proposition@alttheorem question@alttheorem lemma@alttheorem
corollary@alttheorem counterexample@alttheorem definition@alttheorem
example@alttheorem conjecture@alttheorem remark@alttheorem

# Sharp isoperimetric comparison on non collapsed spaces with lower Ricci
bounds

Gioacchino Antonelli  Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126
Pisa, Italy<EMAIL_ADDRESS>,  Enrico Pasqualetto  Scuola
Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy.
<EMAIL_ADDRESS>,  Marco Pozzetta  Dipartimento di Matematica e
Applicazioni, Università di Napoli Federico II, Via Cintia, Monte S. Angelo,
80126 Napoli, Italy<EMAIL_ADDRESS>and  Daniele Semola
Mathematical Institute, University of Oxford, Radcliffe Observatory, Andrew
Wiles Building, Woodstock Rd, Oxford OX2 6GG<EMAIL_ADDRESS>
###### Abstract.

This paper studies sharp isoperimetric comparison theorems and sharp
dimensional concavity properties of the isoperimetric profile for non smooth
spaces with lower Ricci curvature bounds, the so-called $N$-dimensional
$\operatorname{RCD}(K,N)$ spaces $(X,\mathsf{d},\mathscr{H}^{N})$.

The absence of most of the classical tools of Geometric Measure Theory and the
possible non existence of isoperimetric regions on non compact spaces are
handled via an original argument to estimate first and second variation of the
area for isoperimetric sets, avoiding any regularity theory, in combination
with an asymptotic mass decomposition result of perimeter-minimizing
sequences.

Most of our statements are new even for smooth, non compact manifolds with
lower Ricci curvature bounds and for Alexandrov spaces with lower sectional
curvature bounds. They generalize several results known for compact manifolds,
non compact manifolds with uniformly bounded geometry at infinity, and
Euclidean convex bodies. Many other consequences of these results are explored
in a companion paper by the authors.

###### Key words and phrases:

Isoperimetric problem, Isoperimetric profile, Lower Ricci bounds, RCD space

###### 1991 Mathematics Subject Classification:

Primary: 49Q20, 49J45, 53A35. Secondary: 53C23, 49J40.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Convergence and stability results
    2. 2.2 $\rm BV$ functions and sets of finite perimeter in metric measure spaces
    3. 2.3 Sobolev functions, Laplacians and vector fields in metric measure spaces
    4. 2.4 Geometric Analysis on $\operatorname{RCD}$ spaces
    5. 2.5 Localization of the Curvature-Dimension condition
  3. 3 The distance function from isoperimetric sets
  4. 4 Concavity properties of the isoperimetric profile function and consequences
    1. 4.1 Sharp concavity inequalities for the isoperimetric profile
    2. 4.2 Fine properties of the isoperimetric profile
    3. 4.3 Consequences
    4. 4.4 Stability of isoperimetric sets

##  1\. Introduction

### Isoperimetry and lower Ricci curvature bounds

There is a celebrated connection between Ricci curvature and the isoperimetric
problem in Geometric Analysis, going back at least to the Lévy–Gromov
inequality [57, Appendix C]. The primary goal of this paper is to extend
several results about the isoperimetric problem on compact Riemannian
manifolds with lower Ricci curvature bounds to non compact Riemannian
manifolds and non smooth spaces with lower Ricci curvature bounds. In order to
deal with the possible non existence of isoperimetric regions and with the
lack of regularity we develop a series of new tools with respect to the
classical literature. Non smooth spaces enter into play naturally when dealing
with smooth non compact Riemannian manifolds, via the analysis of their
pointed limits at infinity.

We consider the setting of $N$-dimensional $\operatorname{RCD}(K,N)$ metric
measure spaces $(X,\mathsf{d},\mathscr{H}^{N})$, for finite $N\in[1,\infty)$
and $K\in\mathbb{R}$, see [46, 62] after [84, 85, 68, 10, 50, 7, 47, 14, 40].
Here $K\in\mathbb{R}$ plays the role of (synthetic) lower bound on the Ricci
curvature, $N\in[1,\infty)$ plays the role of (synthetic) upper bound on the
dimension and $\mathscr{H}^{N}$ indicates the $N$-dimensional Hausdorff
measure. This class includes (convex subsets of) smooth Riemannian manifolds
with lower Ricci curvature bounds endowed with their volume measure, their
noncollapsing measured Gromov–Hausdorff limits [44], and finite dimensional
Alexandrov spaces with sectional curvature lower bounds [34, 81].

We shall rely on the theory of sets of finite perimeter in
$\operatorname{RCD}(K,N)$ spaces, as developed in [3, 5, 31, 30]. For the sake
of this introduction we just remark that it is fully consistent with the
Euclidean and Riemannian ones. In particular, (reduced) boundaries of sets of
finite perimeter are rectifiable, the perimeter coincides with the restriction
of the $(N-1)$ –dimensional Hausdorff measure to the (reduced) boundary and it
does not charge the boundary of the ambient space.
Given an $\operatorname{RCD}(K,N)$ metric measure space
$(X,\mathsf{d},\mathscr{H}^{N})$ such that $\mathscr{H}^{N}(B_{1}(x))\geq
v_{0}$ for any $x\in X$ for some $v_{0}>0$, we introduce the isoperimetric
profile $I_{X}:[0,\mathscr{H}^{N}(X))\to[0,\infty)$ by

$I_{X}(v):=\inf\left\\{\operatorname{Per}(E)\,:\,E\subset
X\,,\,\,\mathscr{H}^{N}(E)=v\right\\}\,,$ (1.1)

where we drop the subscript $X$ when there is no risk of confusion. When
$E\subset X$ attains the infimum in (1.1) for $v=\mathscr{H}^{N}(E)$, we call
it an isoperimetric region. In this setting we obtain:

  * •

sharp second order differential inequalities for the isoperimetric profile,
corresponding to equalities on the model spaces with constant sectional
curvature. These inequalities are new even in the case of non compact
Riemannian manifolds and use in a crucial way the non smooth approach. The
proof bypasses the possible non existence of isoperimetric regions on the
space, that is classically used for such arguments, employing a concentration-
compactness argument in an original way;

  * •

a sharp Laplacian comparison theorem for the distance function from $\partial
E$, which is a fundamental tool to prove the above items since it corresponds
to the bounds usually obtained via first and second variation of the area in
this low regularity setting;

  * •

Gromov–Hausdorff stability and perimeters’ convergence of isoperimetric
regions along non collapsing sequences of $N$-dimensional
$\operatorname{RCD}(K,N)$ spaces. In order to prove these statements we deduce
uniform regularity estimates for isoperimetric sets from uniform concavity
estimates of the isoperimetric profiles.

Many of the above results are new even for smooth, non compact manifolds with
lower Ricci curvature bounds and for Alexandrov spaces with lower sectional
curvature bounds. They answer several open questions in [25, 69, 66, 78, 23].
We expect the techniques developed in this paper to have a broad range of
applications in Geometric Analysis under lower curvature bounds. For instance,
in the study of more general geometric variational problems, the isoperimetric
problem on weighted Riemannian manifolds with lower bounds on the Bakry-Émery
curvature tensor, other geometric and functional inequalities.

### Main results

On model spaces with constant sectional curvature $K/(N-1)\in\mathbb{R}$ and
dimension $N\geq 2$ the isoperimetric profile $I_{K,N}$ solves the following
second order differential equation on its domain:

$-I_{K,N}^{\prime\prime}I_{K,N}=K+\frac{\left(I^{\prime}_{K,N}\right)^{2}}{N-1}\,.$
(1.2)

Equivalently, setting $\psi_{K,N}:=I_{K,N}^{\frac{N}{N-1}}$, we have

$-\psi_{K,N}^{\prime\prime}=\frac{KN}{N-1}\psi_{K,N}^{\frac{2-N}{N}}\,.$ (1.3)

Combining the existence of isoperimetric regions for any volume, the
regularity theory in GMT, and the second variation of the area (1.4), in [24,
25, 26, 75, 79] it was proved that the isoperimetric profile of a smooth,
compact, $N$-dimensional Riemannian manifold with $\mathrm{Ric}\geq K$
verifies the inequality $\geq$ in (1.2) and (1.3) in a weak sense.

Here we obtain the following far reaching extension to the setting of
$\operatorname{RCD}(K,N)$ metric measure spaces
$(X,\mathsf{d},\mathscr{H}^{N})$ with a uniform lower bound on the volume of
unit balls, without any assumption on the existence of isoperimetric regions.
We stress again that the classical argument to show Theorem 1.1 in the compact
setting uses in a crucial way the existence of isoperimetric regions for every
volume, that we do not have at disposal in the present setting.

######  Theorem 1.1 (cf. with Theorem 4.2).

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ space.
Assume that there exists $v_{0}>0$ such that $\mathscr{H}^{N}(B_{1}(x))\geq
v_{0}$ for every $x\in X$.

Let $I:(0,\mathscr{H}^{N}(X))\to(0,\infty)$ be the isoperimetric profile of
$X$. Then:

  1. (1)

the inequality

$-I^{\prime\prime}I\geq K+\frac{(I^{\prime})^{2}}{N-1}\,\quad\text{holds in
the viscosity sense on $(0,\mathscr{H}^{N}(X))$}\,,$

  2. (2)

if $\psi:=I^{\frac{N}{N-1}}$ then

$-\psi^{\prime\prime}\geq\frac{KN}{N-1}\psi^{\frac{2-N}{N}}\,\quad\text{holds
in the viscosity sense on $(0,\mathscr{H}^{N}(X))$}\,.$

In particular, the above holds for non compact smooth Riemannian manifolds
with Ricci curvature bounded from below and volume of unit balls uniformly
bounded away from zero.
The following direct corollary of Theorem 1.1 was previously known only under
the strong additional assumption of existence of isoperimetric sets. To the
best of our knowledge, this is the first application of the theory of
$\operatorname{RCD}$ spaces to prove a new sharp geometric inequality on
smooth Riemannian manifolds.

######  Corollary .

Let $K\in\mathbb{R}$ and $2\leq N\in\mathbb{N}$. Let $(M^{N},g)$ be a smooth
complete noncompact Riemannian manifold with $\mathrm{Ric}\geq Kg$, and with
volume form $\mathrm{vol}$. Assume that there exists $v_{0}>0$ such that
$\mathrm{vol}(B_{1}(x))\geq v_{0}$ for every $x\in M^{N}$.

Let $I:(0,\mathrm{vol}(M^{N}))\to(0,\infty)$ be the isoperimetric profile of
$M^{N}$. Then:

  1. (1)

the inequality

$-I^{\prime\prime}I\geq K+\frac{(I^{\prime})^{2}}{N-1}\,\quad\text{holds in
the viscosity sense on $(0,\mathrm{vol}(M^{N}))$}\,;$

  2. (2)

if $\psi:=I^{\frac{N}{N-1}}$ then

$-\psi^{\prime\prime}\geq\frac{KN}{N-1}\psi^{\frac{2-N}{N}}\,\quad\text{holds
in the viscosity sense on $(0,\mathrm{vol}(M^{N}))$}\,.$

The proof of Theorem 1.1 combines the generalized existence of isoperimetric
regions (cf. with Theorem 4.1), the interpretation of the differential
inequalities in the viscosity sense and the forthcoming Laplacian comparison
Theorem 1.2 to estimate first and second variation of the area via
equidistants in the non smooth setting.

The main new tool that we develop to prove Theorem 1.1 is a sharp bound on the
Laplacian of the signed distance function from isoperimetric regions inside
$\operatorname{RCD}(K,N)$ metric measure spaces
$(X,\mathsf{d},\mathscr{H}^{N})$.

If $E$ is a (smooth) isoperimetric region inside a smooth Riemannian manifold
$(M^{N},g)$ the first variation formula implies that the mean curvature $H$ of
its boundary $\partial E$ is constant. Moreover, if $t\mapsto E_{t}$ denotes
the parallel deformation of $E$ via equidistant sets, $\nu$ and $\mathrm{II}$
denote a choice of the unit normal to $\partial E$ and its second fundamental
form, respectively, and $\mathrm{Ric}(\nu,\nu)$ indicates the Ricci curvature
of $M$ in the direction of $\nu$, then the second variation formula yields
that

$\frac{\mathop{}\\!\mathrm{d}^{2}}{\mathop{}\\!\mathrm{d}t^{2}}\lvert_{t=0}\operatorname{Per}(E_{t})=\int_{\partial
E}\left(H^{2}-||\mathrm{II}||^{2}-\mathrm{Ric}(\nu,\nu)\right)\mathop{}\\!\mathrm{d}\operatorname{Per}\,,$
(1.4)

where we denoted by $\operatorname{Per}$ the perimeter, which coincides with
the Riemannian surface measure for sufficiently regular sets.
If we further assume that $\mathrm{Ric}\geq Kg$, for some $K\in\mathbb{R}$,
then the Cauchy–Schwarz inequality applied to the eigenvalues of $\mathrm{II}$
yields

$\frac{\mathop{}\\!\mathrm{d}^{2}}{\mathop{}\\!\mathrm{d}t^{2}}\lvert_{t=0}\operatorname{Per}(E_{t})\leq\left(\frac{N-2}{N-1}H^{2}-K\right)\operatorname{Per}(E)\,.$
(1.5)

One of the main technical achievements of this work is to develop a
counterpart of (1.5) for isoperimetric regions in non smooth spaces with lower
Ricci curvature bounds. Our argument departs from all the previous literature,
following the general scheme outlined above, and it requires a different,
global rather than infinitesimal, perspective.

Let us introduce the comparison functions

$s_{k,\lambda}(r):=\cos_{k}(r)-\lambda\sin_{k}(r)\,,$ (1.6)

where

$\cos_{k}^{\prime\prime}+k\cos_{k}=0\,,\quad\cos_{k}(0)=1\,,\quad\cos_{k}^{\prime}(0)=0\,,$
(1.7)

and

$\sin_{k}^{\prime\prime}+k\sin_{k}=0\,,\quad\sin_{k}(0)=0\,,\quad\sin_{k}^{\prime}(0)=1\,.$
(1.8)

######  Theorem 1.2 (cf. with Theorem 3.2).

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ metric
measure space for some $K\in\mathbb{R}$ and $N\geq 2$ and let $E\subset X$ be
an isoperimetric region. Then, denoting by $f$ the signed distance function
from $\overline{E}$, there exists $c\in\mathbb{R}$ such that

$\bm{\Delta}f\geq-(N-1)\frac{s^{\prime}_{\frac{K}{N-1},\frac{c}{N-1}}\circ\left(-f\right)}{s_{\frac{K}{N-1},\frac{c}{N-1}}\circ\left(-f\right)}\quad\text{on
$E$ and
}\quad\bm{\Delta}f\leq(N-1)\frac{s^{\prime}_{\frac{K}{N-1},-\frac{c}{N-1}}\circ
f}{s_{\frac{K}{N-1},-\frac{c}{N-1}}\circ f}\quad\text{on
$X\setminus\overline{E}$}\,.$ (1.9)

The bounds in (1.9) are understood in the sense of distributions, and we
always consider open representatives for isoperimetric regions (see Theorem
2.5). They are sharp, since equalities are attained in the model spaces with
constant sectional curvature.

Notice that the distance function might not be globally smooth even when
$(X,\mathsf{d})$ is isometric to a smooth Riemannian manifold and $E\subset X$
has smooth boundary, in which case (1.9) is equivalent to the requirement that
$\partial E$ has constant mean curvature equal to $c$. We will indicate any
$c\in\mathbb{R}$ such that (1.9) holds as a _mean curvature barrier_ for $E$.

### Consequences and strategies of the proofs

Several consequences of Theorem 1.1 are investigated in the rest of the paper:

  * •

uniform lower bounds, semi-concavity and Lipschitz properties of the
isoperimetric profile in a fixed range of volumes, only depending on the lower
Ricci curvature bound, the dimension and a lower bound on the volume of unit
balls, see subsection 4.2, subsection 4.2, and subsection 4.2;

  * •

the existence of the limit $\lim_{v\to
0}I(v)/v^{\frac{N-1}{N}}\in(0,N\omega_{N}^{\frac{1}{N}}]$ on any
$\operatorname{RCD}(K,N)$ space $(X,\mathsf{d},\mathscr{H}^{N})$ with volume
of unit balls uniformly bounded from below, see subsection 4.2 and subsection
4.2;

  * •

the strict subadditivity of the isoperimetric profile for small volumes (only
depending on $K$, $N$ and the uniform lower bound on the volume of unit
balls), see subsection 4.2. This implies in turn that isoperimetric regions
with small volume are connected, see subsection 4.3. Moreover, in the
asymptotic mass decomposition, minimizing sequences for small volumes do not
split: either they converge to an isoperimetric region, or they drift off to
exactly one isoperimetric region in a pointed limit at infinity, see
subsection 4.3. All the previous conclusions hold for every volume when $K=0$;

  * •

uniform, scale invariant diameter estimates for isoperimetric regions of small
volume, without further assumptions, and for any volume when $K=0$ and
$(X,\mathsf{d},\mathscr{H}^{N})$ has Euclidean volume growth, see subsection
4.3;

  * •

uniform density estimates and uniform almost minimality properties for
isoperimetric sets, see subsection 4.2. They allow to bootstrap
$L^{1}$-convergence to Gromov–Hausdorff convergence and convergence of the
perimeters for sequences of isoperimetric sets and to prove the stability of
mean curvature barriers obtained with Theorem 1.2, see Theorem 4.3. We remark
that uniform almost minimality properties are fundamental in several
circumstances in Geometric Measure Theory, see for instance [87], and that the
classical strategies to achieve them break in the present setting, as they
heavily rely on smoothness.

Let us further comment on Theorem 1.2. When $\partial E$ is a smooth constant
mean curvature hypersurface in a smooth Riemannian manifold, the Laplacian of
the signed distance function from $E$ equals the mean curvature of $\partial
E$ along $\partial E$. Then (1.9) can be proved with a classical computation
using Jacobi fields and one dimensional comparison for Riccati equations away
from the cut locus, finding its roots in [88, 38]. The singular contribution
coming from the cut locus has the right sign, in great generality.

The original proof of the Lévy–Gromov isoperimetric inequality [57, Appendix
C] builds on a variant of (1.9). The key additional difficulty with respect to
smooth constant mean curvature hypersurfaces is that boundaries of
isoperimetric regions might be non smooth when $N\geq 8$ and it is handled
relying on a deep regularity theorem in Geometric Measure Theory [2]. This
strategy seems out of reach in the setting of $\operatorname{RCD}$ spaces.
Our proof of Theorem 1.2 completely avoids the regularity theory even on
smooth Riemannian manifolds. It is inspired by [37, 36, 80] and the recent
study of perimeter minimizing sets in [73].
We start proving a-dimensional versions of (1.9), corresponding to the limit
of (1.9) as $N\to\infty$. Exploiting the equivalence between distributional
and viscosity bounds on the Laplacian from [73], we prove that if the bounds
fail there exists a volume fixing perturbation of $E$ with strictly smaller
perimeter, a contradiction with the isoperimetric condition. The perturbations
are built by sliding simultaneously level sets of distance-like functions with
well controlled Laplacian. The argument can be thought as a highly non linear
version of the moving planes method, with no symmetries on the background and
in a very low regularity setting.
The _stability_ of Laplacian bounds under the Hopf–Lax duality (equivalently,
along solutions of the Hamilton–Jacobi equation), proved in [73] building on
[64, 11], plays the role of the classical computation with Jacobi fields in
Riemannian geometry. It crucially enters into play in the construction of the
perturbations. The a-dimensional versions of (1.9) self improve to sharp
dimensional bounds thanks to the well established localization technique [41,
42]. The additional difficulty with respect to the case of local perimeter
minimizers considered in [73] is that only volume fixing perturbations are
admissible for the isoperimetric problem.

