# On the existence of isoperimetric regions in manifolds with nonnegative
Ricci curvature and Euclidean volume growth

Gioacchino Antonelli  Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126
Pisa, Italy<EMAIL_ADDRESS>,  Elia Bruè  Institute for Advanced
Study, 1 Einstein Dr., Princeton NJ 05840, U.S.A<EMAIL_ADDRESS>,
Mattia Fogagnolo  Centro di Ricerca Matematica Ennio De Giorgi, Scuola Normale
Superiore, Piazza dei Cavalieri, 3, 56126 Pisa, Italy.
<EMAIL_ADDRESS>and  Marco Pozzetta  Dipartimento di Matematica e
Applicazioni, Università di Napoli Federico II, Via Cintia, Monte S. Angelo,
80126 Napoli, Italy<EMAIL_ADDRESS>
###### Abstract.

In this paper we provide new existence results for isoperimetric sets of large
volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean
volume growth. We find sufficient conditions for their existence in terms of
the geometry at infinity of the manifold.

As a byproduct we show that isoperimetric sets of big volume always exist on
manifolds with nonnegative sectional curvature and Euclidean volume growth.

Our method combines an asymptotic mass decomposition result for minimizing
sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the
concavity property of the isoperimetric profile. The latter is new in the
generality of noncollapsed manifolds with Ricci curvature bounded below.

###### Key words and phrases:

Isoperimetric problem, existence of isoperimetric regions, nonnegative Ricci
curvature, Euclidean volume growth.

###### 1991 Mathematics Subject Classification:

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

###### Contents

  1. 1 Introduction
    1. 1.1 Main existence results
    2. 1.2 Concavity of the isoperimetric profile
    3. 1.3 Strategy of proof of the existence results
    4. 1.4 Sharpness and counterexamples
    5. 1.5 Plan of the paper
  2. 2 Definitions and auxiliary results
    1. 2.1 Preliminaries on Riemannian manifolds and metric measure spaces
    2. 2.2 $\mathsf{RCD}$ spaces
    3. 2.3 Finite perimeter sets and minimizing sequences for the isoperimetric problem
    4. 2.4 The collapsed case
    5. 2.5 Sharp Sobolev inequality on $\mathsf{CD}(0,n)$ spaces
  3. 3 Concavity properties of the isoperimetric profile
    1. 3.1 First and second variations of the penalized perimeter
    2. 3.2 Proof of 1.4
    3. 3.3 Consequences of the concavity properties of the isoperimetric profile
  4. 4 Main isoperimetric existence results
    1. 4.1 Proof of 1.1
    2. 4.2 Proof of 1.2
    3. 4.3 Further existence results
    4. 4.4 Proof of 1.3

##  1\. Introduction

Given a Riemannian manifold $(M^{n},g)$ of dimension $n\geq 2$ we denote by
$\mathsf{d}$, $\operatorname{vol}$, and $\mathrm{Ric}$ the Riemannian
distance, the volume measure, and the Ricci tensor of $(M^{n},g)$,
respectively.

For any $V\in[0,\operatorname{vol}(M^{n}))$, the _isoperimetric problem_
consists in studying the minimization problem

(1.1) $I_{(M^{n},g)}(V):=\inf\\{P(\Omega):\text{$\Omega\subset M^{n}$ set of
finite perimeter with $\operatorname{vol}(\Omega)=V$}\\}\,,$

where $P(\Omega)$ denotes the perimeter of $\Omega$, see Section 2. We shall
drop the superscript $n$ in $(M^{n},g)$ and the subscript $(M^{n},g)$ in
$I_{(M^{n},g)}$ when there is no risk of confusion. The function
$I_{(M^{n},g)}$ is the so-called isoperimetric profile of $(M^{n},g)$. Any set
of finite perimeter $E\subset M^{n}$ with $\operatorname{vol}(E)=V$ and
$P(E)=I_{(M^{n},g)}(E)$ is called isoperimetric set or isoperimetric region.

The problem of the existence of isoperimetric regions in the setting of
noncompact manifolds is a hard problem that has seen several important
progresses in the last years. Let us just mention some of the most important
contributions in the field related to the topics of this work. Major results
in the application of a direct method for proving existence of isoperimetric
sets in manifolds with lower bounds on the Ricci curvature are contained [61,
56]. The methods employed there have been generalized in [13]. As we are going
to see, such generalization will be fundamental for the present work. In
dimension $2$, a complete positive answer to the existence issue of
isoperimetric sets under nonnegative curvature has been given in [64]. The
existence of isoperimetric sets in $3$-manifolds with nonnegative scalar
curvature and asymptotically flat asymptotics has been established in [23].
Existence results for isoperimetric sets of large volumes in asymptotically
flat manifolds were also obtained in [37, 38], in asymptotically hyperbolic
spaces in [29], and in the asymptotically conical case in [30]. When the
ambient space is a nonnegatively Ricci curved cone, isoperimetric regions
exist for any given volume and are actually characterized [59]. In Euclidean
solid cones, the problem has been investigated in [66]. In more general convex
bodies, it is treated in [50].

Differential properties of the isoperimetric profile, some of which are
generalized in the present paper, have been studied in [16, 58, 17, 18, 19]. A
regularity theory for isoperimetric sets was established for example in [71,
45, 73, 57], and, recently, it has been partly generalized in [14].

###  1.1. Main existence results

In this paper we provide new existence results for isoperimetric regions of
large volume in the setting of manifolds with nonnegative Ricci curvature and
Euclidean volume growth, i.e., such that

$\mathrm{AVR}(M^{n},g):=\lim_{r\to+\infty}\frac{\operatorname{vol}(B_{r}(x))}{\omega_{n}r^{n}}\in(0,1]\,,$

where $\omega_{n}$ is the volume of the $n$-dimensional unit ball. The above
limit exists as a consequence of the Bishop–Gromov monotonicity (cf. 2.11).
The same notion can be given for $\mathsf{CD}(0,n)$ metric measure spaces
$(X,\mathsf{d},\mathfrak{m})$ (replacing $\operatorname{vol}$ with
$\mathfrak{m}$), a class of nonsmooth spaces with a synthetic notion of
nonnegative Ricci curvature and dimension bounded above by $n>0$, see Section
2.2.

The Euclidean volume growth assumption implies that $(M^{n},g)$ is
_noncollapsed_ , i.e. there exists $v>0$ such that
$\operatorname{vol}(B_{1}(x))\geq v$ for every $x\in M$. The latter can be
shown to be equivalent to $I_{(M^{n},g)}(V)\neq 0$ for some (and in fact for
any) $V>0$ (cf. 2.18). Hence noncollapsedness is necessary for the existence
of isoperimetric regions (see also the counterexamples discussed in [13,
Section 4.3]).

Before stating our first result we recall that $(X,\mathsf{d},\mathfrak{m})$
is said to _split (a line)_ if it is isometric as a metric measure space to
$(\mathbb{R}\times
Y,\mathsf{d}_{\mathbb{R}}\otimes\mathsf{d}_{Y},\mathfrak{m}_{\mathbb{R}}\otimes\mathfrak{m}_{Y})$
for some $(Y,\mathsf{d}_{Y},\mathfrak{m}_{Y})$. Here
$\mathsf{d}_{\mathbb{R}^{k}},\mathfrak{m}_{\mathbb{R}^{k}}$ are just the
Euclidean distance and the Lebesgue measure on $\mathbb{R}^{k}$, respectively,
for any $k\in\mathbb{N}$.

######  Theorem 1.1.

Let $(M^{n},g)$ be a complete noncompact noncollapsed Riemannian manifold with
$\mathrm{Ric}\geq 0$. We write

$(M^{n},g)=(\mathbb{R}^{k}\times N^{n-k},g_{\mathbb{R}^{k}}+g_{N})\,,$

where $(N^{n-k},g_{N})$ does not split a line, and $0\leq k\leq n$. Assume
that there exists $\varepsilon>0$ such that every pmGH limit
$(X_{\infty},\mathsf{d}_{\infty},\mathfrak{m}_{\infty},x_{\infty})$ of
sequences $\\{(M,\mathsf{d},\operatorname{vol},p_{i})\\}_{i\in\mathbb{N}_{\geq
0}}$ with $p_{i}:=(0,x_{i})\in M$, and $\mathsf{d}_{N}(x_{i},x_{0})\to+\infty$
as $i\to+\infty$, satisfies

$\mathrm{AVR}(X_{\infty},\mathsf{d}_{\infty},\mathfrak{m}_{\infty})\geq\mathrm{AVR}(M^{n},g)+\varepsilon.$

Then there exists $V_{0}>0$ such that for every $V\geq V_{0}$ there exists an
isoperimetric region of volume $V$ in $M$.

Recall that any sequence
$\\{(M,\mathsf{d},\operatorname{vol},p_{i})\\}_{i\in\mathbb{N}}$ admits limit
points in the pmGH topology as a consequence of the Gromov compactness theorem
(see 2.4 and Remark 2.7).

The hypothesis in 1.1 is verified when $\mathrm{AVR}(M^{n},g)>0$ and the
manifold $N$ possesses no asymptotic cones that split a line (see 2.14 for the
notion of asymptotic cone). Hence we deduce the following.

######  Theorem 1.2.

Let $(M^{n},g)$ be a complete Riemannian manifold with $\mathrm{Ric}\geq 0$
and $\mathrm{AVR}(M^{n},g)>0$. We write

$(M^{n},g)=(\mathbb{R}^{k}\times N^{n-k},g_{\mathbb{R}^{k}}+g_{N})\,,$

where $(N^{n-k},g_{N})$ does not split a line, and $0\leq k\leq n$. If no
asymptotic cone of $N$ splits a line, then there is $V_{0}>0$ such that for
every $V\geq V_{0}$ there exists an isoperimetric region of volume $V$ in $M$.

It is well-known that when $(M^{n},g)$ has nonnegative sectional curvature the
asymptotic cone is unique and it splits if and only if the manifold splits.
Since we were not able to find an explicit proof of this fact in the
literature, we included one in 4.6. This phenomenon marks an important
difference with the nonnegative Ricci curvature, see [32, Theorem 1.4] and
Section 1.4 below. In turn, we derive the following existence theorem.

######  Corollary 1.3.

Let $(M^{n},g)$ be a complete Riemannian manifold with nonnegative sectional
curvature and Euclidean volume growth. Then there exists $V_{0}>0$ such that
for every $V\geq V_{0}$ there exists an isoperimetric region of volume $V$.

###  1.2. Concavity of the isoperimetric profile

We now state our last result regarding concavity properties of the
isoperimetric profile. This will be an important ingredient in the proof of
the previous existence theorems and has an independent interest.

######  Theorem 1.4.

Suppose that $(M^{n},g)$ is noncollapsed and that $\mathrm{Ric}\geq(n-1)K$ for
$K\leq 0$. Then for any $\widetilde{V}>0$ there is $\delta>0$ and $C\geq 0$
such that the function $V\mapsto I_{(M^{n},g)}(V)-CV^{2}$ is concave on
$(\widetilde{V}-\delta,\widetilde{V}+\delta)$. Moreover, $I_{(M^{n},g)}$ is
twice differentiable almost everywhere and at any point $V$ of twice
differentiability it holds the estimate

(1.2) $I_{(M^{n},g)}^{\prime\prime}(V)\leq\frac{(n-1)|K|}{I_{(M^{n},g)}(V)}.$

If $K=0$ then $I_{(M^{n},g)}$ is concave on $(0,+\infty)$.

The previous result is obtained under the natural assumptions for the study of
the isoperimetric problem, namely noncollapsedness and Ricci bounded below. In
particular, differently from the classical results in [16, 58, 17, 18, 19,
56], the existence of isoperimetric regions is not assumed _a priori_ , and,
in fact, we will employ 1.4 as a tool to provide existence. In order to prove
1.4, we introduce an approximation of the perimeter functional on the manifold
$M$ by a sequence of penalized perimeters $P_{k}$ given by the sum of the
usual perimeter and a potential term. The penalization in the definition of
$P_{k}$, see (3.1), implies the existence of volume constrained minimizers for
$P_{k}$ for any $k$. Also, the profile $I_{k}$ corresponding to $P_{k}$, see
(3.2), converges to the isoperimetric profile $I_{(M^{n},g)}$ of the manifold
locally uniformly. Hence 1.4 eventually follows by studying the concavity
properties of $I_{k}$ by means of the existence of minimizers for $P_{k}$, and
then by passing the estimates to the limit with respect to $k$.

As a consequence of 1.4, we get that under the same assumptions the
isoperimetric profile is a locally Lipschitz function (3.5), which improves
the result of local Hölder continuity contained in [60].

As anticipated, 1.4 has useful consequences for the the existence of
isoperimetric regions in manifolds with nonnegative Ricci curvature. This is
due to the fact that the concavity of the profile provides information on the
asymptotic behavior of $I_{(M^{n},g)}$ and $I^{\prime}_{(M^{n},g)}$, see 3.6.
Under the additional assumption of Euclidean volume growth, we can further
conclude that $I_{(M^{n},g)}$ is strictly increasing (3.8). The latter result
has to be compared with [65], where the author considers hypotheses on the
sectional curvature instead, but without assumptions on the asymptotic volume
ratio.

###  1.3. Strategy of proof of the existence results

Broadly speaking the proof of 1.1 is based on the direct method of the
Calculus of Variations and a compensated compactness argument. Given $V>0$ we
consider a minimizing sequence $E_{i}$ such that $\operatorname{vol}(E_{i})=V$
and $\lim_{i\to\infty}P(E_{i})=I(V)$, and we aim at studying limits in a
generalized sense. Notice that the existence of a limit in
$L^{1}(M^{n},\operatorname{vol})$ would immediately give the existence of
isoperimetric regions of volume $V$, as a consequence of the lower
semicontinuity of the perimeter. The main enemy for the existence of the limit
is the possibility of having pieces of $E_{i}$ escaping at infinity.

The asymptotic decomposition result in [13] (see 2.16 below) tells us that,
for noncollapsed Riemannian manifolds with Ricci curvature bounded from below,
the mass lost at infinity can be recovered by looking at subsets of metric
measure spaces obtained as pmGH limit points of
$(M^{n},\mathsf{d},\operatorname{vol},p_{i})$, where $p_{i}\to\infty$ is a
suitable sequence. The hypotheses in 1.1 are meant to control the
isoperimetric profile of these limit spaces, with the aim of showing that
escaping at infinity is not “isoperimetrically convenient”. To this aim we
employ the recent result [15, Theorem 1.1], showing that the isoperimetric
constant on $\mathsf{CD}(0,n)$ spaces is, roughly speaking, realized by balls
of infinite radius, and thus that it is explicitly related to the asymptotic
volume ratio. This result generalizes earlier analogous inequalities in the
smooth setting [1, 20, 40, 47]. In [15], the isoperimetric inequality is
actually obtained as an a priori weaker estimate on the Minkowksi content of
the sets involved; in this paper, we give a refinement of that in terms of the
perimeter, see 2.21. Notice that the pmGH limits at infinity of a noncollpased
nonnegatively Ricci curved manifold are indeed $\mathsf{CD}(0,n)$ spaces. Our
assumptions in 1.1, in particular, imply that for large enough volumes it is
more convenient for the minimizing sequence not to lose mass at infinity along
$N$. On the other hand, if part of the mass of any minimizing sequence is lost
at infinity along the factor $\mathbb{R}^{k}$, it can be taken back by means
of translations along the Euclidean factor.

To perform such a plan we need to know the exact asymptotic behaviour for
large volumes of the isoperimetric profile $I$ and its derivative
$I^{\prime}$, for an arbitrary manifold with nonnegative Ricci curvature. The
latter piece of information is used to ensure that every minimizing sequence
of sets of sufficiently big volumes can lose at most one piece at infinity
along $N$, see Lemma 4.1.

Loosely speaking, the behaviour at infinity of $I$ is the same as the
isoperimetric profile of a cone with opening $\mathrm{AVR}(M^{n},g)$. The
computation of the asymptotics of $I$ is a direct consequence of Bishop–Gromov
comparison theorem and the sharp isoperimetric inequality, while the
computation of the asymptotics of $I^{\prime}$ is a byproduct of the concavity
property proven in 1.4.

We finally stress that the extremal case in the hypothesis in 1.1, i.e., when
$\mathrm{AVR}(X_{\infty})=1$, happens when every limit $X_{\infty}$ is
isometric to $\mathbb{R}^{n}$. In this simpler case, since the mass of a
minimizing sequence lost along the factor $\mathbb{R}^{k}$ can be taken back
as mentioned above, by a minor variation of the proof of [13, Theorem 5.2] one
even gets existence of isoperimetric regions for every volume.

Let us comment on the proof of 1.2. We show that, under the hypotheses of 1.2,
the hypotheses of 1.1 are satisfied. This is done in two steps, which
correspond to item (i) and (ii) of Lemma 4.2. First, if no asymptotic cone of
$N$ splits a line, then the density of every point at distance $1$ from the
line of the tips of every asymptotic cone of $M$ is uniformly bigger than
$\mathrm{AVR}(M^{n},g)$. The latter is due to a compactness argument and to
the fact that if a point at distance $1$ from the line of the tips of an
asymptotic cone $\mathbb{R}^{k}\times C$ of $M$ has density equal to
$\mathrm{AVR}(M^{n},g)$, then there is a line in $\mathbb{R}^{k}\times C$
which is not contained in $\mathbb{R}^{k}$ and then, by the splitting theorem,
one has a splitting of the cone $C$, which is an asymptotic cone of $N$,
resulting in a contradiction. Second, by means of the volume convergence
theorem, a lower bound on the density of points at distance $1$ from the line
of the tips of the asymptotic cones is readily seen to imply a lower bound on
the $\mathrm{AVR}$ of the pmGH limits at infinity of the manifold along $N$.

The result in item (i) of Lemma 4.2 discussed above allows to get also other
nontrivial existence results for the isoperimetric problem. It can be proved,
see 4.3, that for manifolds as in 1.2, if every point at distance $1$ from the
line of the tips of every asymptotic cone is regular, i.e., has density one,
then isoperimetric regions exist for every volume. In particular this implies,
see 4.4, that if a Riemannian manifold with nonnegative Ricci curvature and
Euclidean volume growth is such that every asymptotic cone has a smooth cross
section, then isoperimetric regions exist for every volume. We point out that
when the dimension is $2$ this partially recovers the aforementioned existence
result of isoperimetric sets in nonnegative curvature due to Ritoré [64].

###  1.4. Sharpness and counterexamples

While the assumptions in 1.1 turn out to be always satisfied in the setting of
manifolds with nonnegative sectional curvature, giving rise to 1.3, the
following example borrowed from [48, pp. 913-914] shows that this is not
always the case on manifolds with nonnegative Ricci curvature. Let us consider
the metric

$g:=f(r)^{2}dt^{2}+dr^{2}+\eta(r)^{2}g_{\mathbb{S}^{k-1}}\quad\text{for
$(t,r,p)\in\mathbb{R}\times[0,+\infty)\times\mathbb{S}^{k-1}\simeq\mathbb{R}^{k+1}$,
$k\geq 3$}\,,$

where

$f(r)=(b+r^{2})^{(\beta+2-k)/2}+c\,\quad\alpha,\beta\in(0,1),\,b,c>1\,,\,k-\beta>2+\alpha\,,$
$\eta(r)=\frac{1}{2}r+\frac{1}{2a}\int_{0}^{r}\int_{t}^{\infty}\xi(s)ds\,,$

for some smooth $\xi:[0,\infty)\to[0,\infty)$ satisfying $\xi(t)=t$ on
$[0,1]$, $\xi>0$ on $[1,2]$ and $\xi(t)=t^{-1-\alpha}$ on $[2,\infty)$. The
parameter $a>0$ is chosen such that $\eta^{\prime}(0)=1$.

It is possible to prove that $g$ has nonnegative Ricci curvature provided $b$
and $c$ are chosen big enough. It is easy to see that $(\mathbb{R}^{k+1},g)$
has Euclidean volume growth and admits a unique asymptotic cone isometric to
$\mathbb{R}\times C(\mathbb{S}^{k-1}_{\rho})$, where $\rho=1/2$ is the radius
of $\mathbb{S}^{k-1}_{\rho}$, which is the cross section of the cone
$C(\mathbb{S}^{k-1}_{\rho})$. Observe that the asymptotic cone splits a line
while $(\mathbb{R}^{k+1},g)$ does not. Moreover, $g$ is invariant under the
translation along the direction $\partial_{t}$. In particular, if
$P_{i}=(t_{i},r_{i},p_{i})$ satisfies $\sup_{i}r_{i}<+\infty$ and
$|t_{i}|\to\infty$, we deduce that
$(\mathbb{R}^{k+1},\mathsf{d}_{g},\operatorname{vol}_{g},P_{i})\to(\mathbb{R}^{k+1},\mathsf{d}_{g},\operatorname{vol}_{g},0)$
in the pmGH topology. Hence the assumptions of 1.1 are _not_ satisfied.

