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

# Positive mass and isoperimetry for continuous metrics with nonnegative
scalar curvature

Gioacchino Antonelli  Courant Institute Of Mathematical Sciences (NYU), 251
Mercer Street, 10012, New York, USA<EMAIL_ADDRESS>,  Mattia Fogagnolo
Dipartimento di Matematica Tullio Levi-Civita, Università di Padova, via
Trieste 63, 35010 Padova, Italy<EMAIL_ADDRESS>,  Stefano Nardulli
Centro de Matemática Cognição e Computação, Universidade Federal do ABC, Av.
Dos Estados 5001, Santo André, SP, Brazil<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.

This paper deals with the positive mass theorem and the existence of
isoperimetric sets on $3$-manifolds endowed with continuous complete metrics
having nonnegative scalar curvature in a suitable weak sense.

We prove that if the manifold has an end that is $C^{0}$-locally
asymptotically flat, and the metric is the local uniform limit of smooth
metrics with vanishing lower bounds on the scalar curvature outside a compact
set, then Huisken’s isoperimetric mass is nonnegative. This addresses a
version of a recent conjecture of Huisken about positive isoperimetric mass
theorems for continuous metrics satisfying $R_{g}\geq 0$ in a weak sense. As a
consequence, any fill-in of a truncation of a Schwarzschild space with
negative ADM mass has nonnegative isoperimetric mass.

Moreover, in case the whole manifold is $C^{0}$-locally asymptotically flat
and the metric is the local uniform limit of smooth metrics with vanishing
lower bounds on the scalar curvature outside a compact set, we prove that, as
a large scale effect, isoperimetric sets with arbitrarily large volume exist.

###### Key words and phrases:

Isoperimetric problem, curvature bounded below, scalar curvature, positive
mass theorem

###### 1991 Mathematics Subject Classification:

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

###### Contents

  1. 1 Introduction
    1. 1.1 Statement of the result
    2. 1.2 Strategy
    3. 1.3 Comments and comparison with related literature
      1. 1.3.1 Other notions of weak scalar curvature bounds, and relations with the works [19, 20]
      2. 1.3.2 Comparison with the literature related to weak PMT
      3. 1.3.3 Another notion of isoperimetric mass
    4. 1.4 The question of rigidity
  2. 2 Preliminaries
    1. 2.1 Basic definitions and properties of $C^{0}$-metrics
    2. 2.2 BV functions and sets of finite perimeter
    3. 2.3 Technical Lemmas on $C^{0}$-Riemannian manifolds
  3. 3 Local inverse mean curvature flow
    1. 3.1 Properties of the weak IMCF
    2. 3.2 Connectedness of level sets of the weak IMCF
  4. 4 Proof of the main results
    1. 4.1 Producing a set satisfying the reverse Euclidean isoperimetric inequality
    2. 4.2 Proof of the main results and consequences
  5. A Auxiliary results

##  1\. Introduction

###  1.1. Statement of the result

The classical positive mass theorem (PMT) in $3$ dimensions, originally due to
Schoen-Yau [77], asserts that the ADM mass of a $3$-dimensional smooth
complete asymptotically flat manifold with nonnegative scalar curvature is
nonnegative. This seminal result has been later rediscovered, generalized,
improved and exploited in a number of ways. Without any attempt to be
complete, we refer to [83, 78, 1, 16] for different proofs and extensions to
higher dimensions, and to [45, 15] for its main refinement, i.e., the
Riemannian Penrose inequality.

In this paper we prove an extension of the PMT for continuous metrics with
weakly nonnegative scalar curvature on $3$-manifolds. First, we introduce the
notions of mass and nonnegative scalar curvature we are going to adopt. We say
that $(M,g)$ is a $C^{0}$-Riemannian manifold if $M$ is a smooth
differentiable manifold, and $g$ is a $C^{0}$-metric on $M$. For a Borel set
$E\subset M$, let $|E|,P(E)$ denote the volume, and perimeter of $E$, see
Section 2.2 for precise definitions. The following notion has been introduced
by Huisken [43].

######  Definition  (Isoperimetric mass, Huisken [43]).

Let $(M,g)$ be a $C^{0}$-Riemannian $3$-manifold, possibly with boundary.
Then, its isoperimetric mass is

$\begin{split}\mathfrak{m}_{\mathrm{iso}}\vcentcolon=\sup\bigg{\\{}\limsup_{j\to+\infty}\frac{2}{P(\Omega_{j})}\left(|\Omega_{j}|-\frac{P(\Omega_{j})^{3/2}}{6\sqrt{\pi}}\right)\,\,\
:\ \,\,&\Omega_{j}\subset M,P(\Omega_{j})<+\infty\,\,\forall\,j,\\\
&P(\Omega_{j})\xrightarrow[]{}+\infty\bigg{\\}}.\end{split}$ (1.1)

For asymptotically flat $3$-manifolds with nonnegative scalar curvature, the
isoperimetric mass coincides with the ADM mass whenever the boundary is
minimal: see, e.g., [47, Theorem 3], [26, Theorem C.2]. In [12, Theorem 4.13]
such an equality is proved in the $C^{1,1/2+\varepsilon}$-asymptotically flat
regime, which is the sharp one to show that the ADM mass is independent of the
chart at infinity [9, 27]. The fact that the ADM mass can be detected through
the isoperimetric deficit of large coordinate spheres was first observed in
the Schwarzschild space in [14], and in asymptotically flat manifolds in [33,
Corollary 2.3].

Huisken has conjectured [44] that a weak (isoperimetric) PMT should hold true
for continuous Riemannian metrics if one interprets the notion of nonnegative
scalar curvature (shortly, $R_{g}\geq 0$) in an appropriate weak sense, see
Section 1.3.1 for more details. In this paper we prove a version of this
conjecture under the following weak notion of nonnegative scalar curvature.

######  Definition  ($R_{g}\geq 0$ in the approximate sense).

Let $(M,g)$ be a complete $C^{0}$-Riemannian manifold without boundary, and
let $\Omega\subset M$ be an open set. We say that $R_{g}\geq 0$ in the
approximate sense on $\Omega$ if there exist smooth complete Riemannian
metrics $g_{j}$ on $M$, such that:

  1. (1)

$g_{j}$ converges to $g$ locally uniformly on $M$;

  2. (2)

There exists a sequence $(\varepsilon_{j})_{j\in\mathbb{N}}$ of positive
numbers such that $\varepsilon_{j}\to 0$, and $R_{g_{j}}\geq-\varepsilon_{j}$
on $\Omega$.

By a result of Gromov, recalled below in Theorem 1.3, a smooth Riemannian
manifold has nonnegative scalar curvature if and only if $R_{g}\geq 0$ holds
in the approximate sense on $M$. Before stating the main theorems of this
paper, we make clear what it means that an end of a $C^{0}$-manifold is
$C^{0}_{\mathrm{loc}}$-asymptotic to $\mathbb{R}^{n}$.

######  Definition .

Let $K\subset M$ be a compact set (possibly empty) of a $C^{0}$-Riemannian
manifold $(M,g)$. We say that an unbounded connected component $\Omega$ of
$M\setminus K$ is $C^{0}_{\mathrm{loc}}$-asymptotic to a $C^{0}$-Riemannian
manifold $(N,h)$ if the following holds. For every diverging sequence
$\Omega\ni p_{i}\to+\infty$ there exists a point $o\in N$ such that
$(M,g_{i},p_{i})\to(N,h,o)$ in the $C^{0}$-sense, see Section 2.1.

######  Theorem 1.1.

Let $(M,g)$ be a complete 3-dimensional $C^{0}$-Riemannian manifold without
boundary. Let $K\subset M$ be a compact set, and let $\Omega$ be an unbounded
connected component $M\setminus K$. Assume that $\Omega$ is
$C^{0}_{\mathrm{loc}}$-asymptotic to $\mathbb{R}^{3}$, see Section 1.1, and
that $R_{g}\geq 0$ in the approximate sense on $\Omega\setminus K^{\prime}$,
see Section 1.1, where $K^{\prime}\subset M$ is a compact set.

Then $\mathfrak{m}_{\mathrm{iso}}\geq 0$.

The PMT for the isoperimetric mass in the
$C^{0}_{\mathrm{loc}}$-asymptotically flat case with nonnegative scalar
curvature only outside a compact set in a given end, is, to our knowledge, a
new result even in the smooth setting. In fact, one can immediately draw out
of Theorem 1.1 the following surprising consequence.

######  Remark .

Any fill-in of the truncated spacelike slice of a Schwarzschild spacetime of
_negative_ ADM mass has _nonnegative_ isoperimetric mass. Namely, any complete
fill-in of the noncompact, scalar-flat, asymptotically flat manifold with
boundary $[s_{0},+\infty)\times\mathbb{S}^{2}$, for $s_{0}>0$, endowed with
the metric

$g=\frac{1}{1-\mathfrak{m}s^{-1}}\,\mathrm{d}s\otimes\,\mathrm{d}s+s^{2}g_{\mathbb{S}^{2}},$

with $\mathfrak{m}<0$, enjoys $\mathfrak{m}_{\mathrm{iso}}\geq 0$. For such
manifolds, we presently do not know neither the shape of isoperimetric regions
nor the exact value of $\mathfrak{m}_{\mathrm{iso}}$.

The observation above shows that without some additional assumption on the
boundary, like minimality (see [12, Theorem 4.13]), the identification between
the two notions of mass in general fails. Moreover, this is coherent with the
cross-sections being unstable constant mean curvature surfaces, a property
showed in [29, Section 5], where the question about the shape of isoperimetric
regions is also raised.

Theorem 1.1 applies also to manifolds possibly having more than one end, for
which there is a distinguished end that has $R_{g}\geq 0$ (outside a compact
set), and that is $C^{0}_{\mathrm{loc}}$-asymptotic to $\mathbb{R}^{3}$.
Positive mass theorems on manifolds with arbitrary ends, except for a globally
asymptotically flat one, have been recently considered e.g. in [57, 55, 25,
28], see also references therein. Our distinguished end might not be globally
asymptotically flat: for example, our result applies to asymptotically conical
ends, and in particular to ALE ends.

On smooth $3$-dimensional asymptotically flat manifolds the condition
$R_{g}\geq 0$, which governs the local isoperimetric structure around each
point, has also a global effect in the existence of isoperimetric sets for
arbitrarily large volumes (see, e.g., [32, Theorem 1.2], [23, Proposition
K.1], [26, Theorem 1.1], and references therein). This can be understood as an
interpolation effect between $R_{g}\geq 0$ and the positivity of the mass at
infinity. The next result confirms this heuristic idea also in the continuous
setting.

######  Theorem 1.2.

Let $(M,g)$ be a complete 3-dimensional $C^{0}$-Riemannian manifold without
boundary, and assume that $R_{g}\geq 0$ in the approximate sense on
$M\setminus\mathcal{C}$, where $\mathcal{C}$ is a compact set, see Section
1.1. Assume in addition that $M$ is $C^{0}_{\mathrm{loc}}$-asymptotic to
$\mathbb{R}^{3}$, see Section 1.1.

Then there exists a sequence of isoperimetric sets $(E_{j})_{j\in\mathbb{N}}$
on $M$ such that $|E_{j}|\to+\infty$.

Very much like Theorem 1.1, even in the smooth setting Theorem 1.2 strongly
weakens the asymptotic assumptions for the existence of isoperimetric sets.
The literature on the subject is very vast, in particular in relation to the
study of canonical foliations of stable CMC: we refer to the seminal [46] and
to [31, 32, 70, 26, 85], as well as to the references therein, for a fairly
complete picture.

In the setting of Theorem 1.2 it is an interesting open problem to analyze
existence of isoperimetric sets for any (large) volume, uniqueness or
foliation properties of such isoperimetric sets, as in [70, 26, 85] (see also
the survey [10]). Under the additional assumption that the isoperimetric
profile $I$ is strictly increasing, we can actually prove existence of
isoperimetric sets for every volume, see Section 4.2.

###  1.2. Strategy

We briefly discuss the strategy of the proof of Theorem 1.1, and Theorem 1.2,
referring the reader to Section 3, and Section 4 for more details.

The starting point in the proof of Theorem 1.1 is the following consequence of
a result due to Shi [79], after important insights by Brendle–Chodosh [18]: in
a smooth $3$-dimensional asymptotically flat manifold with nonnegative scalar
curvature the level sets of the weak inverse mean curvature flow (shortly,
IMCF, see Section 3) issuing from a point satisfy a reverse Euclidean
isoperimetric inequality with the sharp constant, as long as their boundaries
are connected. Hence the isoperimetric deficit appearing on the right hand
side of (1.1) is nonnegative when computed on such sets, and this directly
implies the nonnegativity of the isoperimetric mass by letting the level sets
of the IMCF exhaust the manifold.

In fact, we will push this heuristic to give an isoperimetric PMT for metrics
that are only continuous. The crux of the argument is to show that on the
approximating manifolds $(M,g_{j})$ one can define a well-behaved weak IMCF
$w_{j}$ on arbitrarily large punctured balls. This is done by taking scaled
limits of the logarithms of $p$-Green functions on punctured balls, for $p\to
1^{+}$, as pioneered by R. Moser [66]. In fact, we build on the sharp gradient
estimate obtained in [51], and on the Harnack inequality with explicit
constants in [74], to get a weak IMCF $w_{j}$ on punctured balls $B_{j}$ in
$(M,g_{j})$ that is bounded below explicitly with constants that stay bounded
when $B_{j}$ is $C^{0}$-close enough to a Euclidean ball. This is the content
of the new quantitative existence result for local IMCFs in Theorem 3.1. The
latter allows to pass to the limit the level sets of these IMCFs, obtaining
sets satisfying the (sharp) reverse Euclidean isoperimetric inequality with
arbitrarily large perimeters, see Section 4.1.

The proof of the existence of isoperimetric sets in Theorem 1.2 is not
constructive, and it is based on a novel contradiction argument. One notices
that if after a certain volume threshold isoperimetric sets do not exist, then
the isoperimetric profile is strictly increasing for large volumes: this is a
consequence of a generalized existence theorem for the isoperimetric problem,
see Theorem 2.2. In addition, arguing as in the previous paragraph, we
construct sets that satisfy the reverse sharp Euclidean isoperimetric
inequality with arbitrarily large volumes and perimeters, and that avoid any
compact set. This is enough to show, again using Theorem 2.2 and that the
isoperimetric profile is increasing, that isoperimetric sets with arbitrarily
large volumes must exist, thus resulting in a contradiction.

###  1.3. Comments and comparison with related literature

We collect here some comments and perspectives on the main results of this
paper.

####  1.3.1. Other notions of weak scalar curvature bounds, and relations with
the works [19, 20]

Several notions of scalar curvature lower bounds for smooth manifolds endowed
with $C^{0}$-Riemannian metrics have been proposed in the recent years. In
[37] Gromov proposes a definition based on nonexistence of suitable small
polyhedra on the manifold, see the recent works [58, 17] motivated by this
study; a definition by using a regularization through Ricci flow has been
suggested by Burkhardt-Guim in [19]; and a definition using volumes of small
balls has been hinted by Huisken [44]. Let us compare our result in Theorem
1.1 with the framework in [19, 20].

Assume that a complete $C^{0}$-Riemannian manifold $(M,g)$ is globally
$C^{0}$-asymptotic to $\mathbb{R}^{3}$ outside a compact set $K\subset M$, see
Section 1.4. Hence, for $\beta\in(0,1/2)$, we claim that if $R_{g}\geq 0$ in
the $\beta$-weak sense [20, Definition 2.3] on $M\setminus K$, then $R_{g}\geq
0$ in the approximate sense on $M\setminus K$, up to possibly enlarging $K$.
Indeed, this is due to the fact that under the $C^{0}$-asymptotic hypothesis
at infinity one can first define a Ricci-deTurck flow $(M,g_{t})$ starting
from $g$ which $C^{0}$-converges to $(M,g)$ locally uniformly by using [81,
Theorem 1.1]. Then, since $R_{g}\geq 0$ in the $\beta$-weak sense on
$M\setminus K$, [20, Lemma 5.1] implies that, up to possibly enlarging $K$,
$R_{g_{t}}\geq-o(1)$ on $M\setminus K$ as $t\to 0$, which in turn implies that
$R_{g}\geq 0$ in the approximate sense on $M\setminus K$ by definition. In the
case $M$ is a compact manifold, the previous reasoning has been explicitly
recorded in [19, Corollary 1.6]. Related to this, it would be interesting to
understand whether a non-compact $C^{0}$-manifold $(M,g)$ that has $R_{g}\geq
0$ in the approximate sense on $M$ admits a smooth metric $\tilde{g}$ with
$R_{\tilde{g}}\geq 0$. The compact case has been settled in [19, Corollary
1.8]. Taking into account the discussion above, this seems likely to hold when
$(M,g)$ is, in addition, $C^{0}$-asymptotic to $\mathbb{R}^{3}$ (Section 1.4).