We do not claim that Theorem 1.2 carries all the information provided by the
first and second variation formulas for the area in the smooth setting,
however one of the main novelties of the present work will be to show that it
is a valid replacement in several circumstances. Moreover, its global nature
makes it more suitable for stability arguments. Among its direct consequences
we mention sharp Heintze–Karcher type bounds for perimeters section 3 and
volumes section 3 of variations of isoperimetric sets via equidistant sets,
that can be obtained by integration.

The ability to deduce information about the isoperimetric behaviour of an
$\operatorname{RCD}(K,N)$ metric measure space
$(X,\mathsf{d},\mathscr{H}^{N})$ from Theorem 1.2 is related to the existence
of isoperimetric regions, which is not guaranteed, when
$\mathscr{H}^{N}(X)=\infty$, see, e.g., [17, Example 3.6], or the introduction
of [77].
We overcome this issue thanks to the asymptotic mass decomposition result
recently proved in [18], extending the previous [83, 77, 72, 17] and building
on a concentration-compactness argument. If $(X,\mathsf{d})$ is compact, a
minimizing sequence for the isoperimetric problem (1.1) for volume $V>0$
converges, up to subsequences, to an isoperimetric set, by lower
semicontinuity of the perimeter. This is not true, in general, when
$(X,\mathsf{d})$ is not compact as elementary examples illustrate. However,
[18] shows that, if $(X,\mathsf{d},\mathscr{H}^{N})$ is a non compact
$\operatorname{RCD}(K,N)$ space with $\mathscr{H}^{N}(B_{1}(p))>v_{0}$ for any
$p\in X$ for some $v_{0}>0$, then, up to subsequences, every minimizing
sequence for the isoperimetric problem at a given volume $V>0$ splits into
finitely many pieces. One of them converges to an isoperimetric region in $X$,
possibly of volume less than $V$ (possibly zero). The others converge to
isoperimetric regions inside pointed measured Gromov–Hausdorff limits at
infinity $(Y,\mathsf{d}_{Y},\mathscr{H}^{N},q)$ of sequences
$(X,\mathsf{d},\mathscr{H}^{N},p_{i})$, where $\mathsf{d}(p,p_{i})\to\infty$
as $i\to\infty$ for some reference point $p\in X$. Moreover, there is no loss
of total mass in this process, see Theorem 4.1, hence the result can be seen
as a generalized existence of isoperimetric regions.

An observation going back to [77] is that if $(X,\mathsf{d})$ and all its
pointed limits at infinity are isometric to smooth Riemannian manifolds, then
generalized isoperimetric regions can be used as isoperimetric regions in the
compact case in combination with (1.4) to deduce useful information. However,
the assumption that $(X,\mathsf{d})$ is isometric to a smooth Riemannian
manifold with $\mathrm{Ric}\geq K$ and $\mathscr{H}^{N}(B_{1}(p))>v_{0}$, for
every $p\in X$ and some $v_{0}>0$, does not guarantee any regularity of its
pointed limits at infinity besides them being noncollapsed Ricci limit spaces.
The ability to estimate the second variation of the perimeter via equidistant
sets for isoperimetric sets in general $\operatorname{RCD}(K,N)$ spaces
$(X,\mathsf{d},\mathscr{H}^{N})$ as in Theorem 1.2 makes this heuristic work
without further assumptions in the smooth Riemannian case and in much greater
generality, obtaining Theorem 1.1. This is a fundamental new contribution of
the present work.

We believe that the techniques introduced in this paper will be useful to deal
with general $\operatorname{RCD}(K,N)$ metric measure spaces
$(X,\mathsf{d},\mathfrak{m})$ and, in particular, with smooth weighted
Riemannian manifolds verifying Curvature-Dimension bounds. However, the study
of this more general case is left to the future investigation, due to some
additional difficulties that we outline below.

The two main ingredients for the proof of Theorem 1.1 are the asymptotic mass
decomposition Theorem 4.1 and the Laplacian comparison Theorem 1.2. The
asymptotic mass decomposition Theorem 4.1 has been obtained in [18] by the
first and the third authors together with Nardulli, leveraging on the
techniques developed in [77, 17]. We expect that the results of [18]
generalize to arbitrary $\operatorname{RCD}(K,N)$ metric measure spaces
$(X,\mathsf{d},\mathfrak{m})$. However, the extension to the general case
seems to require some additional ideas. Indeed, when
$\mathfrak{m}=\mathscr{H}^{N}$, in the proof of [18] we exploit the fact that
the density of the measure is $1$ almost everywhere, and the perimeters of the
balls centered at almost every point are infinitesimally equivalent to the
perimeters of the balls in $\mathbb{R}^{N}$. Both these statement fail in
general when $\mathfrak{m}\neq\mathscr{H}^{N}$.
We also expect that Theorem 1.2 holds for arbitrary reference measures
$\mathfrak{m}$. In particular, the Laplacian comparison holds for
isoperimetric sets in smooth weighted Riemannian manifolds verifying the
$\operatorname{CD}(K,N)$ condition. However, the extension would require again
some non trivial new insights. Indeed, the very first step to obtain Theorem
1.2 is a topological regularity result for isoperimetric sets obtained by the
first three authors in [19] when $\mathfrak{m}=\mathscr{H}^{N}$. As in the
previous case, in [19] we heavily exploit the fact that the reference measure
is $\mathscr{H}^{N}$ especially when it comes to prove a local almost-
Euclidean isoperimetric inequality, see in particular [19, Propsition 3.1].
With this non trivial topological regularity at hands, it seems likely that
the proof of Theorem 1.2 could be adapted to general reference measures
$\mathfrak{m}$.

To put things in perspective, we stress that this paper heavily relies on the
results of [19, 18] (mild regularity of isoperimetric sets and asymptotic mass
decomposition, respectively), while it is independent of the existence results
in [16] by the first and third authors together with Bruè and Fogagnolo, and
it is completely independent of the sharp isoperimetric inequality obtained in
[29, 23]. The key contribution of the present work is to develop some refined
tools of Geometric Measure Theory in a low regularity setting and to combine
them in original way with the asymptotic mass decomposition from [18],
obtaining far reaching consequences for the isoperimetric problem under lower
Ricci curvature bounds.

### Comparison with the previous literature

We conclude this introduction with a brief comparison between our results and
the previous literature about the isoperimetric problem under lower curvature
bounds, without the aim of being comprehensive.

  * •

The difficulty of obtaining second order properties for the isoperimetric
profile on _non smooth spaces_ was pointed out in [25, page 99], [66] and in
[69, Appendix]. To the best of our knowledge, Theorem 1.1 is the first
instance of second order properties of the isoperimetric profile in a context
where no approximation with smooth Riemannian manifolds is at disposal.

  * •

The setting of $\operatorname{RCD}(0,N)$ spaces
$(X,\mathsf{d},\mathscr{H}^{N})$ recovers in particular many of the results
for Euclidean convex cones treated in [83] and for cones with non negative
Ricci curvature considered in [75].

  * •

The results of the present paper recover, in a much more general setting, many
of the results proved in [67] for unbounded Euclidean convex bodies of uniform
geometry.
The setting of $\operatorname{RCD}(K,N)$ spaces
$(X,\mathsf{d},\mathscr{H}^{N})$ includes more in general convex subsets of
smooth Riemannian manifolds with Ricci curvature bounded from below,
regardless of any compactness assumption and regularity assumption on the
boundary. Compact convex subsets of Riemannian manifolds with Ricci curvature
lower bounds have been considered in [27].

  * •

The stability of mean curvature barriers in the sense of Theorem 1.2 for
Gromov–Hausdorff converging sequences of boundaries inside measured
Gromov–Hausdorff converging sequences of $\operatorname{RCD}(K,N)$ spaces has
been recently observed in [61] (see also the previous [32]). The main novelty
of our work in this regard is to provide a large and natural class of sets
having mean curvature barriers, namely isoperimetric sets. Moreover, we prove
that $L^{1}$ convergence (which is guaranteed, up to subsequences, for
equibounded isoperimetric sets) self-improves to Gromov–Hausdorff convergence
under very natural assumptions. Therefore the stability of mean curvature
barriers applies in this setting.

### Addendum

This is the first of two companion papers, together with [20]. The joint
version of the two papers appeared on arXiv in [21]. In the second one we are
going to explore several consequences of the main results of this paper to the
study of asymptotic isoperimetry on non collapsed spaces with Ricci lower
bounds, especially in the case where $K=0$:

  * •

exploiting Theorem 1.2, we give a new proof of the sharp isoperimetric
inequality on $\operatorname{RCD}(0,N)$ spaces
$(X,\mathsf{d},\mathscr{H}^{N})$ with Euclidean volume growth that has been
considered, with increasing level of generality, in [1, 48, 29, 23, 39]. We
prove also the rigidity of the inequality in the setting above without any
additional regularity assumption. Namely we prove that equality is achieved if
and only if the ambient space is isometric to a cone and the set saturating
the inequality is isometric to a ball centered at a tip of the cone;

  * •

we explicitly determine the asymptotic behavior of the isoperimetric profile
for volumes tending to zero on $\operatorname{RCD}(K,N)$ spaces
$(X,\mathsf{d},\mathscr{H}^{N})$, see subsection 4.2;

  * •

we analyze the behavior of sequences of isoperimetric sets for volumes tending
to zero on $\operatorname{RCD}(K,N)$ spaces $(X,\mathsf{d},\mathscr{H}^{N})$
and for volumes tending to infinity on Alexandrov spaces with non negative
curvature and Euclidean volume growth.

### Acknowledgements

The first author is partially supported by the European Research Council (ERC
Starting Grant 713998 GeoMeG ‘ _Geometry of Metric Groups_ ’). The second
author is supported by the Balzan project led by Luigi Ambrosio. The first and
the third authors are partially supported by the INdAM - GNAMPA Project
CUP_E55F22000270001 “Isoperimetric problems: variational and geometric
aspects”. The last author is supported by the European Research Council (ERC),
under the European Union Horizon 2020 research and innovation programme, via
the ERC Starting Grant “CURVATURE”, grant agreement No. 802689. He is grateful
to Elia Bruè and Andrea Mondino for several conversations and useful comments
on a preliminary version of this note and to Christian Ketterer for useful
comments. The authors are grateful to Otis Chodosh, Mattia Fogagnolo and
Emanuel Milman for useful feedbacks on a preliminary version of the paper.

##  2\. Preliminaries

In this paper, by a _metric measure space_ (briefly, m.m.s.) we mean a triple
$(X,\mathsf{d},\mathfrak{m})$, where $(X,\mathsf{d})$ is a complete and
separable metric space, while $\mathfrak{m}\geq 0$ is a boundedly-finite Borel
measure on $X$. For any $k\in[0,\infty)$, we denote by $\mathscr{H}^{k}$ the
_$k$ -dimensional Hausdorff measure_ on $(X,\mathsf{d})$. We indicate with
$C(X)$ the space of all continuous functions $f\colon X\to\mathbb{R}$ and
$C_{b}(X)\coloneqq\\{f\in C(X)\,:\,f\text{ is bounded}\\}$. We denote by ${\rm
LIP}(X)\subseteq C(X)$ the space of all Lipschitz functions, while ${\rm
LIP}_{bs}(X)$ (resp. ${\rm LIP}_{c}(X)$) stands for the set of all those
$f\in{\rm LIP}(X)$ whose support ${\rm spt}f$ is bounded (resp. compact). More
generally, we denote by ${\rm LIP}_{loc}(X)$ the space of locally Lipschitz
functions $f\colon X\to\mathbb{R}$. Given $f\in{\rm LIP}_{loc}(X)$,

${\rm lip\,}f(x)\vcentcolon=\limsup_{y\to
x}\frac{|f(y)-f(x)|}{\mathsf{d}(x,y)}$

is the _slope_ of $f$ at $x$, for any accumulation point $x\in X$, and ${\rm
lip\,}f(x)\coloneqq 0$ if $x\in X$ is isolated.

We shall also work with the local versions of the above spaces: given
$\Omega\subseteq X$ open, we will consider the spaces ${\rm
LIP}_{c}(\Omega)\subseteq{\rm LIP}_{bs}(\Omega)\subseteq{\rm
LIP}(\Omega)\subseteq{\rm LIP}_{loc}(\Omega)$. We underline that by ${\rm
LIP}_{bs}(\Omega)$ we mean the space of all $f\in{\rm LIP}(\Omega)$ having
bounded support ${\rm spt}f\subseteq\Omega$ that verifies $\mathsf{d}({\rm
spt}f,\partial\Omega)>0$.

###  2.1. Convergence and stability results

We introduce the pointed measured Gromov–Hausdorff convergence already in a
proper realization even if this is not the general definition. Nevertheless,
the simplified definition of Gromov–Hausdorff convergence via a realization is
equivalent to the standard definition of pmGH convergence in our setting,
because in the applications we will always deal with locally uniformly
doubling measures, see [54, Theorem 3.15 and Section 3.5]. The following
definition is taken from the introductory exposition of [5].

######  Definition  (pGH and pmGH convergence).

A sequence $\\{(X_{i},\mathsf{d}_{i},x_{i})\\}_{i\in\mathbb{N}}$ of pointed
metric spaces is said to converge in the _pointed Gromov–Hausdorff topology,
in the $\mathrm{pGH}$ sense for short,_ to a pointed metric space
$(Y,\mathsf{d}_{Y},y)$ if there exist a complete separable metric space
$(Z,\mathsf{d}_{Z})$ and isometric embeddings

$\begin{split}&\Psi_{i}:(X_{i},\mathsf{d}_{i})\to(Z,\mathsf{d}_{Z}),\qquad\forall\,i\in\mathbb{N}\,,\\\
&\Psi:(Y,\mathsf{d}_{Y})\to(Z,\mathsf{d}_{Z})\,,\end{split}$

such that for any $\varepsilon,R>0$ there is
$i_{0}(\varepsilon,R)\in\mathbb{N}$ such that

$\Psi_{i}(B_{R}^{X_{i}}(x_{i}))\subset\left[\Psi(B_{R}^{Y}(y))\right]_{\varepsilon},\qquad\Psi(B_{R}^{Y}(y))\subset\left[\Psi_{i}(B_{R}^{X_{i}}(x_{i}))\right]_{\varepsilon}\,,$

for any $i\geq i_{0}$, where $[A]_{\varepsilon}\coloneqq\\{z\in Z\ :\
\mathsf{d}_{Z}(z,A)\leq\varepsilon\\}$ for any $A\subset Z$.

Let $\mathfrak{m}_{i}$ and $\mu$ be given in such a way
$(X_{i},\mathsf{d}_{i},\mathfrak{m}_{i},x_{i})$ and $(Y,\mathsf{d}_{Y},\mu,y)$
are m.m.s. If in addition to the previous requirements we also have
$(\Psi_{i})_{\sharp}\mathfrak{m}_{i}\rightharpoonup\Psi_{\sharp}\mu$ with
respect to duality with continuous bounded functions on $Z$ with bounded
support, then the convergence is said to hold in the _pointed measured
Gromov–Hausdorff topology, or in the $\mathrm{pmGH}$ sense for short_.

We need to recall a generalized $L^{1}$-notion of convergence for sets defined
on a sequence of metric measure spaces converging in the pmGH sense. Such a
definition is given in [5, Definition 3.1], and it is investigated in [5]
capitalizing on the results in [12].

######  Definition  ($L^{1}$-strong and $L^{1}_{\mathrm{loc}}$ convergence).

Let $\\{(X_{i},\mathsf{d}_{i},\mathfrak{m}_{i},x_{i})\\}_{i\in\mathbb{N}}$ be
a sequence of pointed metric measure spaces converging in the pmGH sense to a
pointed metric measure space $(Y,\mathsf{d}_{Y},\mu,y)$ and let
$(Z,\mathsf{d}_{Z})$ be a realization as in subsection 2.1.

We say that a sequence of Borel sets $E_{i}\subset X_{i}$ such that
$\mathfrak{m}_{i}(E_{i})<+\infty$ for any $i\in\mathbb{N}$ converges _in the
$L^{1}$-strong sense_ to a Borel set $F\subset Y$ with $\mu(F)<+\infty$ if
$\mathfrak{m}_{i}(E_{i})\to\mu(F)$ and
$\chi_{E_{i}}\mathfrak{m}_{i}\rightharpoonup\chi_{F}\mu$ with respect to the
duality with continuous bounded functions with bounded support on $Z$.

We say that a sequence of Borel sets $E_{i}\subset X_{i}$ converges _in the
$L^{1}_{\mathrm{loc}}$-sense_ to a Borel set $F\subset Y$ if $E_{i}\cap
B_{R}(x_{i})$ converges to $F\cap B_{R}(y)$ in $L^{1}$-strong for every $R>0$.

######  Definition  (Hausdorff convergence).

Let $\\{(X_{i},\mathsf{d}_{i},\mathfrak{m}_{i},x_{i})\\}_{i\in\mathbb{N}}$ be
a sequence of pointed metric measure spaces converging in the pmGH sense to a
pointed metric measure space $(Y,\mathsf{d}_{Y},\mu,y)$.

We say that a sequence of closed sets $E_{i}\subset X_{i}$ converges _in
Hausdorff distance_ (or _in Hausdorff sense_) to a closed set $F\subset Y$ if
there holds convergence in Hausdorff distance in a realization
$(Z,\mathsf{d}_{Z})$ of the pmGH convergence as in subsection 2.1.

It is also possible to define notions of uniform convergence and
$H^{1,2}$-strong and weak convergences for sequences of functions of a
sequence of spaces $X_{i}$ converging in pointed measure Gromov–Hausdorff
sense. We refer the reader to [5, 12] for such definitions.

###  2.2. $\rm BV$ functions and sets of finite perimeter in metric measure
spaces

We begin with the definitions of _function of bounded variation_ and _set of
finite perimeter_ in the setting of m.m.s..

######  Definition  ($\rm BV$ functions and perimeter on m.m.s.).

Let $(X,\mathsf{d},\mathfrak{m})$ be a metric measure space. Given $f\in
L^{1}_{loc}(X,\mathfrak{m})$ we define

$|Df|(A)\vcentcolon=\inf\left\\{\liminf_{i}\int_{A}{\rm
lip\,}f_{i}\,\mathrm{d}\mathfrak{m}\ :\ \text{$f_{i}\in{\rm
LIP}_{loc}(A),\,f_{i}\to f$ in $L^{1}_{loc}(A,\mathfrak{m})$}\right\\}\,,$

for any open set $A\subseteq X$. We declare that a function $f\in
L^{1}_{loc}(X,\mathfrak{m})$ is of _local bounded variation_ , briefly
$f\in{\rm BV}_{loc}(X)$, if $|Df|(A)<+\infty$ for every $A\subseteq X$ open
bounded. A function $f\in L^{1}(X,\mathfrak{m})$ is said to belong to the
space of _bounded variation functions_ ${\rm BV}(X)={\rm
BV}(X,\mathsf{d},\mathfrak{m})$ if $|Df|(X)<+\infty$.