On the other hand, along sequences $Q_{i}=(t_{i},r_{i},p_{i})$ with
$r_{i}\to\infty$, one checks that
$(\mathbb{R}^{k+1},\mathsf{d}_{g},\operatorname{vol}_{g},Q_{i})$ converges to
the Euclidean space $(\mathbb{R}^{k+1},\mathsf{d}_{\rm
eu},\mathcal{L}^{k+1},0)$ in the pmGH topology. Therefore, by the asymptotic
mass decomposition 2.16, one has that the volume of a minimizing sequence lost
at infinity ends up in isoperimetric regions in $(\mathbb{R}^{k+1},g_{\rm
eu})$ or in $(\mathbb{R}^{k+1},g)$. However, the presence of such limit
regions in $(\mathbb{R}^{k+1},g_{\rm eu})$ easily contradicts the minimizing
property of the initial sequence (cf. [13, Theorem 5.1 & 5.2] and [58, Theorem
3.5]), while limit isoperimetric regions in $(\mathbb{R}^{k+1},g)$ can be
obviously brought back into the original $(\mathbb{R}^{k+1},g)$, recovering
the missing volume. Therefore we obtain that $(\mathbb{R}^{k+1},g)$ has
isoperimetric regions for _any_ volume.

All in all, we conclude that our assumptions in 1.1 are not necessary for the
existence of isoperimetric sets in the setting of manifolds with nonnegative
Ricci curvature.

To date we do not know whether manifolds with nonnegative Ricci curvature and
Euclidean volume growth always admit isoperimetric regions for big volumes,
and we plan to investigate further this problem in the future.

###  1.5. Plan of the paper

In Section 2 we introduce the preliminary material. We briefly describe the
class of $\mathsf{CD}$ and $\mathsf{RCD}$ spaces, we recall the notion of pmGH
convergence and the mass decomposition theorem (2.16), and we prove the
mentioned sharp Sobolev inequality (2.21). Section 3 is devoted to the proof
of 1.4. In Section 4 we prove the existence results 1.1, 1.2, 1.3, and further
existence results (Section 4.3).

### Acknowledgments

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 Giorgio and Elena Petronio Fellowship at the
Institute for Advanced Study. He is grateful to Stefano Nardulli for having
introduced him to the isoperimetric problem on noncompact manifolds. The
authors also thank Gian Paolo Leonardi, Aaron Naber, and Manuel Ritoré for
fruitful discussions and comments.

##  2\. Definitions and auxiliary results

###  2.1. Preliminaries on Riemannian manifolds and metric measure spaces

In this preliminary section we introduce basic facts and notations about
Riemannian manifolds and metric measure spaces.

Given two Riemannian manifolds $(M^{m},g),(N^{n},h)$, we denote with
$(M^{m}\times N^{n},g+h)$ the product of the two Riemannian structure. For the
notions of BV and Sobolev spaces on Riemannian manifolds we refer the reader
to [54, Section 1]. For every finite perimeter set $E$ in $\Omega$ we denote
with $P(E,\Omega)$ the perimeter of $E$ inside $\Omega$. When $\Omega=M^{n}$
we simply write $P(E)$. We denote with $\mathcal{H}^{n-1}$ the
$(n-1)$-dimensional Hausdorff measure on $M^{n}$ relative to the distance
induced by $g$. We recall that for every finite perimeter set $E$ one has
$P(E)=\mathcal{H}^{n-1}(\partial^{*}E)$ and the characteristic function
$\chi_{E}$ belongs to $BV_{\rm loc}(M^{n},\operatorname{vol})$ with
generalized gradient $D\chi_{E}=\nu\mathcal{H}^{n-1}\llcorner\partial^{*}E$
for a function $\nu:M\to TM^{n}$ with $|\nu|=1$ at $|D\chi_{E}|$-a.e. point,
where $\partial^{*}E$ is the essential boundary of $E$.

We recall the following terminology.

######  Definition 2.1 (Convergence of finite perimeter sets).

Let $(M^{n},g)$ be a Riemannian manifold. We say that a sequence of measurable
(with respect to the volume measure) sets $E_{i}$ _locally converges_ to a
measurable set $E$ if the characteristic functions $\chi_{E_{i}}$ converge to
$\chi_{E}$ in $L^{1}_{\rm loc}(M^{n},g)$. In such a case we simply write that
$E_{i}\to E$ locally on $M^{n}$.

If the sets $E_{i}$ have also locally finite perimeter, that is,
$P(E_{i},\Omega)<+\infty$ for any $i$ and any bounded open set $\Omega$, we
say that $E_{i}\to E$ _in the sense of finite perimeter sets_ if $E_{i}\to E$
locally on $M^{n}$ and the sequence of measures $D\chi_{E_{i}}$ locally
weakly* converges as measures, that is, with respect to the duality with
compactly supported continuous functions. In such a case, $E$ has locally
finite perimeter and the weak* limit of $D\chi_{E_{i}}$ is $D\chi_{E}$.

######  Remark 2.2 (Approximation of finite perimeter sets with smooth sets).

It can be proved, see [60, Lemma 2.3], that when $M^{n}$ is a complete
Riemannian manifold every finite perimeter set $\Omega$ with
$0<\operatorname{vol}(\Omega)<+\infty$ and $\operatorname{vol}(\Omega^{c})>0$
is approximated by relatively compact sets $\Omega_{i}$ in $M^{n}$ with smooth
boundary such that $\operatorname{vol}(\Omega_{i})=\operatorname{vol}(\Omega)$
for every $i\in\mathbb{N}$, $\operatorname{vol}(\Omega_{i}\Delta\Omega)\to 0$
when $i\to+\infty$, and $P(\Omega_{i})\to P(\Omega)$ when $i\to+\infty$. Thus,
by approximation, one can deduce that

$I(V)=\inf\\{\mathcal{H}^{n-1}(\partial\Omega):\text{$\Omega\Subset M^{n}$ has
smooth boundary, $\operatorname{vol}(\Omega)=V$}\\},$

see [60, Theorem 1.1].

We also need to recall the definition of the simply connected radial models
with constant sectional curvature.

######  Definition 2.3 (Models of constant sectional curvature, cf. [62,
Example 1.4.6]).

Let us define

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

If $K>0$, then
$((0,\pi/\sqrt{K}]\times\mathbb{S}^{n-1},\mathrm{d}r^{2}+\mathrm{sn}_{K}^{2}(r)g_{1})$,
where $g_{1}$ is the canonical metric on $\mathbb{S}^{n-1}$, is the radial
model of dimension $n$ and constant sectional curvature $K$. The metric can be
smoothly extended at $r=0$, and thus we shall write that the the metric is
defined on the ball $\mathbb{B}^{n}_{\pi/\sqrt{K}}\subset\mathbb{R}^{n}$. The
Riemannian manifold
$(\mathbb{B}^{n}_{\pi/\sqrt{K}},g_{K}\vcentcolon=\mathrm{d}r^{2}+\mathrm{sn}_{K}^{2}(r)g_{1})$
is the unique (up to isometry) simply connected Riemannian manifold of
dimension $n$ and constant sectional curvature $K>0$.

If instead $K\leq 0$, then
$((0,+\infty)\times\mathbb{S}^{n-1},\mathrm{d}r^{2}+\mathrm{sn}_{K}^{2}(r)g_{1})$
is the radial model of dimension $n$ and constant sectional curvature $K$.
Extending the metric at $r=0$ analogously yields the unique (up to isometry)
simply connected Riemannian manifold of dimension $n$ and constant sectional
curvature $K\leq 0$, in this case denoted by $(\mathbb{R}^{n},g_{K})$.

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 $K$ of dimension
$n$, and by $s(n,K,r)$ the volume 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
$\mathbb{R}^{n}$.

On $\mathbb{R}^{n}$ we denote with $\mathsf{d}_{\mathbb{R}^{n}}$,
$g_{\mathbb{R}^{n}}$, and $\mathfrak{m}_{\mathbb{R}^{n}}$, respectively, the
Euclidean distance, the Euclidean metric, and the Lebesgue measure.

Let us now briefly recall the main concepts we will need from the theory of
metric measure spaces. We recall that a metric measure space,
$\mathrm{m.m.s.}$ for short, $(X,\mathsf{d}_{X},\mathfrak{m}_{X})$ is a triple
where $(X,\mathsf{d}_{X})$ is a locally compact separable metric space and
$\mathfrak{m}_{X}$ is a Borel measure bounded on bounded sets. A pointed
metric measure space is a quadruple $(X,\mathsf{d}_{X},\mathfrak{m}_{X},x)$
where $(X,\mathsf{d}_{X},\mathfrak{m}_{X})$ is a metric measure space and
$x\in X$ is a point.

For simplicity, and since it will always be our case, we will always assume
that given $(X,\mathsf{d}_{X},\mathfrak{m}_{X})$ a m.m.s. the support ${\rm
spt}\,\mathfrak{m}_{X}$ of the measure $\mathfrak{m}_{X}$ is the whole $X$.

Given two m.m.s. $(X,\mathsf{d}_{X},\mathfrak{m}_{X})$ and
$(Y,\mathsf{d}_{Y},\mathfrak{m}_{Y})$, we denote by $(X\times
Y,\mathsf{d}_{X}\otimes\mathsf{d}_{Y},\mathfrak{m}_{X}\otimes\mathfrak{m}_{Y})$
the product m.m.s., where

$\mathsf{d}_{X}\otimes\mathsf{d}_{Y}((x,y),(x^{\prime},y^{\prime})):=\sqrt{\mathsf{d}_{X}(x,x^{\prime})^{2}+\mathsf{d}_{Y}(y,y^{\prime})^{2}},\qquad\forall
x,x^{\prime}\in X,\quad\forall y,y^{\prime}\in Y,$

and $\mathfrak{m}_{X}\otimes\mathfrak{m}_{Y}$ is the usual product of
measures.

We assume the reader to be familiar with the notion of pointed measured
Gromov–Hausdorff convergence, referring to [72, Chapter 27] and to [22,
Chapter 7 and 8] for an overview on the subject. In the following treatment we
introduce the pmGH-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 [44, Theorem
3.15 and Section 3.5]. The following definition is taken from the introductory
exposition of [4].

######  Definition 2.4 (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 $[A]_{\varepsilon}\vcentcolon=\\{z\in Z\ :\
\mathsf{d}_{Z}(z,A)\leq\varepsilon\\}$ 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 measure
Gromov–Hausdorff topology, or in the $\mathrm{pmGH}$ sense for short_.

We remark that in the setting we will deal with, product structures are stable
under pmGH convergence, i.e., if
$(X_{n},\mathsf{d}_{X_{n}},\mathfrak{m}_{X_{n}},x_{n})\to(X,\mathsf{d}_{X},\mathfrak{m}_{X},x)$
and
$(Y_{n},\mathsf{d}_{Y_{n}},\mathfrak{m}_{Y_{n}},y_{n})\to(Y,\mathsf{d}_{Y},\mathfrak{m}_{Y},y)$
in pmGH then

$(X_{n}\times
Y_{n},\mathsf{d}_{X_{n}}\otimes\mathsf{d}_{Y_{n}},\mathfrak{m}_{X_{n}}\otimes\mathfrak{m}_{Y_{n}},(x_{n},y_{n}))\to(X\times
Y,\mathsf{d}_{X}\otimes\mathsf{d}_{Y},\mathfrak{m}_{X}\otimes\mathfrak{m}_{Y},(x,y)),$

in pmGH.

###  2.2. $\mathsf{RCD}$ spaces

Let us briefly introduce the so-called $\mathsf{RCD}$ condition for m.m.s.,
and discuss some basic and useful properties of it. Since we will use part of
the $\mathsf{RCD}$ theory just as an instrument for our purposes and since we
will never use in the paper the specific definition of $\mathsf{RCD}$ space,
we just outline the main references on the subject and we refer the interested
reader to the survey of Ambrosio [3] and the references therein.

After the introduction, in the independent works [69, 70] and [51], of the
curvature dimension condition $\mathsf{CD}(K,n)$ encoding in a synthetic way
the notion of Ricci curvature bounded from below by $K$ and dimension bounded
above by $n$, the definition of $\mathsf{RCD}(K,n)$ m.m.s. was first proposed
in [43] and then studied in [42, 39, 11], see also [24] for the equivalence
between the $\mathsf{RCD}^{*}(K,n)$ and the $\mathsf{RCD}(K,n)$ condition. The
infinite dimensional counterpart of this notion had been previously
investigated in [9], see also [8] for the case of $\sigma$-finite reference
measures.

######  Remark 2.5 (pmGH limit of $\mathsf{RCD}$ spaces).

We recall that, whenever it exists, a pmGH limit of a sequence
$\\{(X_{i},\mathsf{d}_{i},\mathfrak{m}_{i},x_{i})\\}_{i\in\mathbb{N}}$ of
(pointed) $\mathsf{RCD}(K,n)$ spaces is still an $\mathsf{RCD}(K,n)$ metric
measure space.

Due to the compatibility of the $\mathsf{RCD}$ condition with the smooth case
of Riemannian manifolds with Ricci curvature bounded from below and to its
stability with respect to pointed measured Gromov–Hausdorff convergence,
limits of smooth Riemannian manifolds with Ricci curvature uniformly bounded
from below by $K$ and dimension uniformly bounded from above by $n$ are
$\mathsf{RCD}(K,n)$ spaces. Then the class of $\mathsf{RCD}$ spaces includes
the class of Ricci limit spaces, i.e., limits of sequences of Riemannian
manifolds with the same dimension and with Ricci curvature uniformly bounded
from below [25, 26, 27, 28]. An extension of noncollapsed Ricci limit spaces
is the class of $\mathsf{RCD}(K,n)$ space where the reference measure is the
$n$-dimensional Hausdorff measure relative to the distance, introduced and
studied in [49, 35, 12]. As a consequence of the rectifiability of
$\mathsf{RCD}$ spaces [55, 21], we remark that if
$(X,\mathsf{d},\mathcal{H}^{n})$ is an $\mathsf{RCD}(K,n)$ space then $n$ is
an integer.

We state the volume convergence theorems obtained by Gigli and De Philippis in
[35, Theorem 1.2 and Theorem 1.3], which are the synthetic version of the
celebrated volume convergence of Colding [31].

######  Theorem 2.6.

Let $\\{(X_{i},\mathsf{d}_{i},\mathcal{H}^{n},x_{i})\\}_{i\in\mathbb{N}}$ be a
sequence of pointed $\mathsf{RCD}(K,n)$ m.m.s. with $K\in\mathbb{R}$ and
$n\in[1,+\infty)$. Assume that $(X_{i},\mathsf{d}_{i},x_{i})$ converges in the
pGH topology to $(X,\mathsf{d},x)$. Then precisely one of the following
happens

  * (a)

$\limsup_{i\to\infty}\mathcal{H}^{n}\left(B_{1}(x_{i})\right)>0$. Then the
$\limsup$ is a limit and $(X_{i},\mathsf{d}_{i},\mathcal{H}^{n},x_{i})$
converges in the pmGH topology to $(X,\mathsf{d},\mathcal{H}^{n},x)$. Hence
$(X,\mathsf{d},\mathcal{H}^{n})$ is an $\mathsf{RCD}(K,n)$ m.m.s. endowed with
the $n$-dimensional Hausdorff measure;

  * (b)

$\lim_{i\to\infty}\mathcal{H}^{n}(B_{1}(x_{i}))=0$. In this case we have
$\dim_{H}(X,\mathsf{d})\leq n-1$, where we denoted by $\dim_{H}(X,\mathsf{d})$
the Hausdorff dimension of $(X,\mathsf{d})$.

Moreover, for $K\in\mathbb{R}$ and $n\in[1,+\infty)$, let $\mathbb{B}_{K,n,R}$
be the collection of all equivalence classes up to isometry of closed balls of
radius $R$ in $\mathsf{RCD}(K,n)$ spaces, equipped with the Gromov-Hausdorff
distance. Then the map $\mathbb{B}_{K,n,R}\ni Z\to\mathcal{H}^{n}(Z)$ is real-
valued and continuous.

######  Remark 2.7 (Gromov precompatness theorem for $\mathsf{RCD}$ spaces).

Here we recall the synthetic variant of Gromov’s precompactness theorem for
$\mathsf{RCD}$ spaces, see [35, Equation (2.1)]. Let
$\\{(X_{i},\mathsf{d}_{i},\mathfrak{m}_{i},x_{i})\\}_{i\in\mathbb{N}}$ be a
sequence of $\mathsf{RCD}(K_{i},n)$ spaces with $n\in[1,+\infty)$,
$\operatorname{spt}(\mathfrak{m}_{i})=X_{i}$ for every $i\in\mathbb{N}$,
$\mathfrak{m}_{i}(B_{1}(x_{i}))\in[v,v^{-1}]$ for some $v\in(0,1)$ and for
every $i\in\mathbb{N}$, and $K_{i}\to K\in\mathbb{R}$. Then there exists a
subsequence pmGH-converging to some $\mathsf{RCD}(K,n)$ space
$(X,\mathsf{d},\mathfrak{m},x)$ with $\operatorname{spt}(\mathfrak{m})=X$.

We conclude this part by recalling a few basic definitions and results
concerning the perimeter functional in the setting of metric measure spaces
(see [2, 53, 6]).

######  Definition 2.8 ($BV$ functions and perimeter on m.m.s.).

Let $(X,\mathsf{d},\mathfrak{m})$ be a metric measure space. A function $f\in
L^{1}(X,\mathfrak{m})$ is said to belong to the space of _bounded variation
functions_ $BV(X,\mathsf{d},\mathfrak{m})$ if there is a sequence
$f_{i}\in{\rm Lip}_{\mathrm{loc}}(X)$ such that $f_{i}\to f$ in
$L^{1}(X,\mathfrak{m})$ and $\limsup_{i}\int_{X}{\rm
lip\,}f_{i}\,\mathrm{d}\mathfrak{m}<+\infty$, where ${\rm
lip\,}u(x)\vcentcolon=\limsup_{y\to x}\frac{|u(y)-u(x)|}{\mathsf{d}(x,y)}$ is
the _slope_ of $u$ at $x$, for any accumulation point $x\in X$, and ${\rm
lip\,}u(x):=0$ if $x\in X$ is isolated. In such a case 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}_{\rm
loc}(A),f_{i}\to f$ in $L^{1}(A,\mathfrak{m})$}\right\\},$

for any open set $A\subset X$.

If $E\subset X$ is a Borel set and $A\subset X$ is open, we define the
_perimeter $P(E,A)$ of $E$ in $A$_ by

$P(E,A)\vcentcolon=\inf\left\\{\liminf_{i}\int_{A}{\rm
lip\,}u_{i}\,\mathrm{d}\mathfrak{m}\ :\ \text{$u_{i}\in{\rm Lip}_{\rm
loc}(A),u_{i}\to\chi_{E}$ in $L^{1}_{\rm loc}(A,\mathfrak{m})$}\right\\},$

We say that $E$ has _finite perimeter_ if $P(E,X)<+\infty$, and we denote by
$P(E)\vcentcolon=P(E,X)$. Let us remark that the set functions
$|Df|,P(E,\cdot)$ above are restrictions to open sets of Borel measures that
we denote by $|Df|,|D\chi_{E}|$ respectively, see [6], and [53].

The _isoperimetric profile of $(X,\mathsf{d},\mathfrak{m})$_ is

$I_{X}(V)\vcentcolon=\inf\left\\{P(E)\ :\ \text{$E\subset X$ Borel,
$\mathfrak{m}(E)=V$}\right\\},$

for any $V\in[0,\mathfrak{m}(X))$. If $E\subset X$ is Borel with
$\mathfrak{m}(E)=V$ and $P(E)=I_{X}(V)$, then we say that $E$ is an
_isoperimetric region_.

It follows from classical approximation results (cf. Remark 2.2) that the
above definition yields the usual notion of perimeter on any Riemannian
manifold $(M^{n},g)$ recalled at the beginning of this section.