As a consequence of the above discussion, under the stronger assumption that
$(M,g)$ is $C^{0}$-asymptotic to $\mathbb{R}^{3}$ (Section 1.4), one gets that
$\mathfrak{m}_{\mathrm{iso}}\geq 0$ provided that $R_{g}\geq 0$ in the
$\beta$-weak sense outside a compact set, for some $\beta\in(0,1/2)$ ([20,
Definition 2.3]). It is yet to be understood whether, in such setting, the
isoperimetric mass coincides with the (weak notion of) ADM mass defined in the
work [20], see [20, Question 1]. If the latter holds, then Theorem 1.1 would
answer the question about the nonnegativity of such mass [20, Question 2] in
the positive.

As a side note, we point out that as of today it is not known whether the
definitions in [37] and [19] are equivalent, compare with the discussion [19,
Page 1707-1708], nor any relation with Huisken’s notion in [44] has been
investigated yet.

We conclude this section by mentioning a result according to which the notion
of $R_{g}\geq 0$ in the approximate sense is consistent with $R_{g}\geq 0$ for
smooth metrics. In fact, as a consequence of a result of Gromov [37, page
1118] (see [8] for a proof using the Ricci flow) the following holds.

######  Theorem 1.3 ([37, 8]).

Let $M$ be a smooth manifold and $\kappa:M\to\mathbb{R}$ be a continuous
function on $M$. Suppose $g_{i}$ is a sequence of $C^{2}$-metrics on $M$ that
converges locally uniformly to a $C^{2}$-metric $g$ on $M$. If
$R_{g_{i}}\geq\kappa$ on $M$, then $R_{g}\geq\kappa$ everywhere on $M$ as
well.

####  1.3.2. Comparison with the literature related to weak PMT

Weak positive mass theorems (PMT) have been thoroughly studied in the last two
decades. We briefly discuss some contributions without the aim of being
exhaustive, and refer the interested reader to the references therein.

In [64, Theorem 1] the author proves a PMT for asymptotically flat Lipschitz
metrics that are smooth away from a compact hypersurface, and such that
$R_{g}\geq 0$ in an appropriate weak sense (i.e., a condition on the
exterior/interior mean curvatures of the hypersurface is required). See also
[63, Theorem 1] for a proof of the result in [64, Theorem 1] using Ricci flow.
Then, in [80, Theorem 1.3], the authors prove a PMT for $n$-dimensional
manifolds that carry an asymptotically flat $C^{0}\cap W^{1,n}$ metric that is
smooth away from a compact set $\Sigma$ of codimension at least $2$, and for
which $R_{g}\geq 0$ outside $\Sigma$. See also [48, Theorem 1.1] for an
improvement of [80, Theorem 1.3], where a weaker condition on the codimension
of $\Sigma$ (depending on the integrability of the derivatives of the metric)
is asked. Similar results, but with global $L^{\infty}$ metrics which satisfy
appropriate conditions close to $\Sigma$, are in [59, Theorem 1.8, Theorem
1.9].

In all of the papers discussed above, the manifolds considered are smooth and
with nonnegative scalar curvature outside a compact set, and thus, in
dimension $3$, Theorem 1.1 implies they enjoy nonnegative isoperimetric mass.
Finally, in [54, Theorem 1.1] the authors prove a PMT for a carefully
generalized ADM mass in the setting of $C^{0}\cap W^{1,n}$ metrics with weak
decays at infinity, and for which a notion of distributional scalar curvature
is nonnegative. By means of a mollification procedure, see, e.g., [48, Lemma
2.6], these metrics have nonnegative scalar curvature in the approximate
sense. Moreover, their asymptotic constraints imply $C^{0}$-asymptotic
flatness. In particular, Theorem 1.1 holds in this case as well. To our
knowledge, it is not known whether the isoperimetric mass and the ADM mass
agree in the setting of the results described above.

####  1.3.3. Another notion of isoperimetric mass

On a $C^{0}$-Riemannian $3$-manifold $(M,g)$, it is also common to consider
another notion of isoperimetric mass given in terms of exhaustions [47, 26,
12], namely

$\begin{split}\tilde{\mathfrak{m}}_{\mathrm{iso}}\vcentcolon=\sup\bigg{\\{}\limsup_{j\to+\infty}\frac{2}{P(\Omega_{j})}\left(|\Omega_{j}|-\frac{P(\Omega_{j})^{3/2}}{6\sqrt{\pi}}\right)\,\,\
:\ \,\,&\text{$(\Omega_{j})_{j\in\mathbb{N}}$ is an exhaustion of $M$,}\\\
&\text{$P(\Omega_{j})<+\infty$ $\forall\,j$}\bigg{\\}}.\end{split}$

As a by-product of the proof in [47, Proposition 37] one gets that if $(M,g)$
satisfies a global Euclidean-like isoperimetric inequality, i.e., $P(E)\geq
C|E|^{2/3}$ for some $C>0$ for any $|E|<+\infty$, then
$\tilde{\mathfrak{m}}_{\mathrm{iso}}=\mathfrak{m}_{\mathrm{iso}}$. This holds,
for instance, when $M$ is globally $C^{0}$-asymptotic to $\mathbb{R}^{3}$, see
Section 1.4 for the precise definition. In particular, Theorem 1.1 implies
that on $3$-dimensional $C^{0}$-Riemannian manifolds that are
$C^{0}$-asymptotic to $\mathbb{R}^{3}$ (Section 1.4) and that satisfy
$R_{g}\geq 0$ in the approximate sense out of a compact set, there holds
$\tilde{\mathfrak{m}}_{\mathrm{iso}}\geq 0$.

###  1.4. The question of rigidity

In the smooth PMT [77], one gets that the mass is zero if and only the metric
is flat. In this paper, we do not address the issue of rigidity in Theorem
1.1. We pose it as a problem, which is likely to require the use of new
techniques tailored for the $C^{0}$-setting, or some refined approximation
procedure. We state it in the most basic asymptotically flat case.

######  Definition  (Manifold $C^{0}$-asymptotic to $\mathbb{R}^{n}$).

We say that an $n$-dimensional $C^{0}$-Riemannian manifold $(M,g)$ is
$C^{0}$-asymptotic to $\mathbb{R}^{n}$ if there exists a compact set $K\subset
M$ such that the following holds. There exists $R>0$, and a diffeomorphism
$\Phi:\mathbb{R}^{n}\setminus\overline{B}_{R}^{\mathbb{R}^{n}}(0)\to
M\setminus K$, such that

$|(\Phi^{*}g)_{ij}-\delta_{ij}|_{x}=o(1)\quad\text{as $|x|\to+\infty$.}$

######  Question .

Let $(M,g)$ be a complete $C^{0}$-manifold of dimension $3$ without boundary.
Assume that $M$ has $R_{g}\geq 0$ in the approximate sense everywhere, and
that $M$ is $C^{0}$-asymptotic to $\mathbb{R}^{3}$. If
$\mathfrak{m}_{\mathrm{iso}}=0$, is it true that $(M,g)$ is isometric to
$(\mathbb{R}^{3},g_{\mathrm{eu}})$?

Notice one cannot expect rigidity in the general formulation of Theorem 1.1,
not even in the smooth case: indeed, it can be showed by a direct computation
that any metric that is flat outside a compact set in $\mathbb{R}^{3}$ has
$\mathfrak{m}_{\mathrm{iso}}=0$. We finally point out that Section 1.4 may be
related to the study of the stability of the PMT with respect to nonsmooth
notions of convergence. Stability results in this direction have been proved,
for instance, in [30] with respect to pointed measure Gromov–Hausdorff
convergence, in [56] for rotationally symmetric metrics, and in [41, 42] for
graphs with respect to (intrinsic) convergence in the sense of currents (see
[82] and Conjecture 10.1 therein for more details on the problem).

Acknowledgments. A part of this research was carried out while G.A., M. F.,
and M.P. were visiting S.N. at UFABC. The joyful working conditions and the
hospitality are gratefully acknowledged. Another part of the work was carried
out while G. A. and M. F. were participating in the workshop “Recent Advances
in Comparison Geometry” in Hangzhou. They thank the Banff International
Research Station and the Institute of Advanced Studies in Mathematics for this
enriching opportunity. G.A. is grateful to Paula Burkhardt-Guim for useful
discussions around the topic of the paper. He is also grateful to the Courant
Institute for financial support. M.P. is grateful to Roberto Ladu for
clarifying discussions and stimulating advice. The authors are grateful to
Luca Benatti, Luciano Mari, and Kai Xu for useful discussions around the topic
of this paper. They are also grateful to Virginia Agostiniani, Alessandro
Carlotto, Jeff Jauregui, Dan Lee, Lorenzo Mazzieri, and Christina Sormani for
their interest in this work.

M. F. is supported by the Project funded under the National Recovery and
Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.5 - Call for tender
No. 3277 of 30 dicembre 2021 of Italian Ministry of University and Research
funded by the European Union – NextGenerationEU; Project code: ECS00000043,
Concession Decree No. 1058 of June 23, 2022, CUP C43C22000340006, Project
title “iNEST: Interconnected Nord-Est Innovation Ecosystem”. M. F. and M. P.
are members of INdAM-GNAMPA. S.N. was supported by FAPESP Auxílio Jovem
Pesquisador # 2021/05256-0, FAPESP Sprint # 23/08246-0, CNPq Bolsa de
Produtividade em Pesquisa 1D # 12327/2021-8, 23/08246-0, Geometric Variational
Problems in Smooth and Nonsmooth Metric Spaces # 441922/2023-6. M.P. is
partially supported by the PRIN Project 2022E9CF89 - PNRR Italia Domani,
funded by EU Program NextGenerationEU.

##  2\. Preliminaries

We gather in this section a number of fundamental properties of
$C^{0}$-Riemannian manifolds, possibly endowed with $C^{0}$-asymptotics.

###  2.1. Basic definitions and properties of $C^{0}$-metrics

######  Definition  ($C^{0}$-Riemannian manifold).

A $C^{0}$-Riemannian manifold is a couple $(M,g)$, where $M$ is a smooth
$n$-dimensional differentiable manifold (possibly with boundary) and $g$ is a
continuous metric tensor. More precisely, in any local chart $(U,\varphi)$ on
$M$, the metric tensor in local coordinates is represented by a symmetric
positive definite matrix with components $g_{ij}\in C^{0}(\varphi(U))$.

We briefly recall some metric properties of $C^{0}$-Riemannian manifolds
$(M,g)$, and we refer the reader to [21, 72] for a more detailed discussion.
As in the smooth case, the continuous Riemannian metric $g$ defines a length
structure on absolutely continuous curves on $M$, which in turn gives raise to
a distance $\mathsf{d}$ that induces the manifold topology. We always assume
that $(M,\mathsf{d})$ is a complete metric space, so that by Hopf–Rinow
Theorem the distance $\mathsf{d}$ is geodesic, and every closed ball is
compact.

The volume form induced by $g$ in a local chart induces integration with
respect to $\mathrm{vol}:=\sqrt{\mathrm{det}\,g}\,\mathcal{L}^{n}$, where
$\mathcal{L}^{n}$ denotes Lebesgue measure. It can be proved that
$\mathrm{vol}=\mathcal{H}^{n}$, where $\mathcal{H}^{n}$ is the $n$-dimensional
Hausdorff measure relative to $\mathsf{d}$, and that $\mathcal{H}^{n}$ is a
Radon measure. Hence $(M,\mathsf{d},\mathcal{H}^{n})$ is a complete and
separable metric measure space. We will often denote
$|E|:=\mathcal{H}^{n}(E)$. In the following, if $E$ is a measurable set,
integrals over $E$ are tacitly understood to be taken with respect to
$\mathcal{H}^{n}$.

Given $f\in C^{\infty}(M)$, one can define $\nabla f$ as in the smooth case by
setting $g(\nabla f,X)=\mathrm{d}f(X)$ for every vector field $X$ on $M$.
Hence, in a local chart $\\{\partial_{i}\\}_{i=1,\ldots,n}$, we have $|\nabla
f|^{2}:=g(\nabla f,\nabla f)=g^{ij}\partial_{i}f\partial_{j}f$. Notice that by
the classical Rademacher Theorem applied in chart, if
$f\in\mathrm{Lip}_{\mathrm{loc}}(M)$, then $\nabla f$ exists
$\mathcal{H}^{n}$-almost everywhere.

From now on we will always assume that $C^{0}$-Riemannian manifolds are
complete. We recall that a map between metric spaces
$F:(X,\mathsf{d}_{X})\to(Y,\mathsf{d}_{Y})$ is said to be $L$-biLipschitz,
with $L\geq 1$, when
$L^{-1}\mathsf{d}_{X}(a,b)\leq\mathsf{d}_{Y}(F(a),F(b))\leq
L\mathsf{d}_{X}(a,b)$ for every $a,b\in X$.

######  Lemma .

Fix $n\in\mathbb{N}$ with $n\geq 2$. Then for any $\delta>0$ there exists
$\varepsilon>0$ such that the following holds. Let $(M,g)$ be a
$C^{0}$-Riemannian manifold and let $o\in M$ and $R>0$. Denote by $\mathsf{d}$
the Riemannian distance on $M$. Let $N$ be a differentiable manifold and let
$\Omega\subset N$ be an open set. Suppose that there exists a diffeomorphism
$F:B_{10R}(o)\to\Omega$ and a $C^{0}$-Riemannian metric $h$ on $\Omega$ such
that $|(F^{*}h-g)(v,v)|\leq\varepsilon g(v,v)$ for any $v\in T_{x}M$ and any
$x\in B_{10R}(o)$.

Then, letting

$\tilde{\mathsf{d}}(x,y)\vcentcolon=\inf\left\\{\int_{0}^{1}|\gamma^{\prime}|_{h}\
:\ \gamma:[0,1]\to\Omega,\gamma(0)=x,\gamma(1)=y\right\\},$

the map
$F|_{B_{R}(o)}:(B_{R}(o),\mathsf{d})\to(F(B_{R}(o)),\tilde{\mathsf{d}})$ is
$(1+\delta)$-biLipschitz with its image.

###### Proof.

Denote $o^{\prime}\vcentcolon=F(o)$. Let $x,y\in B_{R}(o)$. A constant speed
geodesic $\gamma:[0,1]\to M$ from $x$ to $y$ has image contained in
$B_{5R}(o)$. Hence we can estimate

$\tilde{\mathsf{d}}(F(x),F(y))\leq\int_{0}^{1}h\left((F\circ\gamma)^{\prime},(F\circ\gamma)^{\prime}\right)^{\frac{1}{2}}\,\mathrm{d}t\leq\sqrt{1+\varepsilon}\int_{0}^{1}g(\gamma^{\prime},\gamma^{\prime})^{\frac{1}{2}}\,\mathrm{d}t\leq\sqrt{1+\varepsilon}\,\mathsf{d}(x,y).$

Denoting $G=F^{-1}$, by assumptions we have that

$\left|G^{*}g(w,w)-h(w,w)\right|\leq\varepsilon G^{*}g(w,w),$

for any $w\in T_{z}N$ and $z\in\Omega$. Taking now $p,q\in F(B_{R}(o))$ and a
constant speed curve $\sigma:[0,1]\to\Omega$ from $p$ to $q$ such that
$\int_{0}^{1}|\sigma^{\prime}|_{h}\leq(1+\eta)\tilde{\mathsf{d}}(p,q)$ for
$\eta\in(0,1)$, we can similarly estimate

$\mathsf{d}(G(p),G(q))\leq\int_{0}^{1}g((G\circ\sigma)^{\prime},(G\circ\sigma)^{\prime})^{\frac{1}{2}}\,\mathrm{d}t\leq\frac{1}{\sqrt{1-\varepsilon}}\int_{0}^{1}h(\sigma^{\prime},\sigma^{\prime})^{\frac{1}{2}}\,\mathrm{d}t\leq\frac{1+\eta}{\sqrt{1-\varepsilon}}\tilde{\mathsf{d}}(p,q).$

Sending $\eta\to 0$, for $\varepsilon$ small enough the claim follows. ∎

######  Definition  ($C^{0}$-convergence).

We say that pointed $C^{0}$-Riemannian manifolds $(M_{i},g_{i},p_{i})$
$C^{0}$-converge to $(M,g,p)$ if the following holds. For every
$R,\varepsilon>0$ there exist $i_{0}:=i_{0}(R,\varepsilon)$ and an open set
$\Omega\subset M$ such that we have:

  1. (1)

$\overline{B}_{R}(p)\subset\Omega$;

  2. (2)

for every $i\geq i_{0}$ there exists an embedding $F_{i}:\Omega\to M_{i}$ such
that

     * •

$F_{i}(p)=p_{i}$;

     * •

$\overline{B}_{R}(p_{i})\subset F_{i}(\Omega)$;

     * •

$|(F_{i}^{*}g_{i}-g)_{x}(v,v)|\leq\varepsilon g_{x}(v,v)$ for every
$x\in\overline{B}_{R}(p)$ and $v\in T_{x}M$.

######  Remark .

In the notation of Section 2.1, arguing as in the proof of Section 2.1, it
follows that for any $R,\varepsilon>0$ the map
$F_{i}:(B_{R}(p),\mathsf{d})\to(F_{i}(B_{R}(p)),\mathsf{d}_{i})$ is
$(1+\varepsilon)$-biLipschitz with its image for any $i$ large enough.