If $E\subseteq{\rm X}$ is a Borel set and $A\subseteq X$ is open, we define
the _perimeter $\operatorname{Per}(E,A)$ of $E$ in $A$_ by

$\operatorname{Per}(E,A)\vcentcolon=\inf\left\\{\liminf_{i}\int_{A}{\rm
lip\,}u_{i}\,\mathrm{d}\mathfrak{m}\ :\ \text{$u_{i}\in{\rm
LIP}_{loc}(A),\,u_{i}\to{\raise 1.29167pt\hbox{$\chi$}}_{E}$ in
$L^{1}_{loc}(A,\mathfrak{m})$}\right\\}\,,$

in other words $\operatorname{Per}(E,A)\coloneqq|D{\raise
1.29167pt\hbox{$\chi$}}_{E}|(A)$. We say that $E$ has _locally finite
perimeter_ if $\operatorname{Per}(E,A)<+\infty$ for every open bounded set
$A$. We say that $E$ has _finite perimeter_ if
$\operatorname{Per}(E,X)<+\infty$, and we denote
$\operatorname{Per}(E)\vcentcolon=\operatorname{Per}(E,X)$.

Let us remark that when $f\in{\rm BV}_{loc}(X,\mathsf{d},\mathfrak{m})$ or $E$
is a set with locally finite perimeter, the set functions
$|Df|,\operatorname{Per}(E,\cdot)$ above are restrictions to open sets of
Borel measures that we still denote by $|Df|,\operatorname{Per}(E,\cdot)$, see
[6], and [70].

In the sequel, we shall frequently make use of the following _coarea formula_
, proved in [70].

######  Theorem 2.1 (Coarea formula).

Let $(X,\mathsf{d},\mathfrak{m})$ be a metric measure space. Let $f\in
L^{1}_{loc}(X)$ be given. Then for any open set $\Omega\subseteq X$ it holds
that $\mathbb{R}\ni
t\mapsto\operatorname{Per}(\\{f>t\\},\Omega)\in[0,+\infty]$ is Borel
measurable and satisfies

$|Df|(\Omega)=\int_{\mathbb{R}}\operatorname{Per}(\\{f>t\\},\Omega)\,\,\mathrm{d}t\,.$

In particular, if $f\in{\rm BV}(X)$, then $\\{f>t\\}$ has finite perimeter for
a.e. $t\in\mathbb{R}$.

######  Remark  (Semicontinuity of the total variation under
$L^{1}_{loc}$-convergence).

Let $(X,\mathsf{d},\mathfrak{m})$ be a metric measure space. We recall (cf.
[70, Proposition 3.6]) that whenever $g_{i},g\in L^{1}_{loc}(X,\mathfrak{m})$
are such that $g_{i}\to g$ in $L^{1}_{loc}(X,\mathfrak{m})$, for every open
set $\Omega$ we have

$|Dg|(\Omega)\leq\liminf_{i\to+\infty}|Dg_{i}|(\Omega)\,.$

###  2.3. Sobolev functions, Laplacians and vector fields in metric measure
spaces

The _Cheeger energy_ on a metric measure space $(X,\mathsf{d},\mathfrak{m})$
is defined as the $L^{2}$-relaxation of the functional
$f\mapsto\frac{1}{2}\int{\rm
lip\,}^{2}f_{n}\mathop{}\\!\mathrm{d}\mathfrak{m}$ (see [9] after [43]).
Namely, for any function $f\in L^{2}(X)$ we define

${\sf Ch}(f)\coloneqq\inf\bigg{\\{}\liminf_{n\to\infty}\int{\rm
lip\,}^{2}f_{n}\mathop{}\\!\mathrm{d}\mathfrak{m}\;\bigg{|}\;(f_{n})_{n\in\mathbb{N}}\subseteq{\rm
LIP}_{bs}(X),\,f_{n}\to f\text{ in }L^{2}(X)\bigg{\\}}\,.$

The _Sobolev space_ $H^{1,2}(X)$ is defined as the finiteness domain $\\{f\in
L^{2}(X)\,:\,{\sf Ch}(f)<+\infty\\}$ of the Cheeger energy. The restriction of
the Cheeger energy to the Sobolev space admits the integral representation
${\sf Ch}(f)=\frac{1}{2}\int|\nabla f|^{2}\mathop{}\\!\mathrm{d}\mathfrak{m}$,
for a uniquely determined function $|\nabla f|\in L^{2}(X)$ that is called the
_minimal weak upper gradient_ of $f\in H^{1,2}(X)$. The linear space
$H^{1,2}(X)$ is a Banach space if endowed with the Sobolev norm

$\|f\|_{H^{1,2}(X)}\coloneqq\sqrt{\|f\|_{L^{2}(X)}^{2}+2{\sf
Ch}(f)}=\sqrt{\|f\|_{L^{2}(X)}^{2}+\||\nabla f|\|_{L^{2}(X)}^{2}},\quad\text{
for every }f\in H^{1,2}(X)\,.$

Following [50], when $H^{1,2}(X)$ is a Hilbert space (or equivalently ${\sf
Ch}$ is a quadratic form) we say that the metric measure space
$(X,\mathsf{d},\mathfrak{m})$ is _infinitesimally Hilbertian_.

For the rest of this paper, the infinitesimal Hilbertianity of
$(X,\mathsf{d},\mathfrak{m})$ will be our standing assumption. The results of
[8] ensure that ${\rm LIP}_{bs}(X)$ is dense in $H^{1,2}(X)$ with respect to
the norm topology. We define the bilinear mapping $H^{1,2}(X)\times
H^{1,2}(X)\ni(f,g)\mapsto\nabla f\cdot\nabla g\in L^{1}(X)$ as

$\nabla f\cdot\nabla g\coloneqq\frac{|\nabla(f+g)|^{2}-|\nabla f|^{2}-|\nabla
g|^{2}}{2},\quad\text{ for every }f,g\in H^{1,2}(X)\,.$

Given $\Omega\subseteq X$ open, we define the _local Sobolev space with
Dirichlet boundary conditions_ $H^{1,2}_{0}(\Omega)$ as the closure of ${\rm
LIP}_{bs}(\Omega)$ in $H^{1,2}(X)$. Notice that $H^{1,2}_{0}(X)=H^{1,2}(X)$.
Moreover, we declare that a given function $f\in L^{2}(\Omega)$ belongs to the
_local Sobolev space_ $H^{1,2}(\Omega)$ provided $\eta f\in H^{1,2}(X)$ holds
for every $\eta\in{\rm LIP}_{bs}(\Omega)$ and

$|\nabla f|\coloneqq{\rm ess\,sup}\big{\\{}\chi_{\\{\eta=1\\}}|\nabla(\eta
f)|\;\big{|}\;\eta\in{\rm LIP}_{bs}(\Omega)\big{\\}}\in L^{2}(X)\,,$

where by ${\rm ess\,sup}_{\lambda\in\Lambda}f_{\lambda}$ we mean the essential
supremum of a set $\\{f_{\lambda}\\}_{\lambda\in\Lambda}$ of measurable
functions.

######  Definition  (Local Laplacian).

Let $(X,\mathsf{d},\mathfrak{m})$ be an infinitesimally Hilbertian space and
$\Omega\subseteq X$ an open set. Then we say that a function $f\in
H^{1,2}(\Omega)$ has _local Laplacian_ in $\Omega$, $f\in D(\Delta,\Omega)$
for short, provided there exists a (uniquely determined) function $\Delta f\in
L^{2}(\Omega)$ such that

$\int_{\Omega}g\Delta f\mathop{}\\!\mathrm{d}\mathfrak{m}=-\int_{\Omega}\nabla
g\cdot\nabla f\mathop{}\\!\mathrm{d}\mathfrak{m},\quad\text{ for every }g\in
H^{1,2}_{0}(\Omega)\,.$

For brevity, we write $D(\Delta)$ instead of $D(\Delta,X)$.

More generally, we will work with functions having a measure-valued Laplacian
in some open set:

######  Definition  (Measure-valued Laplacian).

Let $(X,\mathsf{d},\mathfrak{m})$ be an infinitesimally Hilbertian space and
$\Omega\subseteq X$ an open set. Then we say that a function $f\in
H^{1,2}(\Omega)$ has _measure-valued Laplacian_ in $\Omega$, $f\in
D(\mathbf{\Delta},\Omega)$ for short, provided there exists a (uniquely
determined) locally finite measure $\mathbf{\Delta}f$ on $\Omega$ such that

$\int_{\Omega}g\mathbf{\Delta}f\coloneqq\int_{\Omega}g\mathop{}\\!\mathrm{d}\mathbf{\Delta}f=-\int_{\Omega}\nabla
g\cdot\nabla f\mathop{}\\!\mathrm{d}\mathfrak{m}\,,\quad\text{ for every
}g\in{\rm LIP}_{bs}(\Omega)\,.$

For brevity, we write $D(\mathbf{\Delta})$ instead of $D(\mathbf{\Delta},X)$.
Moreover, given functions $f\in{\rm LIP}(\Omega)\cap H^{1,2}(\Omega)$ and
$\eta\in C_{b}(\Omega)$, we say that _$\mathbf{\Delta}f\leq\eta$ in the
distributional sense_ provided $f\in D(\mathbf{\Delta},\Omega)$ and
$\mathbf{\Delta}f\leq\eta\mathfrak{m}$.

The above two notions of Laplacian are consistent, in the following sense:
given any $f\in H^{1,2}(\Omega)$, it holds that $f\in D(\Delta,\Omega)$ if and
only if $f\in D(\mathbf{\Delta},\Omega)$, $\mathbf{\Delta}f\ll\mathfrak{m}$
and
$\frac{\mathop{}\\!\mathrm{d}\mathbf{\Delta}f}{\mathop{}\\!\mathrm{d}\mathfrak{m}}\in
L^{2}(\Omega)$. If this is the case, then we also have that the
$\mathfrak{m}$-a.e. equality $\Delta
f=\frac{\mathop{}\\!\mathrm{d}\mathbf{\Delta}f}{\mathop{}\\!\mathrm{d}\mathfrak{m}}$
holds.

The _heat flow_ $\\{P_{t}\\}_{t\geq 0}$ on $({\rm X},\mathsf{d},\mathfrak{m})$
is the gradient flow of ${\sf Ch}$ in $L^{2}(X)$. For any $f\in L^{2}(X)$, the
gradient flow trajectory $[0,+\infty]\ni t\mapsto P_{t}f\in L^{2}(X)$ is a
continuous curve with $P_{0}f=f$ that is locally absolutely continuous in
$(0,+\infty)$, and with $P_{t}f\in D(\Delta)$ and
$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}P_{t}f=\Delta P_{t}f$
for a.e. $t>0$. The heat flow provides a linear, continuous and self-adjoint
contraction semigroup in $L^{2}(X)$.

By a _bounded Sobolev derivation_ on $(X,\mathsf{d},\mathfrak{m})$ we mean a
linear operator $v\colon H^{1,2}(X)\to L^{2}(X)$ for which there exists a
function $g\in L^{\infty}(X)$ satisfying $|v(f)|\leq g|\nabla f|$
$\mathfrak{m}$-a.e. for every $f\in H^{1,2}(X)$. The minimal (in the
$\mathfrak{m}$-a.e. sense) function $g$ verifying this condition is denoted by
$|v|\in L^{\infty}(X)$ and called the _pointwise norm_ of $v$. We then define
the space $L^{\infty}(TX)$ of _bounded vector fields_ on
$(X,\mathsf{d},\mathfrak{m})$ as the family of all bounded Sobolev derivations
on $(X,\mathsf{d},\mathfrak{m})$. The space $L^{\infty}(TX)$ is a module over
$L^{\infty}(X)$ if endowed with the multiplication operator
$L^{\infty}(X)\times L^{\infty}(TX)\ni(h,v)\mapsto h\cdot v\in L^{\infty}(TX)$
given by $(h\cdot v)(f)\coloneqq hv(f)$ for every $f\in H^{1,2}(X)$. Moreover,
to any given function $f\in H^{1,2}(X)$ with $|\nabla f|\in L^{\infty}(X)$ we
can associate its _gradient_ $\nabla f\in L^{\infty}(TX)$, which is
characterized as the unique element of $L^{\infty}(TX)$ satisfying $\nabla
f(g)=\nabla f\cdot\nabla g$ for every $g\in H^{1,2}(X)$. In particular, to use
the more descriptive notation

$v\cdot\nabla f\coloneqq v(f)\,,\quad\text{ for every }v\in
L^{\infty}(TX)\text{ and }f\in H^{1,2}(X)$

will cause no ambiguity. Observe also that the pointwise norm of $\nabla f$ de
facto coincides with the minimal weak upper gradient $|\nabla f|$ of $f$.
The above notions are essentially borrowed from [52, 51], up to the technical
subtleties we are going to discuss. In [52, 51] the attention was focused on
the space $L^{2}(TX)$ of $2$-integrable vector fields, which were defined in a
different way (see [52, Definition 2.3.1]) that is in fact equivalent to the
approach via Sobolev derivations (see [52, Theorem 2.3.3]). The space
$L^{\infty}(TX)$ was then introduced at the beginning of [52, Section 3.2] by
starting from $L^{2}(TX)$. In this paper it suffices to work with bounded
vector fields, thus we directly introduce them via bounded Sobolev
derivations, avoiding dealing with $2$-integrable vector fields. The
equivalence of this approach with that of [52] can be shown by slightly
adapting the proof of [52, Theorem 2.3.3].

######  Definition  (Essentially bounded divergence measure vector fields).

Let $(X,\mathsf{d},\mathfrak{m})$ be an infinitesimally Hilbertian space. Then
we say that an element $v\in L^{\infty}(TX)$ is an _essentially bounded
divergence measure vector field_ if there exists a (uniquely determined)
finite Radon measure ${\rm div}(v)$ on $X$ such that

$\int g\,{\rm div}(v)\coloneqq\int g\mathop{}\\!\mathrm{d}({\rm div}(v))=-\int
v\cdot\nabla g\mathop{}\\!\mathrm{d}\mathfrak{m}\quad\text{ for every
}g\in{\rm LIP}_{c}(X)\,.$

We denote by $\mathcal{DM}^{\infty}(X)$ the family of all essentially bounded
divergence measure vector fields in $(X,\mathsf{d},\mathfrak{m})$.

Similarly, one can also define the space $\mathcal{DM}^{\infty}(\Omega)$ of
locally essentially bounded divergence measure vector fields in some open set
$\Omega\subseteq X$. Notice that, given any function $f\in H^{1,2}(X)$ with
$|\nabla f|\in L^{\infty}(X)$, it holds that $\nabla f\in L^{\infty}(TX)$ is
an essentially bounded divergence measure vector field if and only if $f\in
D(\mathbf{\Delta})$ and $\mathbf{\Delta}f$ is finite. If this is the case,
then we also have that ${\rm div}(\nabla f)=\mathbf{\Delta}f$. The analogous
property holds for Sobolev functions defined on an open set $\Omega$.

###  2.4. Geometric Analysis on $\operatorname{RCD}$ spaces

The focus of this paper will be on $\operatorname{RCD}(K,N)$ metric measure
spaces $(X,\mathsf{d},\mathfrak{m})$, i.e. infinitesimally Hilbertian metric
measure spaces with Ricci curvature bounded from below and dimension bounded
from above, in synthetic sense.
The Riemannian Curvature Dimension condition $\operatorname{RCD}(K,\infty)$
was introduced in [10] coupling the Curvature Dimension condition
$\operatorname{CD}(K,\infty)$, previously proposed in [84, 85] and
independently in [68], with the infinitesimally Hilbertian assumption.
The finite dimensional refinements subsequently led to the notions of
$\operatorname{RCD}(K,N)$ and $\operatorname{RCD}^{*}(K,N)$ spaces,
corresponding to $\operatorname{CD}(K,N)$ (resp. $\operatorname{CD}^{*}(K,N)$,
see [22]) coupled with linear heat flow. The class $\operatorname{RCD}(K,N)$
was proposed in [50]. The (a priori more general)
$\operatorname{RCD}^{*}(K,N)$ condition was thoroughly analysed in [47] and
(subsequently and independently) in [14] (see also [40] for the equivalence
between $\operatorname{RCD}^{*}$ and $\operatorname{RCD}$ in the case of
finite reference measure).

We avoid giving a detailed introduction to this notion, addressing the reader
to the survey [4] and references therein for the relevant background. Below we
recall some of the main properties that will be relevant for our purposes.

The $\operatorname{RCD}(K,N)$ condition is compatible with the smooth notion.
In particular, smooth $N$–dimensional Riemannian manifolds with Ricci
curvature bounded from below by $K$ endowed with the canonical volume measure
are $\operatorname{RCD}(K,N)$ spaces. Smooth Riemannian manifolds with smooth
and convex boundary (i.e. non negative second fundamental form with respect to
the interior unit normal) are also included in the theory.
Moreover, $N$-dimensional Alexandrov spaces with sectional curvature bounded
from below by $K/(N-1)$ endowed with the $N$–dimensional Hausdorff measure are
$\operatorname{RCD}(K,N)$ spaces.

A fundamental property of $\operatorname{RCD}(K,N)$ spaces is the stability
with respect to pmGH-convergence, meaning that a pmGH-limit of a sequence of
(pointed) $\operatorname{RCD}(K_{n},N_{n})$ spaces for some $K_{n}\to K$ and
$N_{n}\to N$ is an $\operatorname{RCD}(K,N)$ metric measure space.

Let us define

$\operatorname{sn}_{K}(r):=\begin{cases}(-K)^{-\frac{1}{2}}\sinh((-K)^{\frac{1}{2}}r)&K<0\,,\\\
r&K=0\,,\\\ K^{-\frac{1}{2}}\sin(K^{\frac{1}{2}}r)&K>0\,.\end{cases}$

We denote by $v(N,K,r)$ the volume of the ball of radius $r$ in the (unique)
simply connected Riemannian manifold of sectional curvature $K$ and dimension
$N$, and by $s(N,K,r)$ the surface measure of the boundary of such a ball. In
particular $s(N,K,r)=N\omega_{N}\mathrm{sn}_{K}^{N-1}(r)$ and
$v(N,K,r)=\int_{0}^{r}N\omega_{N}\mathrm{sn}_{K}^{N-1}(r)\,\mathrm{d}r$, where
$\omega_{N}$ is the Euclidean volume of the Euclidean unit ball in
$\mathbb{R}^{N}$.