We will need the following well established approximation result.

######  Lemma 2.9.

Let $(X,\mathsf{d})$ be a complete and separable metric space, and let
$\mathfrak{m}$ be a nonnegative measure, finite on bounded sets. Then, for any
$f\in BV(X,\mathsf{d},\mathfrak{m})$ there exists a sequence
$(f_{k})\subset{\rm Lip}(X,\mathsf{d})$, where $f_{k}$ has bounded support for
any $k$, such that $f_{k}\to f$ pointwise $\mathfrak{m}$-a.e. and in
$L^{1}(X,\mathfrak{m})$, and $\int_{X}{\rm
lip\,}(f_{k})d\mathfrak{m}\to|Df|(X)$ as $k\to\infty$.

###### Proof.

For $f\in BV(X,\mathsf{d},\mathfrak{m})$ it holds that

$\begin{split}|Df|(X)&=|Df|_{w}(X)\\\ &=\inf\left\\{\liminf_{i}\int_{X}{\rm
lip}_{a}\,f_{i}\,\mathrm{d}\mathfrak{m}\ :\ f_{i}\in{\rm Lip}(X)\text{ with
bdd spt, }f_{i}\to f\text{ in }L^{1}(X,\mathfrak{m})\right\\}\\\
&=\inf\left\\{\liminf_{i}\int_{X}{\rm lip}_{a}\,f_{i}\,\mathrm{d}\mathfrak{m}\
:\ f_{i}\in{\rm Lip}_{\rm loc}(X),f_{i}\to f\text{ in
}L^{1}(X,\mathfrak{m})\right\\}\geq|Df|(X),\end{split}$

where $|Df|_{w}$ is as in [6, Section 5.3], that is as in [36, Section 4.4.3],
and ${\rm lip}_{a}\,g$ is the asymptotic Lipschitz constant of a locally
Lispchitz function $g$, which satisfies ${\rm lip}_{a}\,g\geq{\rm lip\,}g$. In
the previous chain, the first equality follows from [6, Theorem 1.1], the
second and third ones from [36, Theorem 4.5.3], and the final inequality from
the estimate ${\rm lip}_{a}\geq{\rm lip\,}$ for locally Lipschitz functions.

Hence [36, Proposition 4.5.6] applies, that yields the claim. ∎

The following general coarea formula will be employed in obtaining the sharp
Sobolev inequality on $\mathsf{CD}(0,n)$ below.

######  Remark 2.10 (Coarea formula on metric measure spaces).

Let $(X,\mathsf{d},\mathfrak{m})$ be a metric measure space. Let us observe
that from the definitions given above, a Borel set $E$ with finite measure has
finite perimeter if and only if the characteristic function $\chi_{E}$ belongs
to $BV(X,\mathsf{d},\mathfrak{m})$.

If $f\in BV(X,\mathsf{d},\mathfrak{m})$, then $\\{f>\alpha\\}$ has finite
perimeter for a.e. $\alpha\in\mathbb{R}$ and the _coarea formula_ holds

$\int_{X}u\,\mathrm{d}|Df|=\int_{-\infty}^{+\infty}\left(\int_{X}u\,\mathrm{d}\lvert
D\chi_{\\{f>\alpha\\}}\rvert\right)\,\mathrm{d}\alpha,$

for any Borel function $u:X\to[0,+\infty]$, see [53, Proposition 4.2]. If $f$
is also continuous and nonnegative, then $|Df|(\\{f=\alpha\\})=0$ for _every_
$\alpha\in[0,+\infty)$ and the _localized coarea formula_ holds

$\int_{\\{a<f<b\\}}u\,\mathrm{d}|Df|=\int_{a}^{b}\left(\int_{X}u\,\mathrm{d}\lvert
D\chi_{\\{f>\alpha\\}}\rvert\right)\,\mathrm{d}\alpha,$

for every Borel function $u:X\to[0,+\infty]$ and every $0\leq a<b<+\infty$,
see [4, Corollary 1.9].

We recall a statement for the classical Bishop–Gromov volume and perimeter
comparison. The conclusions (2.1), (2.2), and the rigidity part of 2.11 are
consequences, e.g., of [41, Theorem 3.101], [67, Theorem 1.2 and Theorem 1.3],
and the arguments within their proofs. The conclusion (2.3) follows from [63,
Corollary 2.22, item (i)] and the coarea formula.

######  Theorem 2.11 (Bishop–Gromov comparison).

Let $(M^{n},g)$ be a complete Riemannian manifold such that
$\mathrm{Ric}\geq(n-1)K$ on $M^{n}$ in the sense of quadratic forms for some
$K\in\mathbb{R}$. Let us set $T_{K}:=+\infty$ if $K\leq 0$, and
$T_{K}:=\pi/\sqrt{K}$ if $K>0$. Then, for every $p\in M$ and for $r\leq T_{K}$
the following hold

(2.1) $\displaystyle\frac{\operatorname{vol}(B_{r}(p))}{v(n,K,r)}\to
1\,\text{as $r\to 0$ and it is nonincreasing},$ (2.2)
$\displaystyle\frac{P(B_{r}(p))}{s(n,K,r)}\to 1\,\text{as $r\to 0$ and it is
almost everywhere nonincreasing},$ (2.3)
$\displaystyle\frac{P(B_{r}(p))}{s(n,K,r)}\leq\frac{\operatorname{vol}(B_{r}(p))}{v(n,K,r)}.$

Moreover, if one has
$\operatorname{vol}(B_{\overline{r}}(p))=v(n,K,\overline{r})$ for some
$\overline{r}\leq T_{K}$, then $B_{\overline{r}}(p)$ is isometric to the ball
of radius $\overline{r}$ in the simply connected model of constant sectional
curvature $K$ and dimension $n$.

######  Remark 2.12 (Bishop–Gromov comparison theorem on m.m.s.).

Let us recall that for an arbitrary $\mathsf{CD}((n-1)K,n)$ space
$(X,\mathsf{d},\mathfrak{m})$ the classical Bishop–Gromov volume comparison
(cf. 2.11) still holds. More precisely, for a fixed $x\in X$, the function
$\mathfrak{m}(B_{r}(x))/v(n,K,r)$ is nonincreasing in $r$ and the function
$P(B_{r}(x))/s(n,K,r)$ is essentially nonincreasing in $r$, i.e.,
$P(B_{R}(x))/s(n,K,R)\leq P(B_{r}(x))/s(n,K,r)$ for almost every radii $R\geq
r$, see [72, Theorem 18.8, Equation (18.8), Proof of Theorem 30.11]. Moreover,
it holds that $P(B_{r}(p))/s(n,K,r)\leq\operatorname{vol}(B_{r}(p))/v(n,K,r)$
for any $r>0$, indeed the last inequality follows from the monotonicity of the
volume and perimeter ratios together with the coarea formula on balls.

Moreover, if $(X,\mathsf{d},\mathcal{H}^{n})$ is an $\mathsf{RCD}((n-1)K,n)$
space, one can conclude that $\mathcal{H}^{n}$-almost every point has a unique
measure Gromov–Hausdorff tangent isometric to $\mathbb{R}^{n}$ ([35, Theorem
1.12]), and thus, from the volume convergence in 2.6, we get

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

where $\omega_{n}$ is the volume of the unit ball in $\mathbb{R}^{n}$.
Moreover, since the density function $x\mapsto\lim_{r\to
0}\mathcal{H}^{n}(B_{r}(x))/\omega_{n}r^{n}$ is lower semicontinuous ([35,
Lemma 2.2]), the latter (2.4) implies that the density is bounded above by the
constant $1$. Hence, from the monotonicity at the beginning of the remark we
deduce that, if $(X,\mathsf{d},\mathcal{H}^{n})$ is an
$\mathsf{RCD}((n-1)K,n)$ space, then for every $x\in X$ we have
$\mathcal{H}^{n}(B_{r}(x))\leq v(n,K,r)$ for every $r>0$.

Let us also recall a classical definition for the convenience of the reader.

######  Definition 2.13 ($\mathrm{AVR}$ and Euclidean volume growth).

Let $(X,\mathsf{d},\mathfrak{m})$ be an arbitrary $\mathsf{CD}(0,n)$ space
with $n\in[1,+\infty)$. From Remark 2.12, the Bishop–Gromov monotonicity
holds, and thus we can define, for an arbitrary $x\in X$,

$\mathrm{AVR}(X,\mathsf{d},\mathfrak{m}):=\lim_{r\to+\infty}\frac{\mathfrak{m}(B_{r}(x))}{\omega_{n}r^{n}},$

the asymptotic volume ratio of $(X,\mathsf{d},\mathfrak{m})$, where
$\omega_{n}:=\pi^{n/2}/\Gamma(n/2+1)$. The previous limit is independent on
$x\in M$. Notice that
$\mathfrak{m}(B_{r}(x))\geq\mathrm{AVR}(X,\mathsf{d},\mathfrak{m})\omega_{n}r^{n}$
for every $r>0$, and every $x\in X$. If
$\mathrm{AVR}(X,\mathsf{d},\mathfrak{m})>0$ we say that
$(X,\mathsf{d},\mathfrak{m})$ has Euclidean volume growth.

Let us recall the basic definition of asymptotic cone in the setting of
$\mathsf{RCD}$ spaces with Euclidean volume growth.

######  Definition 2.14 (Asymptotic cones).

Let $(X,\mathsf{d},\mathfrak{m},x)$ be a pointed $\mathsf{RCD}(0,n)$ space
with $\mathrm{AVR}(X,\mathsf{d},\mathfrak{m})>0$. For every sequence
$\\{r_{i}\\}_{i\in\mathbb{N}}$ with $r_{i}\to+\infty$ the sequence of pointed
metric measure spaces
$\\{(X,r_{i}^{-1}\mathsf{d},r_{i}^{-n}\mathfrak{m},x)\\}_{i\in\mathbb{N}}$ is
precompact in the pmGH topology due to Remark 2.7. Every pmGH limit of such a
sequence is a metric cone, by a slight modification of the proof of [35,
Proposition 2.8]. Any such limit is called an asymptotic cone of $X$ and will
be denoted by $C$ or by $C(Z)$ when we want to higlight the fact that $Z$ is
the metric space that is the basis of the cone. We stress that every such $C$
is an $\mathsf{RCD}(0,N)$ space.

We refer to [35, Definition 2.7] for the precise definition of metric cone.
Notice that the class of asymptotic cones of $X$ is independent on the base
point $x\in X$.

###  2.3. Finite perimeter sets and minimizing sequences for the isoperimetric
problem

In this section we recall the main generalized existence result of [13], which
will be crucially used in this paper. 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 [4,
Definition 3.1], and it is investigated in [4] capitalizing on the results in
[10].

######  Definition 2.15 ($L^{1}$-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 2.4.

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 $R>0$.

Observe that in the above definition it makes sense to speak about the
convergence $\chi_{E_{i}}\mathfrak{m}_{i}\rightharpoonup\chi_{F}\mu$ with
respect to the duality with continuous bounded functions with bounded support
on $Z$ as $(X_{i},\mathsf{d}_{i}),(Y,\mathsf{d}_{Y})$ can be assumed to be
topological subspaces of $(Z,\mathsf{d}_{Z})$ by means of the isometries
$\Psi_{i},\Psi$ of 2.4, and the measures $\mathfrak{m}_{i},\mu$ can be then
identified with the push-forwards
$(\Psi_{i})_{\sharp}\mathfrak{m}_{i},\Psi_{\sharp}\mu$ respectively.

Let us recall here for the reader’s convience the main result of [13], namely
[13, Theorem 1.1]. We will crucially need this result for the proof of our
main results.

######  Theorem 2.16 (Asymptotic mass decomposition [13, Theorem 1.1]).

Let $(M^{n},g)$ be a noncollapsed noncompact complete manifold with infinite
volume, such that $\mathrm{Ric}\geq K$ for some $K\in(-\infty,0]$, and let
$V>0$. For every minimizing (for the perimeter) sequence of sets
$\Omega_{i}\subset M^{n}$ of volume $V$, with $\Omega_{i}$ bounded for any
$i$, up to passing to a subsequence, there exist an increasing 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

  * (i)

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

  * (ii)

$\Omega_{i}^{c}$ converges to $\Omega\subset M^{n}$ in the sense of finite
perimeter sets (2.1), and we have
$\operatorname{vol}(\Omega_{i}^{c})\to_{i}\operatorname{vol}(\Omega)$, and
$P(\Omega_{i}^{c})\to_{i}P(\Omega)$. Moreover $\Omega$ is a bounded
isoperimetric region on $M$;

  * (iii)

for every $j<\overline{N}+1$, $(M^{n},\mathsf{d},\operatorname{vol},p_{i,j})$
converges in the pmGH sense to a pointed $\mathsf{RCD}(K,n)$ space
$(X_{j},\mathsf{d}_{j},\mathfrak{m}_{j},p_{j})$, where $\mathfrak{m}_{j}$ is
the $n$-dimensional Hausdorff measure on $(X_{j},\mathsf{d}_{j})$. Moreover
there are isoperimetric regions $Z_{j}\subset X_{j}$ such that
$\Omega^{d}_{i,j}\to_{i}Z_{j}$ in $L^{1}$-strong (2.15) and
$P(\Omega^{d}_{i,j})\to_{i}P_{X_{j}}(Z_{j})$;

  * (iv)

it holds that

(2.5)
$I_{(M^{n},g)}(V)=P(\Omega)+\sum_{j=1}^{\overline{N}}P_{X_{j}}(Z_{j}),\qquad\qquad
V=\operatorname{vol}(\Omega)+\sum_{j=1}^{\overline{N}}\mathfrak{m}_{j}(Z_{j}).$

We remark that item (ii) in 2.16 is, in fact, proved in [66, Theorem 2.1], and
it consists in the starting point for the rest of the proof of 2.16.

We will need the following result on the boundedness of isoperimetric regions.

######  Proposition 2.17 ([13, Corollary 4.2]).

Let $(M^{n},g)$ be a complete noncollapsed Riemannian manifold with
$\mathrm{Ric}\geq K$ for some $K\in(-\infty,0]$. Then the isoperimetric
regions of $(M^{n},g)$ are bounded.

###  2.4. The collapsed case

As already observed in [13, Remark 4.7], as a nontrivial consequence of 2.16
we have that complete noncollapsed manifolds with a lower bound on the Ricci
curvature have strictly positive isoperimetric profile for any volume.

With an argument partly inspired by the proof of [50, Proposition 3.13], we
are now going to show that, in the nonnegative Ricci case, collapsedness
occurs if and only if the isoperimetric profile vanishes for any volume. In
particular, no isoperimetric sets exist in this situation.

######  Proposition 2.18.

Let $(M^{n},g)$ be a complete noncompact Riemannian manifold with
$\mathrm{Ric}\geq 0$. Then, $\inf_{x\in M}\operatorname{vol}({B_{1}(x)})=0$ if
and only if $I(V)=0$ for any positive volume $V$.

###### Proof.

As observed above, if $M$ is noncollapsed, then $I(V)>0$ for any $V$ by [13,
Remark 4.7]. Then let us assume that $\inf_{x\in
M}\operatorname{vol}(B_{1}(x))=0$. Observe first that $\inf_{x\in
M}\operatorname{vol}(B_{R}(x))=0$ for any $R>0$. Indeed, let
$\\{x_{j}\\}_{j\in\mathbb{N}}$ be a sequence of points such that
$\lim_{j\to\infty}\operatorname{vol}B_{1}(x_{j})=0$. Now, for $R\leq 1$ we
have
$\operatorname{vol}B_{R}(x_{j})\leq\operatorname{vol}(B_{1}(x_{j}))\to_{j}0$,
while for $R>1$, by the Bishop–Gromov volume comparison, we have
$\operatorname{vol}(B_{R}(x_{j}))\leq
R^{n}\operatorname{vol}(B_{1}(x_{j}))\to_{j}0$. Fix now a volume $V>0$, and
consider an arbitrary ball $B_{R}(x)$ such that
$\operatorname{vol}(B_{R}(x))>V$. Since, as we just showed,
$\operatorname{vol}(B_{R}(x_{j}))\to 0$ along the sequence $x_{j}$ above we
can in particular find a point $\overline{x}\in M$ such that
$\operatorname{vol}(B_{R}(\overline{x}))=V$. By (2.3), we have
$P(B_{R}(\overline{x}))\leq nR^{-1}\operatorname{vol}(B_{R}(\overline{x}))$,
and thus

(2.6) $I(V)\leq P(B_{R}(\overline{x}))\leq\frac{n}{R}V.$

Observing that the argument actually holds true for arbitrarily big radii $R$,
our claim is proved by letting $R\to\infty$ in (2.6). ∎

Explicit examples of collapsed manifolds with nonnegative Ricci curvature have
been constructed in dimension $n\geq 4$ by Croke and Karcher in [33, Example 1
and Example 2]. Hence for such examples no isoperimetric regions of positive
volume can exist.

In the same paper, it is shown that the latter examples do not exist in
dimension $n=2$. As a consequence of 2.18 and a former result of Ritoré [64,
Theorem 1.1], asserting that surfaces with nonnegative curvature admit
isoperimetric regions of any given volume, we are able to fully recover such a
positive result proved in [33, Theorem A]. We remark that the proof given here
is completely different from the one given in [33] and recovers the geometric
conclusion of the statement passing through the isoperimetric problem.

######  Corollary 2.19.

Let $(M^{2},g)$ be a noncompact complete Riemannian surface with
$\mathrm{Sect}\geq 0$. Hence there exists $C:=C(M)$ such that

$\operatorname{vol}(B(x,1))\geq C,\qquad\forall x\in M.$

###### Proof.

If the conclusion does not hold, from 2.18 we get that $I_{(M^{2},g)}(V)=0$
for every $V>0$ and hence no isoperimetric regions of positive volume can
exist on $(M^{2},g)$. But this is in contradiction with [64, Theorem 1.1 &
Theorem 2.1]. ∎

######  Remark 2.20.

The proof of the previous result 2.18 builds on uses the Bishop–Gromov
monotonicity results, which hold for $\mathsf{CD}(0,n)$ spaces, cf. Remark
2.12. In fact, one can show with the same argument that also on a
$\mathsf{CD}(0,n)$ space $(X,\mathsf{d},\mathfrak{m})$, if $\inf_{x\in
X}\mathfrak{m}(B_{1}(x))=0$, then the isoperimetric profile identically
vanishes.

###  2.5. Sharp Sobolev inequality on $\mathsf{CD}(0,n)$ spaces

Here, we improve the sharp isoperimetric inequality on $\mathsf{CD}(0,n)$
recently obtained in [15]. Namely, while such result is written in terms of
the Minkowski content of the set involved, we provide one, with the same sharp
constant, in terms of the perimeter. In doing so, we are actually getting also
a sharp Sobolev inequality in these spaces. Earlier versions of inequalities
like these have been obtained in [1, 20, 40].

######  Theorem 2.21 (Sharp Sobolev inequality on $\mathsf{CD}(0,n)$ spaces).

Let $(X,\mathsf{d},\mathfrak{m})$ be a $\mathsf{CD}(0,n)$ space. Then for any
$f\in\text{BV}(X,\mathsf{d},\mathfrak{m})$ it holds

(2.7)
$n\omega_{n}^{\frac{1}{n}}\mathrm{AVR}(X,\mathsf{d},\mathfrak{m})^{\frac{1}{n}}\left(\int_{X}|f|^{\frac{n}{n-1}}\,\mathrm{d}\mathfrak{m}\right)^{\frac{n-1}{n}}\leq|Df|(X)\,.$

In particular, for any set of finite perimeter $E$, with
$\mathfrak{m}(E)<\infty$, we have

(2.8) $P_{X}(E)\geq
n\omega_{n}^{\frac{1}{n}}\mathrm{AVR}(X,\mathsf{d},\mathfrak{m})^{\frac{1}{n}}\mathfrak{m}(E)^{\frac{n-1}{n}}.$

The isoperimetric inequality [15, Theorem 1.1] reads, for any bounded subset
$E\subset X$ of a $\mathsf{CD}(0,n)$ space $(X,\mathsf{d},\mathfrak{m})$,

(2.9) $\mathfrak{m}^{+}(E)\geq
n\omega_{n}^{\frac{1}{n}}\mathrm{AVR}(X,\mathsf{d},\mathfrak{m})^{\frac{1}{n}}\mathfrak{m}(E)^{\frac{n-1}{n}}\,,$

where $\mathfrak{m}^{+}$ denotes the (lower) Minkowski content

(2.10) $\mathfrak{m}^{+}(E):=\liminf_{r\to
0}\frac{\mathfrak{m}(B_{r}(E))-\mathfrak{m}(E)}{r}\,.$

The proof of 2.21 substantially follows from (2.9) together with a relaxation
argument [7]. Observe that (2.8) also allows for unbounded sets.