######  Lemma .

Let $(M,g)$ be a $C^{0}$-Riemannian manifold, let $x_{0}\in M\setminus\partial
M$, and denote by $\mathsf{d}$ the Riemannian distance on $M$. Then the
following holds.

  * •

The metric tangent space of $M$ at $x_{0}$ is isometric to the Euclidean space
$\mathbb{R}^{n}$, i.e., the rescalings $(M,\delta^{-2}g,x_{0})$ converges to
$(\mathbb{R}^{n},g_{\rm eu},0)$ in $C^{0}$-sense as $\delta\searrow 0$.

  * •

For any $\delta>0$ there is $r=r(x_{0},\delta)>0$ and a local chart
$\varphi:(B_{r}(x_{0}),\mathsf{d})\to(\varphi(B_{r}(x_{0})),\mathsf{d}_{\rm
eu})\subset\mathbb{R}^{n}$ such that $\varphi$ is $(1+\delta)$-biLipschitz
with its image, where $\mathsf{d}_{\rm eu}$ denotes Euclidean distance.

###### Proof.

Passing in local coordinates we can identify a neighborhood of $x_{0}$ in $M$
with $(\Omega,g)$, where $\Omega\subset\mathbb{R}^{n}$ is open and
$g=(g_{ij})$ is the metric in local coordinates. Also we can identify $x_{0}$
with the origin $0\in\Omega$. Let $r_{k}\searrow 0$. For the first part of the
statement, it is sufficient to prove that $(\Omega,r_{k}^{-2}g,0)$ $C^{0}_{\rm
loc}$-converges to $(\mathbb{R}^{n},g_{0},0)$, where $g_{0}$ is the constant
metric given by $g$ evaluated at the origin. The diffeomorphism
$\Phi_{k}:(r_{k}^{-1}\Omega,g_{0})\to(\Omega,r_{k}^{-2}g)$ given by
$\Phi_{k}(x)=r_{k}\,x$ satisfies
$(\Phi_{k}^{*}(r_{k}^{-2}g))_{x}(v,v)=g_{r_{k}\,x}(v,v)$ for any tangent
vector $v$. Hence, given any $R>0$ and $\varepsilon>0$, for $k$ large enough
we have that

$1-\varepsilon\leq\frac{(\Phi_{k}^{*}(r_{k}^{-2}g))_{x}(v,v)}{g_{0}(v,v)}\leq
1+\varepsilon,$

for any $x\in B_{R}^{g_{0}}(0)\subset(r_{k}^{-1}\Omega,g_{0})$ and any tangent
vector $v\neq 0$. This establishes the convergence in $C^{0}$ to
$\mathbb{R}^{n}$ as $k\to\infty$.

Concerning the second part of the statement, let $r>0$ to be chosen small and
fix a local chart
$\varphi:B_{10r}(x_{0})\to\varphi(B_{10r}(x_{0}))=:\Omega\subset\mathbb{R}^{n}$
such that $\varphi(x_{0})=0$, and denote $g_{0}:=g_{\rm eu}$. We identify
$B_{10r}(x_{0})$ with $(\Omega,g)$ in the local chart. Let
$\tilde{\mathsf{d}}$ be as in Section 2.1 with $N=\mathbb{R}^{n}$. By
continuity of the metric tensor, for any $\varepsilon>0$ we can take $r$ so
small that $|g_{x}(v,v)-g_{0}(v,v)|\leq\varepsilon g_{0}(v,v)$ for any
$x\in\Omega$ and any $v\in\mathbb{R}^{n}$. For $\varepsilon$ small enough, it
follows from Section 2.1 that the identity ${\rm
id}:(B^{g}_{r}(0),\mathsf{d})\to(B^{g}_{r}(0),\tilde{\mathsf{d}})$ is
$(1+\delta)$-biLipschitz. It remains to observe that
$\tilde{\mathsf{d}}=\mathsf{d}_{\rm eu}$ on $B^{g}_{r}(0)$. Indeed, fix
$p,q\in B^{g}_{r}(0)$ and let $\gamma_{i}:[0,1]\to\Omega$ be a sequence of
constant speed curves such that $\int_{0}^{1}|\gamma_{i}^{\prime}|_{\rm
eu}\to\tilde{\mathsf{d}}(p,q)$. It follows that $\gamma_{i}([0,1])\subset
B^{g}_{9r}(0)$ for large $i$. Indeed, taking $\sigma:[0,1]\to B^{g}_{5r}(0)$ a
constant speed geodesic for $\mathsf{d}$ from $p$ to $q$, we know that

$\tilde{\mathsf{d}}(p,q)\leq\int_{0}^{1}|\sigma^{\prime}|_{\rm
eu}\leq\frac{1}{\sqrt{1-\varepsilon}}\int_{0}^{1}|\sigma^{\prime}|_{g}=\frac{1}{\sqrt{1-\varepsilon}}\mathsf{d}(p,q)\leq
2r\frac{1}{\sqrt{1-\varepsilon}}.$

Therefore, if $\gamma_{i}([0,1])\not\subset B^{g}_{9r}(0)$ for some $i$, we
would get

$\int_{0}^{1}|\gamma_{i}^{\prime}|_{\rm
eu}\geq\frac{1}{\sqrt{1+\varepsilon}}\int_{0}^{1}|\gamma_{i}^{\prime}|_{g}\geq\frac{1}{\sqrt{1+\varepsilon}}16r,$

leading to a contradiction for large $i$, provided $\varepsilon$ is small
enough. Hence we can pass to the limit $\gamma_{i}$ to a curve
$\gamma:[0,1]\to\overline{B}^{g}_{9r}(0)$ such that
$\tilde{\mathsf{d}}(p,q)=\int_{0}^{1}|\gamma^{\prime}|_{\rm eu}$. Hence
$\gamma$ is a critical point for the length functional on
$(B^{g}_{10r}(0),g_{\rm eu})$. Then $\gamma$ is a straight segment from $p$ to
$q$ and in particular $\tilde{\mathsf{d}}(p,q)=\mathsf{d}_{\rm eu}(p,q)$. ∎

###  2.2. BV functions and sets of finite perimeter

Let $(M,g)$ be a complete $C^{0}$-Riemannian manifold of dimension $n$. Then
we can consider the complete and separable metric measure space
$(M,\mathsf{d},\mathcal{H}^{n})$. Following [65, 3], we define the _total
variation_ $|Df|(B)\in[0,+\infty]$ of a function $f\in
L^{1}_{\mathrm{loc}}(M)$ in a Borel set $B\subseteq M$ as

$|Df|(B)\coloneqq\inf_{B\subseteq\Omega\text{
open}}\inf\bigg{\\{}\liminf_{n\to\infty}\int_{\Omega}|\nabla
f_{n}|\;\bigg{|}\;(f_{n})_{n\in\mathbb{N}}\subset\mathrm{Lip}_{\mathrm{loc}}(\Omega),\,f_{n}\to
f\text{ in }L^{1}_{\mathrm{loc}}(\Omega)\bigg{\\}}.$ (2.1)

######  Remark .

On a $C^{0}$-Riemannian manifold, the previous definition of total variation
is equivalent to the usual one adopted on metric measure spaces. More
precisely, defining the slope of a locally Lipschitz function $f$ by

$\mathrm{lip}f(x):=\limsup_{y\to x}\frac{|f(x)-f(y)|}{\mathsf{d}(y,x)},$ (2.2)

then plugging ${\rm lip\,}f_{n}$ in place of $|\nabla f_{n}|$ in (2.1) yields
the same quantity. Indeed, if $f\in\operatorname{Lip}_{\rm loc}(M)$, then
${\rm lip\,}f=|\nabla f|$ almost everywhere, see Appendix A.

If for some open cover $(\Omega_{n})_{n\in\mathbb{N}}$ of $M$ we have that
$|Df|(\Omega_{n})<+\infty$ holds for every $n\in\mathbb{N}$, then $|Df|$ is a
locally finite Borel measure on $M$. We say that a Borel set $E\subseteq M$ is
_of locally finite perimeter_ if $P(E,\cdot)\coloneqq|D\chi_{E}|$ is a locally
finite measure, called the _perimeter measure_ of $E$. When $P(E)\coloneqq
P(E,M)<+\infty$, we say that $E$ is _of finite perimeter_. If $f\in L^{1}(M)$
satisfies $|Df|(M)<+\infty$, then we say that $f\in BV(M)$.

Given any $f\in\mathrm{Lip}_{\mathrm{loc}}(M)$, it holds that $|Df|$ is a
locally finite measure and $|Df|={\rm lip\,}f\mathcal{H}^{n}=|\nabla
f|\mathcal{H}^{n}$, see Appendix A. Moreover, we recall the coarea formula in
our setting,

######  Theorem 2.1 (Coarea formula [65, Proposition 4.2]).

Let $(M,g)$ be a complete $C^{0}$-Riemannian manifold. Let $f\in
L^{1}_{\mathrm{loc}}(M)$ be such that $|Df|$ is a locally finite measure. Fix
a Borel set $E\subseteq M$. Then $\mathbb{R}\ni t\mapsto
P(\\{f<t\\},E)\in[0,+\infty]$ is a Borel measurable function and it holds that

$|Df|(E)=\int_{\mathbb{R}}P(\\{f<t\\},E)\,\mathrm{d}t.$

Let us finally introduce the notion of isoperimetric profile.

######  Definition .

Let $(M,g)$ be a complete $C^{0}$-Riemannian manifold with $|M|=+\infty$. The
isoperimetric profile function is the function $I:(0,+\infty)\to[0,+\infty]$
defined as follows

$I(V):=\inf\\{P(E):E\subset M,\mathcal{H}^{n}(E)=V\\}.$

###  2.3. Technical Lemmas on $C^{0}$-Riemannian manifolds

In this section we prove several technical lemmas about $C^{0}$-Riemannian
manifolds we are going to use throughout the paper. Notably, we prove: a
precompactness theorem for $BV$ functions on converging sequences of
continuous Riemannian manifolds (Section 2.3); the fact that every continuous
Riemannian manifold is PI on every ball (Section 2.3); continuity of the
isoperimetric profile of continuous Riemannian manifolds with
$C^{0}$-controlled geometry at infinity (Section 2.3); and a generalized
existence theorem for the isoperimetric problem on continuous Riemannian
manifolds with $C^{0}$-controlled geometry at infinity (Theorem 2.2).

######  Lemma .

Let $(M,g)$ be a $C^{0}$-Riemannian manifold. For any $p\in M$, $r>0$ and
$\delta>0$ there exists a Riemannian manifold $(N,g^{\delta})$ with smooth
metric $g^{\delta}$ such that the inclusion
$\iota:(B_{r}(p),\mathsf{d})\to(N,\mathsf{d}^{\delta})$ is well defined and it
is $(1+\delta)$-biLipschitz with its image, where
$\mathsf{d},\mathsf{d}^{\delta}$ denote Riemannian distance on $(M,g)$,
$(N,g^{\delta})$ respectively.

###### Proof.

For any $\varepsilon\in(0,1)$ let $g^{\varepsilon}$ be a smooth Riemannian
metric on $\overline{B}_{20r}^{g}(p)$ such that
$|g(v,v)-g^{\varepsilon}(v,v)|\leq\varepsilon g(v,v)$ for any $v\in T_{x}M$
and $x\in B_{20r}(p)$. Gluing the boundary of a smooth connected open domain
$D$ such that $\overline{B}_{10r}^{g}(p)\subset D\subset\subset
B_{20r}^{g}(p)$ with a half-cylinder $[0,+\infty)\times\partial D$ and
suitably extending the metric $g^{\varepsilon}$, we obtain a smooth complete
Riemannian manifold $(N,g^{\varepsilon})$. Denote by
$\mathsf{d}^{\varepsilon}$ the Riemannian distance on $(N,g^{\varepsilon})$
and by $\tilde{\mathsf{d}}$ the distance defined in Section 2.1 with
$F=\iota:B_{10r}^{g}(p)\to B_{10r}^{g}(p)\subset(N,g^{\varepsilon})$. Arguing
as in the second part of the proof of Section 2.1, it follows that
$\tilde{\mathsf{d}}=\mathsf{d}^{\varepsilon}$ on $B_{r}^{g}(p)$. Indeed, as in
the proof of Section 2.1 we find that for any $x,y\in B^{g}_{r}(p)$ there
exists a constant speed curve $\gamma:[0,1]\to B^{g}_{10r}(p)$ such that
$\tilde{\mathsf{d}}(x,y)=\int_{0}^{1}|\gamma^{\prime}|_{g^{\varepsilon}}$,
hence $\gamma$ is a geodesic for the metric $g^{\varepsilon}$. For every $x\in
B^{g}_{r}(p)$, we have that for almost every $y\in B^{g}_{r}(p)$ there exists
a unique, hence minimizing, geodesic in $(N,g^{\varepsilon})$ from $x$ to $y$.
Then, for every $x\in B^{g}_{r}(p)$, we have that for almost every $y\in
B^{g}_{r}(p)$ the curve $\gamma$ just obtained must be the minimizing geodesic
from $x$ to $y$ in $(N,g^{\varepsilon})$. Hence, given $x\in B^{g}_{r}(p)$,
there holds $\tilde{\mathsf{d}}(x,y)=\mathsf{d}^{\varepsilon}(x,y)$ for almost
every $y\in B^{g}_{r}(p)$. Thus by continuity
$\tilde{\mathsf{d}}=\mathsf{d}^{\varepsilon}$ on $B^{g}_{r}(p)$. Therefore
choosing $\varepsilon$ small enough and eventually renaming $g^{\varepsilon}$
into $g^{\delta}$, the conclusion follows from Section 2.1. ∎

######  Remark  (Representation of the perimeter).

Let $(M,g)$ be an $n$-dimensional $C^{0}$-Riemannian manifold, and let $E$ be
a set of finite perimeter. Then
$P(E,\cdot)=\mathcal{H}^{n-1}\llcorner\partial^{e}E$, being
$\partial^{e}E:=M\setminus(E^{(0)}\cup E^{(1)})$ the so-called _essential
boundary_ , where

$E^{(1)}\coloneqq\bigg{\\{}x\in M\;\bigg{|}\;\lim_{r\to 0}\frac{|E\cap
B_{r}(x)|}{|B_{r}(x)|}=1\bigg{\\}},\qquad E^{(0)}\coloneqq\bigg{\\{}x\in
M\;\bigg{|}\;\lim_{r\to 0}\frac{|E\cap B_{r}(x)|}{|B_{r}(x)|}=0\bigg{\\}}.$
(2.3)

Indeed, fix on a compact ball $B\subset M$ a sequence $g_{i}$ of smooth
metrics such that $g_{i}\to g$ uniformly on $B$ (see, e.g., Section 2.3).
Then, up to subsequences, $\partial^{e}E\cap B$ does not depend on the metrics
$g_{i},g$ chosen in the definition (2.3) (compare also with item (1) in
Section 2.3). Moreover, by the very definition of Hausdorff measure,
$(1-o_{i}(1))\mathcal{H}^{n-1}_{g_{i}}\llcorner
B\leq\mathcal{H}^{n-1}_{g}\llcorner
B\leq(1+o_{i}(1))\mathcal{H}^{n-1}_{g_{i}}\llcorner B$. From the classical De
Giorgi-Federer’s theorem,
$|D\chi_{E}|_{i}(\Omega)=\mathcal{H}^{n-1}_{g_{i}}(\partial^{e}E\cap\Omega)$
for every $\Omega\subset\subset B$. Thus, taking into account Appendix A one
has that $|D\chi_{E}|_{i}(\Omega)\to|D\chi_{E}|(\Omega)$ for every Borel
$\Omega\subset\subset B$, and thus we conclude $|D\chi_{E}|\llcorner
B=\mathcal{H}^{n-1}\llcorner(\partial^{e}E\cap B)$. Since $B$ was arbitrary,
we get the sought claim. In the following, if $E$ is a set of locally finite
perimeter, integrals over its essential boundary are tacitly understood to be
taken with respect to the perimeter measure.

######  Lemma .

Let $(M_{i},g_{i},p_{i})$ be a sequence of pointed $C^{0}$-Riemannian
manifolds of dimension $n$ converging in $C^{0}$-sense to a pointed
$C^{0}$-manifold $(M,g,p)$. Denote by $\mathsf{d},\mathsf{d}_{i}$ the
Riemannian distances on $(M,g),(M_{i},g_{i})$ respectively. Let $f_{i}\in
BV(M_{i})$ be such that
$\sup_{i}\|f_{i}\|_{L^{1}(M_{i})}+|Df_{i}|(M_{i})<+\infty$. Then, up to
subsequence, there exist $f\in BV(M)$, $R_{i}\nearrow+\infty$ and
$(1+1/i)$-biLipschitz embeddings
$F_{i}:(B_{R_{i}}(p),\mathsf{d})\to(M_{i},\mathsf{d}_{i})$ with
$F_{i}(p)=p_{i}$ such that the functions $f_{i}\circ F_{i}$ converge to $f$ in
$L^{1}_{\rm loc}(M)$. Moreover

$|Df|(M)\leq\liminf_{i}|Df_{i}|(M_{i}).$

If also ${\rm spt}f_{i}\subset B_{R}(p_{i})$ for some $R>0$ and for every $i$,
then the convergence occurs in $L^{1}(M)$.