For an arbitrary $\operatorname{CD}((N-1)K,N)$ space $({\rm
X},\mathsf{d},\mathfrak{m})$ the classical Bishop–Gromov volume comparison
holds. More precisely, for any $x\in{\rm X}$, the function
$\mathfrak{m}(B_{r}(x))/v(N,K,r)$ is nonincreasing in $r$ and the function
$\operatorname{Per}(B_{r}(x))/s(N,K,r)$ is essentially nonincreasing in $r$,
i.e., the inequality

$\operatorname{Per}(B_{R}(x))/s(N,K,R)\leq\operatorname{Per}(B_{r}(x))/s(N,K,r)\,,$

holds for almost every radii $R\geq r$, see [86, Theorem 18.8, Equation
(18.8), Proof of Theorem 30.11]. Moreover, it holds that

$\operatorname{Per}(B_{r}(x))/s(N,K,r)\leq\mathfrak{m}(B_{r}(x))/v(N,K,r)\,,$

for any $r>0$. The last inequality follows from the monotonicity of the volume
and perimeter ratios together with the coarea formula on balls.

The Bishop–Gromov inequality implies that $\operatorname{RCD}(K,N)$ spaces are
locally uniformly doubling. Then Gromov’s precompactness theorem guarantees
that any sequence of $\operatorname{RCD}(K,N)$ spaces
$(X_{n},\mathsf{d}_{n},\mathfrak{m}_{n},x_{n})$ such that
$0<\inf_{n}\mathfrak{m}_{n}(B_{1}(x_{n}))<\sup_{n}\mathfrak{m}_{n}(B_{1}(x_{n}))<\infty$
is precompact with respect to the pointed measured Gromov-Hausdorff
convergence.

For most of the results of this paper we will consider
$\operatorname{RCD}(K,N)$ spaces of the form $(X,\mathsf{d},\mathscr{H}^{N})$,
for some $K\in\mathbb{R}$ and $N\in\mathbb{N}$. Notice that we are requiring
that the dimension of the Hausdorff measure coincides with the upper
dimensional bound in the $\operatorname{RCD}$ condition. These spaces are
typically called _non-collapsed $\operatorname{RCD}$ spaces_ (${\rm
ncRCD}(K,N)$ spaces for short) or $N$–dimensional $\operatorname{RCD}(K,N)$
spaces (see [62, 46, 59]).

When $(X_{n},\mathsf{d}_{n},\mathscr{H}^{N},x_{n})$ are
$\operatorname{RCD}(K,N)$ spaces and
$\inf_{n}\mathscr{H}^{N}(B_{1}(x_{n}))>0$, up to subsequences they converge to
some $N$-dimensional $\operatorname{RCD}(K,N)$ space
$(Y,\mathsf{d}_{Y},\mathscr{H}^{N},y)$, which amounts to say that the
$N$–dimensional Hausdorff measures converge to the $N$–dimensional Hausdorff
measure of the limit when there is no collapse. This is the so-called volume
convergence theorem, originally proved in [45, 44] for smooth manifolds and
their limits and extended in [46] to the present setting.

If $({\rm X},\mathsf{d},\mathscr{H}^{N})$ is an $\operatorname{RCD}((N-1)K,N)$
space, $\mathscr{H}^{N}$-almost every point has a unique measured
Gromov–Hausdorff tangent isometric to $\mathbb{R}^{N}$ ([46, Theorem 1.12]).
Therefore by volume convergence

$\lim_{r\to 0}\frac{\mathscr{H}^{N}(B_{r}(x))}{v(N,K,r)}=\lim_{r\to
0}\frac{\mathscr{H}^{N}(B_{r}(x))}{\omega_{N}r^{N}}=1\,,\qquad\text{for
$\mathscr{H}^{N}$-almost every $x$}\,,$ (2.1)

where $\omega_{N}$ is the volume of the unit ball in $\mathbb{R}^{N}$.
Moreover, since the density function $x\mapsto\lim_{r\to
0}\mathscr{H}^{N}(B_{r}(x))/\omega_{N}r^{N}$ is lower semicontinuous ([46,
Lemma 2.2]), it is bounded above by the constant $1$. Hence
$\mathscr{H}^{N}(B_{r}(x))\leq v(N,K,r)$ for every $r>0$ and for every $x\in
X$.

In the setting of $\operatorname{RCD}(K,N)$ spaces it is possible to employ a
viscosity interpretation of Laplacian bounds, in addition to the
distributional one, see [73].

######  Definition  (Bounds in the viscosity sense for the Laplacian).

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ metric
measure space and let $\Omega\subset X$ be an open and bounded domain. Let
$f:\Omega\to\mathbb{R}$ be locally Lipschitz and
$\eta\in\operatorname{C_{b}}(\Omega)$. We say that $\Delta f\leq\eta$ in the
viscosity sense in $\Omega$ if the following holds. For any
$\Omega^{\prime}\Subset\Omega$ and for any test function
$\varphi:\Omega^{\prime}\to\mathbb{R}$ such that

  * (i)

$\varphi\in D(\Delta,\Omega^{\prime})$ and $\Delta\varphi$ is continuous on
$\Omega^{\prime}$;

  * (ii)

for some $x\in\Omega^{\prime}$ it holds $\varphi(x)=f(x)$ and $\varphi(y)\leq
f(y)$ for any $y\in\Omega^{\prime}$, $y\neq x$;

it holds

$\Delta\varphi(x)\leq\eta(x)\,.$

We will rely on the equivalence between distributional and viscosity bounds on
the Laplacian. The result is classical in the setting of smooth Riemannian
manifolds and it has been extended in [73, Theorem 3.24] to
$\operatorname{RCD}(K,N)$ metric measure spaces
$(X,\mathsf{d},\mathscr{H}^{N})$.

######  Theorem 2.2.

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ metric
measure space. Let $\Omega\subset X$ be an open and bounded domain,
$f:\Omega\to\mathbb{R}$ be a Lipschitz function and $\eta:\Omega\to\mathbb{R}$
be continuous. Then $\bm{\Delta}f\leq\eta$ in the sense of distributions if
and only if $\Delta f\leq\eta$ in the viscosity sense.

We refer also to [73, Theorem 3.28] for other equivalent characterizations of
bounds on the Laplacian in the setting of $\operatorname{RCD}$ spaces.

In the proof of Theorem 1.2 it will be important to relate the synthetic lower
Ricci curvature bound to the stability of Laplacian bounds through the Hopf-
Lax duality

$f^{c}(x):=\inf_{y\in X}\\{f(y)+\mathsf{d}(x,y)\\}\,.$ (2.2)

The following statement corresponds to [73, Theorem 4.9]. On a smooth
Riemannian manifold, neglecting the regularity issues, it follows from the
two-points Laplacian comparison proved in [15], which is based on a
computation with Jacobi fields. The proof in [73], which works in a much more
general setting, is based on the interplay between the Hopf-Lax semigroup and
the heat flow put forward in [64, 53, 11].

######  Theorem 2.3.

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ metric
measure space for some $K\in\mathbb{R}$ and $1\leq N<\infty$. Let
$f:X\to\mathbb{R}$ be a locally Lipschitz function with polynomial growth. Let
$\Omega,\Omega^{\prime}\subset X$ be open domains and $\eta\in\mathbb{R}$.
Then the following holds. Assume that $f^{c}$ is finite and that, for any
$x\in\Omega^{\prime}$ the infimum defining $f^{c}(x)$ is attained at some
$y\in\Omega$. Assume moreover that

$\Delta f\leq\eta\quad\text{on $\Omega$}\,.$ (2.3)

Then

$\displaystyle\Delta
f^{c}\leq\eta-K\max_{x\in\Omega^{\prime},y\in\Omega}\mathsf{d}(x,y)\quad\text{on
$\Omega^{\prime}$},\quad\text{if $K\leq 0$, }$ $\displaystyle\Delta
f^{c}\leq\eta-K\min_{x\in\Omega^{\prime},y\in\Omega}\mathsf{d}(x,y)\quad\text{on
$\Omega^{\prime}$},\quad\text{if $K\geq 0$,}$

where the Laplacian bounds have to be intended either in the distributional or
in the viscosity sense.

Given a Borel set $E\subseteq X$ in an $\operatorname{RCD}(K,N)$ space
$(X,\mathsf{d},\mathscr{H}^{N})$ and any $t\in[0,1]$, we denote by $E^{(t)}$
the set of _points of density $t$_ of $E$, namely

$E^{(t)}\coloneqq\bigg{\\{}x\in X\;\bigg{|}\;\lim_{r\to
0}\frac{\mathscr{H}^{N}(E\cap
B_{r}(x))}{\mathscr{H}^{N}(B_{r}(x))}=t\bigg{\\}}\,.$

The _essential boundary_ of $E$ is defined as $\partial^{e}E\coloneqq
X\setminus(E^{(0)}\cup E^{(1)})$. We have that $E^{(t)}$ and $\partial^{e}E$
are Borel sets. Furthermore, the _reduced boundary_
$\mathcal{F}E\subseteq\partial^{e}E$ of a given set $E\subseteq X$ of finite
perimeter is defined as the set of those points of $X$ where the unique
tangent to $E$ (up to isomorphism) is the half-space
$\\{x=(x_{1},\ldots,x_{N})\in\mathbb{R}^{N}\,:\,x_{N}>0\\}$ in
$\mathbb{R}^{N}$; see [5, Definition 4.1] for the precise notion of
convergence we are using. We point out that in the classical Euclidean
framework this notion of reduced boundary is not fully consistent with the
usual one, since it allows for non-uniqueness of the blow-ups (in the sense
that one can obtain different half-spaces when rescaling along different
sequences of radii converging to $0$).
It was proved in [31] after [3, 5] that the perimeter measure
$\operatorname{Per}(E,\cdot)$ can be represented as

$\operatorname{Per}(E,\cdot)=\mathscr{H}^{N-1}|_{\mathcal{F}E}.$ (2.4)

As it is evident from (2.4), the notion of perimeter that we are using does
not charge the boundary of the space under consideration, if any. Indeed, by
the very definition, reduced boundary points for $E$ are regular points of the
ambient space $(X,\mathsf{d},\mathscr{H}^{N})$. We refer to [46, 59, 32] for
the relevant background about boundaries of $\operatorname{RCD}(K,N)$ spaces
$(X,\mathsf{d},\mathscr{H}^{N})$ and just point out here that the notion is
fully consistent with the case of smooth Riemannian manifolds and with the
theory of Alexandrov spaces with sectional curvature bounded from below.

Moreover, we recall that, according to [30, Proposition 4.2],

$\mathcal{F}E=E^{(1/2)}=\bigg{\\{}x\in X\;\bigg{|}\;\lim_{r\to
0}\frac{\mathscr{H}^{N}(E\cap
B_{r}(x))}{\mathscr{H}^{N}(B_{r}(x))}=\frac{1}{2}\bigg{\\}}\,,\quad\text{ up
to }\mathcal{H}^{N-1}\text{-null sets}\,.$

We will rely on a Gauss–Green integration by parts formula for essentially
bounded divergence measure fields over sets of finite perimeter. This tool
extends the classical Gauss–Green integration by parts formula in three
directions: the low regularity of the ambient space, of the domain and of the
vector field.

As shown in [30, Section 5] after [33], for any $v\in\mathcal{DM}^{\infty}(X)$
and any set of finite perimeter $E$, there exist measures
$D\chi_{E}(\chi_{E}v)$, $D\chi_{E}(\chi_{E^{c}}v)$ on $X$ such that
$((\chi_{E}v)\cdot\nabla P_{t}\chi_{E})\mathfrak{m}\rightharpoonup
D\chi_{E}(\chi_{E}v)$ and $((\chi_{E^{c}}v)\cdot\nabla
P_{t}\chi_{E})\mathfrak{m}\rightharpoonup D\chi_{E}(\chi_{E^{c}}v)$ as $t\to
0$ with respect to the narrow topology. Moreover, it holds that
$D\chi_{E}(\chi_{E}v),D\chi_{E}(\chi_{E^{c}}v)\ll\operatorname{Per}(E,\cdot)$.
Then we define

$(v\cdot\nu_{E})_{\rm
int}\coloneqq\frac{1}{2}\frac{\mathop{}\\!\mathrm{d}(D\chi_{E}(\chi_{E}v))}{\mathop{}\\!\mathrm{d}\operatorname{Per}(E,\cdot)}\,,\quad(v\cdot\nu_{E})_{\rm
ext}\coloneqq\frac{1}{2}\frac{\mathop{}\\!\mathrm{d}(D\chi_{E}(\chi_{E^{c}}v))}{\mathop{}\\!\mathrm{d}\operatorname{Per}(E,\cdot)}\,.$
(2.5)

We remark that there is full consistency with the classical setting. When
$(X,\mathsf{d})$ is isometric to a smooth Riemannian manifold, $E$ has smooth
boundary and $v$ is a smooth vector field, then the interior and exterior
normal traces defined in (2.5) coincide and they are equal to $v\cdot\nu_{E}$,
where $\nu_{E}$ is the interior unit normal vector field to $\partial E$.

The following result was proved in [30, Theorem 5.2] by building upon [33,
Theorem 6.22].

######  Theorem 2.4 (Gauss–Green).

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ space.
Let $E\subseteq X$ be a set of finite perimeter. Let
$v\in\mathcal{DM}^{\infty}(X)$ be given. Then $(v\cdot\nu_{E})_{\rm
int},(v\cdot\nu_{E})_{\rm ext}\in
L^{\infty}(\mathcal{F}E,\operatorname{Per}(E,\cdot))$ and

$\begin{split}\|(v\cdot\nu_{E})_{\rm
int}\|_{L^{\infty}(\mathcal{F}E,\operatorname{Per}(E,\cdot))}&\leq\|v\|_{L^{\infty}(E)},\\\
\|(v\cdot\nu_{E})_{\rm
ext}\|_{L^{\infty}(\mathcal{F}E,\operatorname{Per}(E,\cdot))}&\leq\|v\|_{L^{\infty}(E^{c})}.\end{split}$

Moreover, the _Gauss–Green integration-by-parts formulas_ hold: given any
$\varphi\in{\rm LIP}_{c}(X)$ it holds

$\begin{split}\int_{E^{(1)}}\varphi\,{\rm
div}(v)+\int_{E}v\cdot\nabla\varphi\mathop{}\\!\mathrm{d}\mathfrak{m}&=-\int_{\mathcal{F}E}\varphi(v\cdot\nu_{E})_{\rm
int}\mathop{}\\!\mathrm{d}\operatorname{Per}(E,\cdot)\,,\\\
\int_{E^{(1)}\cap\mathcal{F}E}\varphi\,{\rm
div}(v)+\int_{E}v\cdot\nabla\varphi\mathop{}\\!\mathrm{d}\mathfrak{m}&=-\int_{\mathcal{F}E}\varphi(v\cdot\nu_{E})_{\rm
ext}\mathop{}\\!\mathrm{d}\operatorname{Per}(E,\cdot)\,.\end{split}$

Next we report on the behaviour of interior/exterior normal traces over the
boundary of the intersection of two finite perimeter sets. In [30] it has been
shown that the unit normal vector and the interior and exterior normal traces
of vector fields over boundaries of sets of finite perimeter have the natural
behaviour with respect to unions and intersections.
Given two sets of finite perimeter $E,F\subseteq X$, it is well-known that
also $E\cap F$ has finite perimeter. The set $\\{\nu_{E}=\nu_{F}\\}$
introduced in [30] can be characterized as

$\\{\nu_{E}=\nu_{F}\\}=\bigg{\\{}x\in\mathcal{F}E\cap\mathcal{F}F\;\bigg{|}\;\lim_{r\to
0}\frac{\mathscr{H}^{N}((E\cap F)\cap
B_{r}(x))}{\mathscr{H}^{N}(B_{r}(x))}=\frac{1}{2}\bigg{\\}},\quad\text{ up to
}\mathscr{H}^{N-1}\text{-null sets}\,,$

as it follows from the results of [30]. The ensuing statement is taken from
[30, Proposition 5.4].

######  Proposition .

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ space.
Let $E,F\subseteq X$ be two sets of finite perimeter. Let
$v\in\mathcal{DM}^{\infty}(X)$ be given. Then it holds that

$\begin{split}(v\cdot\nu_{E})_{\rm int}=(v\cdot\nu_{F})_{\rm
int}&,\quad\mathscr{H}^{N-1}\text{-a.e.\ on }\\{\nu_{E}=\nu_{F}\\}\,,\\\
(v\cdot\nu_{E\cap F})_{\rm int}=(v\cdot\nu_{E})_{\rm
int}&,\quad\operatorname{Per}(E,\cdot)\text{-a.e.\ in }F^{(1)}\,,\\\
(v\cdot\nu_{E\cap F})_{\rm int}=(v\cdot\nu_{E})_{\rm
int}&,\quad\mathscr{H}^{N-1}\text{-a.e.\ on
}\\{\nu_{E}=\nu_{F}\\}\,.\end{split}$

We will apply the previous machinery to the level sets of distance-type
functions obtained through Hopf-Lax duality, see Theorem 2.3. More
specifically, we will need the following result, which is taken from [30,
Proposition 6.1].

######  Proposition .

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ space.
Let $\Omega\Subset\Omega^{\prime}\subseteq X$ be open domains and let
$\varphi\colon\Omega^{\prime}\to\mathbb{R}$ be a $1$-Lipschitz function.
Suppose that $|\nabla\varphi|=1$ holds $\mathscr{H}^{N}$-a.e. on
$\Omega^{\prime}$ and that there esists a constant $L\leq 0$ such that
$\mathbf{\Delta}\varphi\geq L$ in the sense of distributions in
$\Omega^{\prime}$, thus in particular $\varphi\in
D(\mathbf{\Delta},\Omega^{\prime})$. Suppose further that
$\mathbf{\Delta}\varphi$ is finite. Then $\\{\varphi<t\\}$ is a set of locally
finite perimeter in $\Omega$ for a.e. $t\in\mathbb{R}$ such that
$\\{\varphi=t\\}\cap\Omega\neq\emptyset$ and it holds that

$(\nabla\varphi\cdot\nu_{\\{\varphi<t\\}})_{\rm
int}=(\nabla\varphi\cdot\nu_{\\{\varphi<t\\}})_{\rm
ext}=-1\,,\quad\operatorname{Per}(\\{\varphi<t\\},\cdot)\text{-a.e.\ in
}\Omega\,.$

The primary focus of this note will be isoperimetric sets, that as in the
classical Riemannian setting are much more regular than general sets of finite
perimeter.

######  Definition .

Let $(X,\mathsf{d},\mathfrak{m})$ be a metric measure space. We say that a
subset $E\subset X$ is a volume constrained minimizer for compact variations
in $X$ if whenever $F\subset X$ is such that $E\Delta F\subset K\Subset X$,
and $\mathfrak{m}(K\cap E)=\mathfrak{m}(K\cap F)$, then
$\operatorname{Per}(E)\leq\operatorname{Per}(F)$.

We say that a subset $E\subset X$, with $\mathfrak{m}(E)<\infty$, is an
isoperimetric set whenever for any $F\subset X$ with
$\mathfrak{m}(F)=\mathfrak{m}(E)$ we have that
$\operatorname{Per}(E)\leq\operatorname{Per}(F)$.

Notice that an isoperimetric set in $X$ is a fortiori a volume constrained
minimizer for compact variations in $X$.