We will need the following elementary lemma proven in [5, Lemma A.24].

######  Lemma 2.22.

Let $G:[0,\infty)\to[0,\infty)$ a nonincreasing measurable function. Then for
any $\alpha\geq 1$ we have

(2.11)
$\alpha\int_{0}^{\infty}t^{\alpha-1}G(t)\,\mathrm{d}t\leq\left(\int_{0}^{\infty}G^{1/\alpha}(t)\,\mathrm{d}t\right)^{\alpha}\,.$

###### Proof of 2.21.

Let $f:X\to[0,\infty)$ be a Lipschitz function with bounded support. From [7,
Proposition 4.2] we know that

$\int_{X}{\rm
lip\,}(f)\,\mathrm{d}\mathfrak{m}=\int_{0}^{\infty}\mathfrak{m}^{+}(\\{f>t\\})\,\mathrm{d}t,$

hence, from (2.9) we deduce

$\displaystyle\int_{X}{\rm
lip\,}(f)\,\mathrm{d}\mathfrak{m}\,=\int_{0}^{\infty}\mathfrak{m}^{+}(\\{f>t\\})\,\mathrm{d}t\,\geq
n\omega_{n}^{\frac{1}{n}}\mathrm{AVR}(X,\mathsf{d},\mathfrak{m})^{\frac{1}{n}}\int_{0}^{\infty}\mathfrak{m}(\\{f>t\\})^{\frac{n-1}{n}}\,\mathrm{d}t.$

A simple application of Lemma 2.22 coupled with the coarea formula in Remark
2.10 gives

$\int_{0}^{\infty}\mathfrak{m}(\\{f>t\\})^{\frac{n-1}{n}}\,\mathrm{d}t\geq\left(\int_{X}f^{\frac{n}{n-1}}\,\mathrm{d}\mathfrak{m}\right)^{\frac{n-1}{n}}\,,$

for any nonnegative Lipschitz function with bounded support. We can easily
drop the assumption on the sign of $f$ by using the decomposition in positive
and negative part. Let us finally show (2.7) via approximation argument. Let
us fix $f\in BV(X,\mathsf{d},\mathfrak{m})$, and approximate it with $(f_{k})$
as in Lemma 2.9. For any $k\in\mathbb{N}$ it holds

$\int_{X}{\rm lip\,}(f_{k})\,\mathrm{d}\mathfrak{m}\geq
n\omega_{n}^{\frac{1}{n}}\mathrm{AVR}(X,\mathsf{d},\mathfrak{m})^{\frac{1}{n}}\left(\int_{X}|f_{k}|^{\frac{n}{n-1}}\,\mathrm{d}\mathfrak{m}\right)^{\frac{n-1}{n}}\,,$

and passing to the limit as $k\to\infty$ we deduce

$\displaystyle|Df|(X)$
$\displaystyle\geq\liminf_{k\to\infty}n\omega_{n}^{\frac{1}{n}}\mathrm{AVR}(X,\mathsf{d},\mathfrak{m})^{\frac{1}{n}}\left(\int_{X}|f_{k}|^{\frac{n}{n-1}}\,\mathrm{d}\mathfrak{m}\right)^{\frac{n-1}{n}}$
$\displaystyle\geq\omega_{n}^{\frac{1}{n}}\mathrm{AVR}(X,\mathsf{d},\mathfrak{m})^{\frac{1}{n}}\left(\int_{X}|f|^{\frac{n}{n-1}}\,\mathrm{d}\mathfrak{m}\right)^{\frac{n-1}{n}}\,$

where the last inequality follows from Fatou’s lemma.

The last conclusion (2.8) comes by plugging $f=\chi_{E}$ in (2.7). ∎

##  3\. Concavity properties of the isoperimetric profile

In this section we study concavity properties of the isoperimetric profile of
a noncollapsed Riemannian manifold with Ricci bounded from below. We then
apply the obtained results to the case of manifolds with nonnegative Ricci
curvature and Euclidean volume growth. In the first part of this section we
consider a penalized isoperimetric problem and we show that its associated
isoperimetric profile converges locally uniformly to the isoperimetric profile
of the manifold.

Throughout this section let $(M^{n},g)$ be a fixed complete Riemannian
manifold, and let $o\in M$. Fix a sequence $\varepsilon_{k}\searrow 0$ and for
any $k\in\mathbb{N}$ let $U_{k}$ be a bounded open smooth set in $M$
containing the ball $B_{1/\varepsilon_{k}}(o)$ such that $U_{k}\subset
U_{k+1}$; and choose $f_{k}\in C^{\infty}(M)$ such that

$f_{k}\geq 0,\qquad f_{k}|_{U_{k}}\equiv 0,\qquad|\nabla
f_{k}|\leq\varepsilon_{k},\qquad f_{k+1}\leq f_{k},$

for any $k\in\mathbb{N}$. The previous choice can be made in such a way that
for any $k$ and for any $N>0$ there is $R>0$ such that $f_{k}>N$ on
$M\setminus B_{R}(o)$. Define, for any $k\in\mathbb{N}$,

(3.1)
$P_{k}(\Omega)\vcentcolon=P(\Omega)+\int_{\Omega}f_{k}\,\mathrm{d}\operatorname{vol},$

for any set of finite perimeter $\Omega\subset M$, and

(3.2) $I_{k}(V)\vcentcolon=\inf\\{P_{k}(\Omega)\ :\ \text{$\Omega$ is a finite
perimeter set with}\,\operatorname{vol}(\Omega)=V\\},$

for any $V>0$. For simplicity, we shall denote by $I$ the isoperimetric
profile $I_{(M^{n},g)}$.

######  Lemma 3.1.

Suppose that $(M^{n},g)$ is noncollapsed and that $\mathrm{Ric}\geq(n-1)K$ for
some $K\in\mathbb{R}$. Then the following two facts hold.

  1. (1)

For any $k\in\mathbb{N}$ and any $V>0$ there exists $\Omega$ of volume $V$
such that $I_{k}(V)=P_{k}(\Omega)$.

  2. (2)

For any $k\in\mathbb{N}$ the function $I_{k}:(0,+\infty)\to\mathbb{R}$ is
continuous and the sequence $\\{I_{k}\\}_{k\in\mathbb{N}}$ converges from
above to the isoperimetric profile $I_{(M^{n},g)}$ locally uniformly on
$(0,+\infty)$.

###### Proof.

For any $k$ and $V>0$ the existence of a minimizer $\Omega$ for $P_{k}$ at
volume $V$ follows by means of the direct method. Indeed a minimizing sequence
$\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ for $P_{k}$ at volume
$V=\operatorname{vol}(\Omega_{i})$ converges up to subsequence to some set
$\Omega$ in $L^{1}_{\rm loc}(M)$. The functional $P_{k}$ is lower
semicontinuous with respect to $L^{1}_{\mathrm{loc}}$-convergence and if by
contradiction $\operatorname{vol}(\Omega)<V$, since $f_{k}$ diverges at
infinity, we would get that $\int_{\Omega_{i}}f_{k}\to+\infty$ as
$i\to+\infty$.

For any $k$ and $V>0$ denote by $\Omega^{k}_{V}$ a minimizer of $P_{k}$ at
volume $V$. First observe that $I_{k}$ is locally bounded; indeed for any
$V>0$ the value of $I_{k}$ at volumes $v$ in a neighborhood of $V$ is
estimated from above by the value of $P_{k}$ on balls of center $o$ and given
volume $v$.

Let us now prove that $I_{k}$ is continuous. Indeed, fix $V>0$ and let
$V_{i}\to V$ be any sequence. Since $I_{k}(V_{i})=P_{k}(\Omega_{V_{i}}^{k})$,
and thus the perimeters of $\Omega_{V_{i}}^{k}$ are uniformly bounded in $i$,
we get that, up to subsequence, $\\{\Omega_{V_{i}}^{k}\\}_{i\in\mathbb{N}}$
converges in $L^{1}_{\rm loc}$ to some set $\Omega$. As
$\\{P_{k}(\Omega_{V_{i}}^{k})\\}_{i\in\mathbb{N}}$ is uniformly bounded, we
deduce that $\operatorname{vol}(\Omega)=V$ and then the convergence
$\Omega_{V_{i}}^{k}\to_{i}\Omega$ holds in $L^{1}$ on $M$. Moreover

$I_{k}(V)\leq
P_{k}(\Omega)\leq\liminf_{i}P_{k}(\Omega_{V_{i}}^{k})=\liminf_{i}I_{k}(V_{i}).$

Hence $I_{k}$ is lower semicontinuous. Let us now finish the proof of the
continuity. Indeed, suppose by contradiction that
$I_{k}(V)\leq\limsup_{i}I_{k}(V_{i})-\delta$ for some $\delta>0$. There exists
a set $E$ of volume $V$ with $P_{k}(E)\leq I_{k}(V)+\delta/4$, and then for
large $i$ we can find a ball $B_{r_{i}}(x_{i})$ such that $r_{i}\to 0^{+}$ and
either $E\cup B_{r_{i}}(x_{i})$ or $E\setminus B_{r_{i}}(x_{i})$ has volume
$V_{i}$, and, denoting by $E_{i}$ such a set of volume $V_{i}$, we have

(3.3) $P_{k}(E_{i})\leq I_{k}(V)+\delta/2,$

for large $i$. For the last inequality to hold we are using that, as a
consequence of the Bishop–Gromov comparison, $P(E\cup B_{r_{i}}(x_{i}))\leq
P(E)+v(n,K,r_{i})$, together with the fact that $r_{i}\to 0$. Finally,
$P_{k}(E_{i})\geq I_{k}(V_{i})$, thus inserting the absurd hypothesis in (3.3)
and passing to the limsup in $i$ we derive a contradiction.

Therefore $\lim_{i}I_{k}(V_{i})=I_{k}(V)$, and the arbitrariness of the
sequence implies that $I_{k}$ is continuous at $V$, and then $I_{k}$ is
continuous on $(0,+\infty)$.

Since $(M^{n},g)$ is noncollapsed with Ricci bounded below, the isoperimetric
profile $I$ is continuous and strictly positive by [60, Corollary 2] and [13,
Remark 4.7]. Since $I_{k}$ is continuous and $I\leq I_{k+1}\leq I_{k}$ for any
$k\in\mathbb{N}$, if $I_{k}\to_{k}I$ pointwise, then the same convergence
holds locally uniformly by Dini’s Monotone Convergence Theorem. Now for fixed
$V>0$ we show, indeed, that $I_{k}(V)\to_{k}I(V)$. If by contradiction
$\limsup_{k}I_{k}(V)\geq\delta+I(V)$ for some $\delta>0$, then one can
consider a bounded smooth set $\Omega$ of volume $V$ such that $P(\Omega)\leq
I(V)+\delta/2$. Since $\Omega\Subset U_{k}$ for large $k$, we get that

$I(V)+\frac{\delta}{2}\geq P(\Omega)=\limsup_{k}P_{k}(\Omega)\geq\delta+I(V),$

that is impossible. ∎

###  3.1. First and second variations of the penalized perimeter

In this section we derive first and second variations for the potential term
in $P_{k}$, and we collect basic properties of the sets that minimize the
penalized perimeter $P_{k}$ under a volume constraint.

######  Lemma 3.2.

Let $M$ be a complete Riemannian manifold. Let $\Omega\subset M$ be a set of
finite perimeter and suppose that there is a smooth hypersurface $\Sigma$ with
boundary contained in the essential boundary $\partial^{*}\Omega$. Let
$X:\Sigma\to TM$ be a smooth vector field compactly supported in the interior
of $\Sigma$ and denote $\phi_{t}(x)\vcentcolon=\exp(tX(x))$ for $x\in\Sigma$.

For some $\tau>0$, if $\Omega_{t}$ is the set of finite perimeter with
essential boundary $\phi_{t}(\partial^{*}\Omega)$ for $|t|<\tau$ and
$X_{t}\vcentcolon=\partial_{t}\phi_{t}$ is the variation field along
$\phi_{t}(\Sigma)$, then

(3.4)
$\frac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\Omega_{t}}f_{k}\,\mathrm{d}\operatorname{vol}\right)=\int_{\partial^{*}\Omega_{t}}f_{k}\langle
X_{t},-\nu_{\Omega_{t}}\rangle\,\mathrm{d}\mathcal{H}^{n-1}\qquad\forall\,|t|<\tau,$

where $\nu_{\Omega_{t}}$ is the generalized interior unit normal of
$\Omega_{t}$. If also $X$ is normal along $\Sigma$, then

(3.5)
$\begin{split}\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\left(\int_{\Omega_{t}}f_{k}\,\mathrm{d}\operatorname{vol}\right)\bigg{|}_{t=0}=&\int_{\Sigma}\left(f_{k}H_{\Omega}\langle
X,\nu_{\Omega}\rangle^{2}-\langle\nabla f_{k},X\rangle\langle
X,\nu_{\Omega}\rangle\right)\,\mathrm{d}\mathcal{H}^{n-1},\end{split}$

where $\nu_{\Omega}$ is the generalized interior unit normal of $\Omega$ and
$H_{\Omega}$ denotes the mean curvature in direction $\nu_{\Omega}$, i.e.,
$H_{\Omega}\nu_{\Omega}$ is the mean curvature vector of $\Sigma$.

###### Proof.

The first variation (3.4) follows by an application of the area formula as in
[52, Proposition 17.8]. The second variation (3.5) follows by differentiating
(3.4). Indeed, rewriting the varied boundary $\partial^{*}\Omega_{t}$ as an
embedding $F_{t}:\Sigma\to M$, where $F_{t}(\Sigma)=\phi_{t}(\Sigma)$, we have

$\frac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\Omega_{t}}f_{k}\,\mathrm{d}\operatorname{vol}\right)=\int_{\Sigma}[f_{k}\langle
X_{t},-\nu_{\Omega_{t}}\rangle]\circ F_{t}\,\mathrm{d}\operatorname{vol}(t),$

where $\operatorname{vol}(t)$ is the volume form on $\Sigma$ induced by
$F_{t}$. Hence

$\begin{split}\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\left(\int_{\Omega_{t}}f_{k}\,\mathrm{d}\operatorname{vol}\right)=&\int_{\Sigma}-\langle\nabla
f_{k},X_{t}\rangle\langle X_{t},\nu_{\Omega_{t}}\rangle\circ
F_{t}-f_{k}\langle\nabla_{X_{t}}X_{t},\nu_{\Omega_{t}}\rangle\circ F_{t}+\\\
&-f_{k}\langle
X_{t},\nabla_{X_{t}}\nu_{\Omega_{t}}\rangle\,\mathrm{d}\operatorname{vol}(t)+\int_{\Sigma}f_{k}\langle
X_{t},-\nu_{\Omega_{t}}\rangle\circ
F_{t}\,\partial_{t}\,\mathrm{d}\operatorname{vol}(t).\end{split}$

Since $\nabla_{X_{t}}X_{t}=0$ and $\nabla_{X_{t}}\nu_{\Omega_{t}}$ is tangent
along $F_{t}(\Sigma)$, evaluating at $t=0$ we get

(3.6)
$\begin{split}\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\left(\int_{\Omega_{t}}f_{k}\,\mathrm{d}\operatorname{vol}\right)\bigg{|}_{t=0}=&\int_{\Sigma}-\langle\nabla
f_{k},X\rangle\langle
X,\nu_{\Omega}\rangle\,\mathrm{d}\mathcal{H}^{n-1}+\int_{\Sigma}f_{k}\langle
X,-\nu_{\Omega}\rangle\,\partial_{t}\,\mathrm{d}\operatorname{vol}(t)|_{t=0}.\end{split}$

Since $X$ is normal along $\Sigma$, i.e., its tangent projection $X^{\top}$
onto $T\Sigma$ vanishes, the identity (3.5) follows by the fact that

(3.7)
$\partial_{t}\,\mathrm{d}\operatorname{vol}(t)|_{t=0}=\left(\operatorname{div}_{\Sigma}(X^{\top})-H_{\Omega}\langle
X,\nu_{\Omega}\rangle\right)\,\mathrm{d}\mathcal{H}^{n-1}\llcorner\Sigma=-H_{\Omega}\langle
X,\nu_{\Omega}\rangle\,\mathrm{d}\mathcal{H}^{n-1}\llcorner\Sigma.$

∎

In the hypotheses and notation of Lemma 3.2, if $\varphi$ is a smooth function
compactly supported in $\Sigma\subset\partial^{*}\Omega$, we can plug in the
normal field $X=-\varphi\nu_{\Omega}$, so that (3.5) reads

(3.8)
$\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\left(\int_{\Omega_{t}}f_{k}\,\mathrm{d}\operatorname{vol}\right)\bigg{|}_{t=0}=\int_{\partial^{*}\Omega}\left(\varphi^{2}f_{k}H_{\Omega}-\varphi^{2}\langle\nabla
f_{k},\nu_{\Omega}\rangle\right)\,\mathrm{d}\mathcal{H}^{n-1}.$

######  Lemma 3.3.

Suppose that $(M^{n},g)$ is noncollapsed and that $\mathrm{Ric}\geq(n-1)K$ for
$K\in\mathbb{R}$. Let $\Omega$ be a minimizer of $P_{k}$, see (3.1), at volume
$V$. Then $\Omega$ is bounded and there exists a constant
$\lambda=\lambda_{k,V}\in\mathbb{R}$ such that

$\frac{\mathrm{d}}{\mathrm{d}t}P_{k}(\phi_{t}(\Omega))\bigg{|}_{t=0}=\lambda\int_{\partial^{*}\Omega}\langle
X,-\nu_{\Omega}\rangle\,\mathrm{d}\mathcal{H}^{n-1},$

for any smooth compactly supported vector field $X$ on $M$, where $\phi_{t}$
is the flow of $X$ and $\nu_{\Omega}$ the generalized interior normal of
$\Omega$.

In particular $\Omega$ has generalized mean curvature
$H_{\Omega}:\partial^{*}\Omega\to\mathbb{R}$ such that

(3.9) $H_{\Omega}+f_{k}=\lambda\qquad\text{$\mathcal{H}^{n-1}$-a.e. on
$\partial^{*}\Omega$,}$

that is

$\int_{\partial^{*}\Omega}(H_{\Omega}+f_{k})\langle
X,-\nu_{\Omega}\rangle\,\mathrm{d}\mathcal{H}^{n-1}=\lambda\int_{\partial^{*}\Omega}\langle
X,-\nu_{\Omega}\rangle\,\mathrm{d}\mathcal{H}^{n-1},$

for any smooth vector field $X$ on $M$.

Moreover there is an open representative of $\Omega$ and its topological
boundary $\partial\Omega$ is given by the disjoint union
$\partial\Omega=\partial_{r}\Omega\sqcup\partial_{s}\Omega$, where
$\partial_{r}\Omega$ is a smooth hypersurface with topological boundary
$\partial_{s}\Omega$ that is empty if the dimension is $n\leq 7$ and with
Hausdorff dimension less than or equal to $n-8$ when $n\geq 8$.

###### Proof.

The first variation $\partial_{t}P_{k}(\phi_{t}(\Omega))|_{t=0}$ vanishes
whenever $X$ generates a volume preserving variation, i.e., whenever
$\int_{\partial^{*}\Omega}\langle X,\nu_{\Omega}\rangle=0$. Arguing for
example as in [52, Theorem 17.20] it is a classical matter to derive the
existence of the Lagrange multiplier $\lambda$.