###### Proof.

By a diagonal argument, up to passing to a subsequence, by Section 2.1 there
exist $R_{i}\nearrow+\infty$ and $(1+1/i)$-biLipschitz embeddings
$F_{i}:B_{R_{i}}(p)\to M_{i}$ with $F_{i}(p)=p_{i}$. Denote
$h_{i}\vcentcolon=f_{i}\circ F_{i}$. For any $r>0$, for $i$ large we have that
$h_{i}\in BV(B_{r}(p))$ with
$\sup_{i}\|h_{i}\|_{L^{1}(B_{r}(p))}+|Dh_{i}|(B_{r}(p))<+\infty$. If we show
that $h_{i}$ admits a subsequence converging in $L^{1}(B_{r/2}(p))$, the first
part of the statement follows. Indeed, by Section 2.3 we can find a smooth
Riemannian manifold $(N,g^{\delta})$ such that it is well defined the
inclusion $\iota:(B_{r}(p),\mathsf{d})\to(N,\mathsf{d}^{\delta})$ and $\iota$
is $2$-biLipschitz with its image. Hence $h_{i}$ can be seen as an equibounded
sequence in $BV(B_{r}^{g}(p),g^{\delta})$. By classical precompactness we can
extract a subsequence converging in
$L^{1}(B_{r/2}^{g}(p),\mathcal{H}^{n}_{g^{\delta}})$. Since
$\mathcal{H}^{n}_{g^{\delta}}$ and $\mathcal{H}^{n}_{g}$ are equivalent, the
subsequence converges in $L^{1}(B_{r/2}(p))$ as well.

The lower semicontinuity inequality readily follows since, for any $r>0$ for
$i$ large enough we have

$|Dh_{i}|(B_{r}^{g}(p))\leq(1+o(1))|Df_{i}|(B_{2(1+1/i)r}^{g_{i}}(p_{i}))\leq(1+o(1))|Df_{i}|(M_{i}),$

where $o(1)\to 0$ as $i\to\infty$. ∎

######  Lemma .

Let $(M,g)$ be a $C^{0}$-Riemannian manifold of dimension $n$. Fix $R>0$,
$p\in M$. Then there exists $C:=C(p,R)>1$ such that the following hold.

  1. (1)

For any $x\in B_{R}(p)$ and $0<r\leq R$ there holds

$\frac{|B_{2r}(x)|}{|B_{r}(x)|}\leq C,\qquad C^{-1}r^{n}\leq|B_{r}(x)|\leq
Cr^{n}.$

  2. (2)

For any $x\in B_{R}(p)$, any $r\leq R$, and any $f\in\mathrm{Lip}_{\rm
loc}(M)$ there holds

$\fint_{B_{r}(x)}\left|f-\fint_{B_{r}(x)}f\right|\leq
Cr\fint_{B_{2r}(x)}|\nabla f|.$ (2.4)

  3. (3)

For any $E\subset\subset B_{R}(p)$ there holds

$|E|^{\frac{n-1}{n}}\leq CP(E).$ (2.5)

###### Proof.

The claims are well-known to hold true on manifolds with smooth Riemannian
metrics as a consequence of the fact that one can always find a lower bound
for the Ricci curvature on compact sets. In fact, on a smooth manifold, (1) is
a consequence of the Bishop–Gromov comparison theorem [71, Lemma 7.1.4], (2)
follows from [22, 73], and (3) is a consequence of (1) and (2) by [38, Theorem
9.7] and [2, Theorem 4.3, Remark 4.4]. Therefore claims (1) and (3) in the
setting of the statement readily follow from Section 2.3 applied with $r$
sufficiently large.

It remains to prove (2). Let $(N,g^{\delta})$ be given by Section 2.3 with
$\delta=1/4$ and radius equal to $3R$. Letting now
$f\in\operatorname{Lip}_{\rm loc}(M)$, for $x\in B_{R}(p)$ and $r\leq R$, we
have $h\vcentcolon=f\circ\iota^{-1}\in\operatorname{Lip}_{\rm
loc}(\iota(B_{2r}(p)),\mathsf{d}^{\delta})$, for $\iota$ as in Section 2.3
where $\mathsf{d}^{\delta}$ is the distance on $(N,g^{\delta})$. Since
$|\nabla h(y)|_{g^{\delta}}\leq 2|\nabla f(y)|_{g}$ for any $y\in B_{2r}(p)$,
we find

$\begin{split}\int_{B_{r}^{g}(x)}\left|f-\fint_{B_{r}^{g}(x)}f\,\mathrm{d}\mathcal{H}^{n}_{g}\right|\,\mathrm{d}\mathcal{H}^{n}_{g}&\leq
2\int_{B_{r}^{g}(x)}\left|f-\fint_{B_{\frac{5}{4}r}^{g^{\delta}}(x)}h\,\mathrm{d}\mathcal{H}^{n}_{g^{\delta}}\right|\,\mathrm{d}\mathcal{H}^{n}_{g}\\\
&\leq
C\int_{B_{\frac{5}{4}r}^{g^{\delta}}(x)}\left|h-\fint_{B_{\frac{5}{4}r}^{g^{\delta}}(x)}h\,\mathrm{d}\mathcal{H}^{n}_{g^{\delta}}\right|\,\mathrm{d}\mathcal{H}^{n}_{g^{\delta}}\\\
&\leq Cr\int_{B_{\frac{5}{4}r}^{g^{\delta}}(x)}|\nabla
h|_{g^{\delta}}\,\mathrm{d}\mathcal{H}^{n}_{g^{\delta}}\\\ &\leq
Cr\int_{B^{g}_{2r}(x)}|\nabla
f(y)|_{g}\,\mathrm{d}\mathcal{H}^{n}_{g},\end{split}$

where in the third inequality we applied a Poincaré inequality as in (2.4),
recalling that on smooth manifolds it is possible to take the integral on the
right hand side on the ball of the same radius that appears on the left hand
side [76, Corollary 5.3.5]. ∎

What observed so far implies that $C^{0}$-Riemannian manifolds locally
asymptotic at infinity to space forms are PI spaces, see the forthcoming
Section 2.3. We recall their definition, leaving the interested reader to the
seminal [38, 24] and to the survey [50].

######  Definition  (PI space).

Let $({\rm X},\mathsf{d},\mathfrak{m})$ be a complete and separable metric
measure space, where $\mathfrak{m}$ is a Radon measure. We say that
$\mathfrak{m}$ is uniformly locally doubling if for every $R>0$ there exists
$C_{D}(R)>0$ such that the following holds

$\mathfrak{m}(B_{2r}(x))\leq C_{D}(R)\mathfrak{m}(B_{r}(x)),\qquad\forall
x\in{\rm X}\;\forall r\leq R.$

We say that a weak local $(1,1)$-Poincaré inequality holds on $({\rm
X},\mathsf{d},\mathfrak{m})$ if there exists $\lambda$ such that for every
$R>0$ there exists $C_{P}(R)$ such that for every $f\in\mathrm{Lip}(X)$, the
following inequality holds:

$\fint_{B_{r}(x)}|f-\overline{f}(x)|\,\mathrm{d}\mathfrak{m}\leq
C_{P}(R)r\fint_{B_{\lambda r}(x)}\mathrm{lip}f\,\mathrm{d}\mathfrak{m},$

for every $x\in{\rm X}$ and $r\leq R$, where
$\overline{f}(x):=\fint_{B_{r}(x)}f\,\mathrm{d}\mathfrak{m}$, and

$\mathrm{lip}f(x):=\limsup_{y\to x}\frac{|f(x)-f(y)|}{\mathsf{d}(y,x)},$ (2.6)

if $x$ is an accumulation point, or $\mathrm{lip}f(x)=0$ if $x$ is not an
accumulation point.

We say that $({\rm X},\mathsf{d},\mathfrak{m})$ is a PI space when
$\mathfrak{m}$ is uniformly locally doubling and a weak local $(1,1)$-Poincaré
inequality holds on $({\rm X},\mathsf{d},\mathfrak{m})$.

######  Corollary .

Let $(M,g)$ be a $C^{0}$-Riemannian manifold of dimension $n$ that is
$C^{0}_{\rm loc}$-asymptotic to the $n$-dimensional simply connected complete
model $\mathbb{H}^{n}_{K}$ of constant sectional curvature $K\leq
0$111$\mathbb{H}^{n}_{K}=\mathbb{R}^{n}$ if $K=0$, while $\mathbb{H}^{n}_{K}$
is the $n$-dimensional hyperbolic space of constant sectional curvature $K$ if
$K<0$.. Then $(M,\mathsf{d},\mathcal{H}^{n})$ is a PI space.

###### Proof.

Let $R>0$ and fix $o\in M$. By assumptions and recalling Section 2.1, we can
fix $\rho>R$ such that for any $x\in M\setminus B_{\rho}(o)$ there exists a
diffeomorphism $F_{x}:(B_{3R}(x),g)\to(\mathbb{H}^{n}_{K},g_{K})$ that is
$2$-biLipschitz with its image, where $g_{K}$ is the metric on
$\mathbb{H}^{n}_{K}$, and whose image contains $B_{R}(F_{x}(x))$. Since
$\mathbb{H}^{n}_{K}$ is PI, arguing as in the proof of Section 2.3 it follows
that there exist $\overline{C}>0$ such that

$\frac{|B_{2r}(x)|}{|B_{r}(x)|}\leq\overline{C},\qquad\fint_{B_{r}(x)}\left|f-\fint_{B_{r}(x)}f\right|\leq\overline{C}r\fint_{B_{2r}(x)}|\nabla
f|,$

for any $x\in M\setminus B_{\rho}(o)$, any $r\leq R$, and any
$f\in\mathrm{Lip}_{\rm loc}(M)$.

Next apply Section 2.3 with $p=o$ and $R=2\rho$. Hence for any $x\in M$ we
have that either $x\in B_{2\rho}(o)$, or $x\in M\setminus B_{\rho}(o)$. Hence
the fact that $(M,\mathsf{d},\mathcal{H}^{n})$ is PI follows putting together
the previous inequalities with those given by Section 2.3. ∎

The next corollary states the local Hölder continuity of the isoperimetric
profile of $C^{0}$-Riemannian manifolds that are $C^{0}$-locally asymptotic to
a model of constant curvature. The proof essentially follows a classical path,
see e.g. [5, Lemma 2.23]. However in this context we do not have an explicit
asymptotic rate for the perimeter of balls of infinitesimal radii, which are
often used to perturb competitors. In place of ball, we shall employ images of
Euclidean balls through biLipschitz maps into the manifold, so to get a one-
parameter increasing family of sets whose perimeter has an explicit rate as
the parameter goes to zero.

######  Corollary .

Let $(M,g)$ be a $C^{0}$-Riemannian manifold of dimension $n$ that is
$C^{0}_{\rm loc}$-asymptotic to the $n$-dimensional simply connected complete
model $\mathbb{H}^{n}_{K}$ of constant sectional curvature $K\leq 0$. Denote
by $I$ (resp., $I_{K}$) the isoperimetric profile of $M$ (resp.,
$\mathbb{H}^{n}_{K}$).

Then $I\leq I_{K}$, and $I$ is locally $\tfrac{n-1}{n}$-Hölder continuous on
$(0,+\infty)$.

###### Proof.

We start by proving that $I\leq I_{K}$. Let
$B_{t}^{K}(0)\subset\mathbb{H}^{n}_{K}$ be a ball with volume
$V\in(0,+\infty)$ in $\mathbb{H}^{n}_{K}$. If $q_{i}\in M$ is a diverging
sequence of points, for large $i$, up to subsequence, there exist
diffeomorphisms $F:(B_{2t}(0),g_{K})\to F(B_{2t}(0))\subset(M,g)$ such that
$F(0)=q_{i}$ and $F$ is $(1+1/i)$-biLipschitz. Hence, there exists $t_{i}$
such that $t_{i}\to t$, $E_{i}\vcentcolon=F(B_{t_{i}}(0))$ has volume $V$ in
$M$, and $P(E_{i})\to P(B_{t}^{K}(0))=I_{K}(V)$. Therefore
$I(V)\leq\lim_{i}P(E_{i})=I_{K}(V)$.

Fix $o\in M$. Since $M$ is $C^{0}_{\rm loc}$-asymptotic to
$\mathbb{H}^{n}_{K}$, there exists $r>0$ such that for any $p\in M\setminus
B_{r}(o)$ there exists a $2$-biLipschitz map $F_{p}$ from a Euclidean ball of
sufficiently small radius to $M$ with $F_{p}(0)=p$. Combining with Section
2.3, we conclude that there exists $\overline{r}\in(0,1)$ such that for any
$p\in M$ there exists a $2$-biLipschitz map $F_{p}:(B_{\overline{r}}^{\rm
eu}(0),\mathsf{d}_{\rm eu})\to F_{p}(B_{\overline{r}}^{\rm eu}(0))\subset M$
such that $F_{p}(0)=p$. Hence we define the one-parameter family of sets
$E_{t}(p)\vcentcolon=F_{p}(B_{t}^{\rm eu}(0))$, for any $p\in M$ and
$t\in(0,\overline{r}]$. In particular, recalling also the representation of
the perimeter Section 2.3 and the fact that the essential boundary is
biLipschitz invariant,

$P(E_{t}(p))\leq\overline{C}t^{n-1},\qquad\frac{1}{\overline{C}}t^{n}\leq|E_{t}(p)|\leq\overline{C}t^{n},$
(2.7)

for any $p\in M$ and $t\in(0,\overline{r}]$, for some $\overline{C}$
independent of $p,t$.

Combining again the fact that $M$ is $C^{0}_{\rm loc}$-asymptotic to
$\mathbb{H}^{n}_{K}$ with item (1) in Section 2.3 we get that for any $R>0$
there exists a constant $C_{R}>0$ such that $C_{R}^{-1}r^{n}\leq|B_{r}(x)|\leq
C_{R}r^{n}$ for any $x\in M$ and $r\in(0,R)$. Hence recalling that
$(M,\mathsf{d},\mathcal{H}^{n})$ is also PI by Section 2.3, then it is well-
known that there holds a relative isoperimetric inequality in balls of $M$,
see [38, Theorem 5.1] and [2, Remark 4.4]. Since also $\inf_{x\in
M}|B_{1}(x)|>0$ thanks to the asymptotic assumption, it is readily checked
that the proof of [5, Lemma 2.10] can be repeated in our setting, yielding
that: for any $R>0$ there exists $\widetilde{C}=\widetilde{C}(R)>0$ such that
for any $E\subset M$ with $|E|\in(0,+\infty)$ there exists $x_{E}$ such that

$|E\cap
B_{r}(x_{E})|\geq\widetilde{C}\min\left\\{\frac{|E|^{n}}{P(E)^{n}},r^{n}\right\\},$
(2.8)

for any $r\in(0,R]$.

Fix now $\overline{V}\in(0,+\infty)$ and $\eta\in(0,\overline{V})$. Let also
$V_{0}\in(\max\\{\overline{V}-1,\eta\\},\overline{V}+1)$ and for any
$\varepsilon>0$ let $E\subset M$ be a bounded set such that $|E|=V_{0}$ and
$P(E)\leq I(V_{0})+\varepsilon$ (it is readily checked that, arguing as in [4,
Lemma 2.17], the isoperimetric profile is achieved by bounded sets).

Let

$v\vcentcolon=\min\left\\{\min_{x\in
M}|E_{\overline{r}}(x)|,1\right\\}\geq\min\\{\overline{C}^{-1}\overline{r}^{n},1\\}>0.$

Since $E$ is bounded, for any $V\in[V_{0},V_{0}+v)$ there exist $x_{V}\in M$
and $r_{V}\leq\overline{r}$ such that $|E\cup E_{r_{V}}(x_{V})|=V$ and
$|E_{r_{V}}(x_{V})|=V-V_{0}$. Therefore

$I(V)\leq P(E)+P(E_{r_{V}}(x_{V}))\leq
I(V_{0})+\varepsilon+\overline{C}r_{V}^{n-1}.$

Since $r_{V}^{n}\leq\overline{C}(V-V_{0})$, letting $\varepsilon\to 0$ we
conclude that

$I(V)\leq I(V_{0})+\overline{C}^{2-\frac{1}{n}}(V-V_{0})^{\frac{n-1}{n}}.$
(2.9)

We next consider volumes smaller than $V_{0}$. By (2.8) we know that there
exists a constant $\widetilde{C}=\widetilde{C}(\overline{r})$ independent of
$E$ and a point $x_{E}$ such that

$|E\cap E_{t}(x_{E})|\geq|E\cap
B_{\frac{t}{2}}(x_{E})|\geq\frac{\widetilde{C}}{2^{n}}\min\left\\{\frac{|E|^{n}}{P(E)^{n}},t^{n}\right\\}\geq\frac{\widetilde{C}}{2^{n}}\min\left\\{\frac{V_{0}^{n}}{(I_{K}(V_{0})+\varepsilon)^{n}},t^{n}\right\\}=\frac{\widetilde{C}}{2^{n}}t^{n},$

for any $t\leq t_{0}$ for some
$t_{0}=t_{0}(\overline{V},\eta,n,K,\overline{r})\in(0,\overline{r})$. Let
$\widetilde{C}_{2}\vcentcolon=\widetilde{C}2^{-n}t_{0}^{n}>0$. Up to decrease
$t_{0}$, we can assume that $\widetilde{C}_{2}<\overline{V}$.