Let us recall a topological regularity result for volume constrained
minimizers borrowed from [19].

######  Theorem 2.5 ([19, Theorem 1.3 and Theorem 1.4]).

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ space
with $2\leq N<+\infty$ natural number, $K\in\mathbb{R}$. Let $E$ be a volume
constrained minimizer for compact variations in $X$. Then $E^{(1)}$ is open,
$\partial^{e}E=\partial E^{(1)}$, and $\partial E^{(1)}$ is locally uniformly
$(N-1)$-Ahlfors regular in $X$.

Assume further there exists $v_{0}>0$ such that $\mathscr{H}^{N}(B_{1}(x))\geq
v_{0}$ for every $x\in X$, and that $E\subset X$ is an isoperimetric region.
Then $E^{(1)}$ is in addition bounded, and $\partial E^{(1)}$ is
$(N-1)$-Ahlfors regular in $X$.

In the following, when $E$ is an isoperimetric region in a space $X$ as in
Theorem 2.5, we will always assume that $E$ coincides with its open bounded
representative given by $E^{(1)}$.

The proof of Theorem 2.5 builds on the top of a deformation lemma for general
sets of finite perimeter, see [19, Theorem 1.1 and Theorem 2.35].

###  2.5. Localization of the Curvature-Dimension condition

We will rely on the so-called localization of the Curvature-Dimension
condition. We give some basic background about it and address the reader to
[40, 41, 42] for a detailed account about this topic, under much more general
assumptions.

Let us consider an $\operatorname{RCD}(K,N)$ metric measure space
$(X,\mathsf{d},\mathfrak{m})$ for some $K\in\mathbb{R}$ and $1<N<\infty$. Let
$\Omega\subset X$ be an open subset and let $f:X\to\mathbb{R}$ be the signed
distance function from $\Omega$, i.e.,

$f(x):=\mathsf{d}(x,\overline{\Omega})=\inf\\{\mathsf{d}(x,y)\,:\,y\in\Omega\\}\,,\quad\text{if
$x\in X\setminus\Omega$}\,,$ (2.6)

and

$f(x):=-\mathsf{d}(x,X\setminus\Omega)=-\inf\\{\mathsf{d}(x,y)\,:\,y\in
X\setminus\Omega\\}\,,\quad\text{if $x\in\Omega$}\,.$ (2.7)

The signed distance function $f$ induces an $\mathfrak{m}$-almost everywhere
partition of $X$ into geodesics $X_{\alpha}$ indexed over a set $Q$. On a
smooth Riemannian manifold, these geodesics would correspond to gradient flow
lines of the signed distance function, or, equivalently, to integral curves of
$-\nabla f$.
Rays $X_{\alpha}$ are often identified with intervals of the real line via the
ray map $\gamma_{\alpha}:I_{\alpha}\to X_{\alpha}$, where
$I_{\alpha}\subset\mathbb{R}$ is an interval and $\gamma_{\alpha}$ is an
isometry.

The almost-everywhere partition of $X$ into _transport rays_ induced by the
signed distance function $f$ determines the following disintegration formula:

$\mathfrak{m}=\int_{Q}h_{\alpha}\mathscr{H}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
X_{\alpha}\,{\mathfrak{q}}(\mathop{}\\!\mathrm{d}\alpha)\,.$ (2.8)

The non-negative measure $\mathfrak{q}$ in (2.8), defined on the set of
indices $Q$, is obtained in a natural way from the essential partition
$(X_{\alpha})_{\alpha\in Q}$ of $X$, roughly by projecting $\mathfrak{m}$ on
the set $Q$ of equivalence classes (we refer to [42] for the details).

The key property is that, if $(X,\mathsf{d},\mathfrak{m})$ is an
$\operatorname{RCD}(K,N)$ metric measure space, then each $h_{\alpha}$ is a
$\operatorname{CD}(K,N)$ density over the ray $X_{\alpha}$ (see [42, Theorem
3.6]), i.e.,

$(\log h_{\alpha})^{\prime\prime}\leq-K-\frac{1}{N-1}\big{(}(\log
h_{\alpha})^{\prime}\big{)}^{2},$ (2.9)

in the sense of distributions and point-wise except countably many points,
compare with [40, Lemma A.3, Lemma A.5, Proposition A.10]. Equivalently

$\left(h_{\alpha}^{\frac{1}{N-1}}\right)^{\prime\prime}+\frac{K}{N-1}h_{\alpha}^{\frac{1}{N-1}}\leq
0\,,$ (2.10)

in the sense of distributions. This amounts to say that the Curvature-
Dimension condition of the ambient space $(X,\mathsf{d},\mathfrak{m})$ is
inherited by the needles of the partition induced by $f$.

######  Remark .

The localization of the Curvature-Dimension condition in the synthetic setting
finds its roots in earlier works on convex geometry and in [63], where the
case of smooth weighted Riemannian manifolds was considered.

If $(X,\mathsf{d})$ is isometric to a smooth Riemannian manifold, then
neglecting the regularity issues caused by the possible presence of a cut
locus, the change of variables formula shows that

$\Delta f=\left(\log h_{\alpha}\right)^{\prime}$ (2.11)

in the interior of the integral lines of $-\nabla f$, where we denoted by
$\left(\log h_{\alpha}\right)^{\prime}$ the derivative of $\log h_{\alpha}$
along the ray $X_{\alpha}$.
Then we recall that

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\Delta
f(\gamma(t))+\left\lVert\mathrm{Hess}f(\gamma(t))\right\rVert_{\mathrm{HS}}^{2}+\mathrm{Ric}_{\gamma(t)}(\nabla
f,\nabla f)=0\,.$ (2.12)

Hence, using the Cauchy-Schwartz inequality on the eigenvalues of
$\mathrm{Hess}f(\gamma(t))$ and taking into account the fact that
$\mathrm{Hess}f(\nabla f,\nabla f)=0$, if $\mathrm{Ric}\geq K$ then we can
estimate

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\Delta
f(\gamma(t))+\frac{1}{N-1}\left(\Delta f(\gamma(t))\right)^{2}\leq-K\,,$
(2.13)

which corresponds exactly to (2.9).

With the help of the localization technique, we will be able to turn some
estimates into one-dimensional comparison results for solutions of Riccati
equations. We mention here the relevant result for our purposes.

Let us introduce the notation

$s_{k,\lambda}(r):=\cos_{k}(r)-\lambda\sin_{k}(r)\,,$ (2.14)

where

$\cos_{k}^{\prime\prime}+k\cos_{k}=0\,,\quad\cos_{k}(0)=1\,,\quad\cos_{k}^{\prime}(0)=0\,,$
(2.15)

and

$\sin_{k}^{\prime\prime}+k\sin_{k}=0\,,\quad\sin_{k}(0)=0\,,\quad\sin_{k}^{\prime}(0)=1\,.$
(2.16)

Notice that $s_{k,-d}$ is a solution of

$v^{\prime\prime}+kv=0\,,\quad v(0)=1\,,\quad v^{\prime}(0)=d\,.$ (2.17)

Moreover, $s_{0,\lambda}(r)=1-\lambda r$.

######  Lemma  (Riccati comparison, [35, Lemma 4.10]).

Let $k\in\mathbb{R}$. Let $u:[0,b]\to\mathbb{R}$ be such that
$u^{\prime\prime}+ku\leq 0$ in the sense of distributions, $u(0)=1$ and
$u^{\prime}(0)\leq d$. Let $v:[0,\bar{b}]\to\mathbb{R}$ be the maximal non
negative solution of $v^{\prime\prime}+kv=0$ with $v(0)=1$ and
$v^{\prime}(0)=d$, that is $v=s_{k,-d}$. Then $b\leq\bar{b}$, and the
inequalities $u\leq v$ and
$\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}t}\log
u\leq\left(\log v\right)^{\prime}$ hold on $[0,b)$. Notice that
$\frac{\mathop{}\\!\mathrm{d}^{+}}{\mathop{}\\!\mathrm{d}t}\log u$ exists
because $(\log u)^{\prime\prime}\leq-k$ by hypothesis in the sense of
distributions.

Let us fix $N>1$, $H\in\mathbb{R}$ and $K\in\mathbb{R}$. Then we introduce the
Jacobian function

$\mathbb{R}\ni r\mapsto
J_{H,K,N}(r):=\left(\cos_{\frac{K}{N-1}}(r)+\frac{H}{N-1}\sin_{\frac{K}{N-1}}(r)\right)^{N-1}_{+}=\left(s_{\frac{K}{N-1},-\frac{H}{N-1}}(r)\right)^{N-1}_{+}\,.$
(2.18)

Notice that, when $K=0$ the expression for the Jacobian function simplifies
into

$\mathbb{R}\ni r\mapsto
J_{H,N}(r):=\left(1+\frac{H}{N-1}r\right)^{N-1}_{+}\,.$ (2.19)

We stress that the function $J_{H,K,N}$ is precisely the one involved in the
one-dimensional comparison of $\operatorname{CD}(K,N)$ densities, see [60,
Corollary 4.3].

##  3\. The distance function from isoperimetric sets

Let $(M,g)$ be a smooth $N$-dimensional Riemannian manifold with Ricci
curvature bounded from below by $K\in\mathbb{R}$. Let $E\subset M$ be a set of
finite perimeter which is an isoperimetric region. Then, by the classical
regularity theory for constrained perimeter minimizers [2, 56, 74], $E$ has an
open representative and $\partial E$ is smooth away from a set $\partial_{s}E$
of Hausdorff dimension $\dim_{H}(\partial_{s}E)\leq N-8$. Moreover, $\partial
E\setminus\partial_{s}E$ has constant mean curvature $c\in\mathbb{R}$, in the
classical sense and $\partial_{s}E$ can be characterized as the set of those
points in $\partial E$ such that the tangent cone is not included in a half-
space, thanks to [2].

The classical proof of the Lévy–Gromov inequality for manifolds with positive
Ricci curvature [57, Appendix C] combines the regularity results mentioned
above with a Heintze–Karcher type estimate [58]. In particular, the regularity
theorem from [2] is used in a crucial way to overcome the possible lack of
smoothness of the isoperimetric boundary. The proof gives in particular the
following result, valid for any lower curvature bound.

######  Theorem 3.1.

Let $(M,g)$ be a smooth $N$-dimensional Riemannian manifold with Ricci
curvature bounded from below by $K\in\mathbb{R}$ and let $E\subset X$ be an
isoperimetric set. Then, denoting by $f$ the signed distance function from
$\overline{E}$ and by $c$ the value of the constant mean curvature of
$\partial E\setminus\partial_{s}E$, it holds

$\bm{\Delta}f\geq-(N-1)\frac{s^{\prime}_{\frac{K}{N-1},\frac{c}{N-1}}\circ\left(-f\right)}{s_{\frac{K}{N-1},\frac{c}{N-1}}\circ\left(-f\right)}\,\,\,\quad\text{on
$E$ and
}\quad\bm{\Delta}f\leq(N-1)\frac{s^{\prime}_{\frac{K}{N-1},-\frac{c}{N-1}}\circ
f}{s_{\frac{K}{N-1},-\frac{c}{N-1}}\circ f}\quad\text{on
$X\setminus\overline{E}$}\,.$ (3.1)

######  Remark .

The very same conclusion above holds assuming that $E$ is a domain with smooth
boundary and that $\partial E$ has constant mean curvature $c$. The proof of
this variant does not require the deep regularity theorem from [2], but the
assumptions are not natural for the applications to the isoperimetric problem
when $N\geq 8$.

Notice that the bounds in (3.1) make perfectly sense on a metric measure space
$(X,\mathsf{d},\mathfrak{m})$, even though most of the ingredients of the
classical proof that we recalled above do not.

By [19, Theorem 1.3], a volume constrained minimizer $E$ for compact
variations enjoys analogous topological regularity properties of an
isoperimetric region. More precisely, $E^{(1)}$ is open and
$\partial^{e}E=\partial E^{(1)}$. Hence we will always identify such a set
with its representative $E^{(1)}$.

######  Theorem 3.2.

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ metric
measure space for some $K\in\mathbb{R}$ and $N\geq 2$ and let $E\subset X$ be
a set of finite perimeter. Assume that $E$ is a volume constrained minimizer
for compact variations in $X$. Then, denoting by $f$ the signed distance
function from $\overline{E}$, there exists $c\in\mathbb{R}$ such that

$\bm{\Delta}f\geq-(N-1)\frac{s^{\prime}_{\frac{K}{N-1},\frac{c}{N-1}}\circ\left(-f\right)}{s_{\frac{K}{N-1},\frac{c}{N-1}}\circ\left(-f\right)}\quad\text{on
$E$ and
}\quad\bm{\Delta}f\leq(N-1)\frac{s^{\prime}_{\frac{K}{N-1},-\frac{c}{N-1}}\circ
f}{s_{\frac{K}{N-1},-\frac{c}{N-1}}\circ f}\quad\text{on
$X\setminus\overline{E}$}\,.$ (3.2)

######  Remark .

The bounds in (3.2) can be understood in the sense of distributions or in the
sense of viscosity, see Theorem 2.2. The two perspectives will be both
relevant for the sake of the proof while in the applications it will be
important to rely mostly on the distributional perspective.

######  Remark .

Notice that the comparison function in (3.2) are not well defined globally on
$\mathbb{R}$, but only on a maximal interval $I_{K,N,c}\subset\mathbb{R}$. In
the course of the proof we will show that the signed distance function from
$\overline{E}$ can attain only values in $I_{K,N,c}$ where the bounds in (3.2)
perfectly make sense.

The proof of Theorem 3.2 is based on a completely new argument. We outline the
strategy for smooth Riemannian manifolds, avoiding the technicalities and
focusing on the case $K=0$.
We consider a weaker statement, corresponding to the limit of (3.2) as
$N\to\infty$. The self-improvement of the a-dimensional bound to the sharp
dimensional bound is based on a classical computation with Jacobi fields away
from the cut-locus of the distance function.
If this statement fails, then there are points $x\in X\setminus E$ and points
$y\in E$ such that $\Delta f(x)>\Delta f(y)$ in a weak sense. This is already
a subtle point, where we exploit the viscosity theory, since the distance
function is not globally smooth. Thanks to a perturbation argument, we
construct a smooth function $g$, touching $f$ from below only at $x$, and a
smooth function $h$ touching $f$ from above only at $y$ such that $\Delta
g(x)>\Delta h(y)$.
Then we introduce the functions

$\bar{g}(z):=\sup_{w}\\{g(w)-\mathsf{d}(w,z)\\}\,,\quad\,\bar{h}(z):=\inf_{w}\\{h(w)+\mathsf{d}(z,w)\\}\,.$
(3.3)

Heuristically, $\bar{g}$ and $\bar{h}$ behave like distance functions from
their respective level sets. Moreover, the non linear transformation in (3.3)
keeps the bounds on the Laplacian, if the ambient has non negative Ricci
curvature. Notice that a weak perspective on Laplacian bounds needs to be
considered again, since the transformation does not preserve the regularity,
in general. Neglecting the regularity issues, the conclusion would follow from
[15], which is based on a computation with Jacobi fields.
The idea is then to slide the level sets of the functions $\bar{g}$ and
$\bar{h}$ until they start crossing the isoperimetric set $E$. When this
happens, we cut $\partial E$ along the level sets of $\bar{g}$ and $\bar{h}$
making sure to balance the interior and exterior perturbations so that,
globally, the perturbation has the same volume of $E$. Eventually, we estimate
the perimeter of the perturbation and compare it with the perimeter of $E$,
reaching a contradiction.

Notice that we do not need to rely on the full regularity theory for
isoperimetric sets on Riemannian manifolds. For the sake of the proof it is
sufficient to know that the _measure theoretic_ boundary of the isoperimetric
set is closed, which follows from uniform local density estimates.

###### Proof.

The proof will be divided into two steps. In the first one we are going to
prove the weaker adimensional bounds

$\bm{\Delta}f\geq c-Kf\,\quad\text{on $E$ and }\quad\bm{\Delta}f\leq
c-Kf\,\quad\text{on $X\setminus\overline{E}$}\,,$ (3.4)

corresponding to the limit as $N\to\infty$ of the bounds in (3.2).
In the second part of the proof we will show how to obtain the sharp
dimensional bounds with a by now standard application of the localization
technique from [41, 42].

Step 1. We will prove (3.4) in the case $K=0$. The case $K\neq 0$ can be
treated with the modifications explained at the end of [73, Theorem 5.2].

Observe that by Theorem 2.2 the statement is equivalent to the following
Claim: the supremum of the values of the Laplacians of lower supporting
functions as in subsection 2.4 of $f$ at touching points on
$X\setminus\overline{E}$ is lower than the infimum of the values of Laplacians
of upper supporting functions of $f$ at touching points on $E$. Indeed, if
this statement holds, then letting $c$ be any value between the supremum and
the infimum of the two sets, then $\Delta f\leq c$ holds on
$X\setminus\overline{E}$ in the viscosity sense and $\Delta f\geq c$ holds on
$E$, again in the viscosity sense. By Theorem 2.2, (3.4) holds also in the
sense of distributions.
Observe that also the converse implication holds, again by the same kind of
arguments. Since we will not need this implication we omit the proof.

Let us prove the Claim.
We argue by contradiction. If it is not true, then we can find $x\in
X\setminus E$, $y\in E$, $s>0$, $\delta>0$ (that we think to be very small)
and supporting functions $\overline{\psi}:X\to\mathbb{R}$ and
$\overline{\chi}:X\to\mathbb{R}$ with the following properties:

  * ia)

$\overline{\psi}:X\to\mathbb{R}$ is Lipschitz and in the domain of the local
Laplacian on $B_{s}(x)$;

  * iia)

$\overline{\psi}(x)=f(x)$;

  * iiia)

$\overline{\psi}(z)<f(z)$ for any $z\neq x$ and $\overline{\psi}<f-\delta$ on
$X\setminus B_{s}(x)$;

and

  * ib)

$\overline{\chi}:X\to\mathbb{R}$ is Lipschitz and in the domain of the local
Laplacian on $B_{s}(y)$;

  * iib)

$\overline{\chi}(y)=f(y)$;

  * iiib)

$\overline{\chi}(z)>f(z)$ for any $z\neq y$ and $\overline{\psi}>f+\delta$ on
$X\setminus B_{s}(y)$.

Moreover, there exist $c\in\mathbb{R}$ and $\varepsilon>0$ such that

$\Delta\overline{\psi}\geq c+\varepsilon\,\quad\text{on $B_{s}(x)$}\,$ (3.5)

and

$\Delta\overline{\chi}\leq c-\varepsilon\,\quad\text{on $B_{s}(y)$}\,.$ (3.6)

Notice that Theorem 2.2 yields existence of functions $\hat{\psi}$ and
$\hat{\chi}$ verifying the weak inequalities $\hat{\psi}\leq f$ and
$\hat{\chi}\geq f$ in place of the strict ones iiia) and iiib). The functions
$\overline{\psi}$ and $\overline{\chi}$ are obtained with an additional
perturbation argument relying on [73, Proposition 2.63].