Being $\Omega,X,\phi_{t}$ are as in the statement, then from (3.4) we deduce
that

$\frac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\Omega_{t}}f_{k}\,\mathrm{d}\operatorname{vol}\right)\bigg{|}_{t=0}=\int_{\partial^{*}\Omega}f_{k}\langle
X,-\nu_{\Omega}\rangle\,\mathrm{d}\mathcal{H}^{n-1},$

where $\Omega_{t}\vcentcolon=\phi_{t}(\Omega)$. The first variation formula
for the perimeter then yields that

$\frac{\mathrm{d}}{\mathrm{d}t}P_{k}(\phi_{t}(\Omega))\bigg{|}_{t=0}=\int_{\partial^{*}\Omega}\left(\operatorname{div}X-\langle\nabla
X(\nu_{\Omega}),\nu_{\Omega}\rangle-f_{k}\langle
X,\nu_{\Omega}\rangle\right)\,\mathrm{d}\mathcal{H}^{n-1}.$

Letting
$\operatorname{div}_{\Omega}X\vcentcolon=\operatorname{div}X-\langle\nabla
X(\nu_{\Omega}),\nu_{\Omega}\rangle$ the tangential divergence of $X$ along
$\partial^{*}\Omega$, we deduce that

$\int_{\partial^{*}\Omega}\operatorname{div}_{\Omega}X\,\mathrm{d}\mathcal{H}^{n-1}=\int_{\partial^{*}\Omega}(f_{k}-\lambda)\langle
X,\nu_{\Omega}\rangle\,\mathrm{d}\mathcal{H}^{n-1},$

which precisely defines the generalized mean curvature of $\Omega$ via the
identity $H_{\Omega}\nu_{\Omega}=-(f_{k}-\lambda)\nu_{\Omega}$.

The regularity of $\partial\Omega$ in the statement follows by [14, Corollary
1.6 & Remark 1.7]. These results imply that $\Omega$ has an open
representative with topological boundary $\partial\Omega$ given by the
disjoint union $\partial\Omega=\partial_{r}\Omega\sqcup\partial_{s}\Omega$,
where $\partial_{r}\Omega$ is locally $C^{1,\alpha}$ and it has topological
boundary $\partial_{s}\Omega$ with Hausdorff dimension less than or equal to
$n-8$. Since $f_{k}$ is smooth, the Euler–Lagrange equation (3.9) classically
implies that the regular part $\partial_{r}\Omega$ is, in fact, smooth.

Now we can prove that $\Omega$ is bounded by a modification of a classical
argument appeared in [66, Proposition 3.7] in the Euclidean setting and in
[61, Theorem 3] on Riemannian manifolds (see also [13, Appendix B]). Fix
$p_{0}\in\partial_{r}\Omega$. Since $\partial_{s}\Omega$ is closed, there is
$R_{1}>0$ such that $\partial\Omega\cap B_{R_{1}}(p_{0})$ is a nonempty smooth
hypersurface, and it is well defined the inner normal $\nu_{\Omega}$ of
$\Omega$ on $\partial\Omega\cap B_{R_{1}}(p_{0})$. Let $\varphi\in
C^{\infty}_{c}(\partial\Omega\cap B_{R_{1}}(p_{0}))$ be a nonvanishing
nonnegative function and consider the normal vector field
$X=-\varphi\nu_{\Omega}$. Let $\Phi_{t}(x)\vcentcolon=\exp(tX(x))$ and let
$\Omega_{t}$ be the varied set whose essential boundary is
$\Phi_{t}(\partial^{*}\Omega)$, for $|t|<\tau$ and $\tau>0$. Since
$\alpha\vcentcolon=\int_{\partial\Omega}\langle X,-\nu_{\Omega}\rangle>0$,
first variation formulae for perimeter and volume give that

(3.10)
$\begin{split}\operatorname{vol}(\Omega_{t})&=\operatorname{vol}(\Omega)+\alpha
t+O(t^{2}),\\\
P(\Omega_{t},B_{R_{1}}(p_{0}))&=P(\Omega,B_{R_{1}}(p_{0}))+t\int_{\partial\Omega}\varphi
H_{\Omega}\,\mathrm{d}\mathcal{H}^{n-1}+O(t^{2}),\end{split}$

for $|t|<\tau$. Moreover, as $\varphi>0$, we have that

(3.11) $\Omega\subset\Omega_{t}\qquad\text{for $0\leq t<\tau$,}$

up to decreasing $\tau$. As $\alpha>0$ and $|\int\varphi H_{\Omega}|<+\infty$,
(3.10) and (3.11) imply that there exists $\varepsilon_{0}>0$ and $\beta>0$
such that for any $\varepsilon\in(-\varepsilon_{0},\varepsilon_{0})$ there is
a finite perimeter set $E$ with

(3.12) $\begin{split}&E\Delta\Omega\Subset B_{R_{1}}(p_{0}),\\\
&\operatorname{vol}(E)=\operatorname{vol}(\Omega)+\varepsilon,\\\
&P(E,B_{R_{1}}(p_{0}))\leq P(\Omega,B_{R_{1}}(p_{0}))+\beta|\varepsilon|,\\\
&\varepsilon>0\text{ implies }\Omega\subset E.\end{split}$

Now for $r>R_{1}$ let

$V(r)\vcentcolon=\operatorname{vol}(\Omega\setminus B_{r}(p_{0})),\qquad
A(r)\vcentcolon=P(\Omega,M\setminus B_{r}(p_{0})).$

Since $M$ is noncollapsed and the Ricci curvature is bounded from below, it
holds an isoperimetric inequality for sufficiently small volumes, see [46,
Lemma 3.2]. Hence for some $R_{2}>R_{1}$, for $r\geq R_{2}>0$ we can apply
such an isoperimetric inequality on $\Omega\setminus B_{r}(p_{0})$, so that

$|V^{\prime}(r)|+A(r)=\mathcal{H}^{n-1}(\partial
B_{r}(p_{0})\cap\Omega)+P(\Omega,M\setminus B_{r}(p_{0}))=P(\Omega\setminus
B_{r}(p_{0}))\geq c_{0}V(r)^{\frac{n-1}{n}},$

for some $c_{0}>0$ for almost every $r\geq R_{2}$. We want to prove that

(3.13) $A(r)\leq|V^{\prime}(r)|+CV(r),$

for some constant $C$, and for almost every $r$ sufficiently big. Combining
with the previous inequality, in this way we would get

$c_{0}V(r)^{\frac{n-1}{n}}\leq
CV(r)+2|V^{\prime}(r)|\leq\frac{c_{0}}{2}V(r)^{\frac{n-1}{n}}-2V^{\prime}(r),$

because $|V^{\prime}(r)|=-V^{\prime}(r)$ and
$CV(r)\leq\tfrac{c_{0}}{2}V(r)^{\frac{n-1}{n}}$ for almost every sufficiently
big radius. Hence ODE comparison implies that $V(r)$ vanishes at some
$r=\overline{r}<+\infty$, i.e., $\Omega$ is bounded as a set of finite
perimeter.

So we are left to prove (3.13). For some $R_{3}\geq R_{2}$, for $r\geq R_{3}$
we can assume that $V(r)<\varepsilon_{0}$. For fixed $r\geq R_{3}$ let
$\varepsilon\vcentcolon=V(r)$ and assume also that
$\mathcal{H}^{n-1}(\partial\Omega\cap\partial B_{r}(p_{0}))=0$ and
$V^{\prime}(r)$ exists, which hold for almost every $r$. Given such
$\varepsilon$, there exists $E$ as in (3.12). Finally let $F=E\cap
B_{r}(p_{0})$. It follows that
$\operatorname{vol}(F)=\operatorname{vol}(\Omega)$ and (3.12) implies that

(3.14) $\Omega\cap B_{r}(p_{0})\subset F,\qquad F\setminus\Omega\subset
B_{R_{1}}(p_{0}),\qquad\operatorname{vol}(F\setminus\Omega)=\varepsilon.$

Since $\Omega$ is a minimizer for $P_{k}$ at volume
$\operatorname{vol}(\Omega)=\operatorname{vol}(F)$, we deduce

(3.15) $\begin{split}P_{k}(\Omega)&\leq P_{k}(F)=P(E)-P(\Omega,M\setminus
B_{r}(p_{0}))+\mathcal{H}^{n-1}(\partial
B_{r}(p_{0})\cap\Omega)+\int_{F}f_{k}\,\mathrm{d}\operatorname{vol}\\\ &\leq
P(\Omega)+\beta\varepsilon-A(r)+|V^{\prime}(r)|+\int_{F}f_{k}\,\mathrm{d}\operatorname{vol}.\end{split}$

Up to increase $R_{3}$, we can assume that $\inf_{M\setminus
B_{r}(p_{0})}f_{k}>\sup_{B_{R_{1}}(p_{0})}f_{k}$. By (3.14) we then get

$\begin{split}\int_{F}f_{k}\,\mathrm{d}\operatorname{vol}&=\int_{F\cap\Omega}f_{k}\,\mathrm{d}\operatorname{vol}+\int_{F\setminus\Omega}f_{k}\,\mathrm{d}\operatorname{vol}\leq\int_{\Omega\cap
B_{r}(p_{0})}f_{k}\,\mathrm{d}\operatorname{vol}+\varepsilon\sup_{B_{R_{1}}(p_{0})}f_{k}\\\
&\leq\int_{\Omega\cap
B_{r}(p_{0})}f_{k}\,\mathrm{d}\operatorname{vol}+\int_{\Omega\setminus
B_{r}(p_{0})}f_{k}\,\mathrm{d}\operatorname{vol}=\int_{\Omega}f_{k}\,\mathrm{d}\operatorname{vol}\end{split}$

Combining the latter one with (3.15) we obtain (3.13) with $C=\beta$. ∎

###  3.2. Proof of 1.4

We will need the following well known lemma that, roughly speaking, gives an
approximation in $H^{1}$ of the function identically equal to $1$ on the
boundary of a set, provided its singular part is sufficiently small. We will
apply such a result to a volume constrained minimizer of $P_{k}$.

######  Lemma 3.4.

Let $\Omega\subset M^{n}$ be a bounded open set such that its topological
boundary $\partial\Omega$ is given by the disjoint union
$\partial\Omega=\partial_{r}\Omega\sqcup\partial_{s}\Omega$, where
$\partial_{r}\Omega$ is a smooth hypersurface with topological boundary
$\partial_{s}\Omega$ and the Hausdorff dimension of $\partial_{s}\Omega$ is
less than or equal to $n-3$.

Then for any $\delta>0$ there is $\varphi_{\delta}\in
C^{\infty}_{c}(\partial_{r}\Omega)$ such that

$0\leq\varphi_{\delta}\leq 1,\qquad\lim_{\delta\to
0}\int_{\partial\Omega}\varphi_{\delta}^{2}\,\mathrm{d}\mathcal{H}^{n-1}=P(\Omega),\qquad\lim_{\delta\to
0}\int_{\partial\Omega}|\nabla\varphi_{\delta}|^{2}\,\mathrm{d}\mathcal{H}^{n-1}=0,$

where $\nabla\varphi_{\delta}$ here denotes the gradient of $\varphi_{\delta}$
as a function on the submanifold $\partial_{r}\Omega$.

The above lemma has been proved in the Euclidean setting in [68, Lemma 2.4]
under additional regularity assumptions first, and then extended to Riemannian
manifolds in [18, Proposition 2.5, Remark 2.6]. It is observed in [59, Lemma
3.1] that the regularity assumptions on $\partial\Omega$ as in the statement
are sufficient (see also [19, Lemma 3.1]).

In the following proof of 1.4, we are making variations of a volume
constrained minimizer $\Omega$ for the penalized perimeter along the normal
field $X=-\varphi\nu_{\Omega}$, with $\varphi$ given by Lemma 3.4. We point
out that in dimension $n\leq 7$, by means of the regularity result recalled in
Lemma 3.3, one could safely consider $X=-\nu_{\Omega}$.

###### Conclusion of the proof of 1.4.

For the ease of notation we do not explicitly write the volume forms under the
integral sign, since it will be clear from the context.

Let $\Omega=\Omega(k,\widetilde{V})$ be a fixed minimizer of volume
$\widetilde{V}$ for $P_{k}$. Let $\varphi\in
C^{\infty}_{c}(\partial_{r}\Omega)$ be a smooth nonvanishing function with
$0\leq\varphi\leq 1$ supported in the regular part of $\partial\Omega$ and
define the normal field $X=-\varphi\nu_{\Omega}$ and
$\phi_{t}(x)\vcentcolon=\exp(tX(x))$ for $x\in\partial^{*}\Omega$. Denote by
$\Omega_{t}$ the varied set whose reduced boundary is
$\phi_{t}(\partial^{*}\Omega)$, for $|t|<\tau$ and $\tau>0$. Define

$v(t)\vcentcolon=\operatorname{vol}(\Omega_{t}),\qquad
J(v)\vcentcolon=P_{k}(\Omega_{t})\quad\text{for the unique $t$ s.t.
$v=v(t)$}.$

The definitions of $v(t)$ and $J(v)$ are well posed, indeed

$v^{\prime}(t)=\int_{\partial^{*}\Omega_{t}}\langle-\nu_{\Omega_{t}},\partial_{t}\phi_{t}\rangle,$

and then $v^{\prime}(0)=\int_{\partial\Omega}\varphi>0$ and $v^{\prime}>0$ for
every $t$ if $\tau$ is small enough. Hence $v$ is locally invertible and
$J(v)$ is well defined for
$v\in(\widetilde{V}-\widetilde{\delta},\widetilde{V}+\widetilde{\delta})$ for
some $\widetilde{\delta}>0$. Let $v^{-1}$ denote the inverse of $v(t)$. We
thus have

(3.16)
$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}P(\Omega_{t})\bigg{|}_{t=0}&=\int_{\partial\Omega}\varphi
H_{\Omega},\\\
(v^{-1})^{\prime}(v)&=\left(\int_{\partial^{*}\Omega_{t}}\langle-\nu_{\Omega_{t}},\partial_{t}\phi_{t}\rangle\right)^{-1},\\\
(v^{-1})^{\prime}(\widetilde{V})&=\left(\int_{\partial\Omega}\varphi\right)^{-1},\\\
(v^{-1})^{\prime\prime}(\widetilde{V})&=-\frac{1}{\left(\int_{\partial\Omega}\varphi\right)^{2}}(v^{-1})^{\prime}(\widetilde{V})v^{\prime\prime}(0)=-\frac{1}{\left(\int_{\partial\Omega}\varphi\right)^{3}}\int_{\partial\Omega}\varphi^{2}H_{\Omega},\end{split}$

where the last equality follows by computing $v^{\prime\prime}(0)$ analogously
as in (3.6) and (3.7). Observing that $J(v)=P_{k}(\Omega_{v^{-1}(v)})$, we can
compute

$\begin{split}\left(\int_{\partial\Omega}\varphi\right)^{2}&\frac{\mathrm{d}^{2}}{\mathrm{d}v^{2}}J(v)\bigg{|}_{v=\widetilde{V}}\\\
&=\left(\int_{\partial\Omega}\varphi\right)^{2}\frac{\mathrm{d}}{\mathrm{d}v}\left((v^{-1})^{\prime}\left(\frac{\mathrm{d}}{\mathrm{d}t}P_{k}(\Omega_{t})\right)\circ
v^{-1}\right)\bigg{|}_{v=\widetilde{V}}\\\
&=\left(\int_{\partial\Omega}\varphi\right)^{2}\left((v^{-1})^{\prime\prime}(\widetilde{V})\frac{\mathrm{d}}{\mathrm{d}t}P_{k}(\Omega_{t})\bigg{|}_{t=0}+[(v^{-1})^{\prime}(\widetilde{V})]^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}P_{k}(\Omega_{t})\bigg{|}_{t=0}\right).\end{split}$

Using (3.4), (3.8), and (3.16) we get

$\begin{split}\left(\int_{\partial\Omega}\varphi\right)^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}v^{2}}J(v)\bigg{|}_{v=\widetilde{V}}=&-\frac{1}{\int_{\partial\Omega}\varphi}\int_{\partial\Omega}\varphi^{2}H_{\Omega}\left(\int_{\partial\Omega}\varphi
H_{\Omega}+\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega_{t}}f_{k}\bigg{|}_{t=0}\right)+\\\
&+\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}P(\Omega_{t})\bigg{|}_{t=0}+\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\int_{\Omega_{t}}f_{k}\bigg{|}_{t=0}\\\
=&-\frac{1}{\int_{\partial\Omega}\varphi}\int_{\partial\Omega}\varphi^{2}H_{\Omega}\left(\int_{\partial\Omega}\varphi
H_{\Omega}+\int_{\partial\Omega}\varphi f_{k}\right)+\\\
&+\int_{\partial\Omega}\varphi^{2}f_{k}H_{\Omega}-\varphi^{2}\langle\nabla
f_{k},\nu_{\Omega}\rangle+\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}P(\Omega_{t})\bigg{|}_{t=0}.\end{split}$

The second variation formula for the perimeter yields

$\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}P(\Omega_{t})\bigg{|}_{t=0}=\int_{\partial\Omega}|\nabla\varphi|^{2}+\varphi^{2}\left(H_{\Omega}^{2}-|A_{\Omega}|^{2}-\mathrm{Ric}(\nu_{\Omega},\nu_{\Omega})\right),$

where $A_{\Omega}$ is the second fundamental form of $\partial_{r}\Omega$ and
$\nabla\varphi$ is the gradient of $\varphi$ as a function on the submanifold
$\partial_{r}\Omega$. Since $\mathrm{Ric}\geq(n-1)K$, $|\nabla
f_{k}|\leq\varepsilon_{k}$, $\varphi\leq 1$, and
$H_{\Omega}+f_{k}=\lambda\equiv\lambda_{k,\widetilde{V}}\in\mathbb{R}$ at
$\mathcal{H}^{n-1}$ almost every point on $\partial^{*}\Omega$, we estimate

$\begin{split}\left(\int_{\partial\Omega}\varphi\right)^{2}&\frac{\mathrm{d}^{2}}{\mathrm{d}v^{2}}J(v)\bigg{|}_{v=\widetilde{V}}\\\
\leq&-\frac{1}{\int_{\partial\Omega}\varphi}\int_{\partial\Omega}\varphi^{2}H_{\Omega}\left(\int_{\partial\Omega}\varphi(H_{\Omega}+f_{k})\right)+\int_{\partial\Omega}\varphi^{2}f_{k}H_{\Omega}+\varepsilon_{k}P(\Omega)+\\\
&+\int_{\partial\Omega}\varphi^{2}H_{\Omega}^{2}+(n-1)|K|P(\Omega)-\int_{\partial\Omega}\varphi^{2}|A_{\Omega}|^{2}+\int_{\partial\Omega}|\nabla\varphi|^{2}\\\
=&-\lambda\int_{\partial\Omega}\varphi^{2}H_{\Omega}+\int_{\partial\Omega}(f_{k}+H_{\Omega})\varphi^{2}H_{\Omega}+\varepsilon_{k}P(\Omega)+\\\
&+(n-1)|K|P(\Omega)-\int_{\partial\Omega}\varphi^{2}|A_{\Omega}|^{2}+\int_{\partial\Omega}|\nabla\varphi|^{2}\\\
=&(\varepsilon_{k}+(n-1)|K|)P(\Omega)-\int_{\partial\Omega}\varphi^{2}|A_{\Omega}|^{2}+\int_{\partial\Omega}|\nabla\varphi|^{2}.\end{split}$

Hence

(3.17)
$J^{\prime\prime}(\widetilde{V})\leq(\varepsilon_{k}+(n-1)|K|)\frac{P(\Omega)}{\left(\int_{\partial\Omega}\varphi\right)^{2}}+\frac{1}{\left(\int_{\partial\Omega}\varphi\right)^{2}}\left(\int_{\partial\Omega}|\nabla\varphi|^{2}-\int_{\partial\Omega}\varphi^{2}|A_{\Omega}|^{2}\right).$

By definition of $I_{k}$ we have that $I_{k}(v)\leq J(v)$ for any
$v\in(\widetilde{V}-\widetilde{\delta},\widetilde{V}+\widetilde{\delta})$ and
$I_{k}(\widetilde{V})=J(\widetilde{V})$. Then by the latter observation we get

$\begin{split}\overline{D}^{2}I_{k}(\widetilde{V})\vcentcolon=&\limsup_{h\to
0}\frac{I_{k}(\widetilde{V}+h)+I_{k}(\widetilde{V}-h)-2I_{k}(\widetilde{V})}{h^{2}}\leq
J^{\prime\prime}(\widetilde{V})\end{split}$

Now observe that we could take $\varphi=\varphi_{\delta}$ for any $\delta$
with $\varphi_{\delta}$ as in Lemma 3.4. Hence letting $\delta\to 0$, using
that $P(\Omega)\geq I(\widetilde{V})$, and exploiting (3.17), we conclude that

(3.18)
$\overline{D}^{2}I_{k}(\widetilde{V})\leq\frac{\varepsilon_{k}+(n-1)|K|}{P(\Omega)}\leq\frac{\varepsilon_{k}+(n-1)|K|}{I(\widetilde{V})}.$

Recalling that $I$ is continuous and strictly positive ([60, Corollary 2] and
[13, Remark 4.7]), up to taking a smaller $\widetilde{\delta}$, we have that
$I(V)$ is bounded below by a strictly positive constant independent of
$V\in(\widetilde{V}-\widetilde{\delta},\widetilde{V}+\widetilde{\delta})$.
Since $\widetilde{V}$ above is arbitrary, we deduce that
$\overline{D}^{2}I_{k}(V)\leq 2C_{\widetilde{V}}<+\infty$ for every
$V\in(\widetilde{V}-\widetilde{\delta},\widetilde{V}+\widetilde{\delta})$. The
latter, together with the fact that $I_{k}$ is continuous, see Lemma 3.1,
implies that $I_{k}-C_{\widetilde{V}}V^{2}$ is concave on
$(\widetilde{V}-\widetilde{\delta},\widetilde{V}+\widetilde{\delta})$.

