sn_b(r)},\\
u'\frac{\cs_b(r)}{\sn_b(r)}
=-\frac12\bigl(b^2+f^2+(u')^2\bigr).
\end{gathered}
\end{equation}
Combining these identities with \eqref{eq:distance-hessian-comparison},
\[
\Hess u
=du\otimes du-\frac12\bigl(b^2+f^2+|\nabla u|^2\bigr)g
+u'\mathcal H.
\]
The last term is negative semidefinite. Finally, the pole expansions
\[
d\Phi_{p,a}|_p(X)=(aX,0),\qquad
u=\log a-\frac{a^2+b^2}{4}r^2+O(r^4)
\]
show that both estimates in \eqref{eq:map-comparison} also hold at $p$.
\end{proof}
The zero-curvature case also gives three comparisons that will be used
in the equality argument. They express, respectively, convexity of
squared distance, expansion by the exponential map, and contraction
by its inverse.
\begin{corollary}[Distance and metric comparison]\label{lem:distance-comparison}
Let $(M^n,g)$ be complete and simply connected with $\Sec_g\le0$.
For $p\in M$, write $\log_p=\exp_p^{-1}$, $r_p(x)=d(p,x)$, and
$D_p=r_p^2/2$. Then
\begin{equation}\label{eq:distance-metric-comparison}
\Hess_gD_p\ge g,\qquad (\exp_p)^*g\ge g_p,
\qquad |d\log_p(X)|_{g_p}\le |X|_g.
\end{equation}
Here $g_p$ on $T_pM$ denotes its constant Euclidean metric.
\end{corollary}
\begin{proof}
With $b=0$, \eqref{eq:distance-hessian-comparison} gives
\[
\Hess D_p=dr_p\otimes dr_p+r_p\Hess r_p\ge g
\]
away from $p$; smoothness of squared distance and normal coordinates
give equality at $p$. The $b=0$ case of
\eqref{eq:polar-comparison} is exactly $(\exp_p)^*g\ge g_p$,
including at the origin by continuity. Applying this differential
inequality to the inverse map gives the last assertion.
\end{proof}
The map parameters have supplied the geometric estimates. We next
choose the center and scale so that the coordinate functions have zero
mean for a prescribed measure. This use of spherical centering is
related to the conformal variational argument of Li and Yau
\cite[pp.~274--275]{LiYau1982}; the required centering statement is
proved below for these comparison maps.
\begin{lemma}[Balancing a nonatomic measure]\label{lem:balancing}
Let $\mu$ be a compactly supported Borel probability measure on $M$
with $\mu(\{x\})=0$ for every $x$. If $\supp\mu\subset B_D(o)$,
there are $p\in B_D(o)$ and finite $a>0$ such that
\begin{equation}\label{eq:balanced-map}
\int_M\Phi_{p,a}(x)\,d\mu(x)=0\quad\text{in }T_pM\oplus\R.
\end{equation}
\end{lemma}
\begin{proof}
Write $K=\supp\mu$, parameterize $p$ by $\xi=\exp_o^{-1}p$, and let
$P_{p\to o}$ be radial parallel transport. Define
\[
F(\xi,a)=(F_1,F_2)
=\int_M(P_{p\to o}\oplus\mathrm{id}_{\R})\Phi_{p,a}(x)\,d\mu(x).
\]
It is continuous on every cylinder
$\mathcal C=\{\xi\in T_oM:|\xi|\le D\}\times[a_0,a_1]$
with $0<a_0<a_1<\infty$: the integrand is jointly continuous and has
norm one. We choose the two horizontal faces so that $F$ points
strictly inward on the entire boundary.
First suppose $|\xi|=D$. Put $\rho=d(o,\cdot)$. Radial parallel
transport sends $\xi$ to $D\nabla\rho(p)$. Convexity of $\rho$ along
the geodesic from $p$ to $x\in K$ gives
\[
\langle\xi,P_{p\to o}\theta_p(x)\rangle
\le\frac{D(\rho(x)-D)}{d(p,x)}<0.
\]
Here $p\notin K$; the derivative is taken only at $p\ne o$, so a
geodesic passing through $o$ causes no difficulty. The first component
of $\Phi_{p,a}$ is a positive multiple of $\theta_p(x)$ on this face.
Hence
\begin{equation}\label{eq:balance-side}
\langle\xi,F_1(\xi,a)\rangle<0
\qquad(|\xi|=D,\ a>0).
\end{equation}
Set $t_b(r)=\sn_b(r/2)/\cs_b(r/2)$, an increasing positive function
for $r>0$. Choose $a_0>0$ so that $a_0t_b(2D)<1$.
Since $d(p,x)<2D$ for $|\xi|\le D$ and $x\in K$, we obtain
\begin{equation}\label{eq:balance-bottom}
F_2(\xi,a_0)\ge
\frac{1-a_0^2t_b(2D)^2}{1+a_0^2t_b(2D)^2}>0.
\end{equation}
We also have
\begin{equation}\label{eq:uniform-atomless}
\lim_{\delta\downarrow0}\sup_{p\in\overline B_D(o)}
\mu(B_\delta(p))=0.
\end{equation}
Otherwise there would be $\varepsilon>0$, $\delta_j\to0$, and centers
$p_j$ with $\mu(B_{\delta_j}(p_j))\ge\varepsilon$. The closed ball is
compact by completeness, so a subsequence of centers converges to
$p_\infty$. Every $B_s(p_\infty)$ then has mass at least $\varepsilon$.
Continuity of a finite measure from above gives an atom at $p_\infty$,
a contradiction.
Choose $\delta>0$ so that the supremum in
\eqref{eq:uniform-atomless} is less than $1/4$, and choose finite
$a_1>a_0$ with $a_1t_b(\delta)\ge\sqrt3$.
The last coordinate of $\Phi_{p,a_1}$ is at most $-1/2$ outside
$B_\delta(p)$ and at most one everywhere. Thus
\begin{equation}\label{eq:balance-top}
F_2(\xi,a_1)\le-\frac18<0\qquad(|\xi|\le D).
\end{equation}
Let $\Pi_{\mathcal C}$ be Euclidean nearest-point projection onto the
compact convex cylinder. Brouwer's theorem gives a fixed point of
$T(\zeta)=\Pi_{\mathcal C}(\zeta+F(\zeta))$. The projection inequality
at that point is
\begin{equation}\label{eq:projection-fixed-point}
\langle F(\zeta),y-\zeta\rangle\le0
\qquad(y\in\mathcal C).
\end{equation}
On the side face, choosing $y=((1-t)\xi,a)$ with small $t>0$ contradicts
\eqref{eq:balance-side}. On the lower or upper face, moving the last
coordinate inward contradicts \eqref{eq:balance-bottom} or
\eqref{eq:balance-top}. These choices remain admissible at corners.
The fixed point is therefore interior. Taking both signs of every
small displacement in \eqref{eq:projection-fixed-point} gives $F=0$.
Parallel transport is an isomorphism, which proves
\eqref{eq:balanced-map} with the asserted parameters.
\end{proof}
==== END Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//02-maps.tex ====
==== BEGIN Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//03-sobolev.tex ====
\section{The sharp hypersurface Sobolev inequality}
\label{sec:sobolev}
We now turn the comparison maps into a sharp inequality on a possibly
incomplete hypersurface. The proof first rules out a Dirichlet quotient
below the Euclidean constant; a point cutoff then permits tests supported
on an entire compact component.
Let $b\in\{0,1\}$, and let $M^n$ be complete and simply connected,
with $\Sec_g\le-b^2$. Put $m=n-1\ge3$ and
\[
q=\frac{2m}{m-2},\qquad c=\frac4{m-2}=q-2,\qquad
\alpha=\frac{m-2}{2},\qquad Y_m=m s_m^{2/m}.
\]
Let $\Sigma^m$ be a locally $C^{1,1}$ immersed hypersurface in $M$,
with its induced metric and area measure $d\sigma$. It need not be
complete and may have boundary. Compactness is understood in the
manifold topology of $\Sigma$, and $\Sigma^\circ$ denotes its interior.
Ambient functions and maps on $\Sigma$ are composed with the immersion.
For a local unit normal $N$, write $mh=\divg_\Sigma N$ almost everywhere.
The square $h^2$ is independent of the choice of normal. Define
\begin{equation}
\label{eq:sobolev-form}
Q_\Sigma(v)=\int_\Sigma
\left(c|\nabla_\Sigma v|^2+m(h^2-b^2)v^2\right)\,d\sigma.
\end{equation}
For open $U\Subset\Sigma^\circ$, let $H^1_0(U)$ be the closure of
compactly supported Lipschitz functions in the induced $H^1$ norm;
no regularity of $\partial U$ is assumed.
\begin{theorem}[Hypersurface Sobolev inequality]
\label{thm:extrinsic-sobolev}
For every Lipschitz function $v$ with compact support in $\Sigma^\circ$,
\begin{equation}
\label{eq:extrinsic-sobolev}
Q_\Sigma(v)\ge
Y_m\left(\int_\Sigma |v|^q\,d\sigma\right)^{2/q}.
\end{equation}
The inequality also holds for the zero extension of every
$v\in H^1_0(U)$, where $U\Subset\Sigma^\circ$ is open.
\end{theorem}
The constant is sharp: on a geodesic sphere in the space form of
curvature $-b^2$, the constant function gives equality.
By this definition, zero extension satisfies
\begin{equation}
\label{eq:sobolev-zero-extension}
E_0v\in H^1(\Sigma^\circ),\qquad
\nabla_\Sigma(E_0v)=\mathbf1_U\nabla_\Sigma v
\quad\text{almost everywhere}.
\end{equation}
Both statements hold for the approximating functions and pass to their
$H^1$ limits, including when $\partial U$ has positive measure.
In $C^{1,1}$ charts the induced metric is locally Lipschitz and uniformly
positive definite on compact subsets, and $h$ is locally essentially
bounded. Thus the potential in~\eqref{eq:sobolev-form} is bounded
near $\bar U$ and $Q_\Sigma$ is continuous on $H^1_0(U)$.
\subsection{A geometric square}
The comparison maps provide the following lower bound before any
minimization takes place.
\begin{lemma}
\label{lem:square-estimate}
Fix $p\in M$ and $a>0$, and let $\Phi=\Phi_{p,a}$ and
$f=e^u=f_{p,a}$ be the map and functions of
Lemma~\ref{lem:comparison-maps}. For every compactly supported
Lipschitz function $v$ in $\Sigma^\circ$,
\begin{equation}
\label{eq:square-estimate}
\begin{aligned}
Q_\Sigma(v)
&\ge\int_\Sigma\left(
mf^2v^2+m(h+Nu)^2v^2
+c|\nabla_\Sigma v-\alpha v\nabla_\Sigma u|^2\right)\,d\sigma\\
&\ge\int_\Sigma\left(
v^2|d(\Phi|_\Sigma)|^2
+c|\nabla_\Sigma v-\alpha v\nabla_\Sigma u|^2\right)\,d\sigma.
\end{aligned}
\end{equation}
The same inequalities hold on $H^1_0(U)$ for every
$U\Subset\Sigma^\circ$. In particular, $Q_\Sigma$ is nonnegative
on all these functions.
\end{lemma}
\begin{proof}
Write $\nabla=\nabla_\Sigma$ and $\beta=Nu$. The restriction formula is
\begin{equation}
\label{eq:sobolev-restriction-laplacian}
\Delta_\Sigma u=\tr_{T\Sigma}\Hess_g u-mh\beta.
\end{equation}
This holds both almost everywhere and weakly: in a $C^{1,1}$
parametrization, the induced divergence expression has Lipschitz
coefficients, and the restricted smooth ambient function has Lipschitz
first derivatives. The ordinary chain rule therefore gives the same
formula as the distributional divergence.
Taking the tangential trace in Lemma~\ref{lem:comparison-maps}, and
using $|\nabla_g u|^2=|\nabla u|^2+\beta^2$, gives
\[
2\Delta_\Sigma u
\le -(m-2)|\nabla u|^2-m(b^2+f^2)-m\beta^2-2mh\beta.
\]
Equivalently,
\[
m(h^2-b^2)\ge
mf^2+m(h+\beta)^2+(m-2)|\nabla u|^2+2\Delta_\Sigma u.
\]
Multiply by $v^2$ and integrate by parts. Since
$c\alpha=2$ and $c\alpha^2=m-2$, the gradient terms become
\[
c|\nabla v|^2-4v\langle\nabla v,\nabla u\rangle
+(m-2)v^2|\nabla u|^2
=c|\nabla v-\alpha v\nabla u|^2.
\]
This proves the first inequality. The squared differential norm is the
sum over an orthonormal tangent frame, so
$|d(\Phi|_\Sigma)|^2\le mf^2$ by
Lemma~\ref{lem:comparison-maps}, proving the second.
Reversing a local normal reverses both $h$ and $\beta$; all expressions
are therefore globally defined. Finally, all coefficients are bounded
near $\bar U$, so $H^1_0(U)$ approximation proves the last assertion.
\end{proof}
\subsection{Compactness below the Euclidean constant}
We need only local compactness on the hypersurface. The below-threshold
argument follows the critical-quotient application of
Br\'ezis--Lieb~\cite[Section~4(B)]{BrezisLieb1983}; we give the details
for the present metric and potential. The critical concentration
threshold is the same for both curvature bounds.
\begin{lemma}
\label{lem:local-sobolev-threshold}
For every open $U\Subset\Sigma^\circ$ and $\epsilon>0$, there is
$C_{\epsilon,U}<\infty$ such that
\begin{equation}
\label{eq:local-sobolev-threshold}
(Y_m-\epsilon)\|v\|_q^2
\le c\int_\Sigma|\nabla_\Sigma v|^2\,d\sigma
+C_{\epsilon,U}\int_\Sigma v^2\,d\sigma,
\qquad v\in H^1_0(U).
\end{equation}
The inclusion $H^1_0(U)\hookrightarrow L^2(U)$ is compact.
\end{lemma}
\begin{proof}
The sharp Euclidean Sobolev inequality of Aubin and
Talenti~\cite{Aubin1976,Talenti1976}, in our normalization, is
\begin{equation}
\label{eq:talenti-normalization}
c\int_{\mathbb R^m}|D\varphi|^2\,dx
\ge Y_m\left(\int_{\mathbb R^m}|\varphi|^q\,dx\right)^{2/q},
\qquad \varphi\in\Lip_c(\mathbb R^m).
\end{equation}
Before multiplying by $c$, the sharp constant is
$m(m-2)|\mathbb S^m|^{2/m}/4$.
Normalize the metric at a chart center. In a sufficiently small chart,
its matrix $G$ satisfies
$(1-\delta)\mathrm{Id}\le G\le(1+\delta)\mathrm{Id}$.
Comparing inverse metrics and volume densities in
\eqref{eq:talenti-normalization} gives
\begin{equation}
\label{eq:chart-sobolev-distortion}
c\int_\Sigma|\nabla_\Sigma\varphi|^2\,d\sigma
\ge Y_m\left(\frac{1-\delta}{1+\delta}\right)^{m/2}
\|\varphi\|_q^2
\end{equation}
for chart-supported $\varphi$. The gradient integral is at least
$(1-\delta)^{m/2}(1+\delta)^{-1}$ times its Euclidean value,
whereas the squared critical norm is at most
$(1+\delta)^{m/q}$ times its Euclidean value, and $m/q=(m-2)/2$.
Choose $\delta$ so that the coefficient in
\eqref{eq:chart-sobolev-distortion} is at least $Y_m-\epsilon$.
Cover $\bar U$ by finitely many such charts and choose chart-supported
Lipschitz functions $\chi_i$ with $\sum_i\chi_i^2=1$ near $\bar U$.
They are obtained by normalizing a finite family of cutoffs by the
square root of the sum of their squares near $\bar U$, and then
cutting off outside that neighborhood. The triangle inequality in
$L^{q/2}$ and the product rule give
\begin{align}
\label{eq:squared-partition-norm}
\|v\|_q^2
&=\left\|\sum_i(\chi_i v)^2\right\|_{q/2}
\le\sum_i\|\chi_i v\|_q^2,\\
\label{eq:squared-partition-energy}
\sum_i|\nabla_\Sigma(\chi_i v)|^2
&=|\nabla_\Sigma v|^2
+v^2\sum_i|\nabla_\Sigma\chi_i|^2.