Arguing as in the proof of [73, Theorem 5.2] we consider the transform of
$\overline{\psi}$ through the Hopf–Lax duality and introduce
$\varphi:X\to\mathbb{R}$ by letting

$\varphi(z):=\sup_{w\in X}\\{\overline{\psi}(w)-\mathsf{d}(w,z)\\}\,.$ (3.7)

Analogously, we let $\eta:X\to\mathbb{R}$ be defined by

$\eta(z):=\inf_{w\in X}\\{\overline{\chi}(w)+\mathsf{d}(z,w)\\}\,.$ (3.8)

Let $X_{\Sigma}$ and $Y_{\Sigma}$ be the sets of touching points of minimizing
geodesics from $x$ and $y$ respectively to $\Sigma:=\partial E$, i.e.

$X_{\Sigma}:=\\{w\in\partial E\,:\,f(x)-f(w)=\mathsf{d}(x,w)\\},$ (3.9)

and

$Y_{\Sigma}:=\\{w\in\partial E\,:\,f(w)-f(y)=\mathsf{d}(y,w)\\}\,.$ (3.10)

It is easy to verify that $X_{\Sigma}$ and $Y_{\Sigma}$ are compact subsets of
$\partial E$.

Arguing as in the proof of [73, Theorem 5.2] (see in particular the discussion
after equation (5.9) therein) it is possible to verify that $\varphi\leq f$,
$\eta\geq f$ and both $\varphi$ and $\eta$ are $1$-Lipschitz functions.
Moreover, there exist neighbourhoods of $U_{\Sigma}\supset X_{\Sigma}$ and
$V_{\Sigma}\supset Y_{\Sigma}$ such that:

  * a)

$\left\lvert\nabla\varphi\right\rvert=1$ holds $\mathscr{H}^{N}$-a.e. on
$U_{\Sigma}$;

  * b)

$\left\lvert\nabla\eta\right\rvert=1$ holds $\mathscr{H}^{N}$-a.e. on
$V_{\Sigma}$;

  * c)

$\bm{\Delta}\varphi\geq c+\varepsilon$ on $U_{\Sigma}$;

  * d)

$\bm{\Delta}\eta\leq c-\varepsilon$ on $V_{\Sigma}$;

  * e)

$\varphi(z)=f(z)$ for any $z\in X$ such that $f(x)-f(z)=\mathsf{d}(x,z)$;

  * f)

$\eta(z)=f(z)$ for any $z\in X$ such that $f(z)-f(y)=\mathsf{d}(z,y)$.

Above, properties (e) and (f) can be verified as in [73, Equations (5.10),
(5.11), (5.12)]. Properties (a) and (b) can be verified as in [73, Equations
(5.13), (5.14)]. Properties (c) and (d) follow from (3.5) and (3.6) thanks to
Theorem 2.3, see also [73, Corollary 4.10].

Arguing as in Step 4 of the proof of [73, Theorem 5.2], it is possible to
verify that the superlevel set $\\{\varphi>-\delta\\}\cap E$ is a compactly
supported perturbation of $E$ in a neighbourhood of $U_{\Sigma}$ for
$\delta>0$ sufficiently small. By similar reasons, $\\{\eta<\delta\\}\cup E$
is a compactly supported perturbation of $E$ in a neighbourhood of
$V_{\Sigma}$.

We need to consider two cases. The case when $X_{\Sigma}\cap
Y_{\Sigma}\neq\emptyset$ and the case $X_{\Sigma}\cap Y_{\Sigma}=\emptyset$.

Case 1: $X_{\Sigma}\cap Y_{\Sigma}\neq\emptyset$.

Let $p\in U_{\Sigma}\cap V_{\Sigma}$. Let us observe that, by construction,
$\eta\geq f\geq\varphi$. Moreover, $\eta(p)=f(p)=\varphi(p)$, by e), f) and
the assumption that $p\in X_{\Sigma}\cap Y_{\Sigma}$.
Let us set $g:=\eta-\varphi$. Observe that $g\geq 0$ and $g(p)=0$. Moreover,
by c) and d), there exists a neighbourhood of $p$ where

$\bm{\Delta}g\leq(c-\varepsilon)+(-c-\varepsilon)\leq-2\varepsilon\,.$ (3.11)

Therefore we reach a contradiction, since $g$ would be a non constant
superharmonic function attaining its minimum at an interior point, see [55,
Theorem 2.8].

Case 2: $X_{\Sigma}\cap Y_{\Sigma}=\emptyset$.

Let us start observing that in this case we can slightly modify the
construction to make sure that, for $\delta>0$ sufficiently small, the
perturbations of $E$ obtained cutting along the sub/superlevel sets of the
auxiliary potentials $\varphi$ and $\eta$ at values $\left\lvert
t\right\rvert<\delta$ are disjointly supported. This holds because, in
particular, we can make sure that they are supported in small neighbourhoods
of $X_{\Sigma}$ and $Y_{\Sigma}$, which we are now assuming to be disjoint. We
refer again to the proof of Step 4 in [73, Theorem 5.2] for further details.

Let us set now, for any $-\delta\leq s\leq 0<t<\delta$,

$E_{s,t}:=E\setminus\\{\varphi\geq s\\}\cup\\{\eta\leq t\\}\,.$ (3.12)

We claim that there exist values $s,t$ in the range above such that

$\mathscr{H}^{N}(E_{s,t})=\mathscr{H}^{N}(E)$ (3.13)

and

$\mathscr{H}^{N}(E_{s,0})<\mathscr{H}^{N}(E)<\mathscr{H}^{N}(E_{0,t})\,.$
(3.14)

In order to establish the claim it is sufficient to prove that

$(s,t)\mapsto\mathscr{H}^{N}(E_{s,t}),$ (3.15)

is a continuous function. This is a direct consequence of the $1$-Lipschitz
regularity of $\varphi$ and $\eta$, together with the compactness of the
perturbations $E_{s,t}\Delta E$ and the properties
$\left\lvert\nabla\varphi\right\rvert=\left\lvert\nabla\eta\right\rvert=1$-a.e.,
which guarantee that

$\mathscr{H}^{N}\left(\\{\varphi=s\\}\cap
U_{\Sigma}\right)=\mathscr{H}^{N}\left(\\{\eta=t\\}\cap
V_{\Sigma}\right)=0\,,$ (3.16)

for any $-\delta\leq s\leq 0<t<\delta$.

Given the claim, it is easy to find by a continuity argument $s$ and $t$ such
that (3.13) and (3.14) hold. Moreover letting $\Omega$ be the open
neighbourhood of $X_{\Sigma}\cup Y_{\Sigma}$ where the perturbation
$E_{s,t}\Delta E$ is compactly contained, it holds, by [30, Proposition 6.1],
see subsection 2.4,

$\left(\nabla\varphi\cdot\nu_{\\{\varphi<s\\}}\right)_{\mathrm{int}}=\left(\nabla\varphi\cdot\nu_{\\{\varphi<s\\}}\right)_{\mathrm{ext}}=-1\,\quad\operatorname{Per}_{\\{\varphi<s\\}}\text{-a.e.
on $\Omega$},$ (3.17)

and

$\left(\nabla\eta\cdot\nu_{\\{\eta<t\\}}\right)_{\mathrm{int}}=\left(\nabla\eta\cdot\nu_{\\{\eta<t\\}}\right)_{\mathrm{ext}}=-1\,\quad\operatorname{Per}_{\\{\eta<t\\}}\text{-a.e.
on $\Omega$}\,.$ (3.18)

Notice that in order to apply [30, Proposition 6.1] it is necessary to
multiply $\varphi$ and $\eta$ by regular cut-off functions compactly supported
into $\Omega$. This can be easily done thanks to the existence of regular cut-
off functions on $\operatorname{RCD}(K,N)$ spaces, see [13, Lemma 6.7] and
[71, 52]. In the rest of the proof we will assume that $\varphi$ and $\eta$
have compact supports, without changing the notation.

We are going to reach a contradiction comparing the perimeter of $E_{s,t}$
with that of $E$, arguing as in the final part of the proof of [73, Theorem
5.2]. We estimate separately the differences in the perimeter coming from the
perturbation induced by $\varphi$ and $\eta$ by disjointedness:

$\operatorname{Per}(E_{s,t})-\operatorname{Per}(E)=\left(\operatorname{Per}(E_{(s,0)},\Omega)-\operatorname{Per}(E,\Omega)\right)+\left(\operatorname{Per}(E_{(0,t)},\Omega)-\operatorname{Per}(E,\Omega)\right)\,.$
(3.19)

The two contributions can be estimated as follows. Let us set
$F:=E\cap\\{\varphi>s\\}$ and $G:=\\{\eta<t\\}\setminus E$. Observe that by
(3.13) it holds $\mathscr{H}^{N}(F)=\mathscr{H}^{N}(G)$.

On the one hand we can apply [30, Theorem 5.2], see Theorem 2.4, with the
sharp trace estimates, with test function identically equal to $1$, vector
field $V=\nabla\varphi$ and set of finite perimeter $F$. We obtain

$\displaystyle\int_{F^{(1)}}\bm{\Delta}\varphi$
$\displaystyle=-\int_{\mathcal{F}F}\left(\nabla\varphi\cdot\nu_{F}\right)_{\mathrm{int}}\mathop{}\\!\mathrm{d}\operatorname{Per}$
$\displaystyle\leq-\int_{E^{(1)}\cap\mathcal{F}\\{\varphi>s\\}}(\nabla\varphi\cdot\nu_{\\{\varphi>s\\}})_{\mathrm{int}}\mathop{}\\!\mathrm{d}\operatorname{Per}-\int_{\mathcal{F}E\cap\\{\varphi>s\\}^{(1)}}\left(\nabla\varphi\cdot\nu_{E}\right)_{\mathrm{int}}\mathop{}\\!\mathrm{d}\operatorname{Per}$
$\displaystyle=-\mathscr{H}^{N-1}\left(E^{(1)}\cap\mathcal{F}\\{\varphi>s\\}\right)-\int_{\mathcal{F}E\cap\\{\varphi>s\\}^{(1)}}\left(\nabla\varphi\cdot\nu_{E}\right)_{\mathrm{int}}\mathop{}\\!\mathrm{d}\operatorname{Per}$
$\displaystyle\leq-\mathscr{H}^{N-1}\left(E^{(1)}\cap\mathcal{F}\\{\varphi>s\\}\right)+\mathscr{H}^{N-1}\left(\mathcal{F}E\cap\\{\varphi>s\\}^{(1)}\right)\,,$
(3.20)

where the first equality follows from [30, Theorem 5.2], the first inequality
follows from [30, Proposition 5.4], see subsection 2.4, and the fact that

$\mathcal{F}(E\cap\\{\varphi>s\\})\sim\left(E^{(1)}\cap\mathcal{F}(\varphi>s)\right)\sqcup\left(\mathcal{F}E\cap\\{\varphi>s\\}^{(1)}\right)\sqcup\left(E^{(1/2)}\cap
F^{(1/2)}\right),$

as a consequence of [30, Proposition 4.2], while the last inequality follows
from the sharp trace bound
$\left\lvert\left(\nabla\varphi\cdot\nu_{E}\right)_{\mathrm{int}}\right\rvert\leq
1$ in [30, Theorem 5.2], see Theorem 2.4.

The analogous computation with $\nabla\eta$ in place of $\nabla\varphi$ and
$G$ in place of $F$ yields to

$\int_{G^{(1)}}\bm{\Delta}\eta\geq\mathscr{H}^{N-1}\left(E^{(0)}\cap\mathcal{F}\\{\eta<t\\}\right)-\mathscr{H}^{N-1}\left(\mathcal{F}E\cap\\{\eta<t\\}^{(1)}\right)\,.$
(3.21)

Now, the bounds on $\bm{\Delta}\varphi$ and $\bm{\Delta}\eta$ imply

$\int_{F^{(1)}}\bm{\Delta}\varphi\geq(c+\varepsilon)\mathscr{H}^{N}(F)\,$
(3.22)

and

$\int_{G^{(1)}}\bm{\Delta}\eta\leq(c-\varepsilon)\mathscr{H}^{N}(G)\,.$ (3.23)

Hence, by (3.19), (3.20), (3.21), (3.22) and (3.23)

$\displaystyle\operatorname{Per}(E_{s,t})-\operatorname{Per}(E)=$
$\displaystyle\left(\operatorname{Per}(E_{(s,0)},\Omega)-\operatorname{Per}(E,\Omega)\right)+\left(\operatorname{Per}(E_{(0,t)},\Omega)-\operatorname{Per}(E,\Omega)\right)$
$\displaystyle=$
$\displaystyle\left(\mathscr{H}^{N-1}\left(E^{(1)}\cap\mathcal{F}\\{\varphi>s\\}\right)-\mathscr{H}^{N-1}\left(\mathcal{F}E\cap\\{\varphi>s\\}^{(1)}\right)\right)$
$\displaystyle+\left(\mathscr{H}^{N-1}\left(E^{(0)}\cap\mathcal{F}\\{\eta<t\\}\right)-\mathscr{H}^{N-1}\left(\mathcal{F}E\cap\\{\eta<t\\}^{(1)}\right)\right)$
$\displaystyle\leq$
$\displaystyle\int_{G^{(1)}}\bm{\Delta}\eta-\int_{F^{(1)}}\bm{\Delta}\varphi$
$\displaystyle\leq$
$\displaystyle(c-\varepsilon)\mathscr{H}^{N}(G)-(c+\varepsilon)\mathscr{H}^{N}(F)=-2\varepsilon\mathscr{H}^{N}(F)<0\,,$

yielding to the sought contradiction.

Step 2. Let us see how to pass from the adimensional estimates in (3.4) to the
sharp Laplacian comparison in (3.2).

We will rely on the localization technique from [41, 42], following the proof
of [73, Theorem 5.2]. Let us prove the sharp Laplacian comparison on
$X\setminus E$, the bound in the interior follows from an analogous argument.
From [42, Corollary 4.16], we know that

$\bm{\Delta}f\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
X\setminus\overline{E}=(\bm{\Delta}f)^{\text{reg}}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
X\setminus\overline{E}+(\bm{\Delta}f)^{\text{sing}}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits X\setminus\overline{E}\,,$

where the singular part $(\bm{\Delta}f)^{\text{sing}}\perp\mathscr{H}^{N}$
satisfies $(\bm{\Delta}f)^{\text{sing}}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits X\setminus\overline{E}\leq 0$
and the regular part $(\bm{\Delta}f)^{\text{reg}}\ll\mathscr{H}^{N}$ admits
the representation formula

$(\bm{\Delta}f)^{\text{reg}}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits X\setminus\overline{E}=(\log
h_{\alpha})^{\prime}\mathscr{H}^{N}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits X\setminus\overline{E}\,.$
(3.24)

In (3.24), $Q$ is a suitable set of indices, $(h_{\alpha})_{\alpha\in Q}$ are
suitable densities defined on geodesics $(X_{\alpha})_{\alpha\in Q}$, which
are essentially partitioning $X\setminus\overline{E}$ (in the smooth setting,
$(X_{\alpha})_{\alpha\in Q}$ correspond to the integral curves of
$\nabla\mathsf{d}_{E}$; note that here we are using the reverse
parameterization of $X_{\alpha}$ with respect to [42], hence the reversed sign
in the right hand side of (3.24)), such that the following disintegration
formula holds:

$\mathscr{H}^{N}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
X\setminus\overline{E}=\int_{Q}h_{\alpha}\mathscr{H}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
X_{\alpha}\,{\mathfrak{q}}(\mathop{}\\!\mathrm{d}\alpha)\,.$ (3.25)

The non-negative measure $\mathfrak{q}$ in (3.25), defined on the set of
indices $Q$, is obtained in a natural way from the essential partition
$(X_{\alpha})_{\alpha\in Q}$ of $X\setminus\overline{E}$, roughly by
projecting $\mathscr{H}^{N}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits X\setminus\overline{E}$ on the
set $Q$ of equivalence classes (we refer to [42] for the details).
The key point for the proof of Step 2 is that each $h_{\alpha}$ is a
$\operatorname{CD}(K,N)$ density over the ray $X_{\alpha}$ (see [42, Theorem
3.6]), i.e.,

$(\log h_{\alpha})^{\prime\prime}\leq-K-\frac{1}{N-1}\big{(}(\log
h_{\alpha})^{\prime}\big{)}^{2},$ (3.26)

in the sense of distributions and point-wise except countably many points,
compare with [40, Lemma A.3, Lemma A.5, Proposition A.10]. Equivalently

$\left(h_{\alpha}^{\frac{1}{N-1}}\right)^{\prime\prime}+\frac{K}{N-1}h_{\alpha}^{\frac{1}{N-1}}\leq
0\,,$ (3.27)

in the sense of distributions. Moreover, from (3.24) and (3.4) we know that

$(\log h_{\alpha})^{\prime}(\mathsf{d}_{\overline{E}})\leq
c-K\,\mathsf{d}_{\overline{E}}\,\text{ a.e. on $X_{\alpha}$, for
$\mathfrak{q}$-a.e. $\alpha\in Q$}\,.$ (3.28)

The sharp estimates in (3.2) then follow from the standard Riccati comparison
subsection 2.5 applied to the functions
$v(r):=\left(h_{\alpha}(r)/h_{\alpha}(0)\right)^{\frac{1}{N-1}}$ which verify

$v^{\prime\prime}+\frac{K}{N-1}v\leq 0\,,$ (3.29)

in the sense of distributions by (3.27), $v(0)=1$ and $v^{\prime}(0)\leq
c/(N-1)$ by (3.28). ∎

######  Definition  (Mean curvature barriers for isoperimetric sets).

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ space,
let $E\subset X$ be an isoperimetric set. We call any constant $c$ such that
(3.2) holds a _mean curvature barrier_ for $\partial E$.

For discussions concerning the uniqueness of $c$ as in the previous
definition, and comparison with the Riemannian setting, we refer the reader to
the forthcoming section 3 and section 3.

######  Remark .

The above Theorem 3.2 proves in particular that a smooth isoperimetric set on
a Riemannian manifold $M$ (say with Ricci curvature bounded from below) has
constant mean curvature without the use of the first variation formula for the
perimeter.

######  Remark .

If $(M^{n},g)$ is a smooth Riemannian manifold (with Ricci curvature uniformly
bounded from below) and $E\subset M$ is an isoperimetric set, then the
constant $c$ obtained via Theorem 3.2 is unique and equal to the constant mean
curvature of the regular part of $\partial E$.
Notice that, if $n\geq 8$, this conclusion does not follow directly from what
we observed in section 3, since $\partial E$ might have a non empty singular
part (with Hausdorff dimension less than $n-8$). However, the statement can be
verified applying the argument above in a neighbourhood of a regular point of
$\partial E$, where $\partial E$ is smooth.

The validity of a similar statement for isoperimetric sets in
$\operatorname{RCD}(K,N)$ metric measure spaces
$(X,\mathsf{d},\mathscr{H}^{N})$ goes beyond the scope of this note and is
left to the future investigation.