Passing to the limit $k\to+\infty$, as $I_{k}\to_{k}I$ locally uniformly by
Lemma 3.1, we deduce that $I-C_{\widetilde{V}}V^{2}$ is concave on
$(\widetilde{V}-\widetilde{\delta},\widetilde{V}+\widetilde{\delta})$ as well.
Also, if $K=0$, (3.18) implies that $I$ is concave. In any case, the profile
$I$ is twice differentiable almost everywhere, and if $\widetilde{V}$ is such
a point, (3.18) implies that

$I^{\prime\prime}(\widetilde{V})\leq\frac{(n-1)|K|}{I(\widetilde{V})}.$

∎

It would be interesting to understand how the above proof could be improved in
order to get concavity properties on the function
$I_{(M^{n},g)}^{\frac{n}{n-1}}$. Under the a priori assumption of existence of
isoperimetric sets, it has been studied in [16, 17, 18, 19, 56].

###  3.3. Consequences of the concavity properties of the isoperimetric
profile

In this section we collect some consequences of 1.4. First, since concave
functions are locally Lipschitz, we deduce the following useful corollary,
which improves the main result in [60, Theorem 2].

######  Corollary 3.5.

Suppose that $(M^{n},g)$ is noncollapsed and that $\mathrm{Ric}\geq(n-1)K$ for
some $K\in\mathbb{R}$. Then the isoperimetric profile $I_{(M^{n},g)}$ is a
locally Lipschitz function.

In the following result we exploit the concavity of $I$ under nonnegative
Ricci curvature, together with the sharp isoperimetric inequality (2.8), to
deduce the precise asymptotic expansion of $I$ and $I^{\prime}$ at infinity
when $\mathrm{Ric}\geq 0$.

######  Corollary 3.6.

Let $(M^{n},g)$ be a noncompact Riemannian manifold with $\mathrm{Ric}\geq 0$,
and denote by $I$ its isoperimetric profile. Then

(3.19)
$\lim_{v\to+\infty}\frac{I(v)}{v^{(n-1)/n}}=n(\omega_{n}\mathrm{AVR}(M^{n},g))^{1/n}.$

Moreover the right derivative $I^{\prime}_{+}(v)\vcentcolon=\lim_{h\to
0^{+}}\frac{I(v+h)-I(v)}{h}$ exists at any $v>0$ and it satisfies

(3.20)
$\lim_{v\to+\infty}v^{1/n}I^{\prime}_{+}(v)=(n-1)(\omega_{n}\mathrm{AVR}(M^{n},g))^{1/n}.$

###### Proof.

For simplicity, throughout the proof we denote
$\theta:=\mathrm{AVR}(M^{n},g)$. Let us first prove (3.19). We first deal with
the case $\theta>0$. Let us fix $p\in M^{n}$. From Bishop–Gromov comparison
theorem, see 2.11, we know that $P(B_{r}(p))/(n\omega_{n}r^{n-1})$ is almost
everywhere non-increasing on $[0,+\infty)$. Since
$\operatorname{vol}(B_{r}(p))/(\omega_{n}r^{n})\to\theta$, as $r\to+\infty$, a
standard use of the coarea formula for $\operatorname{vol}(B_{r}(p))$,
together with the monotonicity ensured by the Bishop–Gromov comparison result,
imply that the essential limit of $P(B_{r}(p))/(n\omega_{n}r^{n-1})$ as
$r\to+\infty$ is $\theta$ as well. Hence, for every $\varepsilon>0$ there
exists $r_{0}:=r_{0}(\varepsilon)$ such that for almost every $r\geq r_{0}$ we
have

$P(B_{r}(p))\leq(1+\varepsilon)\theta n\omega_{n}r^{n-1}.$

Moreover, by approximating balls from the outside and by using the
semicontinuity of the perimeter, the previous inequality holds for all $r\geq
r_{0}$. Hence for every $V$ sufficiently large, there exists a ball of center
$p$, radius $r:=r(V)\geq r_{0}$ and volume $V$ such that

$I(V)\leq P(B_{r}(p))\leq(1+\varepsilon)\theta
n\omega_{n}r^{n-1}\leq(1+\varepsilon)n(\omega_{n}\theta)^{1/n}V^{(n-1)/n},$

where in the last inequality we used $\theta\omega_{n}r^{n}\leq V$, which
comes from Bishop–Gromov comparison. Since $I(V)\geq
n(\omega_{n}\theta)^{1/n}V^{(n-1)/n}$ for every $V>0$, as an immediate outcome
of 2.21, we get (3.19) in the case $\theta>0$.

Let us now prove (3.19) when $\theta=0$. For every $V>0$ there exists a radius
$r:=r(V)$ such that $\operatorname{vol}(B_{r(V)}(p))=V$. Notice that
$r(V)\to+\infty$ when $V\to+\infty$. Moreover

(3.21)
$\begin{split}\frac{I(V)}{V^{(n-1)/n}}&\leq\frac{P(B_{r}(p))}{\operatorname{vol}(B_{r}(p))^{(n-1)/n}}=\frac{P(B_{r}(p))}{\operatorname{vol}(B_{r}(p))}\left(\frac{\operatorname{vol}(B_{r}(p))}{\omega_{n}r^{n}}\right)^{1/n}r\omega_{n}^{1/n}\\\
&\leq
n\omega_{n}^{1/n}\left(\frac{\operatorname{vol}(B_{r}(p))}{\omega_{n}r^{n}}\right)^{1/n},\end{split}$

where in the first inequality we used the definition of isoperimetric profile,
and in the last inequality we used the comparison in (2.3). Since $\theta=0$
we have that for every $\varepsilon$ there exists $r_{0}:=r_{0}(\varepsilon)$
such that $\operatorname{vol}(B_{r}(p))/(\omega_{n}r^{n})\leq\varepsilon$ for
every $r\geq r_{0}$. Hence from (3.21) we conclude the sought limit in (3.19)
also in the case $\theta=0$.

Now we can prove (3.20). If $M$ is not noncollapsed, then (3.20) is trivially
verified due to the equivalence in 2.18. So let us prove (3.20) when $M^{n}$
is noncollapsed. By 1.4 we know that $I$ is concave, which implies the
existence of $I^{\prime}_{+}$ everywhere and that $I^{\prime}_{+}$ is
nonincreasing.

We claim that, for any $\varepsilon>0$ and $0<\delta<2$ there exists
$\bar{v}=\bar{v}(\varepsilon,\delta)$ such that

(3.22)
$\left|\frac{I(v(1+\delta))-I(v)}{(v(1+\delta))^{\frac{n-1}{n}}-v^{\frac{n-1}{n}}}-n(\omega_{n}\theta)^{1/n}\right|\leq\varepsilon\quad\text{for
any $v\geq\bar{v}$.}$

Indeed the latter follows from (3.19) and the inequality

$\displaystyle\left|\frac{I(v(1+\delta))-I(v)}{(v(1+\delta))^{\frac{n-1}{n}}-v^{\frac{n-1}{n}}}-n(\omega_{n}\theta)^{1/n}\right|$
$\displaystyle\leq\frac{(1+\delta)^{\frac{n-1}{n}}}{(1+\delta)^{\frac{n-1}{n}}-1}\left|\frac{I(v(1+\delta))}{(v(1+\delta))^{\frac{n-1}{n}}}-n(\omega_{n}\theta)^{1/n}\right|+\frac{1}{(1+\delta)^{\frac{n-1}{n}}-1}\left|\frac{I(v)}{v^{\frac{n-1}{n}}}-n(\omega_{n}\theta)^{1/n}\right|\,.$

The monotonicity of $I_{+}^{\prime}$ gives

(3.23)
$I(v(1+\delta))-I(v)=\int_{v}^{v(1+\delta)}I^{\prime}_{+}(t)dt\geq\delta
vI_{+}^{\prime}(v(1+\delta))\,,$

and

(3.24)
$I(v(1+\delta))-I(v)=\int_{v}^{v(1+\delta)}I^{\prime}_{+}(t)dt\leq\delta
vI_{+}^{\prime}(v)\,,$

for all $v>0$, where we also employed the fact that $I$ is Lipschitz, as
observed in 3.5, in order to apply the Fundamental Theorem of Calculus. Hence,
we get

(3.25)
$\begin{split}\frac{I(v(1+\delta))-I(v)}{(v(1+\delta))^{\frac{n-1}{n}}-v^{\frac{n}{n-1}}}&\geq\frac{\delta
v\,I^{\prime}_{+}(v(1+\delta))}{\frac{n-1}{n}\int_{v}^{v(1+\delta)}t^{-1/n}dt}\\\
&\geq(1+\delta)^{-1/n}\frac{n}{n-1}(v(1+\delta))^{1/n}I^{\prime}_{+}(v(1+\delta))\,,\end{split}$

and

(3.26)
$\frac{I(v(1+\delta))-I(v)}{(v(1+\delta))^{\frac{n-1}{n}}-v^{\frac{n}{n-1}}}\leq\frac{\delta
v\,I^{\prime}_{+}(v)}{\frac{n-1}{n}\int_{v}^{v(1+\delta)}t^{-1/n}dt}\leq(1+\delta)^{1/n}\frac{n}{n-1}v^{1/n}I^{\prime}_{+}(v)\,,$

for all $v>0$. The latter, together with (3.22), implies

$\displaystyle(1+\delta)^{-1/n}(n-1)$
$\displaystyle(\omega_{n}\theta)^{1/n}-C(n)\varepsilon$
$\displaystyle\leq\liminf_{v\to\infty}v^{1/n}I_{+}^{\prime}(v)$
$\displaystyle\leq\limsup_{v\to\infty}v^{1/n}I_{+}^{\prime}(v)$
$\displaystyle\leq(1+\delta)^{1/n}(n-1)(\omega_{n}\theta)^{1/n}+C(n)\varepsilon\,.$

Letting $\varepsilon,\delta\to 0$ we get the sought conclusion. ∎

######  Remark 3.7.

We remark that (3.19) holds for arbitrary $\mathsf{CD}(0,n)$ spaces since it
only relies on Bishop–Gromov monotonicity, see Remark 2.12, and 2.21.

In the following corollary we observe that for an arbitrary manifold with
nonnegative Ricci curvature and Euclidean volume growth the isoperimetric
profile is strictly increasing. This should be compared with [65, Theorem 3.3]
where the author shows that on every complete noncompact Riemannian manifold
with strictly positive sectional curvature the isoperimetric profile is
strictly increasing.

######  Corollary 3.8.

Let $(M^{n},g)$ be a noncompact Riemannian manifold with $\mathrm{Ric}\geq 0$
and $\mathrm{AVR}(M^{n},g)>0$. Then the isoperimetric profile $I_{(M^{n},g)}$
is strictly increasing.

###### Proof.

Since $I_{(M^{n},g)}$ is concave due to 1.4, we have that $I^{\prime}_{+}$ is
nonincreasing. Moreover, since we have the asymptotic relation in (3.19) and
$\mathrm{AVR}(M^{n},g)>0$, we conclude that $I^{\prime}_{+}\geq 0$. Hence, if
for some $v_{0}>0$ we have $I^{\prime}_{+}(v_{0})=0$, therefore
$I^{\prime}_{+}(v)=0$ for every $v\geq v_{0}$, which is in contradiction with
(3.19). In conclusion $I^{\prime}_{+}>0$ everywhere, and hence $I$ is strictly
increasing. ∎

##  4\. Main isoperimetric existence results

In this section we prove the main existence result 1.1 and the consequent 1.2
and 1.3.

###  4.1. Proof of 1.1

Let us denote for simplicity $\theta:=\mathrm{AVR}(M^{n},g)$. We divide the
proof in two steps. The first step is the following Lemma in which we prove
that, under the hyptoheses of 1.1, for large volumes $V$ any minimizing
sequence of bounded finite perimeter sets of volume $V$ can lose at most one
piece at infinity along $N$.

######  Lemma 4.1.

Let $k\geq 0$ be a natural number, and let $(M^{n},g)=(\mathbb{R}^{k}\times
N^{n-k},g_{\mathbb{R}^{k}}+g_{N})$ be a complete noncollapsed Riemannian
manifold with $\mathrm{Ric}\geq 0$. Assume that $M$ satisfies the hypothesis
of 1.1. Then there exists $\overline{V}>0$ such that the following holds.

For every $V\geq\overline{V}$, let $\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ be a
minimizing sequence for the volume $V$ of bounded finite perimeter sets. Let
$p_{i,j}:=(t_{i,j},x_{i,j})\in\mathbb{R}^{k}\times N$ be the points for which
2.16 applied to $\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ holds, using the same
notation therein, and for a fixed $x_{0}\in N$ let

(4.1)
$\begin{split}J_{1}\vcentcolon=&\\{j:\sup_{i}\mathsf{d}_{N}(x_{0},x_{i,j})<+\infty\\},\\\
J_{2}\vcentcolon=&\\{j:\limsup_{i}\mathsf{d}_{N}(x_{0},x_{i,j})=+\infty\\}.\end{split}$

Then for every $j\in J_{2}$ we have
$\lim_{i}\operatorname{vol}(\Omega_{i,j}^{d})>V/2$. In particular, $J_{2}$
consists of at most one element for any $V\geq\overline{V}$.

###### Proof.

Let us observe first that (3.20) implies that there exists $\widehat{V}>0$
such that

$\begin{split}I^{\prime}_{+}(v)<\frac{n}{n-1}(n-1)(\omega_{n}(\theta+\varepsilon/2))^{1/n}v^{-1/n}&\qquad\forall
v\geq\widehat{V},\\\ \end{split}$

where the constant $\varepsilon$ is the one provided by the hypothesis in 1.1
. In particular

$\begin{split}I(V)-I(V-\delta)&<\int_{V-\delta}^{V}\frac{n}{n-1}(n-1)(\omega_{n}(\theta+\varepsilon/2))^{1/n}v^{-1/n}\,\mathrm{d}v\\\
&\leq
n(\omega_{n}(\theta+\varepsilon/2))^{1/n}\frac{\delta}{(V-\delta)^{1/n}},\end{split}$

whenever $V>V-\delta\geq\widehat{V}$. Also, we have that
$\delta/(V-\delta)^{1/n}\leq\delta^{1-1/n}$ if $\delta\leq V/2$. Therefore

(4.2)
$I(V)-I(V-\delta)<n(\omega_{n}(\theta+\varepsilon/2))^{1/n}\delta^{1-1/n},$

whenever $V>V-\delta\geq\widehat{V}$ and $\delta\leq V/2$.

So we choose $\overline{V}\vcentcolon=2\widehat{V}$. Let $V\geq\overline{V}$
and $\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ be an arbitrary minimizing sequence of
bounded finite perimeter sets of volume $V$. Let $\Omega_{i,j}^{d}$ be defined
as in 2.16, and we adopt the notation therein. Suppose by contradiction that
$\mathfrak{m}_{j}(Z_{j})=\lim_{i}\operatorname{vol}(\Omega_{i,j}^{d})\leq V/2$
for some $j\in J_{2}$. From 2.16 we have

$I(V)=P_{X_{j}}(Z_{j})+P(\Omega)+\sum_{\ell\neq
j}P_{X_{\ell}}(Z_{\ell})=P_{X_{j}}(Z_{j})+I(\operatorname{vol}(\Omega))+\sum_{\ell\neq
j}I_{X_{\ell}}(\mathfrak{m}_{\ell}(Z_{\ell})).$

By 2.21, by hypothesis, and by [13, Proposition 3.2] we then get

$\begin{split}I(V)&\geq
n(\omega_{n}(\theta+\varepsilon))^{1/n}\mathfrak{m}_{j}(Z_{j})^{(n-1)/n}+I(\operatorname{vol}(\Omega))+\sum_{\ell\neq
j}I_{X_{\ell}}(\mathfrak{m}_{\ell}(Z_{\ell}))\\\ &\geq
n(\omega_{n}(\theta+\varepsilon))^{1/n}\mathfrak{m}_{j}(Z_{j})^{(n-1)/n}+I\left(\operatorname{vol}(\Omega)+\sum_{\ell\neq
j}\mathfrak{m}_{\ell}(Z_{\ell})\right)\\\
&=n(\omega_{n}(\theta+\varepsilon))^{1/n}\mathfrak{m}_{j}(Z_{j})^{(n-1)/n}+I\left(V-\mathfrak{m}_{j}(Z_{j})\right).\end{split}$

By the absurd hypothesis we can take $\delta=\mathfrak{m}_{j}(Z_{j})$ in (4.2)
contradicting the above estimate, and thus completing the proof of the claim.
∎

Let us now conclude the proof of 1.1 by exploiting the previous Lemma. Let us
fix $p:=(0,x_{0})\in\mathbb{R}^{k}\times N$. We now show that there exists
$V_{0}\geq\overline{V}$ such that for every $V\geq V_{0}$ there exists an
isoperimetric region of volume $V$, thus concluding the proof of the Theorem.

To reach the latter conclusion we start proving that there exists
$V_{0}\geq\overline{V}$ such that the following improvement of Lemma 4.1
holds. Let $V\geq V_{0}$, and let $\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ be a
minimizing sequence for the volume $V$ of bounded finite perimeter sets. Let
$p_{i,j}:=(t_{i,j},x_{i,j})\in\mathbb{R}^{k}\times N$ be the points for which
2.16 applied to $\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ holds, using the same
notation therein, and define

$\begin{split}J_{1}\vcentcolon=&\\{j:\sup_{i}\mathsf{d}_{N}(x_{0},x_{i,j})<+\infty\\},\\\
J_{2}\vcentcolon=&\\{j:\limsup_{i}\mathsf{d}_{N}(x_{0},x_{i,j})=+\infty\\},\end{split}$

We claim that $J_{2}$ is actually empty.