\end{align}
Summing~\eqref{eq:chart-sobolev-distortion} proves
\eqref{eq:local-sobolev-threshold} first for compactly supported
Lipschitz functions. Taking, for example, $\epsilon=Y_m/2$ gives
the continuous inclusion into $L^q$, so approximation proves it
for every $v\in H^1_0(U)$.
For compactness, extend a bounded $H^1_0(U)$ sequence by zero.
Each $\chi_i v_j$ is a bounded Euclidean $H^1$ sequence in its chart,
supported in a fixed compact subset. Rellich compactness gives an
$L^2$-convergent subsequence in each of the finitely many charts.
The identity $v_j=\sum_i\chi_i(\chi_i v_j)$ then gives convergence
in $L^2(U)$. No boundary regularity of $U$ is used.
\end{proof}
\begin{lemma}[Attainment below the threshold]
\label{lem:subthreshold-minimizer}
For nonempty open $U\Subset\Sigma^\circ$, put
\[
Y(U)=\inf\{Q_\Sigma(v):v\in H^1_0(U),\ \|v\|_q=1\}.
\]
If $Y(U)<Y_m$, the infimum is attained.
\end{lemma}
\begin{proof}
Write $Y=Y(U)\ge0$, the last inequality following from
Lemma~\ref{lem:square-estimate}. For a normalized minimizing sequence,
H\"older's inequality and $h^2-b^2\ge-b^2$ give
\begin{equation}
\label{eq:minimizing-sequence-bound}
\|v_j\|_2^2\le\sigma(U)^{2/m},\qquad
c\|\nabla_\Sigma v_j\|_2^2
\le Q_\Sigma(v_j)+mb^2\sigma(U)^{2/m}.
\end{equation}
Thus, after a subsequence,
\[
v_j\rightharpoonup\psi\text{ in }H^1_0(U),\qquad
v_j\longrightarrow\psi\text{ in }L^2(U)
\text{ and almost everywhere}.
\]
Set $t=\int_U|\psi|^q\,d\sigma\in[0,1]$ and $z_j=v_j-\psi$.
The Br\'ezis--Lieb lemma~\cite[Theorem~1]{BrezisLieb1983} gives
\begin{equation}
\label{eq:brezis-lieb-splitting}
\int_U|z_j|^q\,d\sigma\longrightarrow1-t.
\end{equation}
Weak convergence splits the quadratic form:
$Q_\Sigma(v_j)=Q_\Sigma(\psi)+Q_\Sigma(z_j)+o(1)$.
The bounded potential and~\eqref{eq:local-sobolev-threshold} give
\[
Q_\Sigma(z_j)\ge
(Y_m-\epsilon)\|z_j\|_q^2
-(C_{\epsilon,U}+mb^2)\|z_j\|_2^2.
\]
Letting $j\to\infty$ and then $\epsilon\downarrow0$, we obtain
\begin{equation}
\label{eq:critical-mass-inequality}
Y\ge Q_\Sigma(\psi)+Y_m(1-t)^{2/q}
\ge Yt^{2/q}+Y_m(1-t)^{2/q}.
\end{equation}
If $t=0$, this contradicts $Y<Y_m$. If $Y=0$, it forces $t=1$.
If $Y>0$ and $0<t<1$, strict concavity gives
$t^{2/q}+(1-t)^{2/q}>1$, and the right side is strictly greater
than $Y$. Thus $t=1$ in every case. Lower semicontinuity of the
gradient energy and strong $L^2$ convergence of the potential term
give $Q_\Sigma(\psi)\le Y$; normalization gives the reverse inequality.
\end{proof}
\subsection{Balanced variations}
A quotient below the sharp threshold would now have a normalized
minimizer $\psi$. After balancing its critical mass with a comparison
map $\Phi$, the spherical-coordinate variations will give
$\int_\Sigma\psi^2|d(\Phi|_\Sigma)|^2\,d\sigma\ge Q_\Sigma(\psi)$.
The geometric square gives the reverse inequality with an additional
nonnegative gradient square. The two bounds force that square to vanish,
and zero extension supplies the contradiction.
\begin{proof}[Proof of Theorem~\ref{thm:extrinsic-sobolev}]
First let $\Sigma$ be connected and boundaryless, and fix a nonempty
open $U\Subset\Sigma$ with $\Sigma\setminus\bar U\ne\varnothing$.
Suppose $Y(U)<Y_m$, and take the normalized minimizer $\psi$ supplied
by Lemma~\ref{lem:subthreshold-minimizer}. Write $Y=Q_\Sigma(\psi)$.
Push the probability measure $|\psi|^q\,d\sigma$, extended by zero,
to $M$ by the immersion. Its support is compact and it has no atoms.
Indeed, an immersion is locally an embedding, so a fiber in the compact
support meets each member of a finite embedding-chart cover in at most
one point; every such fiber has zero area measure.
Lemma~\ref{lem:balancing} supplies fixed parameters $p,a$ such that
the coordinates $w_1,\ldots,w_{n+1}$ of $\Phi_{p,a}|_\Sigma$ satisfy
\begin{equation}
\label{eq:balanced-minimizer}
\int_\Sigma|\psi|^q w_j\,d\sigma=0,\qquad
\sum_{j=1}^{n+1}w_j^2=1.
\end{equation}
These coordinates and $u=u_{p,a}$ have bounded first derivatives
near $\bar U$. Multiplication by $w_j$ or $w_j^2$ preserves $H^1_0(U)$.
Let $B_Q$ be the symmetric bilinear form associated with $Q_\Sigma$.
For a bounded Lipschitz multiplier $\zeta$ near $\bar U$,
differentiate the quotient along $\psi(1+s\zeta)$ to obtain
\begin{equation}
\label{eq:minimizer-first-variation}
B_Q(\psi,\psi\zeta)
=Y\int_\Sigma|\psi|^q\zeta\,d\sigma.
\end{equation}
For small $|s|$, the factor $1+s\zeta$ is bounded away from zero,
and differentiated mass integrands are dominated by a constant times
$|\psi|^q$. No regularity or positivity of the minimizer is needed.
Fix a coordinate $w=w_j$ and put
$B_w=\int_\Sigma|\psi|^q w^2\,d\sigma$. Since $|w|\le1$ and
its weighted mean is zero,
\[
\|\psi(1+sw)\|_q^2=1+(q-1)B_ws^2+o(s^2).
\]
The numerator has zero linear term by
\eqref{eq:minimizer-first-variation}; its second variation at the
minimum therefore gives
\begin{equation}
\label{eq:minimizer-second-variation}
Q_\Sigma(\psi w)\ge(q-1)YB_w.
\end{equation}
The product rule and the first variation with $\zeta=w^2$ give
\begin{equation}
\label{eq:coordinate-product-identity}
\begin{aligned}
Q_\Sigma(\psi w)
&=B_Q(\psi,\psi w^2)
+c\int_\Sigma\psi^2|\nabla_\Sigma w|^2\,d\sigma\\
&=YB_w+c\int_\Sigma\psi^2|\nabla_\Sigma w|^2\,d\sigma.
\end{aligned}
\end{equation}
Subtract and sum over the coordinates. Since $c=q-2$ and
$\sum_jw_j^2=1$, the result is
\begin{equation}
\label{eq:balanced-map-energy}
\int_\Sigma\psi^2|d(\Phi_{p,a}|_\Sigma)|^2\,d\sigma\ge Y.
\end{equation}
Combining this with Lemma~\ref{lem:square-estimate} forces
\begin{equation}
\label{eq:weighted-minimizer-gradient}
\nabla_\Sigma\psi=\alpha\psi\nabla_\Sigma u
\quad\text{almost everywhere on }U.
\end{equation}
Choose a compactly supported Lipschitz cutoff $\chi$ on $\Sigma$
equal to one near $\bar U$. By~\eqref{eq:sobolev-zero-extension},
\[
F=(\chi e^{-\alpha u})E_0\psi\in H^1(\Sigma)
\]
is the zero extension of $e^{-\alpha u}\psi$, and
\eqref{eq:weighted-minimizer-gradient} gives $\nabla_\Sigma F=0$
almost everywhere on all of $\Sigma$. The local Poincar\'e inequality
in coordinate balls makes $F$ constant almost everywhere on each ball.
Overlapping balls have the same constant, so connectedness makes $F$
constant on $\Sigma$. It vanishes on the nonempty open set
$\Sigma\setminus\bar U$ and therefore vanishes everywhere,
contradicting $\|\psi\|_q=1$. Thus $Y(U)\ge Y_m$.
Every proper compact support in a connected boundaryless $\Sigma$
is contained in such a $U$: choose a coordinate ball outside the
support and finitely many relatively compact neighborhoods covering
the support that miss a smaller ball. This proves the inequality
unless the support is the entire compact component.
In that case, let $D_\delta$ be the image of a Euclidean ball of radius
$\delta$ in a fixed chart about a point. Remove that point with a
cutoff $\eta_\delta$ equal to zero in $D_\delta$, equal to one outside
$D_{2\delta}$, and satisfying
$|\nabla_\Sigma\eta_\delta|\le C/\delta$.
For Lipschitz $v$, metric comparability gives
\[
\int_\Sigma|\nabla_\Sigma((1-\eta_\delta)v)|^2\,d\sigma
\le 2\int_{D_{2\delta}}|\nabla_\Sigma v|^2\,d\sigma
+C\|v\|_\infty^2\delta^{m-2}\longrightarrow0.
\]
Also $\eta_\delta v\to v$ in $L^2$ and $L^q$. The potential is
bounded on the compact component, so the inequality for
$\eta_\delta v$ passes to $v$. This is where $m>2$ is needed.
Finally, pass to $\Sigma^\circ$ if $\Sigma$ has boundary.
A compact support meets only finitely many connected components,
because the components are open. If
$a_j=\int_{\Sigma_j}|v|^q\,d\sigma$, the connected result gives
\[
Q_\Sigma(v)\ge Y_m\sum_j a_j^{2/q}
\ge Y_m\left(\sum_j a_j\right)^{2/q}.
\]
For $v\in H^1_0(U)$, use its defining approximation:
Lemma~\ref{lem:local-sobolev-threshold} gives convergence in $L^q$,
and boundedness of the potential near $\bar U$ gives convergence
of the quadratic form. This proves the final assertion.
\end{proof}
==== END Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//03-sobolev.tex ====
==== BEGIN Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//04-profile.tex ====
\section{The confined isoperimetric profile}
\label{sec:profile}
Confinement gives perimeter minimizers at prescribed volume. At volumes
where the confined profile is differentiable, its derivative will determine
the mean curvature on the free boundary and bound it from above on the
contact set. Both portions contribute to full ambient perimeter. These
curvature bounds will enter the hypersurface Sobolev inequality once the
constant test has been justified in Section~\ref{sec:comparison}.
Throughout this section, $M^n$ is a Cartan--Hadamard manifold; only
$\Sec_g\le0$ is needed. Fix an open geodesic ball
$B=B_{R_B}(o)$ of finite radius. Its closure is compact, and its
boundary is a smooth strictly convex hypersurface. For a measurable set
$E$, write $|E|=\vol_g(E)$ and
\[
P(E;U)=|D\chi_E|(U),\qquad P(E)=P(E;M),
\]
where $U\subset M$ is open and $|D\chi_E|$ is the perimeter measure.
In particular, $P(E)$ includes any part of the boundary lying on
$\partial B$. Define
\begin{equation}
\label{eq:confined-profile}
I(v)=\inf\bigl\{P(E):\ |E\setminus B|=0,\ |E|=v\bigr\},
\qquad 0<v<|B|.
\end{equation}
All statements about measurable sets in this definition are understood
up to sets of ambient volume zero.
\input{sections/04a-bv-calculus}
\subsection{Existence, regularity, and variation of the profile}
\begin{proposition}[Confined minimizers and their regularity]
\label{prop:confined-regularity}
For each $v\in(0,|B|)$ there is a minimizer $E$ in
\eqref{eq:confined-profile}. It has a regular representative for which
$\Gamma=\partial E$ is the support of its perimeter measure and is
contained in $\overline B$. There is a compact set $S\subset B$ such
that
\begin{enumerate}
\item $\dim_{\mathcal H}S\le n-8$, with $S=\varnothing$ if $n<8$;
\item $\Sigma=\Gamma\setminus S$ is a boundaryless
$C^{1,1}_{\mathrm{loc}}$ hypersurface, smooth on $\Sigma\cap B$;
\item $\Gamma$ is $C^{1,1}$ in an ambient neighborhood of
$\partial B$, and $S$ is disjoint from a collar of $\partial B$;
\item with $d\sigma$ denoting area on $\Sigma$, for every Borel set
$A\subset M$ one has
\begin{equation}
\label{eq:perimeter-area}
|D\chi_E|(A)=\int_{A\cap\Sigma}d\sigma,
\qquad P(E)=\int_\Sigma d\sigma=I(v).
\end{equation}
\end{enumerate}
The unit normal on $\Sigma$ is the outward normal of $E$.
\end{proposition}
\begin{proof}
Concentric balls provide finite-perimeter competitors at every volume
in $(0,|B|)$. A minimizing sequence has uniformly bounded perimeter
and is supported, up to null sets, in the compact set $\overline B$.
Lemma~\ref{lem:bv-compactness} gives an $L^1(M)$ limit
$\chi_E$ of the same volume with $P(E)\le I(v)$. The limit vanishes
almost everywhere outside $\overline B$; since $\partial B$ has
ambient volume zero, it is admissible in \eqref{eq:confined-profile}.
Thus it attains the infimum.
For the regularity conclusions, choose a minimizing region supplied by
Ghomi--Spruck in the cited arXiv version
\cite[Lemma~7.2(i),(ii), p.~35]{GhomiSpruck2022}, and denote it by $E$.
Set $K=\supp|D\chi_E|$. These unconditional parts of the Lemma give interior
smoothness outside the stated singular set and
$C^{1,1}$ regularity in a neighborhood of the confining sphere in a
Cartan--Hadamard manifold; see also
Morgan~\cite[Section~2 and Corollary~3.8]{Morgan2003} for the interior theory.
For the boundary part, Ghomi and Spruck use the Euclidean obstacle
regularity of Stredulinsky and Ziemer \cite{StredulinskyZiemer1997}
and its local graph extension to the Riemannian setting.
The graph and singular-set description gives $|K|=0$.
Off $K$, $\chi_E$ is locally constant almost everywhere.
Take $E$ to be the union of the components of $M\setminus K$ on which
$\chi_E=1$ almost everywhere. This changes only a null set.
Every ball centered on $K$ has positive perimeter, hence contains both
phases in positive volume, so $\partial E=K=\Gamma$.
At regular points the perimeter measure is the induced area measure.
The singular set has zero $m$-dimensional Hausdorff measure, giving
\eqref{eq:perimeter-area}.
The free portion in the $C^{1,1}$ collar is smooth by interior
regularity for the constant-mean-curvature equation. Hence the actual
interior singular set avoids this collar. It is closed away from the
collar and contained in the compact set $\Gamma$, so it is a compact
subset of $B$. Finally, the $C^{1,1}$ regularity is that of the whole
hypersurface near $\partial B$; the transition between its free and
contact portions does not create a boundary of $\Sigma$.
\end{proof}
We henceforth use this representative whenever discussing the boundary
of a minimizer. Notice that
\begin{equation}
\label{eq:positive-free-perimeter}
P(E;B)>0 \qquad (0<|E|<|B|).
\end{equation}
Indeed, if $D\chi_E$ vanished in the connected ball $B$, then
$\chi_E$ would be constant almost everywhere there, contrary to the
volume condition. In particular $I(v)>0$, and, by
\eqref{eq:perimeter-area}, a minimizing boundary has a nonempty smooth
free patch.
\begin{lemma}[Local Lipschitz continuity]
\label{lem:profile-lipschitz}
The profile $I$ is positive and locally Lipschitz on $(0,|B|)$.
\end{lemma}
\begin{proof}
Positivity was just established. Fix a compact interval
$[v_-,v_+]\subset(0,|B|)$. The radius of a concentric ball varies
continuously with its volume, and its boundary area is continuous in
its radius. Consequently there is $C_0<\infty$ such that
$I(v)\le C_0$ for all $v\in[v_-,v_+]$.