######  Remark .

Let $(D,\mathsf{d},\mathscr{H}^{2})$ be a two dimensional flat disk with
canonical metric measure structure and boundary $\partial D$. Let
$(K,\mathsf{d}_{K},\mathscr{H}^{2})$ be the doubling of $D$ along its boundary
$\partial D$ and set $k>0$ the curvature of $\partial D$. It is a classical
fact that $(K,\mathsf{d}_{K},\mathscr{H}^{2})$ is an Alexandrov space with non
negative curvature, in particular it is an $\operatorname{RCD}(0,2)$ space,
but it is not a smooth Riemannian manifold. Observe that each of the two
isometric copies of $D$ inside $K$ verifies the bound (3.2) for any
$c\in[-k,k]$ (for $K=0$ and $N=2$). Even though $D\subset K$ is not an
isoperimetric set, this example illustrates that the uniqueness of the mean
curvature barrier is a delicate issue in the non smooth setting.

The following mild regularity results are obtained arguing verbatim as in [73,
Proposition 5.4, Theorem 5.5, Proposition 6.14]. This can be done since
Theorem 3.2 is the counterpart of [73, Theorem 5.2] for isoperimetric sets,
while [73, Lemma 6.12] holds for volume constrained minimizers for compact
variations as well, and [73, Lemma 2.42] holds for volume constrained
minimizers for compact variations since they are quasiminimal sets according
to [19, Theorem 3.24].

######  Proposition .

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ space for
some $K\in\mathbb{R}$ and $N\geq 2$. Let $E\subset X$ be a volume constrained
minimizer for compact variations in $X$. Then $\mathsf{d}_{\overline{E}}$ and
$\mathsf{d}_{E}^{s}$ have locally measure valued Laplacian in $X$ and the
following hold

$\begin{split}\Delta\mathsf{d}_{\overline{E}}&=\mathscr{H}^{N-1}\llcorner\partial
E+\Delta\mathsf{d}_{\overline{E}}\llcorner(X\setminus\overline{E}),\\\
\Delta\mathsf{d}_{E}^{s}\llcorner\partial E&=0.\end{split}$ (3.30)

###### Proof.

We only provide an indication of the strategy of the proof, that can be
obtained with minor modifications with respect to the case of local perimeter
minimizers considered in [73].

The proof is divided into two steps: the verification that
$\mathsf{d}_{\overline{E}}$ and $\mathsf{d}^{s}_{E}$ admit locally measure
valued Laplacian and the computation of the singular part of their Laplacian
along $\partial E$.

In order to prove that $\mathsf{d}_{\overline{E}}$ and $\mathsf{d}^{s}_{E}$
admit locally measure valued Laplacian it is sufficient to uniformly bound the
volumes of the $t$-tubular neighbourhoods of $\partial E$ as $Ct$ when $t\to
0$ for some constant $C>0$ and to pass to the limit in the Gauss-Green
integration by parts formulae on super-level sets
$\\{\mathsf{d}_{\overline{E}}>t_{i}\\}$ for suitably chosen sequences
$t_{i}\downarrow 0$.
The uniform volume bound for the tubular neighbourhoods of $\partial E$
follows from [19, Theorem 3.24], where quasiminimality of isoperimetric sets
is proved, and [73, Lemma 2.42]. The conclusion follows arguing as in Step 1
of the proof of [73, Proposition 5.4] (see also the previous [32]).

Thanks to [73, Lemma 2.27],
$\Delta\mathsf{d}_{\overline{E}}\ll\mathscr{H}^{N-1}$ and
$\Delta\mathsf{d}^{s}_{E}\ll\mathscr{H}^{N-1}$. To conclude, it suffices to
compute the densities of these measures with respect to
$\mathscr{H}^{N-1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\partial E$, which is a locally
doubling finite measure. This can be done with a classical blow-up argument,
as in Step 2 of the proof of [73, Proposition 5.4]. In order to prove that at
regular points of $\partial E$ the (signed) distance from the boundary
converges to the (signed) distance from the boundary of a Euclidean space
after the blow-up we rely on the quasiminimality [19, Theorem 3.24] and on
[73, Theorem 2.43].

Finally, in order to prove that $\Delta\mathsf{d}_{E}^{s}\llcorner\partial
E=0$ one argues precisely as in the last part of the proof of [73, Theorem
5.5]. ∎

######  Proposition .

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ space for
some $K\in\mathbb{R}$ and $N\geq 2$. Let $E\subset X$ be a volume constrained
minimizer for compact variations in $X$. Let

$\begin{split}\mu_{\varepsilon}^{+}&:=\varepsilon^{-1}\mathscr{H}^{N}\llcorner\\{0\leq\mathsf{d}_{\overline{E}}<\varepsilon\\},\\\
\mu_{\varepsilon}^{-}&:=\varepsilon^{-1}\mathscr{H}^{N}\llcorner\\{0\leq\mathsf{d}_{E^{c}}<\varepsilon\\}.\end{split}$
(3.31)

Then $\mu_{\varepsilon}^{+}\to\operatorname{Per}_{E}$, and
$\mu_{\varepsilon}^{-}\to\operatorname{Per}_{E}$ weakly as $\varepsilon\to 0$.
In particular, the Minkowski content of $E$ coincides with
$\operatorname{Per}(E)$.

###### Proof.

The proof is analogous to the one of [73, Proposition 6.14].

The fact that the measures $\mu_{\varepsilon}^{+}$ and $\mu_{\varepsilon}^{-}$
have uniformly bounded mass follows from [73, Lemma 2.42] thanks to [19,
Theorem 3.24].

The fact that the perimeter is smaller than any weak limit as
$\varepsilon_{i}\downarrow 0$ of the measures $\mu_{\varepsilon_{i}}^{+}$ and
$\mu_{\varepsilon_{i}}^{-}$ is general and does not require any regularity of
$\partial E$.

In order to prove the converse inequality we rely on Theorem 3.2, which plays
the role of [73, Theorem 5.2] in this setting, on section 3, which plays the
role of [73, Proposition 5.4] in this setting and argue as in the second part
of the proof of [73, Proposition 6.14]. ∎

Our next goal is to turn the Laplacian comparison in Theorem 3.2 into an
estimate for the perimeter of the equidistant sets from the boundary of an
isoperimetric set.

######  Proposition .

Let us consider an $\operatorname{RCD}(K,N)$ metric measure space
$(X,\mathsf{d},\mathscr{H}^{N})$ for some $K\in\mathbb{R}$ and $N\geq 2$. Let
$E\subset X$ be a volume constrained minimizer for compact variations in $X$,
and let $c\in\mathbb{R}$ be given by Theorem 3.2. Then for any $t\geq 0$ it
holds

$\operatorname{Per}(\\{x\in X\,:\,\mathsf{d}(x,\overline{E})\leq t\\})\leq
J_{c,K,N}(t)\operatorname{Per}(E)\,,$ (3.32)

and, for any $t\geq 0$,

$\operatorname{Per}(\\{x\in X\,:\,\mathsf{d}(x,X\setminus E)\leq t\\})\leq
J_{-c,K,N}(t)\operatorname{Per}(E)\,,$ (3.33)

where we recall that the Jacobian function has been introduced in (2.18).

###### Proof.

Let us focus on the estimate for the perimeter of the exterior equidistant
set, the estimate for the interior one can be obtained with a completely
analogous argument.

The bound can be obtained by applying the Gauss–Green integration by parts
formula [30, Theorem 1.6] with vector field the gradient of the distance
function in the slab $E^{t}\setminus E$ (compare with [30, Theorem 5.2], and
Theorem 2.4), where we denoted $E^{t}:=\\{x\in
X\,:\,\mathsf{d}(x,\overline{E})\leq t\\}$ the $t$-enlargement of $E$. Indeed,
taking into account Theorem 3.2 we obtain

$\operatorname{Per}(E_{t})\leq\operatorname{Per}(E)+\int_{E^{t}\setminus
E}(N-1)\frac{s^{\prime}_{\frac{K}{N-1},-\frac{c}{N-1}}\circ\mathsf{d}_{\overline{E}}}{s_{\frac{K}{N-1},-\frac{c}{N-1}}\circ\mathsf{d}_{\overline{E}}}\mathop{}\\!\mathrm{d}\mathscr{H}^{N}\,.$
(3.34)

Arguing as in the proof of [73, Proposition 6.15] via a classical comparison
argument for ODEs (see also subsection 2.5) we obtain that

$\operatorname{Per}(\\{x\in X\,:\,\mathsf{d}(x,\overline{E})\leq t\\})\leq
J_{c,K,N}(t)\operatorname{Per}(E)\,,\quad\text{for any $t\geq 0$}\,,$ (3.35)

as we claimed. ∎

######  Remark .

If $(M^{n},g)$ is a smooth Riemannian manifold and $E\subset M$ is an
isoperimetric set, then

$\lim_{t\to 0^{+}}\frac{\operatorname{Per}(\\{x\in
X\,:\,\mathsf{d}(x,\overline{E})\leq
t\\})-\operatorname{Per}(E)}{t}=c\operatorname{Per}(E)\,$ (3.36)

and an analogous conclusion holds for the perimeters of the interior
equidistant sets. This follows directly from the first variation formula for
the perimeter when $\partial E$ is smooth (since we can consider deformations
induced by a smooth compactly supported extension of the unit normal of
$\partial E$). If $n\geq 8$ an additional approximation argument (relying on
the regularity theory for isoperimetric sets) is required, see for instance
[25, 79].
The validity of an analogous statement for isoperimetric sets in
$\operatorname{RCD}(K,N)$ metric measure spaces goes beyond the scope of this
note and is left to the future investigation.

Let us point out the expression for the bounds above when $K=0$. Under these
assumptions, with the very same notation above we obtain

$\operatorname{Per}(E_{t})\leq\operatorname{Per}(E)\left(1+\frac{ct}{N-1}\right)^{N-1}\,,\quad\text{for
any $t\geq 0$}\,.$ (3.37)

Using the coarea formula we can get volume bounds for the tubular
neighbourhoods of isoperimetric sets integrating the perimeter bounds in
section 3.

######  Corollary .

Let us consider an $\operatorname{RCD}(K,N)$ metric measure space
$(X,\mathsf{d},\mathscr{H}^{N})$ for some $K\in\mathbb{R}$ and $N\geq 2$. Let
$E\subset X$ be a volume constrained minimizer for compact variations in $X$,
and let $c\in\mathbb{R}$ be given by Theorem 3.2. Then for any $t\geq 0$ it
holds

$\mathscr{H}^{N}(\\{x\in X\setminus E\,:\,\mathsf{d}(x,\overline{E})\leq
t\\})\leq\operatorname{Per}(E)\int_{0}^{t}J_{c,K,N}(r)\mathop{}\\!\mathrm{d}r\,,$
(3.38)

and, for any $t\geq 0$,

$\mathscr{H}^{N}(\\{x\in E\,:\,\mathsf{d}(x,X\setminus E)\leq
t\\})\leq\operatorname{Per}(E)\int_{0}^{t}J_{-c,K,N}(r)\mathop{}\\!\mathrm{d}r\,.$
(3.39)

##  4\. Concavity properties of the isoperimetric profile function and
consequences

In Theorem 3.2 we proved sharp bounds on the Laplacian of the distance
function from an isoperimetric set. Such bounds encode information about the
first and second variation of the area of equidistants from the isoperimetric
boundary. As we shall see, this information is sufficient to extend the sharp
concavity properties for the isoperimetric profile known for smooth and
compact Riemannian manifolds with lower Ricci curvature bounds to the setting
of $N$-dimensional compact $\operatorname{RCD}(K,N)$ spaces.
More in general, we are going to join such information together with the
generalized asymptotic mass decomposition Theorem 4.1 to get sharp concavity
properties of the isoperimetric profile for $\operatorname{RCD}(K,N)$ spaces
$(X,\mathsf{d},\mathscr{H}^{N})$ with a lower bound on the volume of unit
balls in Theorem 4.2, which is the main result of this section. In particular,
this will imply that the sharp concavity properties of the isoperimetric
profile hold on complete non compact manifolds with uniform lower Ricci
curvature bounds and uniform lower volume bounds.

Concavity properties of the isoperimetric profile for (weighted) manifolds
with lower Ricci curvature bounds have been considered by various authors, see
for instance [24, 25, 26, 27, 69, 79, 75, 49]. All these works deal with
compact manifolds or with weighted manifolds of finite total measure and they
heavily rely on the existence of isoperimetric regions for any volume. In all
cases smoothness is a relevant assumption, in order to rely on the regularity
theory for isoperimetric regions. The case of non smooth weights in [69] is
handled with a careful approximation procedure. The only previous references
where the problem is considered in non compact manifolds with infinite volume
are [82, 77] and [67]. In [82] the case of surfaces with non negative Gaussian
curvature is treated and existence of isoperimetric regions of any volume is
an intermediate step in order to prove concavity of the isoperimetric profile.
In [77] the case of complete non compact manifolds with $C^{2,\alpha}$ bounded
geometry is considered. In [67] the authors consider unbounded convex bodies
$C\subset\mathbb{R}^{n}$ verifying no further regularity assumptions. Their
proof relies on a generalized existence result for isoperimetric regions and
on an approximation argument, to deal with convex bodies with non smooth
boundary.

The two statements below are proved in [18] building on top of [77, 17, 19].
They will be key ingredients for the proof of Theorem 4.2.

######  Theorem 4.1 (Asymptotic mass decomposition).

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be a non compact
$\operatorname{RCD}(K,N)$ space. Assume there exists $v_{0}>0$ such that
$\mathscr{H}^{N}(B_{1}(x))\geq v_{0}$ for every $x\in X$. Let $V>0$. For every
minimizing (for the perimeter) sequence $\Omega_{i}\subset X$ of volume $V$,
with $\Omega_{i}$ bounded for any $i$, up to passing to a subsequence, there
exist an increasing and bounded sequence
$\\{N_{i}\\}_{i\in\mathbb{N}}\subseteq\mathbb{N}$, disjoint finite perimeter
sets $\Omega_{i}^{c},\Omega_{i,j}^{d}\subset\Omega_{i}$, and points $p_{i,j}$,
with $1\leq j\leq N_{i}$ for any $i$, such that

  * •

$\lim_{i}\mathsf{d}(p_{i,j},p_{i,\ell})=\lim_{i}\mathsf{d}(p_{i,j},o)=\infty$,
for any $j\neq\ell<\overline{N}+1$ and any $o\in X$, where
$\overline{N}:=\lim_{i}N_{i}<\infty$;

  * •

$\Omega_{i}^{c}$ converges to $\Omega\subset X$ in the sense of finite
perimeter sets, and we have
$\mathscr{H}^{N}(\Omega_{i}^{c})\to_{i}\mathscr{H}^{N}(\Omega)$, and
$\operatorname{Per}(\Omega_{i}^{c})\to_{i}\operatorname{Per}(\Omega)$.
Moreover $\Omega$ is a bounded isoperimetric region for its own volume in $X$;

  * •

for every $j<\overline{N}+1$, $(X,\mathsf{d},\mathscr{H}^{N},p_{i,j})$
converges in the pmGH sense to a pointed $\operatorname{RCD}(K,N)$ space
$(X_{j},\mathsf{d}_{j},\mathscr{H}^{N},p_{j})$. Moreover there are
isoperimetric regions $Z_{j}\subset X_{j}$ such that
$\Omega^{d}_{i,j}\to_{i}Z_{j}$ in $L^{1}$-strong and
$\operatorname{Per}(\Omega^{d}_{i,j})\to_{i}\operatorname{Per}(Z_{j})$;

  * •

it holds that

$I_{(X,\mathsf{d},\mathscr{H}^{N})}(V)=\operatorname{Per}(\Omega)+\sum_{j=1}^{\overline{N}}\operatorname{Per}(Z_{j}),\qquad\qquad
V=\mathscr{H}^{N}(\Omega)+\sum_{j=1}^{\overline{N}}\mathscr{H}^{N}(Z_{j}).$
(4.1)

######  Proposition .

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be a non compact
$\operatorname{RCD}(K,N)$ space. Assume there exists $v_{0}>0$ such that
$\mathscr{H}^{N}(B_{1}(x))\geq v_{0}$ for every $x\in X$. Let $\\{p_{i,j}\ :\
i\in\mathbb{N}\\}$ be a sequence of points on $X$, for
$j=1,\ldots,\overline{N}$ where $\overline{N}\in\mathbb{N}\cup\\{+\infty\\}$.
Suppose that each sequence $\\{p_{i,j}\\}_{i}$ is diverging along $X$ and that
$(X,\mathsf{d},\mathscr{H}^{N},p_{i,j})$ converges in the pmGH sense to a
pointed $\operatorname{RCD}(K,N)$ space
$(X_{j},\mathsf{d}_{j},\mathscr{H}^{N},p_{j})$. Defining

$I_{X\sqcup_{j=1}^{\overline{N}}X_{j}}(v):=\inf\left\\{\operatorname{Per}(E)+\sum_{j=1}^{\overline{N}}\operatorname{Per}(E_{j}):E\subseteq
X,E_{j}\subseteq
X_{j},\mathscr{H}^{N}(E)+\sum_{j=1}^{\overline{N}}\mathscr{H}^{N}(E_{j})=v\right\\},$
(4.2)

it holds $I_{X\sqcup_{j=1}^{\overline{N}}X_{j}}(v)=I_{X}(v)$ for any $v>0$.

We need to start with a mild regularity property of the isoperimetric profile,
namely that it is strictly positive and continuous. Later in the paper these
two properties will be sharpened in several directions. We stress that an
argument similar to the one discussed in section 4 had already appeared in
[69, Lemma 6.9], and [49, Lemma 6.2]. By a careful inspection of the proofs,
the argument for proving the local Hölder property in section 4 is likely to
be adapted in the more general case of $\operatorname{CD}(K,N)$ spaces
$(X,\mathsf{d},\mathfrak{m})$ with densities uniformly bounded above and
volume of unit balls uniformly bounded below, as kindly pointed out to the
authors by E. Milman. Since we do not need such level of generality, we will
not give the details of the proof in such a general case.

######  Lemma .

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ space.
Assume that there exists $v_{0}>0$ such that $\mathscr{H}^{N}(B_{1}(x))\geq
v_{0}$ for every $x\in X$. Let $I_{X}:(0,\mathscr{H}^{N}(X))\to\mathbb{R}$ be
the isoperimetric profile of $X$. Then $I_{X}(v)>0$ for every $v>0$ and
$I_{X}$ is continuous.

###### Proof.

The first conclusion can be reached arguing as in [17, Remark 4.7], building
on the the top of Theorem 4.1. The second conclusion can be reached adapting
[76, Theorem 2], which shows that $I$ is locally $(1-\frac{1}{N})$-Hölder, cf.
[18, Lemma 2.23]. ∎

###  4.1. Sharp concavity inequalities for the isoperimetric profile

Given a continuous function $f:(0,\infty)\to(0,\infty)$ and parameters
$K\in\mathbb{R}$ and $1<N<\infty$ we are going to consider second order
differential inequalities of the form

$-f^{\prime\prime}f\geq K+\frac{(f^{\prime})^{2}}{N-1}$ (4.3)

and

$-f^{\prime\prime}\geq\frac{KN}{N-1}f^{\frac{2-N}{N}}\,.$ (4.4)

In general the function $f$ will be continuous but not twice differentiable
everywhere and the inequalities will be understood in the viscosity sense,
i.e. we will require that whenever
$\varphi:(x_{0}-\varepsilon,x_{0}+\varepsilon)\to\mathbb{R}$ is a $C^{2}$
function with $\varphi\leq f$ on $(x_{0}-\varepsilon,x_{0}+\varepsilon)$ and
$\varphi(x_{0})=f(x_{0})$ the corresponding inequality (4.3) or (4.4) holds at
$x_{0}$ with $\varphi$ in place of $f$.