If $V_{0}>\overline{V}-\widetilde{C}_{2}$, let
$V\in(\max\\{\overline{V}-1,\eta,\overline{V}-\widetilde{C}_{2}\\},\overline{V}+1)$
such that $V<V_{0}$. Hence there exists $\overline{t}\in(0,t_{0})$ such that
$|E\setminus E_{\overline{t}}(x_{E})|=V$. Similarly as before, we have

$I(V)\leq P(E)+P(E_{\overline{t}}(x_{E}))\leq
I(V_{0})+\varepsilon+\overline{C}\overline{t}^{n-1}\leq
I(V_{0})+\varepsilon+\frac{2^{n-1}\overline{C}}{\widetilde{C}^{\frac{n-1}{n}}}(V_{0}-V)^{\frac{n-1}{n}},$

which letting $\varepsilon\to 0$ yields

$I(V)\leq
I(V_{0})+\frac{2^{n-1}\overline{C}}{\widetilde{C}^{\frac{n-1}{n}}}(V_{0}-V)^{\frac{n-1}{n}}.$
(2.10)

Putting together (2.9) and (2.10), we have proved that there exist
$v,C_{H}(M)>0$ and $\widetilde{C}_{2}=\widetilde{C}_{2}(M,\overline{V},\eta)$
such that for any
$V_{0}\in(\max\\{\overline{V}-1,\eta,\overline{V}-\widetilde{C}_{2}\\},\overline{V}+1)$,
for any
$V\in(\max\\{\overline{V}-1,\eta,\overline{V}-\widetilde{C}_{2}\\},\min\\{\overline{V}+1,V_{0}+v\\})$
there holds

$I(V)\leq I(V_{0})+C_{H}|V-V_{0}|^{\frac{n-1}{n}}.$ (2.11)

The dependence of the previous constants imply that there exists a
neighborhood $U$ of $\overline{V}$ such that (2.11) holds for any choices of
$V,V_{0}\in U$. This implies the desired local Hölder continuity. ∎

The next theorem is based on a concentration-compactness argument that has
been used several times in the literature, applied to the study of the
isoperimetric problem in noncompact manifolds. In the smooth setting it has
been first obtained in [69, Theorem 2]. See also [75, Theorem 4.48] and
references therein for a complete account. Results analogous to Theorem 2.2
have been worked out also in the setting of nonsmooth spaces with bounds below
on the curvature, see [4, Theorem 4.6] and [5, Theorem 3.3 & Theorem 1.1].

######  Theorem 2.2 (Asymptotic mass decomposition under $C^{0}_{\rm
loc}$-asymptotic assumptions).

Let $(M,g)$ be an $n$-dimensional complete $C^{0}$-Riemannian manifold and
assume that $M$ is $C^{0}_{\rm loc}$-asymptotic to the $n$-dimensional simply
connected complete model $\mathbb{H}^{n}_{K}$ of constant sectional curvature
$K\leq 0$. Fix $o\in M$. Let $V>0$ and let $E_{i}\subset M$ be a sequence of
bounded sets such that $|E_{i}|=V$ for any $i$ and $\lim_{i}P(E_{i})=I(V)$.

Then, up to subsequence, one of the following alternatives holds true.

  * •

The sequence $E_{i}$ converges in $L^{1}(M)$ to an isoperimetric set $E$ of
volume $V$.

  * •

There exist two sequences of radii $R_{i},r_{i}\nearrow+\infty$ with
$R_{i}<r_{i}$ and a diverging sequence of points $p_{i}\in M\setminus
B_{R_{i}(o)}$ such that $E_{i}^{c}\vcentcolon=E_{i}\cap B_{R_{i}}(o)$
converges in $L^{1}(M)$ to a (possibly empty) isoperimetric set $E$, and
$E_{i}^{d}\vcentcolon=E_{i}\cap B_{r_{i}}(p_{i})\setminus B_{R_{i}}(o)$
converges to a ball $B\subset\mathbb{H}^{n}_{K}$ in the sense that

$\lim_{i}P(E_{i}^{d})=P(B),\qquad\lim_{i}|E_{i}^{d}|=|B|.$

Moreover

$V=|E|+|B|,\qquad I(V)=P(E)+P(B).$

###### Proof.

Since the proof of Theorem 2.2 is standard and follows closely the strategy of
[4, 5] we just sketch it.

At first, one gets the analogue of Ritoré–Rosales’ result [5, Theorem 3.3] in
the setting of Theorem 2.2. Indeed, the proof of [5, Theorem 3.3] uses the
coarea formula, the precompactness of $BV$ in $L^{1}_{\mathrm{loc}}$, and the
existence, around every point $p\in M$, of a one-parameter family
$\\{\mathcal{B}_{p,r}\\}_{r\in(0,\varepsilon)}$ of sets such that
$r\mapsto|\mathcal{B}_{p,r}|$ is continuous and vanishing as $r\to 0$, and
$P(\mathcal{B}_{p,r})\to 0$ as $r\to 0$. The first two come from Section 2.3,
and Theorem 2.1, while for the last one it suffices to take pre-images of
small Euclidean balls under the map in the second item of Section 2.1, as it
has been done in the proof of Section 2.3. Once this is done, one follows
verbatim the proof of [4, Theorem 4.6], which additionally needs that
$(M,\mathsf{d},\mathcal{H}^{n})$ is PI, and $|B_{r}(p)|\geq\zeta r^{3}$ for
every $r\in(0,R]$, and every $p\in M$, where $\zeta,R>0$ are constants
depending on $M$: these two properties come from Section 2.3 and from its
proof (compare also with item (1) in Section 2.3).

Following [4, Theorem 4.6] until (4.20), and then jumping to Step 5 in there,
one finally gets the following. There is
$\overline{N}\in\mathbb{N}\cup\\{+\infty\\}$ and radii $R_{i}\to+\infty$,
$T_{i,j}\to_{i}+\infty$ for $1\leq j<\overline{N}+1$, and there are mutually
(with respect to $j$) diverging $p_{i,j}\in M\setminus B_{R_{i}}(o)$ such that

$E_{i}^{c}\xrightarrow[i]{}E\text{ in $L^{1}(M)$,}\qquad(E_{i}\setminus
B_{R_{i}}(o))\cap B_{T_{i,j}}(p_{i,j})\xrightarrow[i]{}B_{j}\text{ in
$L^{1}(M)$,}$

where $B_{j}$ is a ball in $\mathbb{H}^{n}_{K}$ for any $j<\overline{N}+1$,
and $E$ is an isoperimetric set in $M$. The convergence in $L^{1}$ to the
$B_{j}$’s has to be intended as in the statement of Section 2.3, after the
composition with biLipschitz embeddings. Moreover one has

$|E|+\sum_{j=1}^{\overline{N}}|B_{j}|=V,\qquad
P(E)+\sum_{j=1}^{\overline{N}}I_{K}(|B_{j}|)=P(E)+\sum_{j=1}^{\overline{N}}P(B_{j})=I(V),$

where $I_{K}$ is the isoperimetric profile of $\mathbb{H}^{n}_{K}$. Hence
either $\overline{N}=0$ and the first item holds, or $\overline{N}\geq 1$. In
the latter case, we want to show that $\overline{N}=1$, completing the proof
of the second item. By coarea formula we can fix a sequence
$\rho_{i}\nearrow+\infty$ such that $P(E\cap B_{\rho_{i}}(o))\leq P(E)+1/i$.
For any $i$ we find balls $B_{s_{i}}(q_{i})\subset M\setminus
B_{\rho_{i}+1}(o)$ such that $|B_{s_{i}}(q_{i})|=V-|E\cap B_{\rho_{i}}(o)|$
and such that $B_{s_{i}}(q_{i})$ converges to a ball
$B\subset\mathbb{H}^{n}_{K}$ with $\lim_{i}P(B_{s_{i}}(q_{i}))=P(B)$ and
$|B|=V-|E|$. If by contradiction $\overline{N}>1$, since the isoperimetric
profile $I_{K}$ is a strictly subadditive function, we get

$\begin{split}P(E)+\sum_{j=1}^{\overline{N}}I_{K}(|B_{j}|)&=P(E)+\sum_{j=1}^{\overline{N}}P(B_{j})=I(V)\leq\liminf_{i}P(E\cap
B_{\rho_{i}}(o))+P(B_{s_{i}}(q_{i}))\\\ &=P(E)+I_{K}(|B|)\leq
P(E)+I_{K}\left(\sum_{j=1}^{\overline{N}}|B_{j}|\right)<P(E)+\sum_{j=1}^{\overline{N}}I_{K}(|B_{j}|),\end{split}$

which is a contradiction. ∎

##  3\. Local inverse mean curvature flow

We start by recalling the definition of weak inverse mean curvature flow
(IMCF) as introduced in [45].

######  Definition  (Weak IMCF - Level set formulation).

Let $(M,g)$ be a smooth Riemannian manifold. Given a precompact $K\subset M$,
a locally Lipschitz function $u:M\to\mathbb{R}$, and a set of locally finite
perimeter $E$, define

$J_{u}^{K}(E):=P(E,K)-\int_{E\cap K}|\nabla u|.$

Let $\Omega\subset M$ be an open set. A function
$u\in\mathrm{Lip}_{\mathrm{loc}}(\Omega)$ is called a weak solution (resp.,
subsolution, supersolution) to the inverse mean curvature flow (IMCF) in
$\Omega$ if

$J_{u}^{K}(\\{u<t\\})\leq J_{u}^{K}(E),$

for all $t\in\mathbb{R}$, all $K\subset\subset\Omega$, and all sets $E$
(resp., $E\supset\\{u<t\\}$, $E\subset\\{u<t\\}$) such that
$E\Delta\\{u<t\\}\subset K$.

######  Remark .

By virtue of [45, Lemma 1.1], a function
$u\in\mathrm{Lip}_{\mathrm{loc}}(\Omega)$ is a weak solution the IMCF in
$\Omega$ if and only if

$\int_{K}|\nabla u|+u|\nabla u|\leq\int_{K}|\nabla v|+v|\nabla u|,$ (3.1)

for all $K\subset\subset\Omega$, and all
$v\in\mathrm{Lip}_{\mathrm{loc}}(\Omega)$ such that $\\{u\neq v\\}\subset K$.

The aim of this section is to show Theorem 3.1, stating that on every
punctured ball $B$ centered at $o$ on a smooth complete Riemannian manifold
one can define a weak IMCF that is bounded from below explicitly in terms of
constants that will nicely behave on metrics $C^{0}$-close to the flat one. We
also gather useful properties of this flow in Section 3.1, and Section 3.2.

######  Definition .

Let $(M,g)$ be an $n$-dimensional complete $C^{0}$-Riemannian manifold, and
let $1\leq p<n$. Let $\Omega\subset M$ be an open set. We say that $\Omega$
supports a $(p,p^{*})$-Sobolev inequality if there exists a constant $C>0$
such that

$\left(\int_{M}|\psi|^{\frac{np}{n-p}}\right)^{\frac{n-p}{n}}\leq
C\int_{M}|\nabla\psi|^{p}\,\quad\text{for all
$\psi\in\mathrm{Lip}_{\mathrm{c}}(\Omega)$}.$ (3.2)

We denote $C_{p,\mathrm{Sob}}(\Omega)$ the smallest constant $C$ for which the
latter inequality holds.

######  Remark .

It is known that (3.2) with $p=1$ is equivalent to

$|E|^{\frac{n-1}{n}}\leq CP(E),\quad\text{for all bounded measurable
$E\subset\subset\Omega$}.$

Indeed, one implication readily comes from the very definition in (2.1), while
the other comes from an application of the coarea formula in Theorem 2.1. The
latter implication is classical and dates back at least to works of
Federer–Fleming and Maz’ya in the 60s, see, e.g., [34, page 488].

We now provide solutions to the weak IMCF in punctured balls
$B_{R}(o)\setminus\\{o\\}$. The weak IMCF is obtained in the limit, as $p\to
1^{+}$, of functions $w_{p}^{R}:=-(p-1)\log G_{p}^{R}$, where $G_{p}^{R}$, for
$p\in(1,n)$ denotes the $p$-harmonic Green function on $B_{R}(o)$ with
Dirichlet boundary conditions. Namely, $G_{p}^{R}$ is the solution to

$\begin{cases}-\Delta_{p}G_{p}^{R}=\left\lvert\mathbb{S}^{n-1}\right\rvert\left(\frac{n-p}{p-1}\right)^{p-1}\delta_{o}&\text{on
}B_{R}(o),\\\ \phantom{-\Delta_{p}}G_{p}^{R}=0&\text{on }\partial
B_{R}(o),\end{cases}$ (3.3)

where $\delta_{o}$ is the Dirac delta supported at $o$, and
$\left\lvert\mathbb{S}^{n-1}\right\rvert$ is the measure of the
$(n-1)$-dimensional unit sphere. With the above choice of normalization, it
follows from the blow-up procedure leading to [49, Theorem 1.1] that

$\begin{split}\left|\frac{G_{p}^{R}(x)}{r(x)^{-{(n-p)}/{(p-1)}}}-1\right|\to
0,\qquad\left|\frac{\left\lvert\nabla
G_{p}^{R}(x)\right\rvert}{r(x)^{-{(n-1)}/{(p-1)}}}-\frac{n-p}{p-1}\right|\to
0\end{split}$ (3.4)

as $r(x)\to 0$, where we let $r(x)\vcentcolon=\mathsf{d}(o,x)$. We recall that
the relative $p$-capacity of a compact $K\Subset B_{R}(o)$ is defined as

$\mathrm{Cap}_{p}(K,B_{R}(o)):=\inf\left\\{\int_{B_{R}(o)\setminus
K}\left\lvert\nabla v\right\rvert^{p}\ :\
v\in\mathrm{Lip_{c}}(B_{R}(o)),v\geq\chi_{K}\right\\}.$ (3.5)

The following lemma is well known, and consists essentially in [40, Lemma
3.8].

######  Lemma .

Let $(M,g)$ be a smooth complete Riemannian manifold, and let $o\in M$, $R>0$.
Let $E_{t}^{R}:=\\{w_{p}^{R}\leq t\\}$ for $p\in(1,n)$, where
$w_{p}^{R}\vcentcolon=-(p-1)\log G_{p}^{R}$ and $G_{p}^{R}$ solves (3.3). Then

$\mathrm{Cap}_{p}(E_{t}^{R},B_{R}(o))=\left(\frac{n-p}{p-1}\right)^{p-1}\left\lvert\mathbb{S}^{n-1}\right\rvert
e^{t},$ (3.6)

for every $t\in\mathbb{R}$.

###### Proof.

In this proof, let $G_{p}:=G_{p}^{R}$, and $w_{p}:=w_{p}^{R}$ for the ease of
notation. It is well-known that the $p$-capacity defined in (3.5) is attained
computing the $L^{p}$-norm of the gradient of the $p$-harmonic function with
Dirichlet data equal to $1$ on $\partial K$, and equal to $0$ on $\partial
B_{R}(o)$ (see [39] for a thorough account on nonlinear potential theory). In
particular, one gets, for every $t\in(0,+\infty)$,

$\mathrm{Cap}_{p}(\\{G_{p}\geq
t\\},B_{R}(o))=\frac{1}{t^{p}}\int_{\\{G_{p}<t\\}}\left\lvert\nabla
G_{p}\right\rvert^{p}=\frac{1}{t^{p}}\int_{0}^{t}\int_{\\{G_{p}=s\\}}\left\lvert\nabla
G_{p}\right\rvert^{p-1}\,\mathrm{d}\mathcal{H}^{n-1}\,\mathrm{d}s.$ (3.7)

On the other hand, a straightforward application of the divergence theorem
combined with the $p$-harmonicity of $G_{p}$ (see e.g. the computations in
[11, Proposition 2.8]) yields that $\int_{\\{G_{p}=s\\}}\left\lvert\nabla
G_{p}\right\rvert^{p-1}$ attains the same value for almost every
$s\in(0,+\infty)$. Such constant is computed using (3.4) as

$\int_{\\{G_{p}=s\\}}\left\lvert\nabla
G_{p}\right\rvert^{p-1}\,\mathrm{d}\mathcal{H}^{n-1}=\left(\frac{n-p}{p-1}\right)^{p-1}\left\lvert\mathbb{S}^{n-1}\right\rvert,$

for almost every $s\in(0,+\infty)$. Plugging it into (3.7) leaves us with

$\mathrm{Cap}_{p}(\\{G_{p}\geq
t\\},B_{R}(o))=\frac{1}{t^{p-1}}\left(\frac{n-p}{p-1}\right)^{p-1}\left\lvert\mathbb{S}^{n-1}\right\rvert,$
(3.8)

for any $t\in(0,+\infty)$. Rewriting it in terms of $w_{p}$ as stated in (3.6)
completes the proof. ∎

We denote with $C_{P}(B_{R}(o))$ the Poincaré constant of $B_{R}(o)$, defined
as the smallest constant $C$ such that

$\fint_{B_{r}(x)}\left|f-\fint_{B_{r}(x)}f\right|\leq
Cr\fint_{B_{2r}(x)}|\nabla f|$ (3.9)

holds for every $x\in B_{R}(o)$, $r\leq R$, and $f\in\operatorname{Lip}_{\rm
loc}(M)$. We denote with $C_{A}(B_{R}(o))$ the Ahlfors constant of $B_{R}(o)$,
defined as the smallest number $C\geq 1$ such that

$C^{-1}r^{n}\leq\left\lvert B_{r}(x)\right\rvert\leq Cr^{n},$ (3.10)

for every $x\in B_{R}(o)$, and every $0<r\leq R$.