Consider the family
\[
\mathcal K=\bigl\{\chi_F:\ |F\setminus B|=0,
\ v_-\le |F|\le v_+,\ P(F)\le C_0\bigr\}.
\]
BV compactness on a compact neighborhood of $\overline B$, together
with lower semicontinuity and $L^1$ continuity of volume, shows that
$\mathcal K$ is compact in $L^1(M)$. For each $F\in\mathcal K$ there
is a smooth vector field $Z_F$ compactly supported in $B$ for which
\[
A_{Z_F}(F):=\int_F\divg Z_F\,d\vol_g\ne0.
\]
Otherwise $D\chi_F$ would vanish in $B$, contradicting
$v_-\le |F|\le v_+$. Reverse the sign of $Z_F$ if necessary so that
$A_{Z_F}(F)>0$. For each fixed $Z$, the functional $A_Z$ is
$L^1$-continuous. A finite subcover of $\mathcal K$ therefore gives
smooth fields $Z_1,\ldots,Z_k$, compactly supported in $B$, and a
number $\delta>0$ such that
\begin{equation}
\label{eq:uniform-volume-speed}
\max_{1\le j\le k} A_{Z_j}(F)\ge 2\delta
\qquad\text{for every }F\in\mathcal K.
\end{equation}
Let $\Psi_j^t$ be the flow of $Z_j$. These flows preserve $B$ in
both time directions. If $J_j(t,x)$ is the ambient volume Jacobian,
then
\[
G_j(t,F):=|\Psi_j^t(F)|=\int_F J_j(t,x)\,d\vol_g(x),
\qquad \partial_tG_j(0,F)=A_{Z_j}(F).
\]
Smoothness of the finitely many flows gives uniform bounds on
$\partial_t^2J_j$ on a fixed compact set for small $|t|$.
Since $|F|\le |B|$, there is a common $\tau>0$ such that
\[
|\partial_tG_j(t,F)-A_{Z_j}(F)|\le\delta
\quad (|t|\le\tau,\ F\in\mathcal K,\ 1\le j\le k).
\]
For an index satisfying \eqref{eq:uniform-volume-speed}, the function
$t\mapsto G_j(t,F)$ thus has derivative at least $\delta$ on
$[-\tau,\tau]$. It follows that every $w$ with
$|w-|F||<\delta\tau$ is attained for some $t$ satisfying
\begin{equation}
\label{eq:volume-adjustment-time}
|\Psi_j^t(F)|=w,\qquad |t|\le \delta^{-1}|w-|F||.
\end{equation}
By Lemma~\ref{lem:bv-flow}, perimeter transforms by the tangential
Jacobian of a smooth diffeomorphism on the reduced boundary. Applied to these finitely many
flows, this gives
\begin{align*}
P(\Psi_j^t(F))
&=\int_{\partial^*F}
J_m\bigl(D\Psi_j^t(x)|_{T_x\partial^*F}\bigr)
\,d\mathcal H_g^m(x),\\
|P(\Psi_j^t(F))-P(F)|&\le C|t|P(F)
\qquad (|t|\le\tau).
\end{align*}
Here $C$ is uniform over the finitely many flows and all $m$-planes
above the compact set $\overline B$.
Equation~\eqref{eq:volume-adjustment-time} and the bound
$P(F)\le C_0$ show that changing the volume by $\Delta v$ costs at
most $C C_0\delta^{-1}|\Delta v|$ in perimeter.
Apply this construction first to a minimizer at $v$ and then to a
minimizer at $w$, where $v,w\in[v_-,v_+]$ and $|v-w|<\delta\tau$.
Both are in $\mathcal K$, and the resulting competitors give
\[
|I(v)-I(w)|\le C C_0\delta^{-1}|v-w|.
\]
This proves local Lipschitz continuity. All constants here may depend
on the fixed ball and interval; no uniform regularity of the
minimizers is required.
\end{proof}
The profile can now be differentiated at almost every volume. The next
lemma identifies its derivative on free boundary patches and obtains the
needed one-sided bound at contact. Figure~\ref{fig:confined-boundary}
shows why both parts must enter the same ambient perimeter; in the
figure, $\rho=d(o,\cdot)$.
\input{figures/confined-boundary}
\begin{lemma}[Mean curvature at differentiability volumes]
\label{lem:profile-curvature}
Let $v\in(0,|B|)$ be a differentiability point of $I$, and let $E$ be
a minimizer as in Proposition~\ref{prop:confined-regularity}. With
$mh=\divg_\Sigma N$ and $h_0=I'(v)/m$, one has
\[
I'(v)>0,\qquad h=h_0\ \text{on }\Sigma\cap B,
\qquad 0\le h\le h_0\quad\text{almost everywhere on }\Sigma.
\]
In particular, $h^2\le h_0^2$ almost everywhere on $\Sigma$.
\end{lemma}
\begin{proof}
For a smooth flow $\Psi^t$ with $E_t=\Psi^t(E)$, set
$P(t)=P(E_t)$ and $V(t)=|E_t|$. Whenever the flow is admissible,
\begin{equation}
\label{eq:profile-variation-comparison}
P(t)\ge I(V(t)),\qquad P(0)=I(v),\qquad V(0)=v.
\end{equation}
For a field supported on a free regular patch, admissibility holds
for both signs of $t$. Differentiating at the minimum in
\eqref{eq:profile-variation-comparison} gives
$P'(0)=I'(v)V'(0)$. The first variation formulas are
\[
V'(0)=\int_\Sigma\langle Z,N\rangle\,d\sigma,
\qquad
P'(0)=m\int_\Sigma h\langle Z,N\rangle\,d\sigma.
\]
Arbitrary smooth compactly supported normal speeds on the patch
therefore give $mh=I'(v)$. This holds on every free regular patch,
so $h=h_0$ throughout $\Sigma\cap B$. Such patches exist by
\eqref{eq:positive-free-perimeter}.
To determine the sign of $h_0$, put $\rho(x)=d(o,x)$ and
$F(x)=\rho(x)^2/2$. The function $F$ is smooth on $M$, including
at $o$. Hessian comparison gives
\[
\Hess F=d\rho\otimes d\rho+\rho\Hess\rho
\ge d\rho\otimes d\rho
+(g-d\rho\otimes d\rho)
=g,
\]
with the inequality at $o$ understood by smooth extension.
The flow of $-\nabla F$ is
\[
\Psi^t(x)=\exp_o\bigl(e^{-t}\exp_o^{-1}(x)\bigr).
\]
It sends $B$ into itself for $t\ge0$. The volume and perimeter
Jacobian formulas give
\begin{align}
\label{eq:radial-volume-derivative}
V'(0)&=-\int_E\Delta F\,d\vol_g\le -nv<0,\\
\label{eq:radial-perimeter-derivative}
P'(0)&=-\int_{\partial^*E}
\tr_{T_x\partial^*E}(\Hess F)\,d\mathcal H_g^m(x)
\le -mP(E)<0.
\end{align}
These formulas apply to the reduced boundary as a rectifiable set,
so they include its contact portion and do not require smoothness
at singular points. The integrands and their flow derivatives are
bounded on a fixed compact neighborhood of $\overline B$.
The one-sided derivative of
\eqref{eq:profile-variation-comparison} now yields
\[
P'(0)-I'(v)V'(0)\ge0.
\]
Since $V'(0)<0$, division reverses the inequality and gives
\begin{equation}
\label{eq:profile-derivative-positive}
I'(v)\ge\frac{P'(0)}{V'(0)}>0.
\end{equation}
It remains to control the contact set $C=\Gamma\cap\partial B$.
At every point of $C$, the $C^1$ boundary of $E$ is tangent to the
sphere and has the same outward normal $N=\nabla\rho$: the region
lies on the inner side of the sphere. At almost every such point
$x$, its local $C^{1,1}$ graph has a second-order expansion. In
normal coordinates at $x$, with the common outward normal pointing
upward, write the boundary of $E$ and the sphere as graphs $u$ and
$\beta$. They have equal values and first derivatives at $x$, and
$u\le\beta$. Thus $D^2u(x)\le D^2\beta(x)$. In these coordinates their
second fundamental forms for the outward normal are $-D^2u(x)$ and
$-D^2\beta(x)$, respectively. Hessian comparison on the sphere therefore gives
\begin{equation}
\label{eq:contact-curvature-lower}
\mathrm{II}_{\Gamma}(x)\ge\mathrm{II}_{\partial B}(x),
\qquad h(x)\ge R_B^{-1}>0
\quad\text{for almost every }x\in C.
\end{equation}
Choose an open collar $U$ of $\partial B$ whose closure avoids $S$
and $o$ and in which $\Gamma$ is $C^{1,1}$. For arbitrary
$\varphi\in C_c^\infty(U)$ with $\varphi\ge0$, the smooth field
$Z=-\varphi\nabla\rho$, extended by zero, has an admissible flow for
$t\ge0$, because $\rho$ is nonincreasing along its trajectories.
The one-sided variation inequality gives
\begin{equation}
\label{eq:contact-variation}
m\int_\Sigma(h-h_0)\langle Z,N\rangle\,d\sigma\ge0.
\end{equation}
For completeness, the perimeter first variation used here is obtained
by integrating
$\divg_\Sigma Z=\divg_\Sigma Z^{\mathsf T}
+mh\langle Z,N\rangle$ on the whole $C^{1,1}$ hypersurface in
the collar. The tangential divergence integrates to zero, since
its support is compact in $\Sigma$. Thus there is no additional
term on the transition between the free and contact portions.
The integrand in \eqref{eq:contact-variation} vanishes on the free
portion, where $h=h_0$, while $\langle Z,N\rangle=-\varphi$ on $C$.
Consequently
\[
\int_C(h-h_0)\varphi\,d\sigma\le0
\qquad (\varphi\in C_c^\infty(U),\ \varphi\ge0).
\]
Nonnegative smooth ambient tests determine the sign of a Radon
measure, so $h\le h_0$ almost everywhere on $C$, irrespective of
whether $C$ has relative interior. Combining this with
\eqref{eq:contact-curvature-lower}, $h=h_0$ on the free portion,
and \eqref{eq:profile-derivative-positive}, gives
\begin{equation}
\label{eq:profile-curvature-bound}
0\le h\le h_0,\qquad h^2\le h_0^2
\quad\text{almost everywhere on }\Sigma.
\end{equation}
\end{proof}
==== END Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//04-profile.tex ====
==== BEGIN Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//04a-bv-calculus.tex ====
\subsection{Compactness and variation of finite-perimeter sets}
For an integrable function $u$, write $u\in BV(M)$ when its
distributional gradient $Du$ is a finite vector-valued measure, and
write $|Du|$ for its total variation. For $u=\chi_E$, this agrees
with the perimeter measure already defined.
Existence of confined minimizers will follow from compactness, and smooth
flows will change their volume while controlling full ambient perimeter.
We recall these facts in the precise forms used below. In smooth coordinate charts,
De Giorgi's reduced-boundary representation and Gauss--Green formula
\cite[Synopsis, pp.~119--120]{Maggi2012} give the Riemannian identities
\[
D\chi_E=-N_E\,\mathcal H_g^{n-1}\lfloor\partial^*E,
\qquad
\int_E\divg X\,d\vol_g
=\int_{\partial^*E}\langle X,N_E\rangle\,d\mathcal H_g^{n-1}.
\]
Here $\partial^*E$ is countably rectifiable, $N_E$ is its outward
measure-theoretic unit normal, and its tangent plane is $N_E^\perp$
almost everywhere. The formulas apply to every smooth compactly
supported field $X$.
To see the metric conversion, write $g$ for the metric matrix,
$\rho=\sqrt{\det g}$ and $\nu$ for the Euclidean outward normal.
If $a=(\nu^Tg^{-1}\nu)^{1/2}$, then
\[
N_E=g^{-1}\nu/a,\qquad
d\mathcal H_g^{n-1}=\rho a\,d\mathcal H_e^{n-1},\qquad
\divg_gX=\rho^{-1}\divg(\rho X).
\]
The first two identities follow from orthogonality and the tangent
Gram determinant; substitution in Euclidean Gauss--Green gives the
displayed Riemannian formula. Smooth positivity of the metric and
density compares the BV norms on compact subcharts. Chart partitions
then give the invariant measure identity.
\begin{lemma}[Compactness and lower semicontinuity]\label{lem:bv-compactness}
Let $K$ be a compact subset of a smooth Riemannian manifold without
boundary. If functions $u_j$ vanish outside $K$ almost everywhere and
\[
\sup_j\bigl(\|u_j\|_{L^1}+|Du_j|(M)\bigr)<\infty,
\]
then a subsequence converges in $L^1(M)$. Total variation is lower
semicontinuous under this convergence. If the $u_j$ are characteristic
functions, their limit is a characteristic function.
\end{lemma}
\begin{proof}
Choose a finite smooth partition on a neighborhood of $K$, with each
partition function compactly supported in a coordinate chart.
The smooth multiplier formula follows by inserting a multiplier in
the distributional test field. Smooth positivity of the metric and
volume density on the compact chart supports therefore bounds the
Euclidean BV norms of the localized functions. Extend each of them
by zero within its chart.
For a Euclidean BV function $v$, convolution satisfies
$D(v*\rho_\varepsilon)=Dv*\rho_\varepsilon$ and
$\|D(v*\rho_\varepsilon)\|_1\le |Dv|(\R^n)$.
Apply the fundamental theorem of calculus to the convolution and let
its scale tend to zero in $L^1$. This gives
\[
\|v(\cdot+h)-v\|_1\le |h|\,|Dv|(\R^n),\qquad
\|v*\rho_\varepsilon-v\|_1
\le\varepsilon C_\rho |Dv|(\R^n).
\]
For fixed $\varepsilon$, the convolutions have common compact support
and uniformly bounded sup norms and first derivatives, by the uniform
$L^1$ bound on $v$. Approximation on a finite mesh makes this family
precompact in $L^1$. Taking scales to zero and a diagonal subsequence
gives compactness in each chart, hence for the original finite sum.
Total variation is a supremum of continuous divergence integrals, so
is lower semicontinuous. A further almost-everywhere convergent
subsequence of characteristic functions has values in $\{0,1\}$.
\end{proof}
\begin{lemma}[Perimeter under smooth flows]\label{lem:bv-flow}
Let $F:M\to M$ be an orientation-preserving smooth diffeomorphism,
and let $E$ have finite perimeter. If the right side below is finite,
then
\begin{equation}\label{eq:bv-flow-jacobian}
P(FE)=\int_{\partial^*E}
J_{n-1}\bigl(dF_x|_{N_E(x)^\perp}\bigr)\,d\mathcal H_g^{n-1}(x).
\end{equation}
All Jacobians and adjoints use the source and target Riemannian
metrics. For a smooth flow $F_t$ with velocity $Z$ and a set whose
perimeter support is compact,
\begin{equation}\label{eq:bv-flow-first-variation}
\left.\frac d{dt}\right|_{t=0}P(F_tE)
=\int_{\partial^*E}\divg_{N_E^\perp}Z\,d\mathcal H_g^{n-1}.
\end{equation}
For a fixed finite family of flows and perimeter supports contained
in a fixed compact set, there are common $C,\tau>0$ such that
$|P(F_tE)-P(E)|\le C|t|P(E)$ for $|t|<\tau$.
\end{lemma}
\begin{proof}
Write $J_F$ for the positive ambient volume Jacobian. The Piola
pullback of a target test field $X$ is
\[
Q_X(x)=J_F(x)(dF_x)^{-1}X(Fx).
\]
Ordinary smooth change of variables, tested against smooth scalar
functions, gives
$\divg Q_X=J_F(\divg X)\circ F$. Gauss--Green for $E$ implies
\[
\int_{FE}\divg X\,d\vol_g
=\int_{\partial^*E}
\langle X(Fx),J_F(x)(dF_x)^{-*}N_E(x)\rangle_g
\,d\mathcal H_g^{n-1}(x).
\]
Thus the outward vector measure $-D\chi_{FE}$ is the pushforward of
the vector density $a(x)=J_F(x)(dF_x)^{-*}N_E(x)$ against the
perimeter measure of $E$. Since $F$ is injective and $a\ne0$, its
polar direction at $y=F(x)$ is $a(x)/|a(x)|$. Its total variation is
therefore the pushforward of $|a|\,d|D\chi_E|$, with no cancellation.