Let us prove the previous claim by contradiction, after having provided the
precise value of $V_{0}$. Let $\varepsilon$ be the constant given by the
hypothesis of 1.1. Let us first consider the case $\theta>0$. Let us define

$\varepsilon^{*}:=(1+\varepsilon/\theta)^{1/n}-1.$

Let $V_{0}\geq\overline{V}$ be big enough, where $\overline{V}$ is provided by
Lemma 4.1, such that

(4.3) $I(V)\leq
n(\omega_{n}\theta)^{1/n}V^{(n-1)/n}\left(1+2^{-1}\varepsilon^{*}(1/3)^{(n-1)/n}\right),\qquad\text{for
all $V\geq V_{0}$.}$

Such a value of $V_{0}$ exists thanks to 3.6. From now on, we fix $V\geq
V_{0}$ and we fix a minimizing sequence $\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ of
bounded finite perimeter sets of volume $V$ for which, by contradiction, there
exists exactly one $j_{0}\in J_{2}$, by Lemma 4.1. From (2.8) and the
hypothesis we have

(4.4)
$\frac{P_{X_{j_{0}}}(Z_{j_{0}})}{\mathfrak{m}_{j_{0}}(Z_{j_{0}})^{(n-1)/n}}\geq
n\omega_{n}^{1/n}\mathrm{AVR}(X_{j_{0}},\mathsf{d}_{j_{0}},\mathfrak{m}_{j_{0}})^{1/n}\geq
n\omega_{n}^{1/n}(\theta+\varepsilon)^{1/n}.$

From Lemma 4.1 and from item (iii) of 2.16 we have

(4.5)
$\mathfrak{m}_{j_{0}}(Z_{j_{0}})=\lim_{i}\operatorname{vol}(\Omega_{i,{j_{0}}}^{d})\geq
V/3.$

For every $j\in J_{1}$, we have that
$(M,\mathsf{d},p_{i,j},\operatorname{vol})\to_{i}(M,\mathsf{d},\overline{p}_{j},\operatorname{vol})$
in the pmGH topology, for some $\overline{p}_{j}\in M$. Indeed, since $j\in
J_{1}$ we have that, up to subsequences, $x_{i,j}\to_{i}\overline{x}_{j}\in
N$. Hence it suffices to choose $\overline{p}_{j}:=(0,\overline{x}_{j})$
thanks to the homogeneity in the factor $\mathbb{R}^{k}$. Hence, by the result
in 2.16, we have that $\Omega_{i,j}\to_{i}Z_{j}$, where $Z_{j}$ is an
isoperimetric region in $M$. Now, by exploiting item (iv) of 2.16, taking into
account (4.4) and (4.5), exploiting also the isoperimetric inequality in 2.21,
we finally get, recalling that
$\varepsilon^{*}:=(1+\varepsilon/\theta)^{1/n}-1$, that

(4.6) $\begin{split}I(V)&=P(\Omega)+\sum_{j\in
J_{1}}P(Z_{j})+P_{X_{j_{0}}}(Z_{j_{0}})\\\ &\geq
n(\omega_{n}\theta)^{1/n}\left(\operatorname{vol}(\Omega)^{(n-1)/n}+\sum_{j\in
J_{1}}\operatorname{vol}(Z_{j})^{(n-1)/n}+(1+\varepsilon/\theta)^{1/n}\mathfrak{m}_{j_{0}}(Z_{j_{0}})^{(n-1)/n}\right)\\\
&\geq
n(\omega_{n}\theta)^{1/n}\left(\left(\operatorname{vol}(\Omega)+\sum_{j=1}^{\overline{N}}\mathfrak{m}_{j}(Z_{j})\right)^{(n-1)/n}+\varepsilon^{*}\mathfrak{m}_{j_{0}}(Z_{j_{0}})^{(n-1)/n}\right)\\\
&\geq
n(\omega_{n}\theta)^{1/n}V^{(n-1)/n}\left(1+\varepsilon^{*}(1/3)^{(n-1)/n}\right),\end{split}$

which is a contradiction with (4.3). Hence $J_{2}$ is empty.

Let us now consider the case $\theta=0$. From (3.19) we have that there exists
$V_{0}\geq\overline{V}$ such that for all $V\geq V_{0}$ we have

(4.7) $I(V)\leq
n(\omega_{n}2^{-1}(1/3)^{(n-1)/n}\varepsilon)^{1/n}V^{(n-1)/n}.$

Arguing similarly as before we have, for every $V\geq V_{0}$, and for every
minimizing sequence $\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ of bounded sets of
finite perimeter of volume $V$, by using the same notation as before,

(4.8) $\begin{split}I(V)&=P(\Omega)+\sum_{j\in
J_{1}}P(Z_{j})+P_{X_{j_{0}}}(Z_{j_{0}})\geq P_{X_{j_{0}}}(Z_{j_{0}})\\\ &\geq
n(\omega_{n}\varepsilon)^{1/n}\mathfrak{m}_{j_{0}}(Z_{j_{0}})^{(n-1)/n}\geq
n(\omega_{n}(1/3)^{(n-1)/n}\varepsilon)^{1/n}V^{(n-1)/n},\end{split}$

which is a contradiction with (4.7), and thus $J_{2}$ is empty also in this
case.

Let us now take $V_{0}\geq\overline{V}$ such that the previous claim about
$J_{2}$ to be empty (for every minimizing sequence of bounded sets of finite
perimeter of volume greater than $V_{0}$) holds. Let us now conclude that for
every $V\geq V_{0}$ there exists an isoperimetric region of volume $V$ in $M$.
Indeed, let us take a minimizing sequence $\\{\Omega_{i}\\}_{i\in\mathbb{N}}$
of bounded finite perimeter sets of volume $V$. We have already shown that for
every $j\in J_{1}$, and thus for all $j$’s since $J_{2}$ is empty as $V\geq
V_{0}$, we have that
$(M,\mathsf{d},p_{i,j},\operatorname{vol})\to_{i}(M,\mathsf{d},\overline{p}_{j},\operatorname{vol})$
in the pmGH topology, for some $\overline{p}_{j}\in M$. From 2.17 we get that
$Z_{j}$ is bounded for every $j$ since it is an isoperimetric region in $M$.
Hence, by properly translating $Z_{j}$ along the coordinate $\mathbb{R}^{k}$
into some $Z_{j}^{\prime}$, we can define
$\Omega^{\prime}:=\sqcup_{j=1}^{\overline{N}}Z^{\prime}_{j}$, where the
$Z_{j}^{\prime}$s are mutually disjoint. Moreover, from (2.5) we get that
$\operatorname{vol}(\Omega^{\prime})=V$. Hence, from (2.5), we conclude that

(4.9) $I(V)=P(\Omega)+\sum_{j=1}^{\overline{N}}P(Z_{j})=P(\Omega^{\prime})\geq
I(V),$

from which $\Omega^{\prime}$ is the sought isoperimetric region of volume $V$.

###  4.2. Proof of 1.2

In order to complete the proof of 1.2 we will use 1.1 together with the
following Lemma, that we state and prove in the general $\mathsf{RCD}$
setting. In item (i) of Lemma 4.2 we show that when the hypothesis of 1.2
holds, then the density of the points at distance one from the tips of the
asymptotic cones of $M^{n}$ is uniformly greater than $\mathrm{AVR}(M^{n},g)$.
In item (ii) of Lemma 4.2 we show that when the previous conclusion on the
density holds, then every pmGH limit at infinity has $\mathrm{AVR}$ uniformly
greater than the $\mathrm{AVR}(M^{n},g)$.

######  Lemma 4.2.

Let $k\geq 0$ be a natural number, let $(Y,\mathsf{d}_{Y},\mathfrak{m}_{Y})$
be a metric measure space, and assume

$(X:=\mathbb{R}^{k}\times
Y,\mathsf{d}:=\mathsf{d}_{\mathbb{R}^{k}}\otimes\mathsf{d}_{Y},\mathfrak{m}:=\mathfrak{m}_{\mathbb{R}^{k}}\otimes\mathfrak{m}_{Y}),$

is an $\mathsf{RCD}(0,n)$ space. Assume

(4.10) $\mathrm{AVR}(X,\mathsf{d},\mathfrak{m})>0.$

Then the following two statements hold.

  * (i)

If there exists no asymptotic cone of $(Y,\mathsf{d}_{Y},\mathfrak{m}_{Y})$
that splits a line, then there exists an ${\varepsilon}>0$ such that for every
asymptotic cone

$(\mathbb{R}^{k}\times
C,\mathsf{d}_{\mathbb{R}^{k}}\otimes\mathsf{d}_{C},\mathfrak{m}_{\mathbb{R}^{k}}\otimes\mathfrak{m}_{C},(0,v_{\infty})),$

of $(X,\mathsf{d},\mathfrak{m})$, and for every $p\in\mathbb{R}^{k}\times C$
with
$\mathsf{d}_{\mathbb{R}^{k}}\otimes\mathsf{d}_{C}(p,\mathbb{R}^{k}\times\\{v_{\infty}\\})=1$,
we have

$\,\,\,\lim_{r\to
0}\frac{\mathfrak{m}_{\mathbb{R}^{k}}\otimes\mathfrak{m}_{C}(B_{r}(p))}{\omega_{n}r^{n}}=:\vartheta[(\mathbb{R}^{k}\times
C,\mathsf{d}_{\mathbb{R}^{k}}\otimes\mathsf{d}_{C},\mathfrak{m}_{\mathbb{R}^{k}}\otimes\mathfrak{m}_{C},p)]\geq\mathrm{AVR}(X,\mathsf{d},\mathfrak{m})+{\varepsilon}.$

  * (ii)

Assume that there exists $\alpha>0$ such that

$\lim_{r\to
0}\frac{\mathfrak{m}_{\mathbb{R}^{k}}\otimes\mathfrak{m}_{C}(B_{r}(p))}{\omega_{n}r^{n}}\geq\alpha,$

for every asymptotic cone

$(\mathbb{R}^{k}\times
C,\mathsf{d}_{\mathbb{R}^{k}}\otimes\mathsf{d}_{C},\mathfrak{m}_{\mathbb{R}^{k}}\otimes\mathfrak{m}_{C},(0,v_{\infty})),$

of $(X,\mathsf{d},\mathfrak{m})$ and for every $p\in\mathbb{R}^{k}\times C$
with
$\mathsf{d}_{\mathbb{R}^{k}}\otimes\mathsf{d}_{C}(p,\mathbb{R}^{k}\times\\{v_{\infty}\\})=1$.
Fix $x_{0}:=(0,y_{0})\in\mathbb{R}^{k}\times Y$. Fix
$\\{x_{i}:=(t_{i},y_{i})\\}_{i\geq 1}\subset X$ a sequence with
$\mathsf{d}_{Y}(y_{i},y_{0})\to_{i}+\infty$, and let $X_{\infty}$ be a pmGH
limit of a subsequence of $(X,\mathsf{d},\mathfrak{m},x_{i})$, namely

$(X,\mathsf{d},\mathfrak{m},x_{i})\to_{i}(X_{\infty},\mathsf{d}_{\infty},\mathfrak{m}_{\infty},x_{\infty}),$

in the pmGH sense up to a subsequence. Hence,
$\mathrm{AVR}(X_{\infty},\mathsf{d}_{\infty},\mathfrak{m}_{\infty})\geq\alpha$.

###### Proof.

In the proof for simplicity we denote
$\theta_{X}:=\mathrm{AVR}(X,\mathsf{d},\mathfrak{m})$. Inductively using [42,
Theorem 7.4], one deduces that $(Y,\mathsf{d}_{Y},\mathfrak{m}_{Y})$ is an
$\mathsf{RCD}(0,n-k)$ space. Moreover, one can check that ${\rm
AVR}(Y,\mathsf{d}_{Y},\mathfrak{m}_{Y})>0$ as well.

Let us first prove item (i). Let us assume by contradiction that there exist,
for every $i\in\mathbb{N}$, asymptotic cones
$(C_{i}^{\prime},\mathsf{d}_{i},\mathfrak{m}_{i},(0,v_{\infty,i}))$ of
$(X,\mathsf{d},\mathfrak{m})$, and points $p_{i}\in C_{i}^{\prime}$ such that
$\mathsf{d}_{i}(p_{i},\mathbb{R}^{k}\times\\{v_{\infty,i}\\})=1$ and

$\vartheta[(C_{i}^{\prime},\mathsf{d}_{i},\mathfrak{m}_{i},p_{i})]\leq\theta_{X}+1/i.$

Recall that $C_{i}^{\prime}=\mathbb{R}^{k}\times C_{i}$, for some cone $C_{i}$
with tip $v_{\infty,i}$. Notice that by translating along $\mathbb{R}^{k}$ we
may assume that the $\mathbb{R}^{k}$ coordinate of $p_{i}$ is $0$. Let us
first notice that the sequence
$\\{(C_{i}^{\prime},\mathsf{d}_{i},\mathfrak{m}_{i},(0,v_{\infty,i}))\\}_{i\in\mathbb{N}}$
is precompact in the pmGH topology. Indeed, since any
$(C_{i}^{\prime},\mathsf{d}_{i},\mathfrak{m}_{i},(0,v_{\infty,i}))$ is the
pmGH limit of a rescaling of $X$, the $C_{i}^{\prime}$’s are
$\mathsf{RCD}(0,n)$ spaces with
$\mathfrak{m}_{i}(B_{1}(0,v_{\infty,i}))=\theta_{X}$ for every
$i\in\mathbb{N}$, due to the volume convergence result in 2.6. Hence
precompactness follows from Remark 2.7. Moreover, any limit of such a sequence
of asymptotic cones must be an asymptotic cone of
$(X,\mathsf{d},\mathfrak{m})$, due to the fact that the pmGH-topology, in our
case, is metrizable, see [44, Theorem 3.15]. Indeed, by exploiting the latter
information, a diagonal argument implies that any pmGH limit point of the
sequence
$\\{(C_{i}^{\prime},\mathsf{d}_{i},\mathfrak{m}_{i},(0,v_{\infty,i}))\\}_{i\in\mathbb{N}}$
is also an asymptotic cone of $X$.

Hence, let us take
$(C^{\prime},\mathsf{d}_{C^{\prime}},\mathfrak{m}_{C^{\prime}},(0,v_{\infty}))$
an asymptotic cone of $(X,\mathsf{d},\mathfrak{m})$, that is also a pmGH limit
of the sequence
$\\{(C_{i}^{\prime},\mathsf{d}_{i},\mathfrak{m}_{i},(0,v_{\infty,i}))\\}_{i\in\mathbb{N}}$.
In a proper realization of such a pmGH limit, we can assume that, up to
further subsequences, $p_{i}\to p\in C^{\prime}$, and since
$\mathsf{d}_{i}(p_{i},\mathbb{R}^{k}\times\\{v_{\infty,i}\\})=1$, we get by
continuity of the distance that
$\mathsf{d}(p,\mathbb{R}^{k}\times\\{v_{\infty}\\})=1$, and hence
$p\notin\mathbb{R}^{k}\times\\{v_{\infty}\\}$. Since $(0,v_{\infty})$ is one
tip of the asymptotic cone $C^{\prime}$, from the volume convergence result in
2.6, we conclude that

(4.11)
$\mathfrak{m}_{C^{\prime}}(B_{r}(0,v_{\infty}))=\theta_{X}\omega_{n}r^{n},$

for every $r>0$. By an immediate consequence of Bishop–Gromov volume
comparison and the volume convergence we also have that

(4.12)
$\vartheta[(C^{\prime},\mathsf{d}_{C^{\prime}},\mathfrak{m}_{C^{\prime}},p)]\geq\theta_{X}.$

Moreover, since the density $\vartheta$ is lower semicontinuous with respect
to the pmGH convergence, see [35, Lemma 2.2 (i)], we also conclude that

(4.13)
$\vartheta[(C^{\prime},\mathsf{d}_{C^{\prime}},\mathfrak{m}_{C^{\prime}},p)]\leq\liminf_{i\to+\infty}\vartheta[(C_{i}^{\prime},\mathsf{d}_{i},\mathfrak{m}_{i},p_{i})]\leq\theta_{X}.$

Hence, from (4.12) and (4.13), we deduce that

(4.14) $\lim_{r\to
0}\frac{\mathfrak{m}_{C^{\prime}}(B_{r}(p))}{\omega_{n}r^{n}}=:\vartheta[(C^{\prime},\mathsf{d}_{C^{\prime}},\mathfrak{m}_{C^{\prime}},p)]=\theta_{X}.$

Since $p$ and $(0,v_{\infty})$ are at finite distance, from (4.11), a simple
argument involving the triangle inequality, and the monotonicity of
Bishop–Gromov ratios (see Remark 2.12) we also deduce that

(4.15)
$\lim_{r\to+\infty}\frac{\mathfrak{m}_{C^{\prime}}(B_{r}(p))}{\omega_{n}r^{n}}=\theta_{X}.$

From (4.14), (4.15), and the monotonicity of the Bishop–Gromov ratios, see
Remark 2.12, we deduce that
$\mathfrak{m}_{C^{\prime}}(B_{r}(p))=\theta_{X}\omega_{n}r^{n}$ for every
$r>0$. Hence, from the result in [34, Theorem 1.1], we conclude that
$C^{\prime}$ is a metric cone with tip
$p\notin\mathbb{R}^{k}\times\\{v_{\infty}\\}$. Hence
$C^{\prime}=\mathbb{R}^{k}\times C$ is also metric cone over a tip that is not
in $\mathbb{R}^{k}\times\\{v_{\infty}\\}$, and then from [12, Proposition
1.18] we get that $C$ splits a line. But this is not possible, since $C$ is an
asymptotic cone of $Y$ that by hypothesis does not split a line, thus giving
the sought contradiction.

Let us now prove item (ii). Since $(X,\mathsf{d},\mathfrak{m})$ is an
$\mathsf{RCD}(0,n)$ space we conclude by stability that also
$(X_{\infty},\mathsf{d}_{\infty},\mathfrak{m}_{\infty})$ is an
$\mathsf{RCD}(0,n)$ space. Let us denote $\widetilde{x}_{i}:=(0,y_{i})$ and
observe that $(X,\mathsf{d},\mathfrak{m},x_{i})$ is isomorphic to
$(X,\mathsf{d},\mathfrak{m},\widetilde{x}_{i})$. Then we may suppose that the
pmGH limit $(X_{\infty},\mathsf{d}_{\infty},\mathfrak{m}_{\infty},x_{\infty})$
is obtained through (a sub)sequence of
$(X,\mathsf{d},\mathfrak{m},\widetilde{x}_{i})$.

Since we have that $\mathsf{d}_{Y}(y_{i},y_{0})\to_{i}+\infty$, if we set
$\rho_{i}:=\mathsf{d}(x_{0},\widetilde{x}_{i})$, we get that, up to
subsequences in $i$,
$(X,\rho_{i}^{-1}\mathsf{d},\rho_{i}^{-n}\mathfrak{m},x_{0})$ pmGH-converges
to an asymptotic cone
$(C^{\prime},\mathsf{d}_{C^{\prime}},\mathfrak{m}_{C^{\prime}},(0,v_{\infty}))$.
Notice that $C^{\prime}=\mathbb{R}^{k}\times C$ for some cone $C$ with tip
$v_{\infty}$. Moreover, in a realization of such a convergence,
$\widetilde{x}_{i}\to_{i}p$ such that
$\mathsf{d}_{C^{\prime}}(p,\mathbb{R}^{k}\times\\{z_{\infty}\\})=1$ from the
fact that both $x_{0}$ and $\widetilde{x}_{i}$ have zero component along
$\mathbb{R}^{k}$. From the hypothesis we get that, for some $\alpha>0$, we
have

$\lim_{r\to
0}\frac{\mathfrak{m}_{C^{\prime}}(B_{r}(p))}{\omega_{n}r^{n}}\geq\alpha.$

Hence, for every $\eta$ there exists some $\delta:=\delta(\eta)>0$ for which

(4.16)
$\frac{\mathfrak{m}_{C^{\prime}}(B_{\delta}(p))}{\omega_{n}\delta^{n}}\geq\alpha-\eta.$

Since
$(X,\rho_{i}^{-1}\mathsf{d},\rho_{i}^{-n}\mathfrak{m},\widetilde{x}_{i})$
pmGH-converges to
$(C^{\prime},\mathsf{d}_{C^{\prime}},\mathfrak{m}_{C^{\prime}},p)$, the volume
convergence result in 2.6 implies that

(4.17)
$\lim_{i\to\infty}\frac{\mathfrak{m}(B_{\rho_{i}\delta}(\widetilde{x}_{i}))}{\omega_{n}(\rho_{i}\delta)^{n}}=\frac{\mathfrak{m}_{C^{\prime}}(B_{\delta}(p))}{\omega_{n}\delta^{n}}.$

Hence, from (4.16) and (4.17) we get that

(4.18)
$\frac{\mathfrak{m}(B_{\rho_{i}\delta}(\widetilde{x}_{i}))}{\omega_{n}(\rho_{i}\delta)^{n}}\geq\alpha-2\eta,\qquad\text{for
all $i\geq i_{0}(\eta)$ large enough}.$

Let us fix $R>0$. From the volume convergence result in 2.6 we have that

$\lim_{i\to\infty}\frac{\mathfrak{m}(B_{R}(\widetilde{x}_{i}))}{\omega_{n}R^{n}}=\frac{\mathfrak{m}_{\infty}(B_{R}(x_{\infty}))}{\omega_{n}R^{n}}.$

By Bishop–Gromov volume comparison on $X$ and (4.18) applied with $i\geq
i_{1}(\eta,R)$ large enough, we have

$\frac{\mathfrak{m}_{\infty}(B_{R}(x_{\infty}))}{\omega_{n}R^{n}}\geq\alpha-3\eta.$

Since $R,\eta>0$ are arbitrary we get the sought conclusion taking
$R\to+\infty$ and then $\eta\to 0$. ∎

###### Conclusion of the proof of 1.2.

It is a direct consequence of Lemma 4.2, and 1.1. ∎

###  4.3. Further existence results

As a consequence of item (ii) of Lemma 4.2, we can derive further existence
results on manifolds with nonnegative Ricci curvature and Euclidean volume
growth. The next theorem states that if points located at distance $1$ from
the tips of every asymptotic cone of a manifold have density $1$, then there
exist isoperimetric regions of every volume. This is ultimately due to the
fact that item (ii) of Lemma 4.2 implies that the essentially relevant pGH
limits at infinity are $\mathbb{R}^{n}$, allowing us to conclude by the main
existence result in [13].