Finally, denoting
$A_{\rho_{1},\rho_{2}}(o)\vcentcolon=B_{\rho_{2}}(o)\setminus\overline{B}_{\rho_{1}}(o)$
for $\rho_{2}>\rho_{1}$, we denote

$\begin{split}C_{\mathrm{cov}}(B_{\rho}(o))\vcentcolon=\min\Big{\\{}N\in\mathbb{N}\
:\ &\text{$A_{3r/4,5r/4}(o)$ is covered by $N$ open balls of radius $r/2$}\\\
&\text{with centers in $A_{3r/4,5r/4}(o)$ for any
$0<r\leq\rho$}\Big{\\}}.\end{split}$ (3.11)

Observe that on any complete smooth Riemannian manifold $(M,g)$ of dimension
$n\geq 2$, for any $o\in M$ and $R>0$ there exists $\rho\in(0,R/2]$ such that

$\forall 0<r\leq\rho,\,\,\forall p,q\in\partial B_{r}(o)\,\,\exists$
continuous curve $\gamma\subset A_{3r/4,5r/4}(o)$ connecting $p$ and $q$.
(3.12)

######  Theorem 3.1.

Let $(M,g)$ be a complete smooth Riemannian manifold of dimension $n\geq 2$.
Fix $o\in M$, and $R>0$, and let $p\in(1,n)$. Let $w_{p}^{2R}=-(p-1)\log
G_{p}^{2R}$, with $G_{p}^{2R}$ as in (3.3). Let $\rho\in(0,R/2]$ be such that
(3.12) is satisfied. Then, the following hold.

  1. (1)

The sequence of functions $w_{p}^{2R}$ converges, up to subsequence, locally
uniformly in $B_{2R}(o)\setminus\\{o\\}$ as $p\to 1^{+}$ to a weak solution
$w$ of the IMCF on $B_{2R}(o)\setminus\\{o\\}$.

  2. (2)

The function $w$ satisfies

$w(x)\geq(n-1)\log r(x)-C,\qquad\text{for all
$x\in\overline{B}_{R}(o)\setminus\\{o\\}$},$ (3.13)

where
$C=C\big{(}n,C_{1,\mathrm{Sob}}(B_{2R}(o)),C_{P}(B_{2R}(o)),C_{A}(B_{2R}(o)),R/\rho,C_{\mathrm{cov}}(B_{\rho}(o))\big{)}$,
and $r(x)\vcentcolon=\mathsf{d}(o,x)$.

  3. (3)

It holds $w(x)\to-\infty$ as $x\to o$.

  4. (4)

For every $r\leq R$, letting $T_{r}:=(n-1)\log r-C-1$, there holds $\\{w\leq
T_{r}\\}\subset B_{r}(o)$.

######  Remark .

If $(M,g)$ in Theorem 3.1 supports a $(1,1^{*})$-Sobolev inequality, then the
existence of a _global_ proper weak IMCF issuing from a point follows from a
careful modification of the proof of [84]. Yet, we do not know how to obtain
the quantitative estimate (3.13) through the procedure introduced there.

The proof of Theorem 3.1 requires some preliminary steps. The following result
is a direct consequence of [51, Theorem 1.1].

######  Theorem 3.2.

Let $(M,g)$ be a complete smooth Riemannian manifold, and let $u_{p}$ be a
positive $p$-harmonic function defined in an open set $U\subset M$, for
$p\in(1,2)$. Let $w_{p}=-(p-1)\log u_{p}$. Then, for any compact subset
$K\subset U$, we have

$\left\lvert\nabla w_{p}\right\rvert\leq C(K),$ (3.14)

where $C(K)$ does not depend on $p\in(1,2)$.

The constant $C(K)$ in Theorem 3.2 depends on sectional curvature bounds for
$g$ on $K$. The following observation will be useful also for estimating the
asymptotic behavior of the Hawking mass at $o$ along the IMCF, see Section 3.1
below.

######  Remark .

For $p\in(1,2)$, if $u_{p}$ is a positive $p$-harmonic function on
$B_{2R}(o)\setminus\\{o\\}$, and $\mathrm{Sec}\geq-k^{2}$ on $B_{2R}(o)$, then
there exist two constants $\eta:=\eta(k,n)$ and $\zeta:=\zeta(k,n)$ such that
for every $x\in B_{2R}(o)\setminus\\{o\\}$ with
$\mathsf{d}(x,o)<\min\\{\eta,R\\}$ there holds

$|\nabla w_{p}|(x)\leq\frac{\zeta}{\mathsf{d}(o,x)}.$ (3.15)

Indeed, it suffices to apply [51, Equation (1.5)] on balls
$B(x,\mathsf{d}(o,x)/2)$. We remark that both $\eta,\zeta$ can be chosen to be
uniform with respect to $p\to 1^{+}$.

The following result yields the uniform lower bound on $w_{p}^{2R}$ that will
result in (3.13). Its core is in the Harnack inequality for $p$-harmonic
functions that comes with the sharp dependence with respect to $p$. The
estimate (3.17) below was suggested to the authors by Luca Benatti.

######  Theorem 3.3.

Let $(M,g)$ be a complete smooth Riemannian manifold of dimension $n\geq 2$.
Fix $o\in M$, and $R>0$, and let $p\in(1,n)$. Let $w_{p}^{2R}=-(p-1)\log
G_{p}^{2R}$, with $G_{p}^{2R}$ as in (3.3). Let $\rho\in(0,R/2]$ be such that
(3.12) holds.

Then

$w_{p}^{2R}(x)\geq(n-p)\log r(x)-C,\qquad\text{for all $x\in
B_{3R/2}(o)\setminus\\{o\\}$},$ (3.16)

for
$C=C\big{(}n,C_{1,\mathrm{Sob}}(B_{2R}(o)),C_{P}(B_{2R}(o)),C_{A}(B_{2R}(o)),R/\rho,C_{\mathrm{cov}}(B_{\rho}(o))\big{)}$,
where $r(x)\vcentcolon=\mathsf{d}(o,x)$.

###### Proof.

Within this proof let $G_{p}^{2R}=:G_{p}$, and $w_{p}^{2R}=:w_{p}$ for the
ease of notation. Let $m(r)=\max_{\partial B_{r}(o)}w_{p}$ for $r\in(0,3R/2)$.
Notice that $w_{p}(x)\to-\infty$ as $x\to o$. Then, by the maximum principle,
$B_{r}(o)\subset\\{w_{p}\leq m(r)\\}$, the monotonicity of the $p$-capacity
with respect to inclusion and (3.6) give

$\mathrm{Cap}_{p}(\overline{B}_{r}(o),B_{2R}(o))\leq\mathrm{Cap}_{p}(\\{w_{p}\leq
m(r)\\},B_{2R}(o))=\left(\frac{n-p}{p-1}\right)^{p-1}\left\lvert\mathbb{S}^{n-1}\right\rvert
e^{m(r)},$

that results in

$m(r)\geq\log(\mathrm{Cap}_{p}(\overline{B}_{r}(o),B_{2R}(o)))-(p-1)\log\left(\frac{n-p}{p-1}\right)-\log\left\lvert\mathbb{S}^{n-1}\right\rvert.$
(3.17)

It is now well-known that capacities can be estimated exploiting isoperimetric
inequalities. More precisely, setting $C=1/C_{1,{\rm Sob}}(B_{2R}(o))$, we can
apply [36, Eq. (7)] to get

$\begin{split}\mathrm{Cap}_{p}(\overline{B}_{r}(o),B_{2R}(o))&\geq\left(\int_{\left\lvert
B_{r}(o)\right\rvert}^{\left\lvert
B_{2R}(o)\right\rvert}\frac{1}{Cv^{\frac{p(n-1)}{n(p-1)}}}\right)^{1-p}\\\
&=\left(C\frac{(n-p)}{n(p-1)}\right)^{p-1}\left\lvert
B_{r}(o)\right\rvert^{\frac{n-p}{n}}\left[1-\left(\frac{\left\lvert
B_{r}(o)\right\rvert}{\left\lvert
B_{2R}(o)\right\rvert}\right)^{\frac{n-p}{n(p-1)}}\right]^{1-p}\\\
&\geq\left(C\frac{(n-p)}{n(p-1)}\right)^{p-1}\left\lvert
B_{r}(o)\right\rvert^{\frac{n-p}{n}}.\end{split}$

We can now estimate $\left\lvert B_{r}(o)\right\rvert^{\frac{n-p}{n}}\geq
C_{A}(B_{2R}(o))^{\frac{n-p}{n}}r^{n-p}$, and so we get from (3.17) that

$m(r)\geq(n-p)\log r-C$ (3.18)

where now $C=C\big{(}n,C_{1,\mathrm{Sob}}(B_{2R}(o)),C_{A}(B_{2R}(o))\big{)}$,
and it is independent of $p$ as $p\to 1^{+}$.

Let us consider now $y\in\partial B_{r}(o)$ for $r\in(0,\tfrac{3}{2}R)$. We
aim at estimating $w_{p}(y)$ combining (3.18) with a Harnack inequality. The
Harnack inequality for positive $p$-harmonic functions [74, Theorem 1.28]
applied to $G_{p}$ reads

$G_{p}(y)\leq C_{H}^{\frac{1}{p-1}}(B_{s}(z))G_{p}(x),$ (3.19)

for any $x,y\in B_{s}(z)\subset\subset B_{2R}(o)\setminus\\{o\\}$, and the
Harnack constant $C_{H}(B_{s}(z))$ can be estimated from above in terms of
$n,C_{P}(B_{2R}(o)),C_{A}(B_{2R}(o))$ and $C_{p,\mathrm{Sob}}(B_{2R}(o))$. The
interested reader might consult [62, Theorem 3.4 and Remark 3.5] for the full
computations leading to the explicit constant in the Harnack inequality. Since
$C_{p,\mathrm{Sob}}(B_{2R}(o))\to C_{1,\mathrm{Sob}}(B_{2R}(o))$ as $p\to
1^{+}$, then

$C_{H}(B_{s}(z))\leq
C=C\big{(}n,C_{P}(B_{2R}(o)),C_{A}(B_{2R}(o)),C_{1,\mathrm{Sob}}(B_{2R}(o))\big{)}$

for any $B_{s}(z)\subset\subset B_{2R}(o)\setminus\\{o\\}$.

Fix $x,y\in\partial B_{r}(o)$ such that

$G_{p}(x)=\min_{\partial B_{r}(o)}G_{p},\qquad G_{p}(y)=\max_{\partial
B_{r}(o)}G_{p}.$

We first consider the case $r\leq\rho$, where $\rho$ is as in the assumptions.
Hence by (3.12) we can connect $x$ and $y$ with a curve $\gamma\subset
A_{3r/4,5r/4}(o)$. Letting $N\vcentcolon=C_{\rm cov}(B_{\rho}(o))$, we can fix
a family of at most $N$ balls $\\{B_{r/2}(z_{j})\\}$ such that $z_{j}\in
A_{3r/4,5r/4}(o)$ and $A_{3r/4,5r/4}(o)\subset\cup_{j}B_{r/2}(z_{j})$. The
existence of $\gamma$ implies that we can find a sequence of points
$y:=x_{1},\ldots,x_{N^{\prime}}=:x$ such that $N^{\prime}\leq N+1$,
$x_{i},x_{i+1}\in\overline{B}_{r/2}(z_{j_{i}})$ for every
$i=1,\ldots,N^{\prime}-1$ for some $z_{j_{i}}$. Thus applying iteratively
(3.19) we deduce (the value of the constant $C$ might change from line to
line)

$\max_{\partial B_{r}(o)}G_{p}\leq C^{\frac{N^{\prime}}{p-1}}\min_{\partial
B_{r}(o)}G_{p}\leq C^{\frac{1}{p-1}}\min_{\partial B_{r}(o)}G_{p},$ (3.20)

for
$C=C\big{(}n,C_{P}(B_{2R}(o)),C_{A}(B_{2R}(o)),C_{1,\mathrm{Sob}}(B_{2R}(o)),C_{\rm
cov}(B_{\rho}(o))\big{)}$.
If instead $r\in(\rho,\tfrac{3}{2}R)$, then we consider
$p_{1},p_{2}\in\partial B_{\rho}(o)$ such that $p_{1}$ (resp. $p_{2}$) belongs
to the intersection of $\partial B_{\rho}(o)$ with a minimizing geodesic from
$y$ to $o$ (resp. from $o$ to $x$). By (3.20) we already know that

$G_{p}(p_{1})\leq C^{\frac{1}{p-1}}G_{p}(p_{2}).$

Applying (3.19) with $s=\rho/4$ iteratively along the geodesic from $y$ to $o$
we find

$G_{p}(y)\leq C^{\frac{N^{\prime\prime}}{p-1}}G_{p}(p_{1}),$

with $N^{\prime\prime}\in\mathbb{N}$ such that
$N^{\prime\prime}\leq\tfrac{3}{2}R/(\tfrac{\rho}{8})+1$. Arguing analogously
along the geodesic from $o$ to $x$, we finally get that

$\max_{\partial B_{r}(o)}G_{p}\leq C^{\frac{1}{p-1}}\min_{\partial
B_{r}(o)}G_{p},$ (3.21)

for
$C=C\big{(}n,C_{P}(B_{2R}(o)),C_{A}(B_{2R}(o)),C_{1,\mathrm{Sob}}(B_{2R}(o)),C_{\rm
cov}(B_{\rho}(o)),R/\rho\big{)}$. Rewriting (3.21) in terms of $w_{p}$ and
combining with (3.18) yields

$\min_{\partial B_{r}(o)}w_{p}=w_{p}(y)\geq w_{p}(x)-C=\max_{\partial
B_{r}(o)}w_{p}-C\geq(n-p)\log r-C,$

for
$C=C\big{(}n,C_{1,\mathrm{Sob}}(B_{2R}(o)),C_{P}(B_{2R}(o)),C_{A}(B_{2R}(o)),R/\rho,C_{\mathrm{cov}}(B_{\rho}(o))\big{)}$.
∎

In order to pass to the limit $w_{p}=-(p-1)\log G_{p}$ as $p\to 1^{+}$ on
compact sets $K\subset B_{2R}(o)\setminus\\{o\\}$, we need also some uniform
upper bound on $w_{p}$. This is a well-known consequence of the Laplacian
comparison theorem and of the asymptotics of the Green function at the pole in
(3.4), see [52, Theorem 1.2].

######  Proposition .

Let $(M,g)$ be an $n$-dimensional smooth complete Riemannian manifold, and let
$p\in(1,n)$. Let $G_{p}$ be the solution of (3.3) with $\Omega$ in place of
$B_{R}(o)$. Let $r(x):=\mathsf{d}(o,x)$. Fix $R>0$, and assume
$B_{R}(o)\subset\Omega$. Suppose that $\mathrm{Ric}\geq-(n-1)a$ holds on
$B_{R}(o)$, for some $a>0$.

Then

$G_{p}(x)\geq\int_{r(x)}^{R}(v_{a}(t))^{-\frac{1}{p-1}}\,\mathrm{d}t,$ (3.22)

for any $x\in B_{R}(o)\setminus\\{o\\}$, where
$v_{a}(t):=c_{n,p}\left(\left(\sqrt{a}\right)^{-1}\sinh(\sqrt{a}t)\right)^{n-1}$,
for suitable $c_{n,p}>0$. Moreover $c_{n,p}\to c_{n}>0$ as $p\to 1^{+}$.

######  Remark .

In Section 3, the function

$\int_{r(x)}^{R}(v_{a}(t))^{-\frac{1}{p-1}}\,\mathrm{d}t$

is the solution of (3.3) in the space form of constant curvature $a$.