Linear algebra in orthonormal bases gives
$|a|=J_{n-1}(dF|_{N_E^\perp})$, proving
\eqref{eq:bv-flow-jacobian}. Differentiation of the Gram determinant
at $t=0$ gives tangential divergence. The Jacobians and their time
derivatives are uniformly bounded over all tangent planes above the
fixed compact set. This justifies differentiation under the integral
and proves the final estimate.
\end{proof}
The compactness statement supplies minimizers and compact families of
competitors. The flow formula counts contact perimeter as well as free
perimeter, both for volume adjustment and for one-sided variations.
==== END Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//04a-bv-calculus.tex ====
==== BEGIN Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//05-comparison.tex ====
\section{From the hypersurface inequality to model comparison}
\label{sec:comparison}
Fix $b\in\{0,1\}$, assume $\Sec_M\le-b^2$, and let $n=m+1\ge4$.
We use the confined profile $I$ in a geodesic ball $B=B_{R_B}(o)$
from Section~\ref{sec:profile}. A minimizing boundary has the
decomposition $\Gamma=\Sigma\cup S$ of
Proposition~\ref{prop:confined-regularity}: $\Gamma$ is compact,
$S\Subset B$ is closed, and $\Sigma=\Gamma\setminus S$ is a
boundaryless $C^{1,1}_{\mathrm{loc}}$ hypersurface. Its area measure
records the entire perimeter, including obstacle contact:
\begin{equation}
\label{eq:regular-perimeter-measure}
P(E;D)=\sigma(\Sigma\cap D)
\qquad\text{for every Borel set }D\subset M.
\end{equation}
At a differentiability volume $v$, Lemma~\ref{lem:profile-curvature} gives
$0\le h\le I'(v)/m$ almost everywhere on the regular boundary. The constant Sobolev test
would therefore bound the profile derivative from below in terms of
$I(v)$. Although $\Gamma$ is compact, its regular part
$\Sigma=\Gamma\setminus S$ need not be compact, so compactness of
$\Gamma$ alone does not justify that test. We will approximate it by compactly
supported cutoffs whose Dirichlet energy tends to zero. This requires
area growth near $S$, which we obtain directly from constrained minimality.
\subsection{Area growth by deletion and volume repair}
Deleting a small ball removes boundary near a singular point and creates
at most the area of the distance sphere. A variation on a fixed free
patch will restore the lost volume. We first record the deletion estimate
for finite-perimeter sets, so neither operation presupposes regularity
at the singular point.
\begin{lemma}[Deleting a geodesic ball]\label{lem:bv-deletion}
Let $E$ have finite volume and perimeter in a complete smooth
Riemannian manifold, let $x\in M$, and let
$0<r<\operatorname{inj}(x)$. Then
\begin{equation}\label{eq:bv-deletion}
P(E\setminus B_r(x))\le
P(E;M\setminus\overline B_r(x))
+\mathcal H_g^{n-1}(\partial B_r(x)).
\end{equation}
For all but countably many radii in any bounded interval,
$P(E;\partial B_r(x))=0$, so the perimeters on the two open sides
sum to $P(E)$.
\end{lemma}
\begin{proof}
For a bounded Lipschitz scalar $\varphi$, distributional testing gives
\begin{equation}\label{eq:bv-multiplier}
D(\varphi\chi_E)=\varphi D\chi_E+
\chi_E\nabla\varphi\,d\vol_g.
\end{equation}
First prove this for smooth multipliers. On a compact neighborhood of
a test field's support, approximate $\varphi$ smoothly, uniformly and
in $W^{1,1}$. Uniform convergence controls the measure term and
$W^{1,1}$ convergence controls the second term, since $|\chi_E|\le1$.
This proves the displayed distributional identity.
Take
\[
\varphi_\varepsilon(y)=
\min\{1,\max\{0,(d(x,y)-r)/\varepsilon\}\},
\qquad r+\varepsilon<\operatorname{inj}(x).
\]
Its derivative is supported in a compact annulus. The products
$\varphi_\varepsilon\chi_E$ converge in $L^1$ to
$\chi_{E\setminus B_r(x)}$, because the smooth distance sphere has
ambient volume zero. Lower semicontinuity and
\eqref{eq:bv-multiplier} give
\[
\begin{split}
P(E\setminus B_r(x))
&\le\liminf_{\varepsilon\downarrow0}
\left(\int\varphi_\varepsilon\,d|D\chi_E|
+\frac{|E\cap(B_{r+\varepsilon}(x)\setminus B_r(x))|}
{\varepsilon}\right)\\
&\le P(E;M\setminus\overline B_r(x))
+\left.\frac d{ds}\right|_{s=r}|B_s(x)|.
\end{split}
\]
Polar coordinates identify the last derivative with the area of the
distance sphere. Finally, disjoint spheres can carry positive mass
for a finite measure at only countably many radii.
\end{proof}
\begin{lemma}[Area growth at singular points]
\label{lem:area-growth}
Let $0<v<\vol(B)$, and let $E$ be a confined minimizing region of
volume $v$. There are constants $C<\infty$ and $r_0>0$, depending on
$E$ and $B$, such that
\begin{equation}
\label{eq:singular-area-growth}
\sigma\bigl(\Sigma\cap B_r(x)\bigr)\le Cr^m
\qquad(x\in S,\ 0<r<r_0).
\end{equation}
\end{lemma}
\begin{proof}
Suppose $S\ne\varnothing$, since otherwise the assertion is empty.
By \eqref{eq:positive-free-perimeter} and
\eqref{eq:perimeter-area}, there is a free smooth boundary point.
Choose a neighborhood $W$ of that point with
$\overline W\subset B\setminus S$ so small that $\Gamma\cap W$ is
smooth. Extend a nonnegative, nonzero normal variation of a smaller
patch to a smooth field $Z$ compactly supported in $W$. With $N$ the
outward normal, this gives
\[
a:=\int_{\Sigma\cap W}\langle Z,N\rangle\,d\sigma>0.
\]
Let $\phi_t$ be its flow and put
$G(t)=\vol(\phi_t(E))-\vol(E)$. The volume Jacobian formula gives
$G'(0)=a$. Choose $t_0>0$ so that $G'(t)\ge a/2$ for
$0\le t\le t_0$. The flow preserves $W$ and $B$ and fixes
$M\setminus W$. Whenever a finite-perimeter set $F$ agrees with $E$
almost everywhere on $W$, the volume Jacobian minus one is supported
in $W$, so
\begin{equation}
\label{eq:fixed-patch-volume}
\vol(\phi_t(F))-\vol(F)=G(t).
\end{equation}
The tangential Jacobian formula and locality of the reduced boundary
likewise give
\[
P(\phi_t(F))-P(F)
=\int_{\partial^*E\cap W}
\left(J_m\bigl(D\phi_t|_{T_x\partial^*E}\bigr)-1\right)
\,d\mathcal H_g^m(x).
\]
On the compact support of the flow, the tangential Jacobians satisfy
$|J_m-1|\le Lt$ uniformly over all tangent $m$-planes for
$0\le t\le t_0$. Hence
\begin{equation}
\label{eq:fixed-patch-perimeter}
\bigl|P(\phi_t(F))-P(F)\bigr|\le L P(E;W)t.
\end{equation}
Every volume deficit $0\le\delta\le at_0/2$ can therefore be
repaired by a time $0\le t\le2\delta/a$, at perimeter cost at most
$C_R\delta$, where $C_R=2LP(E;W)/a$. The bound is independent of
$F$ as long as $F=E$ almost everywhere on $W$.
Compactness of $S$ and its disjointness from
$\overline W\cup\partial B$ give $\rho_*>0$ such that
\[
B_{2\rho_*}(x)\subset B\setminus\overline W
\qquad(x\in S).
\]
After decreasing $\rho_*$, smoothness of the ambient metric on a
compact neighborhood of $S$ gives uniform constants $C_V,C_A$ with
\begin{equation}
\label{eq:small-ambient-balls}
\vol(B_r(x))\le C_Vr^n,
\qquad
\mathcal H_g^m(\partial B_r(x))\le C_Ar^m
\quad(x\in S,\ 0<r<\rho_*).
\end{equation}
Decrease $\rho_*$ again so that $C_V\rho_*^n\le at_0/2$.
Fix $x\in S$. For almost every $r\in(0,\rho_*)$, one has
$P(E;\partial B_r(x))=0$. At such radii set $F=E\setminus B_r(x)$;
Lemma~\ref{lem:bv-deletion}, applied to $E$, gives
\begin{equation}
\label{eq:deleted-perimeter}
P(F)\le P(E;M\setminus\overline{B_r(x)})
+\mathcal H_g^m(\partial B_r(x)).
\end{equation}
The volume deficit is
$\delta=\vol(E\cap B_r(x))\le C_Vr^n$, and $F=E$ on $W$.
Use \eqref{eq:fixed-patch-volume} to repair this deficit by a flow
time $t\le2\delta/a$. The repaired set remains in $B$ and has
volume $v$. Minimality, \eqref{eq:fixed-patch-perimeter}, and
\eqref{eq:deleted-perimeter} imply
\[
\begin{split}
P(E;B_r(x))
&\le\mathcal H_g^m(\partial B_r(x))+C_R\delta\\
&\le C_Ar^m+C_RC_Vr^n\le C_0r^m,
\end{split}
\]
where $C_0=C_A+C_RC_V\rho_*$ because $n=m+1$.
The constants are uniform in $x\in S$.
For an arbitrary $0<r<\rho_*/2$, choose a good radius
$\rho\in(r,2r)$. Monotonicity of the perimeter measure gives
\[
P(E;B_r(x))\le P(E;B_\rho(x))\le2^m C_0r^m.
\]
Together with \eqref{eq:regular-perimeter-measure}, this proves the
claim with $r_0=\rho_*/2$.
\end{proof}
\subsection{Removing the singular set from the constant test}
The area bound turns a cutoff gradient of order $r^{-1}$ on a ball of
radius $r$ into an energy cost of order $r^{m-2}$. The small Hausdorff
dimension of the singular set will make the sum of these costs tend to
zero. We isolate the cutoff construction because the same conclusion
will later apply to the boundary of an equality set.
\begin{lemma}[Cutoffs around a set of zero capacity]\label{lem:capacity-cutoffs}
Let $\Gamma$ be a compact subset of a smooth Riemannian manifold, and
let $S\subset\Gamma$ be closed. Suppose that
$\Sigma=\Gamma\setminus S$ is an embedded, boundaryless, locally
$C^{1,1}$ hypersurface of dimension $m\ge3$ and finite area.
Assume $\mathcal H^{m-2}(S)=0$ and that, for some $C,r_0>0$,
\[
\sigma(\Sigma\cap B_r(x))\le Cr^m
\qquad(x\in S,\ 0<r<r_0).
\]
There are $\eta_j\in\Lip_c(\Sigma)$ such that
\begin{equation}\label{eq:singular-cutoff-properties}
0\le\eta_j\le1,\qquad \eta_j(x)\longrightarrow1\quad(x\in\Sigma),
\qquad \int_\Sigma|\nabla_\Sigma\eta_j|^2\,d\sigma\longrightarrow0.
\end{equation}
If $S=\varnothing$, the choice $\eta_j=1$ works in every dimension.
\end{lemma}
\begin{proof}
If $S=\varnothing$, then $\Sigma=\Gamma$ is compact and we take
$\eta_j=1$, including when $\Sigma$ has several components.
Otherwise, use $\mathcal H^{m-2}(S)=0$ as follows.
For each
positive integer $j$, choose a finite covering
\begin{equation}
\label{eq:singular-cover}
\begin{gathered}
S\subset\bigcup_{i=1}^{N_j}B_{r_{ji}}(x_{ji}),
\qquad x_{ji}\in S,\\
0<r_{ji}<\min(2^{-j},r_0/4),
\qquad\sum_{i=1}^{N_j}r_{ji}^{m-2}<2^{-j}.
\end{gathered}
\end{equation}
Indeed, Hausdorff nullity first gives a countable open-ball cover
with arbitrarily small radii and sum. Recenter the balls at points
of $S$ with a fixed enlargement, starting with tolerances small enough
to absorb this enlargement, and take a finite subcover by compactness.
Put $\theta(t)=\min\{1,\max\{0,t-1\}\}$ for $t\ge0$, and define
\[
\eta_{ji}(x)=\theta\!\left(\frac{d(x,x_{ji})}{r_{ji}}\right),
\qquad
\eta_j=\min_{1\le i\le N_j}\eta_{ji}\big|_\Sigma.
\]
Each ambient cutoff vanishes on $B_{r_{ji}}(x_{ji})$, is one
outside $B_{2r_{ji}}(x_{ji})$, and has Lipschitz constant at most
$r_{ji}^{-1}$. A finite minimum is Lipschitz, and almost everywhere
\[
|\nabla_\Sigma\eta_j|^2
\le\sum_{i=1}^{N_j}|\nabla_\Sigma\eta_{ji}|^2.
\]
There is no restriction on the overlap of the balls. Applying the assumed area bound at their doubled radii gives
\begin{equation}
\label{eq:singular-cutoff-energy}
\begin{split}
\int_\Sigma|\nabla_\Sigma\eta_j|^2\,d\sigma
&\le\sum_i r_{ji}^{-2}
\sigma(\Sigma\cap B_{2r_{ji}}(x_{ji}))\\
&\le2^m C\sum_i r_{ji}^{m-2}<2^m C\,2^{-j}.
\end{split}
\end{equation}
The open balls in \eqref{eq:singular-cover} form a neighborhood of
$S$ on which $\eta_j$ vanishes. Its support is therefore contained
in a compact subset of $\Gamma\setminus S$. The embedded graph
charts identify the subspace and manifold topologies on $\Sigma$,
so this support is compact in $\Sigma$. For each fixed
$x\in\Sigma$, closedness of $S$ gives $d(x,S)>0$. As the maximum
radius tends to zero, the doubled balls eventually miss $x$, and
$\eta_j(x)=1$. This proves
\eqref{eq:singular-cutoff-properties}, without assuming completeness
of $\Sigma$ or finiteness of its components.
\end{proof}
\begin{lemma}[The constant test]
\label{lem:constant-test}
Let $v\in(0,\vol(B))$ be a differentiability point of $I$, and let
$E$ minimize the confined perimeter at volume $v$. Then
\begin{equation}
\label{eq:constant-sobolev}
m\int_\Sigma(h^2-b^2)\,d\sigma
\ge Y_m I(v)^{(m-2)/m}.
\end{equation}
\end{lemma}
\begin{proof}
Lemma~\ref{lem:profile-curvature} gives
$0\le h\le h_0:=I'(v)/m<\infty$ almost everywhere on $\Sigma$.
If $S\ne\varnothing$, the singular-dimension bound gives
\[
\dim_{\mathcal H}S\le n-8=m-7<m-2,
\qquad \mathcal H^{m-2}(S)=0.
\]
Lemma~\ref{lem:area-growth} and the compact embedded boundary structure
therefore verify the hypotheses of Lemma~\ref{lem:capacity-cutoffs}.
That lemma supplies cutoffs with \eqref{eq:singular-cutoff-properties};
when $S=\varnothing$ take $\eta_j=1$.
Apply Theorem~\ref{thm:extrinsic-sobolev} to these cutoffs:
\[
c\int_\Sigma|\nabla_\Sigma\eta_j|^2\,d\sigma
+m\int_\Sigma(h^2-b^2)\eta_j^2\,d\sigma
\ge Y_m\left(\int_\Sigma\eta_j^q\,d\sigma\right)^{2/q}.
\]
The area is $I(v)<\infty$, and
$|h^2-b^2|\le h_0^2+b^2$ almost everywhere. Dominated convergence
and \eqref{eq:singular-cutoff-properties} prove
\eqref{eq:constant-sobolev}, since $2/q=(m-2)/m$.
The same construction applies in every dimension under consideration:
when $4\le n\le7$ the singular set is empty, and when $n=8$ its
possible infinite zero-dimensional compact set still admits the
covers in \eqref{eq:singular-cover}.