######  Theorem 4.3.

Let $k\geq 0$ be a natural number, and let $(M^{n},g)=(\mathbb{R}^{k}\times
N^{n-k},g_{\mathbb{R}^{k}}+g_{N})$ be a complete Riemannian manifold such that
$\mathrm{Ric}\geq 0$ and $\mathrm{AVR}(M^{n},g)>0$. Assume that for every
asymptotic cone $(\mathbb{R}^{k}\times C,(0,v_{\infty}))$ of $M$ and every
$p\in\mathbb{R}^{k}\times C$ with
$\mathsf{d}_{\mathbb{R}^{k}}\otimes\mathsf{d}_{C}(p,\mathbb{R}^{k}\times\\{z_{\infty}\\})=1$,
we have

(4.19) $\lim_{r\to
0}\frac{\mathfrak{m}_{\mathbb{R}^{k}}\otimes\mathfrak{m}_{C}(B_{r}(p))}{\omega_{n}r^{n}}=:\vartheta[(\mathbb{R}^{k}\times
C,\mathsf{d}_{\mathbb{R}^{k}}\otimes\mathsf{d}_{C},\mathfrak{m}_{\mathbb{R}^{k}}\otimes\mathfrak{m}_{C},p)]=1.$

Then for every volume $V>0$ there exists an isoperimetric region of volume $V$
on $M^{n}$.

###### Proof.

We are in the setting of item (ii) of Lemma 4.2 with $\alpha=1$. From this we
deduce that for any pmGH-limit $X_{\infty}$ at infinity as in the statement of
item (ii) of Lemma 4.2 we have that
$\mathrm{AVR}(X_{\infty},\mathsf{d}_{\infty},\mathfrak{m}_{\infty})\geq 1$.
But since
$\mathrm{AVR}(X_{\infty},\mathsf{d}_{\infty},\mathfrak{m}_{\infty})\leq 1$ due
to Bishop–Gromov comparison and the fact that
$\mathfrak{m}_{\infty}=\mathcal{H}^{n}_{d_{\infty}}$ (cf. 2.6, and Remark
2.12), we have
$\mathrm{AVR}(X_{\infty},\mathsf{d}_{\infty},\mathfrak{m}_{\infty})=1$.
Therefore
$(X_{\infty},\mathsf{d}_{\infty},\mathfrak{m}_{\infty})=(\mathbb{R}^{n},\mathsf{d}_{\mathbb{R}^{n}},\mathfrak{m}_{\mathbb{R}^{n}})$
as a consequence of [35, Corollary 1.7] and the Bishop–Gromov comparison
discussed in Remark 2.12.

Now if $k=0$, this implies that $(M^{n},g)$ is GH-asymptotic to flat
$\mathbb{R}^{n}$ (see [13, Definition 1.2]), and therefore on $M^{n}$ we have
isoperimetric regions for every volume by [13, Theorem 5.2].

More generally, if $k>0$, we can still conclude that on $(M^{n},g)$ we have
isoperimetric regions of every volume $V>0$ arguing like at the end of the
proof of 1.1. Indeed, pGH limits along sequences $(x_{i},y_{i})$ with
$\\{y_{i}\\}_{i}$ bounded in $N$ are isometric to $(M^{n},g)$, and then
possible mass of a minimizing sequence lost along these sequence can be
obviously recovered as in the proof of 1.1. On the other hand we obtained that
pGH limits obtained along sequences diverging along $N^{n-k}$ are flat
$\mathbb{R}^{n}$, and then a minor modification in the proof of [13, Theorem
5.2] immediately shows that leak of the mass of a minimizing sequence along
such sequences would lead to a contradiction of the minimality assumption on
the sequence. ∎

We record a straightforward consequence of the above statement.

######  Corollary 4.4.

Let $(M^{n},g)$ be a complete Riemannian manifold such that $\mathrm{Ric}\geq
0$ and $\mathrm{AVR}(M^{n},g)>0$.

Assume that every asymptotic cone of $(M^{n},g)$ is _smooth outside the tip_ ,
i.e., it is of the form $C(Z)$ where $Z$ is a smooth closed manifold. Then for
every volume $V>0$ there exists an isoperimetric region of volume $V$ on
$M^{n}$.

###### Proof.

If every asymptotic cone of $M$ is smooth outside the tip, then (4.19) is
clearly satisfied, and thus the existence of isoperimetric regions follows. ∎

Let us observe that, in the setting of 4.4, if $n=2$ then every asymptotic
cone is (isometric to) a cone over a circle of some radius, which is obviously
smooth outside the tip. Hence the latter statement, when specialized to $n=2$,
recovers a particular case of the existence result of [64] on surfaces with
nonnegative curvature.

###  4.4. Proof of 1.3

As anticipated in the Introduction, for the proof of 1.3 we need to show that
manifolds with nonnegative sectional curvature have a unique asymptotic cone
that splits if and only if the manifold splits. This is a standard result in
the field, but since we were not able to find a satisfactory reference in the
literature, we add a short proof for the readers’ convenience. We first prove
an auxiliary lemma.

######  Lemma 4.5.

Let $(X,\mathsf{d}_{X})$ and $(Y,\mathsf{d}_{Y})$ be compact metric spaces. If
there exist maps $\Phi:X\to Y$ and $\Psi:Y\to X$ surjective and $1$-Lipschitz,
then $(X,\mathsf{d}_{X})$ and $(Y,\mathsf{d}_{Y})$ are isometric.

###### Proof.

It is enough to check that $T:=\Psi\circ\Phi$ is an isometry, indeed from

$\mathsf{d}_{X}(x,y)=\mathsf{d}_{X}(T(x),T(y))\leq\mathsf{d}_{Y}(\Phi(x),\Phi(y))\leq\mathsf{d}_{X}(x,y)$

we deduce that $\Phi$ is an isometry.

Fix distinct $x,y\in X$ and $\varepsilon\in(0,\mathsf{d}_{X}(x,y))$. Since $T$
is $1$-Lipschitz, it is enough to prove that
$\mathsf{d}_{X}(T(x),T(y))\geq\mathsf{d}_{X}(x,y)-\varepsilon$ and then let
$\varepsilon\to 0$. Set $D:=\mathsf{d}_{X}(x,y)-\varepsilon/2$. Given an
$\varepsilon/4$-dense set $S\subset X$, i.e., such that for any $z\in X$ there
exists $s\in S$ with $\mathsf{d}_{X}(z,s)\leq\varepsilon/4$, we define
$N(S):=\\#\\{(s_{1},s_{2})\in S\times S\,:\mathsf{d}_{X}(s_{1},s_{2})\geq
D\\}$. Let $S_{0}$ be an $\varepsilon/4$-dense set that minimizes the function
$N(\cdot)$ among $\varepsilon/4$-dense sets. Since $T$ is $1$-Lispchitz, then
$\mathsf{d}_{X}(T(s_{1}),T(s_{2}))\leq\mathsf{d}_{X}(s_{1},s_{2})$ for any
$s_{1},s_{2}\in S_{0}$, and since $T$ is also surjective, then $T(S_{0})$ is
$\varepsilon/4$-dense, therefore $N(T(S_{0}))\geq N(S_{0})$. This forces
$\mathsf{d}_{X}(T(s_{1}),T(s_{2}))\geq D$ for any $s_{1},s_{2}\in S_{0}$ with
$\mathsf{d}_{X}(s_{1},s_{2})\geq D$. In particular, if we pick $s_{1},s_{2}\in
S_{0}$ such that $\mathsf{d}_{X}(x,s_{1})\leq\varepsilon/4$ and
$\mathsf{d}_{X}(y,s_{2})\leq\varepsilon/4$ we have
$\mathsf{d}_{X}(s_{1},s_{2})\geq-\varepsilon/2+\mathsf{d}(x,y)=D$, and
therefore, as $T$ is $1$-Lipschitz, we conclude

$\mathsf{d}_{X}(T(x),T(y))\geq\mathsf{d}_{X}(T(s_{1}),T(s_{2}))-\varepsilon/2\geq
D-\varepsilon/2=\mathsf{d}(x,y)-\varepsilon\,.$

∎

######  Theorem 4.6.

If $(M^{n},g)$ has nonnegative sectional curvature and Euclidean volume
growth, then there exists a unique asymptotic cone
$(C,\mathsf{d},\mathcal{H}^{n})$. Moreover $C$ splits if and only if $M$
splits.

###### Proof.

Let us begin by proving the uniqueness of asymptotic cones.

Thanks to Lemma 4.5, it is enough to show that, given two asymptotic cones
$(C_{1},\mathsf{d}_{1},\mathfrak{m}_{1})$ and
$(C_{2},\mathsf{d}_{2},\mathfrak{m}_{2})$, there exists a surjective
$1$-Lipschitz map $\Psi:\overline{B}_{1}(z_{1})\to\overline{B}_{1}(z_{2})$,
where $z_{1}\in C_{1}$ and $z_{2}\in C_{2}$ are tip points.

Fix $p\in M$. Given $1<R_{1}<R_{2}$ we consider the set $E\subset
B_{R_{2}}(p)$ of those points $x\in B_{R_{2}}(p)$ such that there exists a
unique unit speed geodesic $\gamma_{p,x}:[0,\mathsf{d}(x,p)]\to M$ with
$\gamma_{p,x}(0)=p$, $\gamma_{p,x}(\mathsf{d}(x,p))=x$. Notice that
$\operatorname{vol}(B_{R_{2}}\setminus E)=0$, hence $E\subset B_{R_{2}}(p)$ is
dense.

We then define $T:E\subset B_{R_{2}}(p)\to B_{R_{1}}(p)$ as
$T(x):=\gamma_{p,x}(\mathsf{d}(x,p)R_{1}/R_{2})$. Toponogov’s theorem implies

(4.20)
$\mathsf{d}(T(x),T(y))R_{1}^{-1}\geq\mathsf{d}(x,y)R_{2}^{-1}\quad\text{for
any $x,y\in E$},$

in particular $T^{-1}:(T(E)\subset
B_{R_{1}}(p),\mathsf{d}/R_{1})\to(B_{R_{2}}(p),\mathsf{d}/R_{2})$ is
$1$-Lipschitz and has dense image, hence it can be extended to a surjective
$1$-Lipschitz map
$\Psi_{R_{1},R_{2}}:(K_{R_{1},R_{2}},\mathsf{d}/R_{1})\to(\overline{B}_{R_{2}}(p),\mathsf{d}/R_{2})$,
where $K_{R_{1},R_{2}}:=\overline{T(E)}$.

Let us now consider two sequences $r_{i}\to\infty$ and $s_{i}\to\infty$
realizing $(C_{1},\mathsf{d}_{1},\mathfrak{m}_{1})$ and
$(C_{2},\mathsf{d}_{2},\mathfrak{m}_{2})$, respectively. We assume without
loss of generality that $s_{i}<r_{i}<s_{i+1}<r_{i+1}$ for any $i\geq 1$. By
the Gromov–Hausdorff convergence of the sequences of the rescaled space to the
asymptotic cones, by a classical Ascoli–Arzelà-type argument we can pass to
the limit the maps
$\Psi_{s_{i},r_{i}}:((K_{s_{i},r_{i}},\mathsf{d}/s_{i})\to(\overline{B}_{r_{i}}(p),\mathsf{d}/r_{i})$
and we derive the existence of a $1$-Lipschitz surjective map

(4.21) $\Psi:K\subset\overline{B}_{1}(z_{1})\to\overline{B}_{1}(z_{2})\,,$

where $z_{1}\in C_{1}$ and $z_{2}\in C_{2}$ are tip points, and $K$ is
compact. Since $(M^{n},g)$ has Euclidean volume growth, the volume convergence
theorem, see 2.6, guarantees that the reference measures on the asymptotic
cones is the Hausdorff measure (with respect to their own distance), and
$\mathcal{H}^{n}((B_{1}(z_{1}))=\mathfrak{m}_{1}(B_{1}(z_{1}))=\omega_{n}{\rm
AVR}(M^{n},g)=\mathfrak{m}_{2}(B_{1}(z_{2}))=\mathcal{H}^{n}(B_{1}(z_{2}))$,
hence

$\displaystyle\mathcal{H}^{n}((B_{1}(z_{1}))\setminus K)$
$\displaystyle=\mathcal{H}^{n}(B_{1}(z_{1}))-\mathcal{H}^{n}(K)$
$\displaystyle=\mathcal{H}^{n}(B_{2}(z_{2}))-\mathcal{H}^{n}(K)$
$\displaystyle\leq\mathcal{H}^{n}(B_{2}(z_{2}))-\mathcal{H}^{n}(\Psi(K))$
$\displaystyle=0\,,$

which yields $K=\overline{B}_{1}(z_{1})$ since $K$ is closed. In particular
(4.21) provides the sought map.

Let us now prove that the asymptotic cone $(C,\mathsf{d},\mathcal{H}^{n})$
splits if and only if $M$ splits. In view of the splitting theorem it is
enough to show the following implication

(4.22) $C\,\text{contains a line}\implies M\,\text{contains a line,}$

since the other one is readily verified by stability of the product structure
under GH-convergence. Fix $p\in M$. It is enough to show that for any
$0<\varepsilon<1/9$ there exist $p_{1},p_{2}\in M$ satisfying
$\mathsf{d}(p_{1},p)=\mathsf{d}(p_{2},p)=1/\varepsilon$ and

(4.23)
$\mathsf{d}(p,\gamma_{p,p_{1}}(s))+\mathsf{d}(p,\gamma_{p,p_{2}}(s))\leq\mathsf{d}(\gamma_{p,p_{1}}(s),\gamma_{p,p_{2}}(s))+\varepsilon\quad\text{for
any $s\in(0,1)$}\,,$

where $\gamma_{p,p_{i}}:[0,1]\to M$ is a minimizing constant speed geodesic
such that $\gamma(0)=p$ and $\gamma(1)=p_{i}$, for $i=1,2$.

Indeed the curve $\gamma_{\varepsilon}:(-1/\varepsilon,1/\varepsilon)\to M$,
defined as

(4.24)
$\gamma_{\varepsilon}:(-1/\varepsilon,1/\varepsilon):=\begin{cases}\gamma_{p,p_{1}}(-t\varepsilon)&\text{for
$t\in(-1/\varepsilon,0)$}\\\ \gamma_{p,p_{2}}(t\varepsilon)&\text{for
$t\in(0,1/\varepsilon)$}\,,\end{cases}$

is $1$-Lipschitz and satisfies $\gamma_{\varepsilon}(0)=p$. Hence, up to
extracting a subsequence, $\gamma_{\varepsilon}\to\gamma$ locally uniformly,
where $\gamma:\mathbb{R}\to M$. We claim that $\gamma$ is a line. To see this,
we fix $t>0$ and we show that $\mathsf{d}(\gamma(t),\gamma(-t))=2t$. By (4.23)
we have

$\begin{split}2t&=\mathsf{d}(p,\gamma_{\varepsilon}(-t))+\mathsf{d}(p,\gamma_{\varepsilon}(t))\leq\mathsf{d}(\gamma_{\varepsilon}(-t),\gamma_{\varepsilon}(t))+\varepsilon\leq\mathsf{d}(\gamma_{\varepsilon}(-t),p)+\mathsf{d}(p,\gamma_{\varepsilon}(t))+\varepsilon\\\
&\leq 2t+\varepsilon.\end{split}$

Hence letting $\varepsilon\to 0$ implies
$2t\leq\mathsf{d}(\gamma(-t),\gamma(t))\leq 2t$ as claimed.

So we are left to prove (4.23). Given $0<\varepsilon<1/9$ we can find
$R=R(\varepsilon,n,p)>5/\varepsilon$ such that

(4.25) $\mathsf{d}_{GH}(B_{R}(p),B_{R}(z))\leq\varepsilon^{2}R\,,$

where $z\in C(Z)$ is a tip. Since $C$ splits a Euclidean factor, there exist
$q_{1},q_{2}\in B_{R}(p)$ satisfying

(4.26)
$\mathsf{d}(p,q_{1})=R,\quad\mathsf{d}(p,q_{2})=R,\quad\text{and}\quad\mathsf{d}(p,q_{1})+\mathsf{d}(p,q_{2})\leq\mathsf{d}(q_{1},q_{2})+\varepsilon^{2}R\,.$

Let $t\in(0,1)$ be such that
$\mathsf{d}(p,\gamma_{p,q_{1}}(t))=\mathsf{d}(p,\gamma_{p,q_{2}}(t))=1/\varepsilon$,
where where $\gamma_{p,q_{i}}:[0,1]\to M$ is a minimizing constant speed
geodesic such that $\gamma(0)=p$ and $\gamma(1)=q_{i}$, for $i=1,2$. We set
$p_{i}:=\gamma_{p,q_{i}}(t)$ and check that

(4.27)
$\mathsf{d}(p,\gamma_{p,q_{1}}(s))+\mathsf{d}(p,\gamma_{p,q_{2}}(s))\leq\mathsf{d}(\gamma_{p,q_{1}}(s),\gamma_{p,q_{2}}(s))+\varepsilon\quad\text{for
any $s\in(0,t)$}\,,$

which amounts to our conclusion.

Toponogov’s theorem implies that
$\mathsf{d}(\gamma_{p,q_{1}}(s),\gamma_{p,q_{2}}(s))\geq|\tilde{\gamma}_{\tilde{p},\tilde{q}_{1}}(s)-\tilde{\gamma}_{\tilde{p},\tilde{q}_{2}}(s)|$,
where $(\tilde{p},\tilde{q}_{1},\tilde{q}_{2})$ is a comparison triangle in
$\mathbb{R}^{2}$ corresponding to the triangle $(p,q_{1},q_{2})$, and
$\tilde{\gamma}_{\tilde{p},\tilde{q}_{i}}$ is the constant speed Euclidean
segment from $\tilde{p}$ to $\tilde{q}_{i}$, for $i=1,2$. The similarity of
the triangles $(\tilde{p},\tilde{q}_{1},\tilde{q}_{2})$ and
$(\tilde{p},\tilde{\gamma}_{\tilde{p},\tilde{q}_{1}}(s),\tilde{\gamma}_{\tilde{p},\tilde{q}_{2}}(s))$,
together with (4.26), give

$\displaystyle\mathsf{d}(\gamma_{p,q_{1}}(s),\gamma_{p,q_{2}}(s))$
$\displaystyle\geq|\tilde{\gamma}_{\tilde{p},\tilde{q}_{1}}(s)-\tilde{\gamma}_{\tilde{p},\tilde{q}_{2}}(s)|=|\tilde{q}_{1}-\tilde{q}_{2}|\frac{|\tilde{p}-\tilde{\gamma}_{\tilde{p},\tilde{q}_{1}}(s)|}{|\tilde{p}-\tilde{q}_{1}|}$
$\displaystyle=\mathsf{d}(q_{1},q_{2})\frac{\mathsf{d}(p,\gamma_{p,q_{1}}(s))+\mathsf{d}(p,\gamma_{p,q_{2}}(s))}{2R}$
$\displaystyle\geq(\mathsf{d}(p,q_{1})+\mathsf{d}(p,q_{2})-\varepsilon^{2}R)\frac{\mathsf{d}(p,\gamma_{p,q_{1}}(s))+\mathsf{d}(p,\gamma_{p,q_{2}}(s))}{2R}$
$\displaystyle=(1-\frac{\varepsilon^{2}}{2})(\mathsf{d}(p,\gamma_{p,q_{1}}(s))+\mathsf{d}(p,\gamma_{p,q_{2}}(s)))$
$\displaystyle\geq\mathsf{d}(p,\gamma_{p,q_{1}}(s))+\mathsf{d}(p,\gamma_{p,q_{2}}(s))-\varepsilon\,,$

where in the last step we used
$\mathsf{d}(p,\gamma_{p,q_{1}}(s))=\mathsf{d}(p,\gamma_{p,q_{2}}(s))\leq
1/\varepsilon$. ∎

######  Remark 4.7.

The statement of 4.6 holds in the class of Alexandrov spaces with nonnegative
sectional curvature. Indeed, the proof only builds on Toponogov’s theorem and
metric arguments.

###### Conclusion of the proof of 1.3.

The proof immediately follows from 4.6 and 1.2. ∎

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