######  Theorem 4.2.

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ space.
Assume that there exists $v_{0}>0$ such that $\mathscr{H}^{N}(B_{1}(x))\geq
v_{0}$ for every $x\in X$.

Let $I:(0,\mathscr{H}^{N}(X))\to(0,\infty)$ be the isoperimetric profile of
$X$. Then

  1. (1)

The inequality

$-I^{\prime\prime}I\geq K+\frac{(I^{\prime})^{2}}{N-1}\,\quad\text{holds in
the viscosity sense on $(0,\mathscr{H}^{N}(X))$}\,;$ (4.5)

  2. (2)

Let $N\leq\alpha<+\infty$ and $\xi:=I^{\frac{\alpha}{\alpha-1}}$. Hence the
inequality

$\begin{split}-\xi^{\prime\prime}&\geq\frac{\alpha}{\alpha-1}\xi^{\frac{2-\alpha}{\alpha}}\left(\left(\frac{1}{N-1}-\frac{1}{\alpha-1}\right)(I^{\prime})^{2}+K\right)\\\
&=\frac{\alpha}{\alpha-1}\xi^{\frac{2-\alpha}{\alpha}}\left(\left(\frac{1}{N-1}-\frac{1}{\alpha-1}\right)\frac{(\alpha-1)^{2}}{\alpha^{2}}\xi^{-\frac{2}{\alpha}}(\xi^{\prime})^{2}+K\right),\end{split}$
(4.6)

holds in the viscosity sense on $(0,\mathscr{H}^{N}(X))$.
In particular, in the case $\alpha=N$, if $\psi:=I^{\frac{N}{N-1}}$ then

$-\psi^{\prime\prime}\geq\frac{KN}{N-1}\psi^{\frac{2-N}{N}}\,\quad\text{holds
in the viscosity sense on $(0,\mathscr{H}^{N}(X))$}\,.$ (4.7)

###### Proof.

Let us assume $\mathscr{H}^{N}(X)=+\infty$, the compact case being completely
analogous. Let us prove (4.5) first. Let $v\in(0,\infty)$ be fixed. Take
$\Omega_{i}$ a minimizing sequence of bounded sets of volume $v$ and let
$\Omega,Z_{j}$ be the isoperimetric regions in $X,X_{j}$ respectively,
according to the notation of Theorem 4.1.

Let us consider a smooth function $\varphi$ such that $\varphi\leq I$ in a
neighbourhood of $v$ and $\varphi(v)=I(v)$. We wish to prove that

$-\varphi^{\prime\prime}(v)\varphi(v)\geq
K+\frac{\left(\varphi^{\prime}(v)\right)^{2}}{N-1}\,.$ (4.8)

Let $E$ be one of the isoperimetric sets $\Omega,Z_{j}$, and let $E_{t}$ be
the $t$-enlargements, in the associated space $X,X_{j}$, of $E$ for
$t\in(-\varepsilon,\varepsilon)$. To be more precise, for $t\geq 0$, the
points in $E_{t}$ are those points of $X$ that have distance $\leq t$ from
$\overline{E}$, while, for $t\leq 0$, the points in $E_{t}$ are those points
of $X$ such that the distance from $X\setminus E$ is $\geq-t$. Let $c$ be a
mean curvature barrier for $E$ provided by Theorem 3.2. We now show that
$c=\varphi^{\prime}(v)$.
As we observed in (3.32),

$\operatorname{Per}(E_{t})\leq
J_{c,K,N}(t)\operatorname{Per}(E),\quad\text{for any
$t\in(-\varepsilon,\varepsilon)$}\,.$ (4.9)

By simple computations,

$J^{\prime}_{c,K,N}(0)=c\,\quad\text{and}\quad
J^{\prime\prime}_{c,K,N}(0)=-K+\frac{N-2}{N-1}c^{2}\,.$ (4.10)

Let us show that $t\mapsto\operatorname{Per}(E_{t})$ is differentiable at
$t=0$.

Setting
$\beta(t):=\mathscr{H}^{N}(E_{t})+\sum_{T\in\\{\Omega,Z_{1},\dots,Z_{\overline{N}}\\},T\neq
E}\mathscr{H}^{N}(T)$, by the coarea formula we get that $\beta(t)$ is
continuous at $t=0$. Hence, by (4.2), and (4.9), we have that, for any
$t\in(-\varepsilon,\varepsilon)$,

$\begin{split}J_{c,K,N}(t)\operatorname{Per}(E)&+\sum_{T\in\\{\Omega,Z_{1},\dots,Z_{\overline{N}}\\},T\neq
E}\operatorname{Per}(T)\geq\operatorname{Per}(E_{t})+\sum_{T\in\\{\Omega,Z_{1},\dots,Z_{\overline{N}}\\},T\neq
E}\operatorname{Per}(T)\\\ &\geq
I_{X\sqcup_{j=1}^{\overline{N}}X_{j}}(\beta(t))=I(\beta(t))\geq\varphi(\beta(t))\,.\end{split}$
(4.11)

From (4.11), the continuity of $\beta(t)$ at $t=0$, the fact that

$\varphi(v)=I(v)=\operatorname{Per}(E)+\sum_{T\in\\{\Omega,Z_{1},\dots,Z_{\overline{N}}\\},T\neq
E}\operatorname{Per}(T),$ (4.12)

we first deduce that $\operatorname{Per}(E_{t})$ is continuous at $t=0$.
Hence, by the continuity of $t\mapsto\operatorname{Per}(E_{t})$ at $t=0$ and
the coarea formula, $t\mapsto\beta(t)$ is differentiable at $t=0$. Exploiting
this information again in (4.11) we obtain that
$t\mapsto\operatorname{Per}(E_{t})$ is differentiable at $t=0$.

By using the latter differentiability together with (4.9) we get that

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\operatorname{Per}(E_{t})|_{t=0}=c\operatorname{Per}(E)\,.$
(4.13)

By coarea, this implies that

$t\mapsto\mathscr{H}^{N}(E_{t})\,,$ (4.14)

is twice differentiable at $t=0$ with

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\mathscr{H}^{N}(E_{t})|_{t=0}=\operatorname{Per}(E)\,\quad\text{and}\quad\frac{\mathop{}\\!\mathrm{d}^{2}}{\mathop{}\\!\mathrm{d}t^{2}}\mathscr{H}^{N}(E_{t})|_{t=0}=c\operatorname{Per}(E)\,.$
(4.15)

Moreover, from (4.1) and the discussion above, $\beta(0)=v$,
$\beta^{\prime}(0)=\operatorname{Per}(E)$,
$\beta^{\prime\prime}(0)=c\operatorname{Per}(E)$ and
$\varphi^{\prime}(v)\beta^{\prime}(0)=c\operatorname{Per}(E)$, therefore
$\varphi^{\prime}(v)=c$, which is the sought claim.

Hence, we showed that the barrier $c$ given by Theorem 3.2 is unique and
$c:=\varphi^{\prime}(v)$ for every isoperimetric set $\Omega,Z_{j}$ in any of
the limit spaces $X,X_{j}$.

Let us define
$\widetilde{\beta}(t):=\mathscr{H}^{N}(\Omega_{t})+\sum_{j=1}^{\overline{N}}\mathscr{H}^{N}((Z_{j})_{t})$
for every $t\in(-\varepsilon,\varepsilon)$ with $\varepsilon>0$ small enough.

Arguing as in (4.11) for each isoperimetric region separately, we get that
$\widetilde{\beta}(t)$ is twice differentiable at $t=0$. Moreover

$\varphi(\widetilde{\beta}(t))\leq
J_{c,K,N}(t)\left(\operatorname{Per}(\Omega)+\sum_{j=1}^{\overline{N}}\operatorname{Per}(Z_{j})\right)\,\quad\text{for
any $t\in(-\varepsilon,\varepsilon)$}\,,$ (4.16)

$\widetilde{\beta}(0)=v$, $\widetilde{\beta}^{\prime}(0)=I(v)$, and
$\widetilde{\beta}^{\prime\prime}(0)=cI(v)$. Using (4.16), we can pass to the
limit as $t\to 0$ in

$\begin{split}\frac{\varphi(\widetilde{\beta}(t))+\varphi(\widetilde{\beta}(-t))-2\varphi(\widetilde{\beta}(0))}{t^{2}}&=\frac{\varphi(\widetilde{\beta}(t))+\varphi(\widetilde{\beta}(-t))-2I(v)}{t^{2}}\\\
&\leq\frac{J_{c,K,N}(t)+J_{c,K,N}(-t)-2}{t^{2}}I(v)\,,\end{split}$ (4.17)

to obtain

$\varphi^{\prime\prime}(\widetilde{\beta}(0))\widetilde{\beta}^{\prime}(0)^{2}+\varphi^{\prime}(\widetilde{\beta}(0))\widetilde{\beta}^{\prime\prime}(0)\leq
I(v)\left(c^{2}\frac{N-2}{N-1}-K\right)\,,$ (4.18)

Using the previous identities, we obtain the sought conclusion

$\varphi^{\prime\prime}(v)\varphi(v)=\varphi^{\prime\prime}(v)I(v)\leq-\frac{\varphi^{\prime}(v)^{2}}{N-1}-K\,.$
(4.19)

Let us prove (4.6). We need to prove that for every
$v\in(0,\mathscr{H}^{N}(X))$, if $\varphi$ is a smooth function in a
neighbourhood $U$ of $v$ such that $\varphi\leq\xi$ on $U$ and
$\varphi(v)=\xi(v)$, then (4.6) holds with $\varphi$ in place of $\xi$. The
latter inequality is obtained noticing that
$\varphi^{\frac{\alpha-1}{\alpha}}\leq I$ on $U$ and
$\varphi^{\frac{\alpha-1}{\alpha}}(v)=I(v)$, using (4.5) with some easy
algebraic computations. ∎

######  Remark .

Let us point out that (4.6), and thus (4.7), holds also in the sense of second
order incremental quotients considered in [25, 26]. Given a continuous
function $f:(0,\mathscr{H}^{N}(X))\to(0,\infty)$ and
$x\in(0,\mathscr{H}^{N}(X))$ we denote

$\overline{D^{2}}f(x):=\limsup_{h\downarrow
0}\frac{f(x+h)+f(x-h)-2f(x)}{h^{2}}\,.$ (4.20)

Then, for example, the very same proof of (4.7), where the bound was
considered in the viscosity sense, shows that

$\overline{D^{2}}\psi\leq-\frac{KN}{N-1}\psi^{\frac{2-N}{N}}\,\quad\text{on
$(0,\mathscr{H}^{N}(X))$}\,.$ (4.21)

###  4.2. Fine properties of the isoperimetric profile

In this subsection we derive further regularity properties and asymptotics for
small volumes of the isoperimetric profile. They will be particularly useful
to study the stability of isoperimetric regions under non collapsed (pointed)
Gromov–Hausdorff convergence, diameter bounds and connectedness properties for
isoperimetric regions, and the asymptotic isoperimetric behaviour of non
collapsed spaces with lower Ricci curvature bounds (see also the forthcoming
[20]).
The arguments essentially rely only on the sharp concavity properties of the
isoperimetric profile and on qualitative isoperimetric inequalities.

The following corollary is a standard consequence of Theorem 4.2, and
therefore we omit its proof.

######  Corollary .

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ space
with $K\leq 0$. Assume that there exists $v_{0}>0$ such that
$\mathscr{H}^{N}(B_{1}(x))\geq v_{0}$ for every $x\in X$.

For every $v\in(0,\mathscr{H}^{N}(X))$ and for every
$\delta\in(v,\mathscr{H}^{N}(X)-v)$ there exists $C>0$ such that the function
$I^{\frac{N}{N-1}}(x)-Cx^{2}$ is concave on $(v-\delta,v+\delta)$. Hence the
isoperimetric profile function $I$ has right derivative $I_{+}^{\prime}(v)$
and left derivative $I_{-}^{\prime}(v)$ defined for every
$v\in(0,\mathscr{H}^{N}(X))$. Moreover the isoperimetric profile $I$ is
differentiable in all $(0,\mathscr{H}^{N}(X))$ except at most countably many
values, it is locally Lipschitz, and it is twice differentiable almost-
everywhere. Moreover (4.5), (4.6), and (4.7) also hold pointwise almost
everywhere.

The following result has been proved in [19, Proposition 3.20] (cf. also [19,
Remark 3.21]).

######  Lemma  (Isoperimetric inequality for small volumes).

Let $(X,\mathsf{d},\mathfrak{m})$ be a $\operatorname{CD}(K,N)$ space with
$N\geq 2$ and $K\in\mathbb{R}$. Assume that there exists $v_{0}>0$ such that
$\mathfrak{m}(B_{1}(x))\geq v_{0}$ holds for every $x\in X$. Then there exist
constants $v_{1}=v_{1}(K,N,v_{0})>0$ and $\vartheta=\vartheta(K,N,v_{0})>0$
such that the implication

$\mathfrak{m}(E)\leq
v_{1}\qquad\Longrightarrow\qquad\vartheta\mathfrak{m}(E)^{\frac{N-1}{N}}\leq\operatorname{Per}(E)$
(4.22)

is verified for any set of finite perimeter $E\subset X$.

Let us now give a slight improvement of subsection 4.2. For this aim we will
need the following lemma of independent interest, that is an immediate
consequence of the isoperimetric inequality for small volumes subsection 4.2.

######  Lemma .

Let $K\in\mathbb{R}$ and $N\geq 2$. Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an
$\operatorname{RCD}(K,N)$ space. Let us assume there exists $v_{0}>0$ such
that $\mathscr{H}^{N}(B_{1}(x))\geq v_{0}$ for every $x\in X$. Then there
exist $v_{1}:=v_{1}(K,N,v_{0})$, and $\vartheta:=\vartheta(K,N,v_{0})$ such
that

$I(v)\geq\vartheta v^{\frac{N-1}{N}},\qquad\text{for all $v\leq v_{1}$}.$
(4.23)

######  Remark .

Under the assumptions of subsection 4.2, there exist constants
$C=C(K,N,v_{0})>0$ and $\bar{v}=\bar{v}(K,N,v_{0})>0$ such that

$\frac{I(v)}{v^{\frac{N-1}{N}}}\leq C,\quad\text{ for every }v\leq\bar{v}.$
(4.24)

Indeed, we can find (cf. [19, Remarks 3.21 and 3.25]) $c=c(K,N,v_{0})>0$ and
$R=R(K,N,v_{0})>0$ such that $\operatorname{Per}(B_{r}(x))\leq cr^{N-1}$ for
every $r\leq R$ and $x\in X$, while $\mathscr{H}^{N}(B_{r}(x))\geq
c^{-1}r^{N}$ if $r\leq R$. Set $\bar{v}\coloneqq c^{-1}R^{N}$ and $C\coloneqq
c^{\frac{2N-1}{N}}$. Given any $v\leq\bar{v}$ and $x\in X$, there exists
$r\leq R$ with $\mathscr{H}^{N}(B_{r}(x))=v$. Hence, we have
$I(v)\leq\operatorname{Per}(B_{r}(x))\leq c(r^{N})^{\frac{N-1}{N}}\leq
C\mathscr{H}^{N}(B_{r}(x))^{\frac{N-1}{N}}=Cv^{\frac{N-1}{N}}$, which yields
(4.24).

Joining the second order differential inequalities derived in Theorem 4.2 with
the isoperimetric inequality for small volumes we derive further analytical
properties of the isoperimetric profile.

######  Proposition .

Let $K\leq 0$ and $N\geq 2$. Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an
$\operatorname{RCD}(K,N)$ space such that there exists $v_{0}\geq 0$ with
$\mathscr{H}^{N}(B_{1}(x))\geq v_{0}$ for every $x\in X$. Then the following
hold.

  1. (1)

There exists $C:=C(K,N,v_{0})>0$ and $v_{1}:=v_{1}(K,N,v_{0})>0$ such that the
function $\eta(v):=I^{\frac{N}{N-1}}(v)-Cv^{\frac{2+N}{N}}$ is concave on the
interval $[0,v_{1}]$. Moreover, if $N=2$, we can choose $C=-K$ and the claim
holds on $[0,\mathscr{H}^{N}(X)]$, if $\mathscr{H}^{N}(X)<\infty$, or on
$[0,\mathscr{H}^{N}(X))$, if $\mathscr{H}^{N}(X)=\infty$.
As a consequence the function

$[0,\mathscr{H}^{N}(X))\ni v\mapsto\frac{I(v)}{v^{\frac{N-1}{N}}},$

has a finite strictly positive limit as $v\to 0$.

  2. (2)

There exist $\tilde{C}=\tilde{C}(K,N,v_{0})>0$ such that the function
$\tilde{\eta}(v)\coloneqq I(v)-\tilde{C}v^{\frac{1+N}{N}}$ is concave on
$[0,v_{1}]$.

###### Proof.

Let us prove item (1). Let us first deal with the case $N>2$.
We claim that it suffices to take

$C:=\frac{-KN^{3}\vartheta^{\frac{2-N}{N-1}}}{2(N-1)(N+2)},\,\text{where
$\vartheta$ and $v_{1}$ are the constants in
\eqref{eqn:BoundBelowIsopProfile}}\,.$ (4.25)

We aim at proving that $-\eta^{\prime\prime}\geq 0$ holds in the viscosity
sense on $(0,v_{1})$. The conclusion will follow since $\eta$ is continuous on
$(0,v_{1})$ thanks to section 4.

Let us take $v^{\prime}\in(0,v_{1})$ and let $\varphi$ be smooth in a
neighbourhood $U$ of $v^{\prime}$ such that $\varphi\leq\eta$ on $U$ and
$\varphi(v^{\prime})=\eta(v^{\prime})$. We need to check that
$-\varphi^{\prime\prime}(v^{\prime})\geq 0$.
From $\varphi\leq\eta$ on $U$ we get $\varphi+Cv^{\frac{2+N}{N}}\leq
I^{\frac{N}{N-1}}$ on $U$. Moreover
$\varphi(v^{\prime})+Cv^{\prime\frac{2+N}{N}}=I(v^{\prime})^{\frac{N}{N-1}}$.
Hence from item (2) of Theorem 4.2 we obtain that

$-\left(\varphi+Cv^{\frac{2+N}{N}}\right)^{\prime\prime}(v^{\prime})\geq\frac{KN}{N-1}\left(\varphi(v^{\prime})+Cv^{\prime\frac{2+N}{N}}\right)^{\frac{2-N}{N}}=\frac{KN}{N-1}I(v^{\prime})^{\frac{2-N}{N-1}},$

therefore

$-\varphi^{\prime\prime}(v^{\prime})\geq\frac{KN}{N-1}I(v^{\prime})^{\frac{2-N}{N-1}}+\frac{2C(N+2)}{N^{2}}v^{\prime\frac{2-N}{N}}.$