\end{proof}
\subsection{The profile differential inequality and its integral}
The singular set has now been removed from the constant test without
assuming completeness of the regular locus. The curvature bound from
the confined profile can therefore be inserted in the sharp Sobolev
inequality. This is the step that reduces the geometric problem to one
scalar differential inequality.
\begin{proposition}[Profile differential inequality]
\label{prop:profile-ode}
For almost every $v\in(0,\vol(B))$,
\begin{equation}
\label{eq:profile-differential}
I'(v)\ge m\sqrt{b^2+\left(\frac{s_m}{I(v)}\right)^{2/m}}.
\end{equation}
\end{proposition}
\begin{proof}
The profile is locally Lipschitz by
Lemma~\ref{lem:profile-lipschitz}, hence differentiable almost
everywhere. At a differentiability volume choose a minimizer.
Lemma~\ref{lem:profile-curvature} gives
$h_0=I'(v)/m>0$ and $h^2\le h_0^2$ almost everywhere on its regular
boundary. Lemma~\ref{lem:constant-test} therefore gives
\[
m(h_0^2-b^2)I(v)\ge Y_m I(v)^{(m-2)/m}.
\]
Since $I(v)>0$ and $Y_m=m s_m^{2/m}$, division yields
\[
h_0^2\ge b^2+\left(\frac{s_m}{I(v)}\right)^{2/m}.
\]
Taking the positive square root proves
\eqref{eq:profile-differential}.
\end{proof}
\begin{theorem}[Comparison in the normalized curvatures]
\label{thm:normalized-comparison}
Let $n\ge4$ and $b\in\{0,1\}$. If $M^n$ is complete, simply
connected and smooth, with $\Sec_M\le-b^2$, then every bounded measurable
set $E\subset M$ of finite perimeter and positive volume satisfies
\[
P(E)\ge\mathcal I_b(\vol(E)).
\]
\end{theorem}
\begin{proof}
Choose radii $0<R_0<R$ such that $E\subset B_{R_0}(o)$ up to a
null set, and put $B=B_R(o)$. The annulus
$B_R(o)\setminus\overline{B_{R_0}(o)}$ has positive volume, so
$\vol(B)>\vol(E)$. Recall the model area and volume functions
\[
\mathcal A_b(r)=s_m\sn_b(r)^m,
\qquad
\mathcal V_b(r)=s_m\int_0^r\sn_b(t)^m\,dt.
\]
Both are strictly increasing from zero to infinity. For $A>0$ put
\begin{equation}
\label{eq:inverse-model-profile}
R_b(A)=\mathcal A_b^{-1}(A),
\qquad
\mathcal W_b(A)=\mathcal V_b(R_b(A)).
\end{equation}
Thus $\mathcal W_b$ is strictly increasing. The identities
$\sn_b'=\cs_b$ and $\cs_b^2-b^2\sn_b^2=1$ give
\begin{equation}
\label{eq:inverse-profile-derivative}
\begin{split}
\mathcal W_b'(A)
&=\frac{\sn_b(R_b(A))}{m\cs_b(R_b(A))}\\
&=\frac{1}{m\sqrt{b^2+(s_m/A)^{2/m}}}.
\end{split}
\end{equation}
On every interval $[\varepsilon,v]\subset(0,\vol(B))$, the
positive Lipschitz function $I$ has image in a compact subset of
$(0,\infty)$, where $\mathcal W_b$ is continuously differentiable.
Thus $\mathcal W_b\circ I$ is absolutely continuous there.
Proposition~\ref{prop:profile-ode}, the chain rule, and
\eqref{eq:inverse-profile-derivative} imply
\[
(\mathcal W_b\circ I)'(s)
=\mathcal W_b'(I(s))I'(s)\ge1
\quad\text{for almost every }s\in[\varepsilon,v].
\]
Integrating yields
\[
\mathcal W_b(I(v))
\ge\mathcal W_b(I(\varepsilon))+v-\varepsilon
\ge v-\varepsilon.
\]
Letting $\varepsilon\downarrow0$, we obtain
\begin{equation}
\label{eq:integrated-profile}
\mathcal W_b(I(v))\ge v
\qquad(0<v<\vol(B)).
\end{equation}
This endpoint passage uses only nonnegativity of
$\mathcal W_b(I(\varepsilon))$.
Set $V=\vol(E)$. The choice of $B$ gives $0<V<\vol(B)$, and $E$
is an admissible competitor in \eqref{eq:confined-profile}. Consequently
\[
P(E)\ge I(V).
\]
This step uses the full ambient perimeter of $E$, with no regularity
assumption on its boundary.
If $r>0$ is determined by $\mathcal V_b(r)=V$, then
$\mathcal W_b(\mathcal A_b(r))=V$.
Equation~\eqref{eq:integrated-profile} and strict monotonicity of
$\mathcal W_b$ give $I(V)\ge\mathcal A_b(r)=\mathcal I_b(V)$,
which proves the assertion.
\end{proof}
==== END Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//05-comparison.tex ====
==== BEGIN Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//06-dimensions.tex ====
\section{All dimensions and finite-volume sets}\label{sec:dimensions}
Theorem~\ref{thm:normalized-comparison} already gives the comparison for
bounded sets of finite ambient perimeter in dimensions at least four.
We first obtain the same statement in dimensions two and three by smooth
approximation. A single cutoff argument then removes boundedness, and
metric rescaling restores every negative curvature bound.
\subsection{The classical lower-dimensional inequalities}
We first check that the classical smooth comparisons allow disconnected
domains. For $b\ge0$ and $r=\mathcal V_b^{-1}(V)>0$,
\[
\mathcal I_b'(V)=m\frac{\cs_b(r)}{\sn_b(r)}>0,
\qquad
\frac{d}{dr}\frac{\cs_b(r)}{\sn_b(r)}=-\frac1{\sn_b(r)^2}<0.
\]
Thus $\mathcal I_b$ is concave. Since $\mathcal I_b(0)=0$, concavity
between $0$ and $u+v$ gives
\[
\mathcal I_b(u)\ge\frac{u}{u+v}\mathcal I_b(u+v),\qquad
\mathcal I_b(v)\ge\frac{v}{u+v}\mathcal I_b(u+v).
\]
Adding proves subadditivity. Consequently, comparisons for the finitely
many connected components of a relatively compact smooth domain imply
the comparison at its total volume.
In dimension three, Kleiner's Theorem~2~\cite[p.~38]{Kleiner1992} gives
the model comparison for every upper sectional-curvature bound
$\kappa\le0$ on a complete simply connected manifold, for arbitrary
compact smooth domains. Apply this result to each connected component
and use subadditivity if necessary.
In dimension two, the classical Weil--Bol inequality
\cite{Weil1926,Bol1941}, in the smooth disk formulation of Chavel
and Feldman \cite[Isoperimetric Inequality~(I), p.~83]{ChavelFeldman1980},
gives, for a smooth disk of area $V$ and boundary length $L$ under
$K\le\kappa\le0$,
\[
L^2\ge4\pi V-\kappa V^2;
\]
This disk statement suffices for arbitrary relatively compact smooth
domains. The Cartan--Hadamard theorem identifies the surface with $\R^2$.
Direct evaluation of the two-dimensional model functions gives
$\mathcal I_{\sqrt{-\kappa}}(V)=\sqrt{4\pi V-\kappa V^2}$.
Fill the holes in each connected component, retaining its outer boundary.
The resulting smooth disk has at least the original area and no greater
boundary length. The disk inequality and monotonicity therefore bound
the original component's boundary length below by the model profile at
its original area. Summing and using subadditivity proves the comparison
for the original domain, even when the filled disks overlap.
\subsection{Smooth approximation in a fixed neighborhood}
Recall that $BV(M)$ consists of integrable functions with finite
distributional variation. Strict convergence means $L^1$ convergence
together with convergence of the total masses $|Du|(M)$. For
$u=\chi_E$, that mass is the full ambient perimeter.
The following form of approximation preserves ambient perimeter. No
correction to impose exactly equal volumes is needed.
\begin{lemma}[Smooth approximation]\label{lem:smooth-bv-approximation}
Let $M$ be a complete connected smooth Riemannian manifold without
boundary. Let $E$ have finite ambient perimeter, and suppose that
$|E\setminus K|=0$ for a compact set $K\subset M$. For every open
neighborhood $U\supset K$ with compact closure, there are relatively
compact open sets $E_j\Subset U$ with smooth boundary such that
\[
|E_j\mathbin{\triangle}E|\longrightarrow0,
\qquad P(E_j)\longrightarrow P(E).
\]
The sets $E_j$ need not be connected.
\end{lemma}
\begin{proof}
Strict smooth approximation on complete manifolds
\cite[Theorem~2.12 in the author preprint]{GuneysuPallara2015}
provides smooth compactly supported functions $f_j$ with
\[
\|f_j-\chi_E\|_{L^1(M)}\longrightarrow0,
\qquad \int_M|\nabla f_j|\,d\vol_g\longrightarrow P(E).
\]
The cited result assumes completeness, smoothness, connectedness, and
absence of boundary; it imposes no curvature condition. We may take real
parts, since doing so cannot increase the $L^1$ error or gradient
integral, and lower semicontinuity gives the reverse limiting bound.
Choose $\eta\in C_c^\infty(U)$ with $0\le\eta\le1$ and $\eta=1$
near $K$, and put $u_j=\eta f_j$. Then $u_j\to\chi_E$ in $L^1(M)$,
and $\chi_E\nabla\eta=0$ almost everywhere. Hence
\[
\int_M|\nabla u_j|\,d\vol_g
\le\int_M|\nabla f_j|\,d\vol_g
+\|\nabla\eta\|_\infty\|f_j-\chi_E\|_1.
\]
Lower semicontinuity shows that the left side also converges to $P(E)$.
Thus all approximating functions now have support in the same compact
subset of $U$.
Write $e_j=\|u_j-\chi_E\|_1$, and choose $0<\delta_j<1/2$ with
$\delta_j\to0$ and $e_j/\delta_j\to0$. For
$\delta_j<t<1-\delta_j$, pointwise comparison with $\chi_E$ gives
\begin{equation}\label{eq:threshold-volume-error}
\bigl|\{u_j>t\}\mathbin{\triangle}E\bigr|
\le\delta_j^{-1}e_j.
\end{equation}
The ordinary coarea formula for the smooth function $u_j$ gives
\[
\int_{\delta_j}^{1-\delta_j}P(\{u_j>t\})\,dt
\le\int_M|\nabla u_j|\,d\vol_g.
\]
By Sard's Theorem, almost every $t$ is a regular value. We can therefore
choose such a $t_j\in(\delta_j,1-\delta_j)$ with
\[
P(\{u_j>t_j\})
\le\frac{1}{1-2\delta_j}\int_M|\nabla u_j|\,d\vol_g+\frac1j.
\]
Set $E_j=\{u_j>t_j\}$. Its boundary is smooth and compact in $U$.
Equation~\eqref{eq:threshold-volume-error} gives $L^1$ convergence of
the characteristic functions. The displayed perimeter bound and lower
semicontinuity give the claimed perimeter convergence.
\end{proof}
Apply Lemma~\ref{lem:smooth-bv-approximation} to a bounded
finite-perimeter set in a two- or three-dimensional Cartan--Hadamard
manifold. The smooth comparison just proved applies to every $E_j$,
including its disconnected cases. Since $|E_j|\to|E|$ and
$\mathcal I_b$ is continuous, it passes to $E$.
Together with Theorem~\ref{thm:normalized-comparison}, this establishes
the normalized comparison for bounded finite-perimeter sets in every
dimension $n\ge2$.
\subsection{Removing boundedness}
We record the cutoff argument separately, since it applies to any
continuous increasing perimeter lower bound.
\begin{lemma}[Passage to finite volume]\label{lem:finite-volume-cutoff}
Let $M$ be a complete connected smooth Riemannian manifold without
boundary. Let $J:[0,\infty)\to[0,\infty)$ be continuous and increasing,
with $J(0)=0$. Suppose that $P(F)\ge J(|F|)$ for every bounded
finite-perimeter set $F\subset M$. Then the same inequality holds for
every measurable $E$ of finite volume and finite ambient perimeter.
\end{lemma}
\begin{proof}
We use two elementary BV identities. For a compactly supported
Lipschitz function $\varphi$ and bounded $u\in BV(M)$, the product rule is
\[
D(\varphi u)=\varphi\,Du+u\nabla\varphi\,d\vol_g.
\]
It follows from the distributional product rule for smooth multipliers
by smoothing $\varphi$ locally in charts: the approximations converge
uniformly, and their gradients converge in $L^1$ on the fixed compact
support. The boundedness of $u$ permits passage in the second term.
For $0\le u\le1$, the coarea identity is
\begin{equation}\label{eq:bv-coarea}
|Du|(M)=\int_0^1P(\{u>t\})\,dt.
\end{equation}
The integrand is measurable: the defining perimeter supremum can be
taken over a countable dense family of compactly supported test fields.
For completeness, apply the strict smooth approximation theorem
\cite[Theorem~2.12 in the author preprint]{GuneysuPallara2015} to
$u\in BV(M)$. It applies to arbitrary integrable functions.
The ordinary coarea formula then proves the inequality ``$\ge$'': take a subsequence whose
level sets converge in $L^1$ for almost every $t$, and use lower
semicontinuity and Fatou's Lemma. Such a subsequence exists because
the integral over $t\in\R$ of the $L^1$ distance between the two level
indicators equals the $L^1$ distance between the functions. For the
reverse inequality, write $u=\int_0^1\chi_{\{u>t\}}\,dt$, test against
$\divg X$ for $X\in C_c^\infty(TM)$ with $|X|\le1$, and take the supremum
in the definition of total variation. This also shows that almost every
level in \eqref{eq:bv-coarea} has finite ambient perimeter.
Fix $o\in M$ and, for $R>0$, define
\[
\varphi_R(x)=\min\{1,\max\{0,R+1-d(o,x)\}\}.
\]
Completeness makes its support compact. It equals one on $B_R(o)$,
vanishes outside $B_{R+1}(o)$, and has Lipschitz constant at most one.
The product rule and \eqref{eq:bv-coarea} give
\begin{equation}\label{eq:cutoff-coarea-perimeter}
\begin{split}
\int_0^1P(\{\varphi_R\chi_E>t\})\,dt
&=|D(\varphi_R\chi_E)|(M)\\
&\le\int_M\varphi_R\,d|D\chi_E|
+\int_E|\nabla\varphi_R|\,d\vol_g\\
&\le P(E)+|E\setminus B_R(o)|.
\end{split}
\end{equation}
Almost every level is a bounded finite-perimeter set, and for every
$0<t<1$ its volume is at least $v_R=|E\cap B_R(o)|$. The hypothesis
and monotonicity of $J$ therefore bound the left side of
\eqref{eq:cutoff-coarea-perimeter} below by $J(v_R)$.
Since $v_R\to|E|$ and $|E\setminus B_R(o)|\to0$, continuity of $J$
proves the assertion.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Null sets have zero perimeter and $\mathcal I_b(0)=0$.
The preceding bounded-set comparison and
Lemma~\ref{lem:finite-volume-cutoff}, with $J=\mathcal I_b$, prove the
assertion for all $n\ge2$ and normalized curvature parameters
$b\in\{0,1\}$.
For an arbitrary negative curvature bound, put $b=\sqrt{-\kappa}>0$
and $\widetilde g=b^2g$. Then
\[
\Sec_{\widetilde g}=b^{-2}\Sec_g\le-1,
\qquad |E|_{\widetilde g}=b^n|E|_g,
\qquad P_{\widetilde g}(E)=b^{n-1}P_g(E).
\]
The last identity holds for ambient BV perimeter: constant metric
rescaling leaves divergence unchanged, while
$d\vol_{\widetilde g}=b^n d\vol_g$ and $|X|_{\widetilde g}=b|X|_g$.
In the dual definition of perimeter, the substitution $Y=bX$ therefore
multiplies the supremum by $b^{n-1}$.
The model functions have the matching relations
\[
\mathcal V_b(r)=b^{-n}\mathcal V_1(br),\qquad
\mathcal A_b(r)=b^{-(n-1)}\mathcal A_1(br),\qquad
\mathcal I_b(V)=b^{-(n-1)}\mathcal I_1(b^nV).