###### Proof of Theorem 3.1..

Within this proof let $G_{p}^{2R}=:G_{p}$, and $w_{p}^{2R}=:w_{p}$ for the
ease of notation. Let $K\subset B_{2R}(o)\setminus\\{o\\}$ be compact. We
first show that $w_{p}$ is equibounded on $K$ as $p\to 1^{+}$. Indeed by
(3.16) we know that $w_{p}\geq(n-p)\log(r(x))-C$ on
$B_{3R/2}(o)\setminus\\{o\\}$ for $C$ as in Theorem 3.3. On the other hand,
taking $\overline{r}\in(0,2R)$ such that $K\subset
B_{\overline{r}}(o)\setminus\\{o\\}\subset\subset B_{2R}(o)$, if
$\mathrm{Ric}\geq-(n-1)a$ on $B_{2R}(o)$ for some $a>0$, by (3.22) we have
that for all $x\in B_{\overline{r}}(o)\setminus\\{o\\}$ there holds

$w_{p}(x)\leq-\log\left(\int_{r(x)}^{\overline{r}}(v_{a}(t))^{-\frac{1}{p-1}}\,\mathrm{d}t\right)^{(p-1)}\leq-\log\left(\left\lvert
r(x)-\overline{r}\right\rvert^{p-2}\int_{r(x)}^{\overline{r}}(v_{a}(t))^{-1}\,\mathrm{d}t\right).$
(3.23)

Hence the above upper bound is uniform with respect to $p\to 1^{+}$ for any
$x\in K$. Thus $w_{p}$ is bounded on $K\cap B_{3R/2}(o)$ uniformly with
respect to $p\to 1^{+}$. Therefore, applying the gradient bound (3.14) on $K$,
we deduce that $w_{p}$ is bounded on $K$ uniformly with respect to $p\to 1$.
Exhausting $B_{2R}(o)\setminus\\{o\\}$ with a sequence of increasing compact
sets, by Ascoli–Arzelà and by a diagonal argument, we get that $w_{p}$
converges to some function $w$ locally uniformly on
$B_{2R}(o)\setminus\\{o\\}$, up to passing to a subsequence with respect to
$p$.

We claim that $w$ satisfies the weak formulation of the IMCF on
$B_{2R}(o)\setminus\\{o\\}$. Indeed, arguing as in [66, Equation (9)], one
gets that $|\nabla w_{p}|^{p}\mathcal{H}^{n}$ converges to $|\nabla
w|\mathcal{H}^{n}$ in duality with bounded continuous functions on compact
subsets contained in $B_{2R}(o)\setminus\\{o\\}$, along the suitable sequence
$p_{i}\to 1^{+}$ for which $w_{p_{i}}$ converges. Using again [66, Equation
(9)] this is enough to conclude that $w$ is a weak solution of the IMCF on
$B_{2R}(o)\setminus\\{o\\}$.

The lower bound (3.16) passes to the limit and gives (3.13). The upper bound
(3.23) is preserved on compact subsets of $B_{3R/2}(o)\setminus\\{o\\}$ for
the limit $w$ as well, hence it shows that $w\to-\infty$ as $x\to o$.

Therefore we proved items (1), (2) and (3) of the statement. It remains to
show that $\\{w\leq(n-1)\log r-C-1\\}\subset B_{r}(o)$ for every $r\leq R$.
Fix $r\in(0,R]$. By Theorem 3.3 there exists

$C=C\big{(}n,C_{1,\mathrm{Sob}}(B_{2R}(o)),C_{P}(B_{2R}(o)),C_{A}(B_{2R}(o)),R/\rho,C_{\mathrm{cov}}(B_{\rho}(o))\big{)},$

such that $w_{p}\geq(n-p)\log r-C$ on $\partial B_{r}(o)$. Denoting
$T_{r}\vcentcolon=(n-1)\log r-C-1$, it follows that $w_{p}\geq
T_{r}+\tfrac{1}{2}$ on $\partial B_{r}(o)$ for any $p$ sufficiently close to
$1$. By (3.13), it follows that $\\{w\leq T_{r}\\}\cap\partial
B_{r}(o)=\emptyset$. Suppose by contradiction that there exists $z\in
B_{2R}(o)\setminus B_{r}(o)$ such that $w(z)\leq T_{r}$. Then
$w_{p_{i}}(z)\leq T_{r}+\tfrac{1}{4}$ for a sequence $p_{i}\to 1^{+}$ such
that $w_{p_{i}}$ converges to $w$, and for $i$ large enough. Since
$w_{p_{i}}\geq T_{r}+\tfrac{1}{2}$ on $\partial B_{r}(o)$ and
$w_{p_{i}}(x)\to+\infty$ as $r(x)\to 2R$, then $w_{p_{i}}$ would have an
interior minimum on $B_{2R}(o)\setminus\overline{B}_{r}(o)$, which is a
contradiction to the maximum principle for $p$-harmonic functions. ∎

###  3.1. Properties of the weak IMCF

Let us collect in this section few known properties on the weak IMCF
constructed in Theorem 3.1.

Let $(M,g)$ be a smooth complete Riemaniann manifold. Let $o\in M,R>0$ and let
$w$ be given by Theorem 3.1. Let $T\in\mathbb{R}$ be such that $\\{w\leq
T\\}\subset B_{R}(o)$ (such a $T$ exists by Theorem 3.1). For every
$t\in(-\infty,T]$, set $\Omega_{t}:=\\{w<t\\}$. Then the following hold.

  1. (1)

If $n\leq 7$, then for every $t\in(-\infty,T]$ both the set $\Omega_{t}$ and
the set $\\{w\leq t\\}$ have $C^{1,\alpha}$ boundary, for some $\alpha>0$.
Indeed, as a direct consequence of Section 3, the previous sets are local
$(\Lambda,r_{0})$-minimizers in $B_{R}(o)\setminus\\{o\\}$, see, e.g., [61,
Example 21.2]; thus by (the Riemannian analogue of) [61, Theorem 21.8] one
gets the sought claim.
It follows that $\partial\Omega_{t}=\partial\\{w\leq t\\}=\\{w=t\\}$ for
almost every $t\leq T$, and that $\partial\Omega_{s}\to\partial\Omega_{t}$ in
$C^{1}$ as $s\nearrow t$, for every $t\leq T$, cf. [45, Eq. (1.10)].
Moreover, it is meaningful to speak about the weak mean curvature $H$ on
$\partial\Omega_{t}$, see, e.g., [45, page 16]. Moreover, $H=|\nabla w|>0$
$\mathcal{H}^{n-1}$-almost everywhere on $\partial\Omega_{t}$ for almost every
$t\in(-\infty,T]$, see [45, Equation (1.12)], and [45, Lemma 5.1].

  2. (2)

For almost every $t\in(-\infty,T]$, the boundary $\partial\Omega_{t}$ has weak
second fundamental form $A$ (see [67, Definition 1.3], or [45, pages
401–405]), and

$\int_{\partial\Omega_{t}}|A|^{2}<+\infty.$

The last inequality is a consequence of [67, Theorem 1.1(vi)].

The following Proposition gathers some key properties of the weak IMCF in
dimension $3$ constructed through the procedure of the previous section; most
notably the Geroch monotonicity formula of Huisken–Ilmanen through such flow.

######  Proposition .

Let $(M,g)$ be a smooth complete Riemaniann manifold of dimension $n=3$. Let
$o\in M,R>0$ and let $w$ be given by Theorem 3.1. Denote
$\Omega_{t}\vcentcolon=\\{w<t\\}$. Let $T$ be such that $\\{w\leq T\\}\subset
B_{R}(o)$. Let $H$ be the weak mean curvature of the boundary
$\partial\Omega_{t}$. Then the following hold.

  * •

There exist $\bar{t}\in\mathbb{R}$, $C_{1}>0$ such that

${P(\Omega_{t})=4\pi e^{t}}\,\quad\text{for all
$t\in(-\infty,T]$},\quad\text{and}\quad\int_{\partial\Omega_{t}}H^{2}\leq
C_{1},\quad\text{for almost every $t\in(-\infty,\bar{t})$}.$ (3.24)

  * •

For $t\in(-\infty,T]$, define the _Hawking mass_

$\mathfrak{m}_{H}(\partial\Omega_{t}):=\frac{P(\Omega_{t})^{1/2}}{(16\pi)^{3/2}}\left(4\pi-\int_{\partial\Omega_{t}}\frac{H^{2}}{4}\right).$
(3.25)

Then $\mathfrak{m}_{H}(\partial\Omega_{t})\to 0$ for almost every
$t\to-\infty$. Moreover, for every $-\infty<r<s\leq T$, if
$\partial\Omega_{t}$ is connected for every $t\in[r,s]$, then

$\begin{split}\mathfrak{m}_{H}(\partial\Omega_{s})&\geq\mathfrak{m}_{H}(\partial\Omega_{r})+\frac{1}{(16\pi)^{3/2}}\int_{r}^{s}P(\Omega_{t})^{1/2}\int_{\partial\Omega_{t}}R_{g}\,\mathrm{d}t\end{split}.$
(3.26)

###### Proof.

Denote $E_{t}\vcentcolon=\\{w\leq t\\}$ and let
$E^{p}_{t}\vcentcolon=\\{w^{2R}_{p}\leq t\\}$ for $w^{2R}_{p}$ as in Theorem
3.1. We shall omit the superscript $2R$ in the sequel, as $R$ is fixed. We
know from Theorem 3.1 that $w_{p}$ converges to $w$ locally uniformly on
$B_{2R}(o)\setminus\\{o\\}$ along a sequence $p_{i}\to 1$.

We claim that for any $\sigma\in(-\infty,T),\varepsilon>0$ there exists
$\tilde{p}>1$ such that $E^{p_{i}}_{t-\varepsilon}\subset E_{t}\subset
E^{p_{i}}_{t+\varepsilon}$ for any $p_{i}\in(1,\tilde{p})$ and
$t\in[\sigma,T]$.
We prove the containment $E^{p_{i}}_{t-\varepsilon}\subset E_{t}$ first.
Assume by contradiction that there exist $\sigma\in(-\infty,T)$,
$\varepsilon>0$ such that, up to subsequence, there exist $t_{i}\in[\sigma,T]$
such that $E^{p_{i}}_{t_{i}-\varepsilon}\not\subset E_{t_{i}}$ for any $i$.
Then there exist points $x_{i}\in B_{2R}(o)\setminus\\{o\\}$ such that
$w_{p_{i}}(x_{i})\leq t_{i}-\varepsilon$ but $w(x_{i})>t_{i}$ for any $i$.
Hence $\liminf_{i}\mathsf{d}(x_{i},o)>0$, for otherwise
$-\infty=\liminf_{i}w(x_{i})\geq\sigma$ by Theorem 3.1. Moreover, there exists
$\eta>0$ such that $w_{p_{i}}>T+\eta$ on $\partial B_{R}(o)$ for large $i$,
for otherwise $\\{w\leq T\\}\cap\partial B_{R}(o)$ would be nonempty. Hence
$x_{i}\in B_{R}(o)$ for large $i$, for if $x_{i}\in
B_{2R}(o)\setminus\overline{B}_{R}(o)$, since $w_{p_{i}}(x_{i})\leq
T-\varepsilon$ and $w_{p_{i}}(x)\to+\infty$ as $r(x)\to 2R$, then $w_{p_{i}}$
would have an interior minimum on $B_{2R}(o)\setminus\overline{B}_{R}(o)$,
contradicting the maximum principle for $p$-harmonic functions. Hence, up to
subsequence, $t_{i}\to\tau\in[\sigma,T]$ and $x_{i}\to
x\in\overline{B}_{R}(o)\setminus\\{o\\}$. But this contradicts the local
uniform convergence, as this implies
$\tau-\varepsilon\geq\lim_{i}w_{p_{i}}(x_{i})=w(x)=\lim_{i}w(x_{i})\geq\tau$.
The containment $E_{t}\subset E^{p_{i}}_{t+\varepsilon}$ follows by an
analogous contradiction argument. This time the contradicting sequence of
points $x_{i}\in B_{2R}(o)\setminus\\{o\\}$ satisfies
$w_{p_{i}}(x_{i})>t_{i}+\varepsilon$ and $w(x_{i})\leq t_{i}$, for
$t_{i}\in[\sigma,T]$. Hence $x_{i}\in\\{w\leq T\\}\subset B_{R}(o)$. Also
$x_{i}\not\to o$ because of the uniform upper bound (3.23). Hence we have
again the convergence $x_{i}\to x\in\overline{B}_{R}(o)\setminus\\{o\\}$, up
to subsequence, and one derives a contradiction as in the previous case.

Recalling Section 3, for any $\sigma\in(-\infty,T),\varepsilon>0$ there exists
$\tilde{p}>1$ such that

$\begin{split}4\pi\left(\frac{3-p_{i}}{p_{i}-1}\right)^{p_{i}-1}e^{t-\varepsilon}&=\mathrm{Cap}_{p_{i}}(E^{p_{i}}_{t-\varepsilon},B_{2R}(o))\leq\mathrm{Cap}_{p_{i}}(E_{t},B_{2R}(o))\leq\mathrm{Cap}_{p_{i}}(E^{p_{i}}_{t+\varepsilon},B_{2R}(o))\\\
&=4\pi\left(\frac{3-{p_{i}}}{{p_{i}}-1}\right)^{{p_{i}}-1}e^{t+\varepsilon},\end{split}$

for any $p_{i}\in(1,\tilde{p})$ and $t\in[\sigma,T]$. Letting first $p_{i}\to
1$ and then $\varepsilon\to 0$ and $\sigma\to-\infty$, we find that

$\lim_{i}\mathrm{Cap}_{p_{i}}(E_{t},B_{2R}(o))=4\pi e^{t},$

for any $t\in(-\infty,T]$. Then the first equality in (3.24) will follow if we
prove that

$\lim_{p\to 1}\mathrm{Cap}_{p}(E_{t},B_{2R}(o))=P(\Omega_{t}),$ (3.27)

for any $t\in(-\infty,T]$.
Fix $t\in(-\infty,T]$. It follows from the very definition of weak IMCF as in
[45, Minimizing Hull Property 1.4] that $E_{t}$ is strictly outward minimizing
relatively to $B_{2R}(o)$; meaning that whenever $E_{t}\subsetneq
F\subset\subset B_{2R}(o)$, then $P(E_{t})<P(F)$. It is readily checked that
the proof of [35, Theorem 1.2] can be localized in the ball $B_{2R}(o)$, thus
yielding that

$\lim_{p\to 1}\mathrm{Cap}_{p}(E_{t},B_{2R}(o))=P(E_{t}).$

Finally, as in [45, Minimizing Hull Property 1.4(iv)], there holds
$P(E_{t})=P(\Omega_{t})$, so that (3.27) follows.

We show the second property in (3.24). By applying (3.23) with
$\overline{r}=2r(x)$ and sending $p=p_{i}\to 1^{+}$, we get that there exists
$\tilde{\eta},\vartheta>0$ such that if $r(x)<\tilde{\eta}$, then $w(x)\leq
2\log(r(x))+\vartheta$. On the other hand, for $\bar{t}$ small enough we have
$\Omega_{t}\subset B_{e^{(t+C)/2}}(o)$ for every $t<\bar{t}$, see item (4) in
Theorem 3.1. Up to possibly taking a smaller $\bar{t}$, for every $t<\bar{t}$
we have $\partial\Omega_{t}\subset\\{w=t\\}\subset M\setminus
B_{e^{(t-\vartheta)/2}}$. Thus

$\operatorname{esssup}_{\partial\Omega_{t}}|\nabla
w|\leq\operatorname{esssup}_{\partial\Omega_{t}}\frac{\zeta}{\mathsf{d}(o,x)}\leq\zeta
e^{\vartheta/2}e^{-t/2},$

for almost every $t<\bar{t}$, where the first inequality comes from the fact
that (3.15) passes to the limit as $p_{i}\to 1^{+}$. Thus, for almost every
$t\in(-\infty,\bar{t})$, there holds

$\int_{\partial\Omega_{t}}|\nabla
w|^{2}\leq|\partial\Omega_{t}|\zeta^{2}e^{\vartheta}e^{-t}=4\pi\zeta^{2}e^{\vartheta}=:C_{1}<+\infty,$

and then the first item is proved recalling that $H=|\nabla w|$
$\mathcal{H}^{n-1}$-almost everywhere on $\partial\Omega_{t}$ for almost every
$t\in(-\infty,\bar{t})$.

The fact that $\mathfrak{m}_{H}(\partial\Omega_{t})\to 0$ for almost every
$t\to-\infty$ is a direct consequence of the first item and the definition of
Hawking mass. The last part of the second item is a consequence of the
analogue of [45, Geroch Monotonicity Formula 5.8] in our setting. A detailed
proof the Geroch Monotonicity Formula for weak IMCFs defined through limits of
$p$-harmonic functions as in our setting will be provided in the forthcoming
[13]. ∎

###  3.2. Connectedness of level sets of the weak IMCF

In the following lemma we collect some facts about connectedness of level set
of the weak IMCF. This is analogous to [45, Connectedness Lemma 4.2], we
provide a proof for the convenience of the reader.

######  Lemma .

Let $\Omega$ be an open set with Lipschitz boundary in a smooth complete
$n$-dimensional Riemannian manifold $(M,g)$ with $n\leq 7$, and let
$o\in\Omega$. Let $\Omega\subset\subset U$, where $U$ is an open set, and let
$w\in\mathrm{Lip}_{\mathrm{loc}}(U\setminus\\{o\\})$ be a weak solution of the
IMCF on $U\setminus\\{o\\}$, see Section 3. Then the following hold.