\]
Applying the normalized result to $\widetilde g$ and dividing by
$b^{n-1}$ gives the claimed inequality for $g$ and $\kappa$.
The case $\kappa=0$ was proved directly, without a curvature limit.
\end{proof}
==== END Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//06-dimensions.tex ====
==== BEGIN Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//07-equality-regularity.tex ====
\section{Regularity of equality sets}
\label{sec:equality-regularity}
The comparison established in \Cref{thm:main} turns an equality
set into a local perimeter almost-minimizer. We first identify its
regular boundary and prove the differentiability needed for the
curvature formulas. We then remove the singular set from the integral
identities that will locate the equality region.
\subsection{The perimeter support and its curvature}
\begin{proposition}[Boundary of an equality set]
\label{prop:equality-boundary}
Let $M^n$ be complete, simply connected and smooth, with $n\ge2$ and
$\Sec_M\le0$. Suppose that a bounded measurable set $E\subset M$ has
positive volume $V$, finite ambient perimeter $A$, and
\[
A=n\omega_n^{1/n}V^{(n-1)/n}.
\]
Put $m=n-1$, $R=(V/\omega_n)^{1/n}$, and $H_0=m/R$. The support
$\Gamma=\supp|D\chi_E|$ is compact and has a decomposition
$\Gamma=\Sigma\mathbin{\dot\cup}Z$ with the following properties:
\begin{enumerate}
\item $\Sigma$ is a relatively open, boundaryless, embedded
$C^{1,1}_{\mathrm{loc}}$ hypersurface, and $Z$ is compact, with
$\dim_{\HH}Z\le n-8$. In particular, $Z=\varnothing$ when $n<8$.
\item The side occupied by $E$ defines an outward unit normal $N$
on $\Sigma$, and
\begin{equation}
\label{eq:equality-perimeter-area}
|D\chi_E|=d\sigma\lfloor\Sigma,
\qquad \sigma(\Sigma)=A=s_mR^m,
\qquad \overline\Sigma=\Gamma.
\end{equation}
Here $d\sigma$ denotes induced area on $\Sigma$, extended by zero
to $M$.
\item There are $C<\infty$ and $r_0>0$ such that
\begin{equation}
\label{eq:equality-area-growth}
\sigma\bigl(\Sigma\cap B_r(x)\bigr)\le Cr^m
\qquad(x\in\Gamma,\ 0<r<r_0).
\end{equation}
\item The outward trace mean curvature satisfies
$H=\divg_{T\Sigma}N=H_0$ almost everywhere on $\Sigma$.
\end{enumerate}
These assertions concern the perimeter support and the measurable
class of $E$, not the topological boundary of an arbitrary
representative.
\end{proposition}
\begin{proof}
\emph{Almost-minimality and the regular part.}
Write $\theta=(n-1)/n$ and $\Lambda=A/V$. For every bounded
finite-perimeter set $F$, \Cref{thm:main} gives
\begin{equation}
\label{eq:equality-almost-minimality}
\begin{split}
P(E)-P(F)
&\le A\left[1-\left(\frac{|F|}{V}\right)^\theta\right]
\le\Lambda|E\triangle F|.
\end{split}
\end{equation}
Indeed $z^\theta\ge\min\{z,1\}$ for $z\ge0$, and the difference
of the volumes is bounded by the volume of the symmetric difference.
If $E\triangle F$ is contained, up to a null set, in a compact subset
of a bounded open set $O$, locality of perimeter cancels the part
outside $O$. Thus
\[
P(E;O)\le P(F;O)+\Lambda|E\triangle F|.
\]
It suffices to consider competitors with $P(F;O)<\infty$; agreement
with $E$ near the complement of $O$ then gives finite global
perimeter as well.
We apply the local smooth-manifold conclusion of
\cite[Corollary~1.6]{AntonelliPasqualettoPozzetta2022}. To check its
quasi-perimeter hypothesis, on measurable subsets of $O$ define
\[
G(F)=\Lambda|F\triangle(E\cap O)|.
\]
This functional is finite at the empty set and satisfies
\[
|G(F_1)-G(F_2)|\le\Lambda|F_1\triangle F_2|.
\]
Its continuity exponent is therefore $1>1-1/n$, as required in
Condition~(1.3) of that reference. The localized inequality says that
$E$ minimizes $P+G$ under compact modifications; in particular it
does so under the volume-preserving modifications of Definition~1.2.
For a ball $O$ centered on $\Gamma$, the remaining hypothesis
$P(E;O)>0$ follows from the definition of support. The cited local
conclusion applies in every dimension $n\ge2$ and requires no global
lower Ricci bound.
It gives a $C^{1,\alpha}_{\mathrm{loc}}$ regular part of full
perimeter, a relatively closed singular part of Hausdorff dimension
at most $n-8$, and local upper area growth. Here and below the local
exponent may be decreased so that $0<\alpha<1$. These are statements
about the perimeter support: one may take the density-one
representative supplied by the theorem, whose boundary equals that
support. For completeness, the perimeter measure is concentrated on
the reduced boundary, giving one inclusion. Conversely, a boundary
point of this representative is an essential boundary point; if the
perimeter vanished in a neighborhood, the characteristic function
would be constant almost everywhere there, a contradiction.
The support is unchanged by modifying $E$ on a null set. It is a
closed subset of a compact set containing $E$, since completeness
makes closed bounded sets compact. Hence $\Gamma$ and its closed
singular subset $Z$ are compact. The local area estimates become
\Cref{eq:equality-area-growth} with uniform constants by a finite
cover of $\Gamma$.
Near a regular point, the whole support is a graph dividing a
coordinate neighborhood into two connected sides. On either side
$D\chi_E=0$, so $\chi_E$ is constant almost everywhere. One way to
see this last fact is to express the distributional derivatives in
coordinates, compensating for the volume density, and mollify on
smaller coordinate balls. Overlapping balls give the same constant.
The two side values must differ: otherwise the characteristic
function would be constant across the neighborhood, since the graph
has zero ambient volume. The occupied side consequently defines a
consistent outward normal. The divergence formula on a graph
identifies its perimeter with its induced area. Together with the
full-perimeter conclusion, this proves
\Cref{eq:equality-perimeter-area}. In particular, every neighborhood
of a support point meets the regular part, so $\Sigma$ is dense in
$\Gamma$.
\emph{The graph equation.}
Choose graph coordinates $(y,z)\in\R^m\times\R$ with
$z=u(y)$ on $\Sigma$ and $E$ below the graph, up to a null set.
Let $G(y,z)$ be the ambient metric matrix and
$W(y,z)=\sqrt{\det G(y,z)}$. The graph area integrand is
\[
L_g(y,z,p)
=W(y,z)\left[(-p,1)^T G(y,z)^{-1}(-p,1)\right]^{1/2}.
\]
For a smooth compactly supported function $\zeta$ in the graph
domain, the graph $u+t\zeta$ gives admissible sets for both signs of
small $t$. The sharp deficit
$P(F)-n\omega_n^{1/n}|F|^\theta$ is nonnegative and vanishes at
$F=E$. Its volume derivative has coefficient
$n\omega_n^{1/n}\theta V^{\theta-1}=H_0$. Define
\[
\mathcal B(y,z,p)=\partial_zL_g(y,z,p)-H_0W(y,z).
\]
Differentiating area and the volume between the two graphs gives
\begin{equation}
\label{eq:equality-graph-equation}
\int\left[
\partial_pL_g(y,u,Du)\cdot D\zeta
+\mathcal B(y,u,Du)\zeta\right]dy=0.
\end{equation}
The derivative $D$ here is Euclidean differentiation in $y$.
The matrix $\partial_{pp}L_g$ is positive definite. Indeed, the
Hessian of a positive definite norm is positive in directions
transverse to its radial kernel, and a nonzero slope variation
$(-v,0)$ is not radial at $(-p,1)$. On compact sets of arguments the
equation is therefore uniformly elliptic.
The local graph-regularity result in \Cref{lem:equality-graph-regularity}
applies to this weak equation: a $C^{1,\alpha}$ graph stationary for
area minus $H_0$ times volume is locally $C^{1,1}$. Thus $Du$ is
locally Lipschitz. The proof of that result follows in the next
subsection; it obtains this conclusion without an advance bound on
second derivatives.
\emph{The signed curvature.}
We can now interpret the graph equation geometrically. On a
$C^{1,1}$ graph the induced metric and its volume density are locally Lipschitz.
For a tangential field $X=X^iF_i$ in a parametrization $F$, with
induced metric $(a_{ij})$, metric compatibility gives, almost
everywhere,
\[
\divg_\Sigma X
=\partial_iX^i+\tfrac12X^i a^{jk}\partial_i a_{jk}
=\frac{1}{\sqrt{\det a}}\,
\partial_i\bigl(\sqrt{\det a}\,X^i\bigr).
\]
The identities are also distributional: Lipschitz coefficients obey
the product rule, and the weak mixed derivatives of $F$ commute.
Consequently compactly supported tangential Lipschitz fields have
zero integral divergence. Splitting a field into its tangential
and normal parts now gives
\begin{equation}
\label{eq:equality-tangential-divergence}
\divg_{T\Sigma}Y
=\divg_\Sigma(Y^{\mathsf T})+H\langle Y,N\rangle,
\qquad H=\divg_{T\Sigma}N,
\end{equation}
almost everywhere and weakly on each regular chart.
For the graph variation used in
\Cref{eq:equality-graph-equation}, take $Y=\zeta\,\partial_z$.
The area derivative is $\int\divg_{T\Sigma}Y\,d\sigma$, and the
volume derivative is its outward flux. By
\Cref{eq:equality-tangential-divergence}, the graph equation becomes
\[
\int_\Sigma (H-H_0)\langle\partial_z,N\rangle
\zeta\,d\sigma=0.
\]
The factor $\langle\partial_z,N\rangle$ is strictly positive,
because $E$ occupies the lower side; equivalently its product with
the graph area density is $W$. Arbitrary compactly supported
$\zeta$ therefore give $H=H_0$ almost everywhere. The coefficient
$H_0$ came from the same global deficit on every patch, so this is
the same signed outward curvature on every component. This
completes the proof.
\end{proof}
\subsection{Local regularity of the graph equation}
The local input used above concerns the graph equation itself. Its
proof first bounds the energy of difference quotients, uniformly in
their step, and then controls their gradients by comparison with a
constant-coefficient equation.
\begin{lemma}[Regularity of a stationary graph]
\label{lem:equality-graph-regularity}
Let $\Omega\subset\R^m$ be open, with $m\ge1$, and let
$u\in C^{1,\alpha}_{\mathrm{loc}}(\Omega)$ for $0<\alpha<1$.
Let $G(y,z)$ be the matrix of a smooth Riemannian metric on a
neighborhood of the graph of $u$ in $\R^m\times\R$, and put
\[
W(y,z)=\sqrt{\det G(y,z)},\qquad
L_g(y,z,p)=W(y,z)
\left[(-p,1)^TG(y,z)^{-1}(-p,1)\right]^{1/2}.
\]
Fix $H_0>0$ and write
$\mathcal B(y,z,p)=\partial_zL_g(y,z,p)-H_0W(y,z)$.
If
\[
\int_\Omega\left[
\partial_pL_g(y,u,Du)\cdot D\zeta
+\mathcal B(y,u,Du)\zeta\right]dy=0
\qquad(\zeta\in C_c^\infty(\Omega)),
\]
then $u\in C^{1,1}_{\mathrm{loc}}(\Omega)$.
\end{lemma}
\begin{proof}
\emph{Difference quotients and an energy bound.}
The positivity of $\partial_{pp}L_g$ was verified after
\Cref{eq:equality-graph-equation}. Shrink the graph chart so that
its arguments and the segments between nearby arguments lie in one
compact ellipticity region. Choose a Euclidean
ball $B_{4\rho}$ compactly contained in its parameter
domain. For a coordinate unit vector $e$ and $0<\delta<\rho$, put
\[
v(y)=\frac{u(y+\delta e)-u(y)}{\delta},
\qquad y\in B_{3\rho}.
\]
Each $v$ is $C^1$, though no uniform bound on $Dv$ is yet known.
The identity
\[
v(y)=\int_0^1\partial_eu(y+s\delta e)\,ds
\]
does give a uniform $C^{0,\alpha}$ bound for $v$.
Subtract the translated equations and divide by $\delta$. To
describe the coefficients explicitly, set
\[
\begin{aligned}
T_s(y)&=(1-s)(y,u(y),Du(y))\\
&\quad+s(y+\delta e,u(y+\delta e),Du(y+\delta e)).
\end{aligned}
\]
The resulting weak equation is
\begin{equation}
\label{eq:equality-quotient-equation}
\int\left[(aDv+f)\cdot D\zeta
+(b\cdot Dv+d)\zeta\right]dy=0,
\end{equation}
where
\begin{align*}
a&=\int_0^1\partial_{pp}L_g(T_s)\,ds,
\qquad b=\int_0^1\partial_p\mathcal B(T_s)\,ds,\\
f&=\int_0^1
\bigl(\partial_e\partial_pL_g(T_s)
+\partial_z\partial_pL_g(T_s)v\bigr)\,ds,\\
d&=\int_0^1
\bigl(\partial_e\mathcal B(T_s)
+\partial_z\mathcal B(T_s)v\bigr)\,ds.
\end{align*}
In these expressions $\partial_e$ differentiates only the explicit
$y$ argument. The known $C^{1,\alpha}$ norm of $u$ bounds the
$C^{0,\alpha}$ norms of all $T_s$, uniformly in $s$ and $\delta$.
Smooth composition on the compact argument set, together with the
bound for $v$, shows that $a,f$ have uniform $C^{0,\alpha}$ bounds
and $b,d$ are uniformly bounded. The matrices $a$ are symmetric and
uniformly elliptic.
Test \Cref{eq:equality-quotient-equation} with $\chi^2v$, where
$\chi$ is supported in $B_{3\rho}$ and equals one on $B_{2\rho}$.
Such tests are justified by smooth approximation. Ellipticity gives
a multiple of $\int\chi^2|Dv|^2$ on the left. All other terms are
bounded by Young's inequality and the uniform bounds on
$v,f,b,d$, absorbing the terms linear in $Dv$ into that left side.
In particular, the first-order term involving $b\cdot Dv$ is
absorbed in this way. The remaining bound is
$C\int(\chi^2+|D\chi|^2)$, so
\begin{equation}
\label{eq:equality-quotient-energy}
\int_{B_{2\rho}}|Dv|^2\,dy\le C,
\end{equation}
uniformly in the difference step. No bound on a second derivative of
$u$ has been used.
\emph{Harmonic comparison and the gradient bound.}
Fix $x\in B_\rho$ and $0<r\le\min\{\rho,1\}$. On $B_r(x)$,
let $w-v\in H^1_0(B_r(x))$ solve
\[
\divg(a(x)Dw)=0.
\]
Existence and uniqueness follow by minimizing the positive quadratic
Dirichlet energy in $v+H^1_0(B_r(x))$. Put $z=v-w$. Subtract the
equations and test with $z$; the constant vector $f(x)$ integrates
to zero against $Dz$. With all norms on $B_r(x)$, ellipticity and
the coefficient bounds give
\[
\begin{split}
\lambda\|Dz\|_2^2
&\le Cr^\alpha
\bigl(\|Dv\|_2+|B_r|^{1/2}\bigr)\|Dz\|_2\\
&\quad+C\bigl(\|Dv\|_2+|B_r|^{1/2}\bigr)\|z\|_2.
\end{split}
\]
The second line includes the first-order term $b\cdot Dv$.
Zero-boundary Poincar\'e gives $\|z\|_2\le Cr\|Dz\|_2$.
Since $r\le r^\alpha$ for $r\le1$, it follows that
\begin{equation}
\label{eq:equality-frozen-comparison}
\left(\fint_{B_r(x)}|Dv-Dw|^2\right)^{1/2}
\le Cr^\alpha\left(
1+\left(\fint_{B_r(x)}|Dv|^2\right)^{1/2}\right).
\end{equation}
Energy minimization after subtracting an affine function also gives,
for every constant vector $p$,
\begin{equation}
\label{eq:equality-harmonic-energy}
\left(\fint_{B_r(x)}|Dw-p|^2\right)^{1/2}
\le C\left(\fint_{B_r(x)}|Dv-p|^2\right)^{1/2}.
\end{equation}
We use the following elementary interior estimate for this constant
coefficient equation. For $0<\gamma\le1/4$,
\begin{equation}
\label{eq:equality-harmonic-oscillation}
\left(\fint_{B_{\gamma r}(x)}
|Dw-(Dw)_{B_{\gamma r}(x)}|^2\right)^{1/2}
\le C\gamma
\left(\fint_{B_r(x)}|Dw-p|^2\right)^{1/2}.
\end{equation}
Here subscripts on a function denote its average. To justify the
estimate at the weak regularity in use, mollify $Dw-p$ in an
interior ball. Its components solve the same constant coefficient
equation. A linear coordinate change, with distortion controlled by
ellipticity, makes them harmonic. The mean value formula reproduces
each such function by convolution with a smooth radial averaging
kernel of radius comparable to $r$. Differentiating that kernel
and applying Cauchy--Schwarz bounds its gradient on $B_{r/2}(x)$ by
\[
Cr^{-1}\left(\fint_{B_r(x)}|Dw-p|^2\right)^{1/2}.
\]
The radius of the averaging kernel is chosen small enough that its
support remains in the transformed interior ball. Letting the
mollification scale tend to zero proves the same estimate for a
representative of $Dw-p$, and its oscillation on $B_{\gamma r}(x)$
is at most this bound times $2\gamma r$. This proves
\Cref{eq:equality-harmonic-oscillation}.
We control the mean gradient and its oscillation together: harmonic
comparison contracts the oscillation, while the coefficient errors
are summable on geometric scales. Define
\[
p_r=\fint_{B_r(x)}Dv,
\qquad
J(r)=\left(\fint_{B_r(x)}|Dv-p_r|^2\right)^{1/2}.
\]
Using $p=p_r$ in the preceding estimates and comparing averages on
the two balls gives
\begin{align}
\label{eq:equality-oscillation-iteration}
J(\gamma r)
&\le C\gamma J(r)
+C\gamma^{-m/2}r^\alpha\bigl(1+|p_r|+J(r)\bigr),\\
\label{eq:equality-mean-iteration}
|p_{\gamma r}-p_r|&\le\gamma^{-m/2}J(r).
\end{align}
Fix $\gamma$ so that $C\gamma\le1/2$, and choose a fixed
$\ell\ge1$ with $\ell/2\ge\gamma^{-m/2}$. Then
\[
S(r)=1+|p_r|+\ell J(r)
\quad\hbox{satisfies}\quad
S(\gamma r)\le(1+C'r^\alpha)S(r).
\]
Choose one starting radius $r_*\le\min\{\rho,1\}$. Its starting
averages are bounded uniformly for every $x\in B_\rho$ by
\Cref{eq:equality-quotient-energy}. On the geometric scales
$r_j=\gamma^jr_*$, the product of $1+C'r_j^\alpha$ is finite,
because $\sum_jr_j^\alpha<\infty$. Thus $|p_{r_j}|$ is uniformly
bounded. Each fixed-step quotient is $C^1$, so these averages tend
to $Dv(x)$ as $j\to\infty$. We obtain a bound for $Dv(x)$ uniform
in $x\in B_\rho$, in $e$, and in the sufficiently small step
$\delta$.
Finally,
\[
Dv(y)=\frac{Du(y+\delta e)-Du(y)}{\delta}
\]
holds classically. The uniform bound just proved bounds increments
of $Du$ along every coordinate direction. On a smaller coordinate
cube, join two points by coordinate segments and add the increment
bounds. The sum of their lengths is at most $\sqrt m$ times the
Euclidean distance between the points. Therefore $Du$ is locally
Lipschitz and $u\in C^{1,1}_{\mathrm{loc}}(\Omega)$.
\end{proof}
\subsection{Integral identities across the singular set}
The regular boundary now has constant mean curvature. To use it in
the rigidity argument, we must justify integral tests that need not
vanish near its singular set.
\begin{lemma}[Cutoffs for an equality boundary]
\label{lem:equality-cutoffs}
Under the hypotheses of \Cref{prop:equality-boundary}, there are
$\eta_j\in\Lip_c(\Sigma)$ such that
\[
0\le\eta_j\le1,
\qquad \eta_j(x)\longrightarrow1\quad(x\in\Sigma),
\qquad
\int_\Sigma|\nabla_\Sigma\eta_j|^2\,d\sigma\longrightarrow0.
\]
\end{lemma}
\begin{proof}
If $Z=\varnothing$, then $\Sigma=\Gamma$ is compact, and we take
$\eta_j=1$. Otherwise $n\ge8$, $m\ge7$, and
\[
\dim_{\HH}Z\le m-7<m-2,
\qquad \HH^{m-2}(Z)=0.
\]
The compactness, finite area, embedding and uniform growth
hypotheses of \Cref{lem:capacity-cutoffs} follow from
\Cref{prop:equality-boundary}. That lemma supplies the stated
cutoffs. In particular their supports are compact in the manifold
topology of $\Sigma$, and the construction does not assume that
$\Sigma$ is complete or has finitely many components.
\end{proof}
For every smooth ambient field $Y$ defined near $\Gamma$, the
cutoffs and \Cref{eq:equality-tangential-divergence} imply
\begin{equation}
\label{eq:global-first-variation}
\int_\Sigma\divg_{T\Sigma}Y\,d\sigma
=H_0\int_\Sigma\langle Y,N\rangle\,d\sigma.
\end{equation}
Indeed, apply the compact-support identity to $\eta_jY$ and expand:
\begin{align*}
&\int_\Sigma\eta_j\divg_{T\Sigma}Y\,d\sigma
+\int_\Sigma\langle\nabla_\Sigma\eta_j,Y^{\mathsf T}\rangle\,d\sigma\\
&\hspace{35mm}=H_0\int_\Sigma\eta_j\langle Y,N\rangle\,d\sigma.
\end{align*}
The middle integral has absolute value at most
$\|Y\|_\infty A^{1/2}\|\nabla_\Sigma\eta_j\|_2$, which tends
to zero. All other terms converge by dominated convergence, since
the ambient data are bounded near compact $\Gamma$. The same
identity holds on any compact boundaryless component of $\Sigma$
directly, by integrating
\Cref{eq:equality-tangential-divergence} on that component.
When $n\ge4$, the same cutoffs extend
\Cref{thm:extrinsic-sobolev} to every smooth ambient function $v$
restricted to $\Sigma$. Here the signed normalized mean curvature is
$h=H/m=1/R$, and we use the case $b=0$. With $q=2m/(m-2)$ and $c=4/(m-2)$, the result is
\begin{equation}
\label{eq:equality-sobolev}
c\int_\Sigma|\nabla_\Sigma v|^2\,d\sigma
+\frac{m}{R^2}\int_\Sigma v^2\,d\sigma
\ge m s_m^{2/m}
\left(\int_\Sigma|v|^q\,d\sigma\right)^{2/q}.
\end{equation}
To justify passage from $\eta_jv$ to $v$, boundedness of $v$ and
its gradient near $\Gamma$ gives convergence in $L^2$ and $L^q$,
and
\[
\nabla_\Sigma(\eta_jv)-\nabla_\Sigma v
= (\eta_j-1)\nabla_\Sigma v
+v\nabla_\Sigma\eta_j
\longrightarrow0\quad\hbox{in }L^2(\Sigma).
\]
The first term tends to zero by dominated convergence and the
second by \Cref{lem:equality-cutoffs}. Thus the whole Dirichlet
energy, including its cross term, converges. In particular the
constant function and fixed ambient perturbations of it are now
legitimate tests; no completeness of the regular locus has been
introduced.
==== END Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//07-equality-regularity.tex ====
==== BEGIN Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//08-rigidity.tex ====
\section{Rigidity of the equality case}\label{sec:rigidity}
We prove Theorem~\ref{thm:equality}. Let $M^n$ and $E$ satisfy its
hypotheses, and assume equality in the Euclidean perimeter bound.
In particular, $E$ is bounded and has positive finite volume and
finite ambient perimeter. Retain the notation
\[
m=n-1,\qquad V=\vol(E),\qquad
R=(V/\omega_n)^{1/n},\qquad A=P(E)=s_mR^m.
\]
By \Cref{prop:equality-boundary}, the perimeter support is a compact set
$\Gamma=\Sigma\cup Z$, its regular part carries all the perimeter,
and its outward trace curvature is $H_0=m/R$.
We first show that $\Gamma$ is contained in the exponential image of a
Euclidean sphere of radius $R$. The argument depends on the dimension.
A single final argument then recovers both the set and the metric on its
interior.
\subsection{A barycenter and a flux inequality}
We record two observations used in the higher-dimensional and curve cases.
If $\mu$ is a nonzero finite positive measure of compact support in $M$,
the function
\[
\mathcal F(p)=\int_M D_p(x)\,d\mu(x),\qquad
D_p(x)=\tfrac12 d(p,x)^2,
\]
is continuous and proper. Indeed, if the support lies in $B_a(o)$, then
$d(p,x)\ge d(p,o)-a$, so $\mathcal F(p)\to\infty$ as $p$ leaves compact
sets. Completeness and Hopf--Rinow give a minimizer. Squared distance is
smooth on $M\times M$, and differentiation on compact sets gives
\begin{equation}\label{eq:barycenter}
\int_M\log_p x\,d\mu(x)=0
\end{equation}
at any minimizer. Uniqueness of the minimizer is not needed.
Let $T$ be either $\Sigma$ or a compact boundaryless component of $\Sigma$,
and put $a=\area(T)$. The global first-variation identity from
\Cref{sec:equality-regularity} applies on $T$. The distance comparison
\Cref{lem:distance-comparison} yields, for every $p\in M$,
\begin{align}
ma
&\le \int_T \divg_{T\Sigma}\nabla D_p\,d\sigma
=\frac mR\int_T\langle\nabla D_p,N\rangle\,d\sigma \notag\\
&\le \frac mR\left(a\int_T r_p^2\,d\sigma\right)^{1/2}.
\label{eq:flux-bound}
\end{align}
All fields may be multiplied by an ambient cutoff equal to one near
$\Gamma$, so their lack of compact support on $M$ causes no difficulty.
In particular,
\begin{equation}\label{eq:lower-moment}
\int_T r_p^2\,d\sigma\ge R^2a.
\end{equation}
If the opposite inequality also holds, all inequalities in
\Cref{eq:flux-bound} are equalities. Consequently
\begin{equation}\label{eq:radial-normal}
\nabla D_p=RN\quad\text{almost everywhere on }T.
\end{equation}
For example, the squared $L^2$ norm of the difference is
\[
\int_T r_p^2\,d\sigma+R^2a
-2R\int_T\langle\nabla D_p,N\rangle\,d\sigma=0.
\]
Continuity upgrades \Cref{eq:radial-normal} to every point of $T$:
a nonempty open regular patch has positive area. Since $|N|=1$,
this puts $T$ on the distance sphere $r_p=R$.
\subsection{Dimensions at least four}
Here $m\ge3$. The cutoff passage in \Cref{sec:equality-regularity}
has established \Cref{eq:equality-sobolev} for every smooth ambient
function restricted to $\Sigma$. Retain
$q=2m/(m-2)$ and $c=4/(m-2)=q-2$.
The constant function $v=1$ attains equality, since $A=s_mR^m$.
Let $\phi$ be such a restriction with
$\int_\Sigma\phi\,d\sigma=0$. Boundedness permits Taylor expansion for
$v=1+\tau\phi$:
\begin{equation}\label{eq:critical-expansion}
\left(\int_\Sigma|1+\tau\phi|^q\,d\sigma\right)^{2/q}
=A^{2/q}\left[1+(q-1)\frac{\tau^2}{A}
\int_\Sigma\phi^2\,d\sigma+o(\tau^2)\right].
\end{equation}
The constant terms in \Cref{eq:equality-sobolev} agree, and the linear
terms vanish. Since $m s_m^{2/m}A^{2/q-1}=m/R^2$ and $c=q-2$,
comparison of the quadratic terms proves
\begin{equation}\label{eq:equality-spectral-gap}
\int_\Sigma|\nabla_\Sigma\phi|^2\,d\sigma
\ge \frac m{R^2}\int_\Sigma\phi^2\,d\sigma.
\end{equation}
The cutoff limit has already been taken before the expansion; no
mean-zero condition on cutoff approximations is asserted.
Choose a barycenter $p$ for the measure $d\sigma$ on $\Sigma$ and an
orthonormal basis $e_1,\ldots,e_n$ of $T_pM$.
The smooth functions
$\phi_i(x)=\langle\log_p x,e_i\rangle$ have zero mean by
\Cref{eq:barycenter}. Because $d\log_p$ is contracting,
\[
\sum_{i=1}^n|\nabla_\Sigma\phi_i|^2
=\sum_{j=1}^m|d\log_p(v_j)|^2\le m
\]
for any orthonormal basis $v_1,\ldots,v_m$ of $T_x\Sigma$.
Summing \Cref{eq:equality-spectral-gap} gives
\begin{equation}\label{eq:upper-moment}
\int_\Sigma r_p^2\,d\sigma\le R^2A.
\end{equation}
Together with \Cref{eq:lower-moment}, this yields
\Cref{eq:radial-normal} on $\Sigma$. Its density in $\Gamma$ therefore
implies
\begin{equation}\label{eq:central-sphere}
\Gamma\subset\{x\in M:d(p,x)=R\}.
\end{equation}
Neither connectedness nor completeness of the regular part was used.
\subsection{Dimension two}
Now $m=1$ and $\Gamma=\Sigma$ is a compact embedded $C^{1,1}$
curve without boundary, with total length $A=2\pi R$.
Choose a connected component $T$ and let $L=\operatorname{length}(T)>0$.
A connected compact one-dimensional manifold without boundary admits a
periodic arclength parametrization of period $L$. Here this can also be
seen by continuing an oriented arclength chart: a traverse of arbitrarily
large length cannot stay injective in a component of finite length, and
local uniqueness gives a least positive period. Its image is open and
closed in the component, and one least period traverses it once.
The periodic Wirtinger inequality says that
\begin{equation}\label{eq:wirtinger}
\int_T |\nabla_T\phi|^2\,d\sigma
\ge \left(\frac{2\pi}{L}\right)^2
\int_T\phi^2\,d\sigma,
\qquad \int_T\phi\,d\sigma=0.
\end{equation}
For completeness, in arclength coordinates the assertion follows by
expanding a trigonometric polynomial in Fourier modes; the zero mode is
absent. Periodic convolution approximates a $C^1$ function and its
derivative uniformly, preserving the zero mean, so the same inequality
holds for all the tests used below.
Take a barycenter of the area measure on $T$. Apply
\Cref{eq:wirtinger} to its logarithm coordinates and use the contraction
of $d\log_p$ as above. The result is
\[
\int_T r_p^2\,d\sigma\le \frac{L^3}{(2\pi)^2}.
\]
On the other hand, \Cref{eq:lower-moment} gives
$\int_T r_p^2\,d\sigma\ge R^2L$. Thus $L\ge2\pi R=A$.
Since $L\le A$, this component has all the length, and every inequality
just used is an equality. Another component would contain a regular open
arc of positive length, which is impossible. Hence $T=\Gamma$, and
\Cref{eq:radial-normal} again proves \Cref{eq:central-sphere}.
\subsection{Dimension three}
Here $m=2$ and $\Sigma=\Gamma$ is a compact embedded $C^{1,1}$ surface,
with $A=4\pi R^2$. Write $h=1/R$, so its trace curvature is $2h$.
The critical Sobolev argument used for $m\ge3$ is replaced by a punctured
radial flux calculation.
Fix $p\in\Sigma$ and, away from $p$, define
\[
r=r_p,\qquad Y=r^{-2}\nabla D_p=\frac{\nabla r}{r},
\qquad t=\langle Y,N\rangle.
\]
Since $\Hess D_p\ge g$ and $|\nabla r|=1$,
\begin{align}
\divg_{T\Sigma}Y
&=r^{-2}\tr_{T\Sigma}\Hess D_p
-2r^{-2}|\nabla_\Sigma r|^2 \notag\\
&\ge2r^{-2}\bigl(1-|\nabla_\Sigma r|^2\bigr)=2t^2.
\label{eq:puncture-divergence}
\end{align}
Choose a smooth nondecreasing function $\chi:[0,\infty)\to[0,1]$
that is zero on $[0,1]$ and one on $[2,\infty)$.
The field $\chi(r/\varepsilon)Y$ extends smoothly by zero near $p$.
Apply the global first-variation identity and differentiate the cutoff.
The term containing $\chi'$ is nonnegative, and
\Cref{eq:puncture-divergence} gives
\begin{equation}\label{eq:puncture-square}
\int_\Sigma\chi(r/\varepsilon)
\left[\frac{h^2}{2}-2\left(t-\frac h2\right)^2\right]d\sigma
\ge J_\varepsilon,
\quad
J_\varepsilon=
\int_\Sigma\frac{\chi'(r/\varepsilon)}{\varepsilon r}
|\nabla_\Sigma r|^2\,d\sigma.
\end{equation}
Indeed the left integrand before completing the square is
$2ht-2t^2=h^2/2-2(t-h/2)^2$.
We claim that
\begin{equation}\label{eq:annular-flux}
\lim_{\varepsilon\downarrow0}J_\varepsilon=2\pi.
\end{equation}
Use exponential coordinates at $p$, with the first two coordinate axes
in $T_p\Sigma$. In a neighborhood of $p$ the entire support is a graph
$(y,b(y))$, where $b(0)=0$ and $Db(0)=0$.
For small $\varepsilon$ this one graph contains the whole annulus
contributing to $J_\varepsilon$; no other sheet or component enters it.
Set $y=\varepsilon z$. For fixed $z\ne0$,
\[
\frac r\varepsilon\longrightarrow |z|,\qquad
\frac{d\sigma}{\varepsilon^2}\longrightarrow dz,
\qquad |\nabla_\Sigma r|\longrightarrow1.
\]
Here $r=|(y,b(y))|$ exactly, by the radial distance formula.
On the support of $\chi'$ one has $1\le r/\varepsilon\le2$.
Thus $z$ stays in a fixed bounded disk and, using the vanishing slope of
$b$, away from zero. The area density is uniformly bounded there,
$|\nabla_\Sigma r|\le1$, and
$|\chi'(r/\varepsilon)/(r/\varepsilon)|\le\|\chi'\|_\infty$.
Dominated convergence after extension by zero to the fixed disk gives
\[
\lim_{\varepsilon\downarrow0}J_\varepsilon
=\int_{\R^2}\frac{\chi'(|z|)}{|z|}\,dz
=2\pi\int_1^2\chi'(s)\,ds=2\pi.
\]
This proves \Cref{eq:annular-flux}.
Because $h^2A/2=2\pi$, \Cref{eq:puncture-square} implies
\[
0\le\int_\Sigma\chi(r/\varepsilon)(t-h/2)^2\,d\sigma
\le \frac{h^2}{4}\int_\Sigma\chi(r/\varepsilon)\,d\sigma
-\frac12J_\varepsilon\longrightarrow0.
\]
Fatou's lemma yields $t=h/2$ almost everywhere off $p$, without any
prior integrability assumption on $t^2$ at the pole. Continuity makes
this identity valid at every point of $\Sigma\setminus\{p\}$.
This everywhere off-diagonal identity has been proved for each fixed
$p$. We may therefore now fix $x_0\in\Sigma$ and evaluate it at $x_0$
for every $p\ne x_0$; there is no intersection of uncountably many
exceptional null sets. The identity
$\nabla D_p(x_0)=-\log_{x_0}p$ gives
\[
-\langle\log_{x_0}p,N(x_0)\rangle
=\frac{|\log_{x_0}p|^2}{2R}.
\]
Completing the square, including $p=x_0$ by continuity, we obtain
\begin{equation}\label{eq:offset-sphere}
\Gamma\subset\exp_{x_0}
\left\{v\in T_{x_0}M:|v+RN(x_0)|=R\right\}.
\end{equation}
The Euclidean sphere in \Cref{eq:offset-sphere} passes through the
origin of $T_{x_0}M$; its center is $-RN(x_0)$.
\Cref{fig:offset-sphere} depicts this distinction.
\input{figures/offset-sphere}
\subsection{Recovering the whole ball and its metric}
The three dimension regimes give the same conclusion: for some $o\in M$,
$\Gamma$ is contained in $\exp_o(S)$, where $S$ is a Euclidean sphere
of radius $R$ in $T_oM$. Let $U_0$ be the open ball bounded by $S$ and
put $U=\exp_o(U_0)$. The ball need not be centered at the tangent-space
origin.
The global exponential diffeomorphism maps the two connected components
of $T_oM\setminus S$ onto those of $M\setminus\exp_o(S)$.
Their interior component $U$ is bounded, since its closure is a compact
image. The exterior component is unbounded: vectors escaping to infinity
in the Euclidean exterior have image distances $d(o,\exp_o v)=|v|$
escaping to infinity. Exterior connectedness uses $n\ge2$.
The smooth hypersurface $\exp_o(S)$ has zero ambient volume.
The distributional derivative of $\chi_E$ is supported on $\Gamma$,
so it vanishes on each complementary component. A function with all
weak coordinate derivatives zero is locally constant almost everywhere,
as follows by mollification in coordinate balls. Overlapping balls
then make $\chi_E$ almost everywhere constant on each connected component.
Its two constants belong to $\{0,1\}$. Boundedness of $E$ forces the
exterior constant to be zero, because the open exterior contains
positive-volume balls arbitrarily far from $E$. Since $V>0$ and the
separating sphere has zero volume, the interior constant is one.
Therefore
\begin{equation}\label{eq:phase-recovery}
E=U\quad\text{up to a set of volume zero}.
\end{equation}
Only containment of the perimeter support in the sphere was needed.
By \Cref{lem:distance-comparison}, the pullback metric
$\widetilde g=(\exp_o)^*g$ dominates the constant Euclidean metric
$g_o$ on all of $T_oM$, hence also on the offset ball $U_0$.
On that ball, \Cref{eq:phase-recovery} gives
\[
\vol_{\widetilde g}(U_0)=V=\omega_nR^n=\vol_{g_o}(U_0).
\]
The smooth relative density $\sqrt{\det_{g_o}\widetilde g}$ is at least
one. Equality of its integral with $\vol_{g_o}(U_0)$ and continuity
make it identically one on $U_0$. All eigenvalues of $\widetilde g$
relative to $g_o$ are at least one and their product is one.
Each eigenvalue is therefore one, and
\[
(\exp_o)^*g=g_o\quad\text{on }U_0.
\]
Thus $\exp_o:U_0\to U$ is a Riemannian isometry, proving the necessity
in \Cref{thm:equality}.
For the converse, let $F:B_R(0)\subset\R^n\to U\subset M$ be a
Riemannian isometry onto an open region with its induced metric.
Put $p=F(0)$ and $L=dF_0$. For $|v|<R$, the image of the segment
$s\mapsto sv$ is an ambient geodesic with initial data $(p,Lv)$:
the Levi--Civita connection on an open region is the restriction of
the ambient connection. Geodesic uniqueness gives
\[
F(v)=\exp_p(Lv).
\]
Consequently $U=B_R(p)$. This formula extends smoothly to the closed
Euclidean ball through the ambient exponential map. Its pullback metric
is Euclidean in the interior and therefore on the boundary by continuity.
The boundary is a smooth sphere of area $s_{n-1}R^{n-1}$, which equals
its ambient perimeter by the divergence formula. Its volume is
$\omega_nR^n$, so equality holds. Perimeter and volume are unchanged
by null modifications. This proves the sufficiency and completes
\Cref{thm:equality}. No flatness outside the equality region is implied.
==== END Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//08-rigidity.tex ====
==== BEGIN Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//09-consequences.tex ====
\section{Spectral and torsional comparison}\label{sec:consequences}
We now derive the model-ball variational comparisons stated in
Corollary~\ref{cor:model-ball-spectral}. Theorem~\ref{thm:main}
allows the modulus of each compactly supported smooth test on the
original domain to be replaced by an equimeasurable radial test on its
model ball with no greater energy.
The point to check is that the radial rearrangement belongs to the
model ball's Dirichlet Sobolev space.
\begin{proof}[Proof of Corollary~\ref{cor:model-ball-spectral}]
Let $M$, $D$, $\kappa$ and $p$ satisfy the hypotheses of the corollary.
Fix $0\ne v\in C_c^\infty(D)$ and extend $u=|v|$ by zero to $M$;
this is a compactly supported Lipschitz function.
Put $b=\sqrt{-\kappa}$ and take the interval density
$a(r)=\mathcal A_b(r)$ on $(0,\infty)$, whose cumulative mass is
$F_a(r)=\mathcal V_b(r)$. Thus $a(F_a^{-1}(s))=\mathcal I_b(s)$.
Apply Nobili and Violo's P\'olya--Szeg\H{o} theorem
\cite[Theorem~1.1]{NobiliViolo2025} on $M$ with constant one.
Its metric BV perimeter is the ambient Riemannian perimeter
\eqref{eq:ambient-perimeter}, as recalled after Definition~2.8 of
\cite{AntonelliBrueFogagnoloPozzetta2022}. Hence Theorem~\ref{thm:main}
gives the required support-neighborhood bound at every finite volume,
trivially also for sets of infinite perimeter. The positive superlevels
of $u$ have finite volume, and its relaxed metric $p$-energy is at most its Riemannian
energy \cite[Definition~2.1]{NobiliViolo2025}. The theorem and
\cite[Lemma~3.1]{NobiliViolo2025} give the decreasing equimeasurable
rearrangement $u^*$ for $a(r)\,dr$, bounded and locally absolutely
continuous, with weighted derivative energy at most
$\int_D|\nabla_gv|_g^p\,d\vol_g$.
Write $B=B_\kappa(|D|)$, with center $o$. Since $\supp u\Subset D$, its volume is strictly
below $|D|$, so $u^*$ vanishes on an interval before
$R=\mathcal V_b^{-1}(|D|)$.
We use the radial construction in
\cite[proof of Theorem~1.6, Section~7.1]{NobiliViolo2025}.
The polar density is $a(r)=n\omega_n r^{n-1}w_b(r)$, where
$w_b(r)=(\sn_b(r)/r)^{n-1}$ extends smoothly and positively at zero
and is log-convex. Define
$\widetilde u(x)=u^*(d_\kappa(o,x))$. In normal coordinates it is bounded
and locally $W^{1,p}$ away from the origin, with finite gradient
$p$-energy. Its gradient is therefore also locally integrable across
the origin. Integrating by parts outside the radius-$\varepsilon$ ball,
the inner boundary term is $O(\varepsilon^{n-1})$ because $u^*$ is
bounded. This term tends to zero since $n\ge2$, so the weak gradient
extends across the origin. Equivalence of Sobolev norms in smooth
coordinates and vanishing near $R$ now give
$\widetilde u\in W^{1,p}_0(B)$.
The model radial gradient is the radial derivative, and therefore
\[
\int_B|\nabla_\kappa\widetilde u|_\kappa^p\,d\vol_\kappa
\le\int_D|\nabla_gv|_g^p\,d\vol_g,
\qquad
\int_B|\widetilde u|^q\,d\vol_\kappa
=\int_D|v|^q\,d\vol_g\quad(q=1,p).
\]
Taking the Rayleigh infimum and the torsion-quotient supremum proves the
claims. The defining density of $C_c^\infty(D)$ and the finite volume of
$D$ allow both variational values to be computed on the tests used above.
\end{proof}
==== END Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//09-consequences.tex ====
==== BEGIN Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//confined-boundary.tex ====
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=.95,>=Latex]
\draw[gray!65,dashed] (0,0) circle (2.2);
\fill[blue!7] (75:2.2) arc (75:105:2.2)
.. controls (-1.65,1.84) and (-1.6,.5) .. (-1.05,-.45)
.. controls (-.55,-1.35) and (.75,-1.4) .. (1.2,-.55)
.. controls (1.7,.4) and (1.65,1.84) .. (75:2.2) -- cycle;
\draw[thick,blue!70!black] (105:2.2)
.. controls (-1.65,1.84) and (-1.6,.5) .. (-1.05,-.45)
.. controls (-.55,-1.35) and (.75,-1.4) .. (1.2,-.55)
.. controls (1.7,.4) and (1.65,1.84) .. (75:2.2);
\draw[very thick,orange!80!black] (75:2.2) arc (75:105:2.2);
\node at (0,.3) {$E$};
\node[gray!75!black] at (2.55,-1.8) {$\partial B$};
\draw[->,gray!65] (2.2,-1.75)--(1.66,-1.45);
\draw[->,thick] (0,2.2)--(0,2.85) node[right] {$N=\nabla\rho$};
\node[orange!80!black,anchor=west] at (1.05,2.45) {$C=\Gamma\cap\partial B$};
\draw[->,orange!80!black] (1,2.4)--(.42,2.17);
\node[blue!70!black,anchor=west] at (2.3,.45) {free part $\Sigma\cap B$};
\draw[->,blue!70!black] (2.3,.32)--(1.47,.15);
\end{tikzpicture}
\caption{The confined problem minimizes full ambient perimeter, including
contact with the confining sphere. The occupied phase fixes the outward
normal. Free variations give $mh=I'(v)$; inward variations give
$mh\le I'(v)$ almost everywhere on the contact set. The planar drawing is schematic:
the proof does not assume that the contact set contains an open patch.}
\label{fig:confined-boundary}
\end{figure}
==== END Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//confined-boundary.tex ====
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==== BEGIN Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//main.tex ====
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\title{Generalized Cartan--Hadamard isoperimetry\\
and Euclidean equality rigidity}
\author{OpenAI}
\date{September 23, 2026}
\input{glyph-mappings}
\begin{document}
\maketitle
\begin{abstract}
We resolve the generalized Cartan--Hadamard isoperimetric conjecture
in every dimension. In a complete simply connected smooth manifold
with sectional curvature at most $\kappa\le0$, every finite-volume set
of finite ambient perimeter has perimeter at least that of the
equal-volume ball in curvature $\kappa$. For bounded positive-volume
sets, equality in the Euclidean comparison holds precisely when the set
agrees up to null sets with an open region isometric, with its induced
metric, to a round Euclidean ball.
\end{abstract}
\tableofcontents
\clearpage
\input{sections/01-introduction}
\input{sections/02-maps}
\input{sections/03-sobolev}
\input{sections/04-profile}
\input{sections/05-comparison}
\input{sections/06-dimensions}
\input{sections/07-equality-regularity}
\input{sections/08-rigidity}
\input{sections/09-consequences}
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//main.tex ====
==== BEGIN Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//offset-sphere.tex ====
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=0.95,>=Latex]
\fill[blue!6] (-2,0) circle (2);
\draw[thick,blue!65!black] (-2,0) circle (2);
\draw[->,gray!65] (-4.5,0)--(1.8,0);
\draw[->,gray!65] (0,-2.35)--(0,2.5);
\fill (-2,0) circle (1.5pt) node[below left] {$-RN(x_0)$};
\fill (0,0) circle (1.7pt) node[below right] {$0\in T_{x_0}M$};
\draw[thick,->] (0,0)--(1.25,0) node[above] {$N(x_0)$};
\draw[thick] (-2,0)--(-2,2) node[midway,left] {$R$};
\node at (-2,-1.1) {$U_0$};
\node[align=center] at (-2,2.65) {A planar section of the tangent-space sphere};
\end{tikzpicture}
\caption{In dimension three, the open ball $U_0\subset T_{x_0}M$
has radius $R$ and center $-RN(x_0)$, so the logarithm origin lies on
its boundary. At this stage the perimeter support is known to lie on
$\exp_{x_0}(\partial U_0)$. The phase-recovery argument identifies the occupied
phase and then proves that the induced metric on the enclosed region
is Euclidean.}
\label{fig:offset-sphere}
\end{figure}
==== END Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//offset-sphere.tex ====
==== BEGIN Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//references.bib ====
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}
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@article{GhomiStavroulakis2026,
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archivePrefix = {arXiv},
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note = {arXiv:2609.17093v2, revised 21 September 2026; \url{https://arxiv.org/abs/2609.17093v2}},
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==== END Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity//references.bib ====