  1. (1)

Let $t\in\mathbb{R}$. Then every connected component of $\\{w<t\\}\cap\Omega$
(resp., $\\{w>t\\}\cap\Omega$) is not relatively compact in
$\Omega\setminus\\{o\\}$.

  2. (2)

Assume further that $\partial\Omega$ is connected, and there exist
$t_{0}<t_{1}$ such that:

     * •

$\\{w<t_{1}\\}\subset\subset\Omega$;

     * •

there exists an open set $\Omega^{\prime\prime}\ni o$, such that
$\Omega^{\prime\prime}\setminus\\{o\\}$ is connected, and
$\Omega^{\prime\prime}\setminus\\{o\\}\subset\\{w<t_{0}\\}$.

Then for every $t\in(t_{0},t_{1})$ we have that both $\\{w<t\\}$ and
$\\{w>t\\}\cap\Omega$ are connected.

  3. (3)

Let the hypotheses of (2) above be satisfied. Assume further that $\Omega$ is
connected, and $H_{1}(\Omega;\mathbb{Z})=\\{0\\}$. Then $\partial\\{w<t\\}$ is
connected for every $t\in(t_{0},t_{1})$.

$M$$U$$\Omega$$o$$\Omega_{t_{1}}$$\Omega_{t_{0}}$$\Omega^{\prime\prime}$
Figure 1. The picture sketches the assumptions of Section 3.2.

###### Proof of Section 3.2.

The proof of item (1) follows verbatim as in [45, Connectedness Lemma 4.2(i)].
Let us repeat it here for the ease of the reader. Assume by contradiction a
connected component $C$ of $\\{w>t\\}\cap\Omega$ is relatively compact in
$\Omega\setminus\\{o\\}$. Then consider the function $v:=w$ on
$U\setminus(C\cup\\{o\\})$ and $v:=t$ on $C$. By (3.1) we get

$\int_{C}|\nabla w|+w|\nabla w|\leq\int_{C}t|\nabla w|.$

Since $C\subset\\{w>t\\}$, the latter implies that $|\nabla w|=0$ on $C$, and
thus $w=t$ on $C$, which is a contradiction.

Now assume by contradiction a connected component $C^{\prime}$ of
$\\{w<t\\}\cap\Omega$ is relatively compact in $\Omega\setminus\\{o\\}$. Take
$\bar{t}:=\min_{\overline{C^{\prime}}}w$. Then for $0<\eta<1$ small enough
there is a connected component $C^{\prime\prime}$ of
$\\{w<\bar{t}+\eta\\}\cap\Omega$ inside $C^{\prime}$. Notice that
$w\geq\bar{t}>\bar{t}+\eta-1$ on $C^{\prime\prime}$. Repeating the previous
argument with $\bar{t}+\eta$ in place of $t$ and $C^{\prime\prime}$ in place
of $C$ gives again a contradiction.

The item (2) follows from item (1). First, by the assumption in the first
bullet, for $t\in(t_{0},t_{1})$ every connected component of $\\{w<t\\}$ stays
away from $\partial\Omega$. Then, from item (1), for every
$t\in(t_{0},t_{1})$, $o$ is in the closure of any connected component of
$\\{w<t\\}$. Then every connected component of $\\{w<t\\}$ intersects
$\Omega^{\prime\prime}\setminus\\{o\\}$. Since
$\Omega^{\prime\prime}\setminus\\{o\\}$ is connected, and
$\Omega^{\prime\prime}\setminus\\{o\\}\subset\\{w<t_{0}\\}\subset\\{w<t\\}$,
thus every connected component of $\\{w<t\\}$ contains
$\Omega^{\prime\prime}\setminus\\{o\\}$. Thus, there exists at most one
connected component of $\\{w<t\\}$, because every connected component contains
$\Omega^{\prime\prime}\setminus\\{o\\}$, which is itself connected. Similarly,
from the hypotheses of the second bullet, for every $t\in(t_{0},t_{1})$, we
have $\partial\Omega\subset\\{w>t\\}$, and
$\\{w>t\\}\cap\Omega\subset\Omega\setminus\overline{\Omega^{\prime\prime}}$.
Thus, every connected component of $\\{w>t\\}\cap\Omega$ avoids $o$, and then,
from item (1), its closure must then intersect $\partial\Omega$. Since
$\partial\Omega$ is connected, there exists at most one connected component of
$\\{w>t\\}\cap\Omega$, arguing as before.

The item (3) is inspired by [45, Connectedness Lemma 4.2(ii)]. Analogous
arguments have appeared in [68, Lemma 2.3], [60, Lemma 6.1], and [53, Lemma
4.46]. We give here a self-contained proof using item (2) and the
Mayer–Vietoris sequence.

Recall from item (1) of Section 3.1 that $\partial\\{w<t\\}$ is $C^{1,\alpha}$
for every $t\in(t_{0},t_{1})$, and that $\partial\\{w<t\\}=\\{w=t\\}$ for
almost every $t\in(t_{0},t_{1})$. It is sufficient to prove that
$\partial\\{w<t\\}$ is connected for $t$ such that
$\partial\\{w<t\\}=\\{w=t\\}$. Indeed, for $\tau\in(t_{0},t_{1})$ such that
$\partial\\{w<\tau\\}\neq\\{w=\tau\\}$, there exists a sequence
$t_{i}\nearrow\tau$ such that $\partial\\{w<t_{i}\\}=\\{w=t_{i}\\}$. From item
(1) of Section 3.1 we know that $\partial\\{w<t_{i}\\}\to\partial\\{w<\tau\\}$
in $C^{1}$, hence connectedness will be preserved in the limit.

Hence we can assume by contradiction that there exists $t\in(t_{0},t_{1})$
such that $\partial\\{w<t\\}=\\{w=t\\}$ is not connected. Since
$\partial\\{w<t\\}$ is $C^{1,\alpha}$, it has a finite number $m\geq 2$ of
connected components. Since
$\bigcap_{\eta>0}\\{t-\eta<w<t+\eta\\}=\\{w=t\\}=\partial\\{w<t\\}$, there
exists $\eta$ small enough such that $[t-\eta,t+\eta]\subset(t_{0},t_{1})$ and
$\\{t-\eta<w<t+\eta\\}$ has $m^{\prime}\geq 2$ connected components. Call
$A:=\\{w<t+\eta\\}\cup\\{o\\}$ and $B:=\\{w>t-\eta\\}\cap\Omega$. Notice that
$A$ and $B$ are open, and by item (2) they are both connected.

Finally notice that $A\cap B=\\{t-\eta<w<t+\eta\\}$ is not connected, and
$A\cup B=\Omega$. The Mayer–Vietoris exact sequence (where homology is
understood with integer coefficients) ends with

$\ldots\rightarrow H_{1}(\Omega)\rightarrow H_{0}(A\cap B)\rightarrow
H_{0}(A)\oplus H_{0}(B)\rightarrow H_{0}(\Omega)\rightarrow 0.$

Recall that for a topological space $X$ there holds
$H_{0}(X;\mathbb{Z})\cong\mathbb{Z}^{\ell}$, where $\ell$ is the number of
connected components of $X$. Thus by using the assumptions of item (3) the
previous exact sequence becomes

$\ldots\rightarrow
0\rightarrow\mathbb{Z}^{m^{\prime}}\rightarrow\mathbb{Z}^{2}\rightarrow\mathbb{Z}\rightarrow
0,$

from which $\mathbb{Z}\cong\mathbb{Z}^{2}/\mathbb{Z}^{m^{\prime}}$, and thus
$m^{\prime}=1$, which results in a contradiction. ∎

##  4\. Proof of the main results

In this section we prove the main theorems Theorem 1.2, Theorem 1.1.

###  4.1. Producing a set satisfying the reverse Euclidean isoperimetric
inequality

In this section we show how, in the hypotheses of Theorem 1.1, we can produce
sets with arbitrarily large perimeter and volume that satisfy the Euclidean
reserve isoperimetric inequality with sharp constant, see Section 4.1.

######  Lemma .

Let $(M,g)$ be a smooth complete Riemannian manifold of dimension $3$. Let
$o\in M$ and $R>0$. Let $w$ be given by Theorem 3.1. Let $T\in\mathbb{R}$ be
such that $\\{w<T\\}\subset\subset B_{\rho}(o)$ for some $\rho\leq R$. Suppose
that $\partial\\{w<t\\}$ is connected for any $t<T$. If $R_{g}\geq-\delta$ on
$B_{\rho}(o)$, for some $\delta>0$, then

$|\Omega_{t}|\geq\frac{1}{\sqrt{1+\frac{2}{3}\delta
e^{T}}}\frac{P(\Omega_{t})^{3/2}}{6\sqrt{\pi}}\qquad\forall t<T,$

where $\Omega_{t}\vcentcolon=\\{w<t\\}$.

###### Proof.

We recall that $\Omega_{t}$ has $C^{1,\alpha}$ boundary, $\partial\Omega_{t}$
admits weak mean curvature $H$ for all $t<T$, and $H=|\nabla w|>0$
$\mathcal{H}^{2}$-a.e. for a.e. $t\in(-\infty,T)$, see item (1) of Section
3.1. The Hawking mass of a set $\Omega$ with $C^{1}$ boundary $\partial\Omega$
possessing weak mean curvature $H$ is given by

$\mathfrak{m}_{H}(\partial\Omega)=\frac{P(\Omega)^{1/2}}{(16\pi)^{3/2}}\left(4\pi-\int_{\partial\Omega}\frac{H^{2}}{4}\right).$

As $\\{w\leq T^{\prime}\\}\subset\subset B_{\rho}(o)$ for any $T^{\prime}<T$,
we can apply Section 3.1, which yields that $P(\Omega_{t})=4\pi e^{t}$ for any
$t<T$, and that $\mathfrak{m}_{H}(\partial\Omega_{t})\to 0$ for almost every
$t\to-\infty$. Hence the Geroch Monotonicity formula (3.26) with $r\to-\infty$
and $s=t$, together with the fact that $R_{g}\geq-\delta$ on $B_{\rho}(o)$,
gives that for all $t\in(-\infty,T)$ there holds

$\mathfrak{m}_{H}(\partial\Omega_{t})\geq-\frac{\delta}{(16\pi)^{3/2}}\int_{-\infty}^{t}P(\Omega_{t})^{3/2}=-\frac{\delta}{(16\pi)^{3/2}}\int_{-\infty}^{t}(4\pi)^{3/2}e^{3t/2}=-\frac{\delta}{12}e^{3t/2}.$

Since
$\mathfrak{m}_{H}(\partial\Omega_{t})=\frac{2\pi^{1/2}e^{t/2}}{64\pi^{3/2}}\left(4\pi-\int_{\partial\Omega_{t}}\frac{H^{2}}{4}\right)$,
for every $t\in(-\infty,T)$, we find

$\int_{\partial\Omega_{t}}H^{2}\leq 16\pi+\frac{32}{3}\delta\pi e^{t}.$ (4.1)

By Hölder inequality we get that for almost every $t\in(-\infty,T)$ there
holds

$P(\Omega_{t})\leq\left(\int_{\partial\Omega_{t}}|\nabla
w|^{2}\right)^{1/3}\left(\int_{\partial\Omega_{t}}\frac{1}{|\nabla
w|}\right)^{2/3}.$ (4.2)

Hence, by recalling that $\int_{\partial\Omega_{t}}|\nabla
w|^{2}=\int_{\partial\Omega_{t}}H^{2}$ is finite by (4.1) for almost all
$t\in(-\infty,T)$, recalling that $|\nabla w|>0$ $\mathcal{H}^{2}$-a.e. on
$\partial\Omega_{t}$ for a.e. $t\in(-\infty,T)$, and by using the coarea
formula together with (4.2) and (4.1), for any $t<T$ we obtain

$\begin{split}|\Omega_{t}|&\geq\int_{\Omega_{t}\cap\\{|\nabla
w|>0\\}}\frac{1}{|\nabla w|}|\nabla
w|=\int_{-\infty}^{t}\left(\int_{\partial\Omega_{\tau}\cap\\{|\nabla
w|>0\\}}\frac{1}{|\nabla w|}\right)\,\mathrm{d}\tau\\\
&=\int_{-\infty}^{t}\left(\int_{\partial\Omega_{\tau}}\frac{1}{|\nabla
w|}\right)\,\mathrm{d}\tau\geq\int_{-\infty}^{t}\left(P(\Omega_{\tau})^{3/2}\left(\int_{\partial\Omega_{\tau}}|\nabla
w|^{2}\right)^{-1/2}\right)\\\ &\geq\int_{-\infty}^{t}\frac{2\pi
e^{3t/2}}{\sqrt{1+\frac{2}{3}\delta e^{t}}}\geq\int_{-\infty}^{t}\frac{2\pi
e^{3t/2}}{\sqrt{1+\frac{2}{3}\delta e^{T}}}=\frac{1}{\sqrt{1+\frac{2}{3}\delta
e^{T}}}\,\frac{4}{3}\pi e^{3t/2}=\frac{1}{\sqrt{1+\frac{2}{3}\delta
e^{T}}}\,\frac{P(\Omega_{t})^{3/2}}{6\sqrt{\pi}}.\end{split}$

∎

In the following proposition we construct sets with arbitrarily large
perimeter and volume that satisfy the reverse Euclidean isoperimetric
inequality. This represents the crucial step for the proof of the main
results.

######  Proposition .

Let $(M,g)$ be a $3$-dimensional complete $C^{0}$-Riemannian manifold without
boundary, let $K\subset M$ be a compact set, and let $\Omega$ be an unbounded
connected component of $M\setminus K$. Assume that $\Omega$ is
$C^{0}_{\mathrm{loc}}$-asymptotic to $\mathbb{R}^{3}$ (Section 1.1), and that
$R_{g}\geq 0$ in the approximate sense on $\Omega\setminus K^{\prime}$
(Section 1.1), where $K^{\prime}\subset M$ is a compact set. Then, there
exists a universal constant $\vartheta<1$ such that the following holds.

For every $\mathscr{P}>0$, there exists a set of finite perimeter
$E\subset\subset\Omega\setminus K^{\prime}$ such that

$\vartheta\mathscr{P}\leq P(E)\leq\mathscr{P},$

and

$|E|\geq\frac{1}{6\sqrt{\pi}}\mathscr{P}^{3/2}\geq\frac{1}{6\sqrt{\pi}}P(E)^{3/2}.$
(4.3)

###### Proof.

Let $\rho>1$, $\eta<1/2$ to be chosen. The choice of $\rho$, only depending on
$\mathscr{P}$ and on geometric constants on $\mathbb{R}^{3}$, will be made
clear during the proof in (4.8). We do not insist on the precise choice of
$\eta$ for the sake of readability; however, it will be clear from the proof
that choosing $\eta<10^{-3}$ is sufficient. In this proof we will repeatedly
use the elementary metric results recorded in Section 2.1, and in the last
part of the proof of Section 2.3.

By applying a contradiction argument and Section 2.1 (see also the beginning
of the proof of Section 2.3) we can find a compact set
$\mathcal{C}\supset(K\cup K^{\prime})$ such that for every
$x\in\Omega\setminus\mathcal{C}$ there exists
$F_{x}:(B_{64\rho+64}(x),g)\to(\mathbb{R}^{3},g_{\mathrm{eu}})$ which is a
$(1+\eta)$-biLipschitz diffeomorphism with its image, with $F_{x}(x)=0$, and
whose image contains $B_{32\rho+32}^{\mathbb{R}^{3}}(0)$. Let us now fix $o$
such that

$B_{64\rho+64}(o)\subset\subset\Omega\setminus\mathcal{C}.$

Let $g_{i}$ be a sequence of smooth Riemannian metrics on $M$ converging to
$g$ locally uniformly with $R_{g_{i}}\geq-\varepsilon_{i}$ on
$\Omega\setminus\mathcal{C}$. We will frequently pass to subsequences with
respect to $i$ in the course of the proof, without relabeling. We will denote
$B_{s}(p),B_{s}^{i}(p)$ the open balls of radius $s$ and center $p$ in the
metrics $g,g_{i}$, respectively. Up to passing to a subsequence with respect
to $i$, we can assume that

$B^{i}_{\rho-1}(o)\subset\subset B_{\rho}(o)\subset
F_{o}^{-1}(B_{2\rho}^{\mathbb{R}^{3}}(0))\subset\subset
B_{4\rho+4}^{i}(o)\subset\subset\Omega\setminus\mathcal{C}\,\,\text{for all
$i\in\mathbb{N}$}.$ (4.4)

Notice that (see Section 2.1) the compact $\mathcal{C}$ can be chosen such
that, additionally, we have

$|(g-F_{o}^{*}g_{\mathrm{eu}})_{x}(v,v)|\leq\eta(F_{o}^{*}g_{\mathrm{eu}})_{x}(v,v),$
(4.5)

for every $x\in B_{64\rho+64}(o)$, and every $v\in T_{x}M$. Then, for $i$
large enough, we have

$|(g^{i}-F_{o}^{*}g_{\mathrm{eu}})_{x}(v,v)|\leq
2\eta(F_{o}^{*}g_{\mathrm{eu}})_{x}(v,v),$ (4.6)

for every $x\in B_{32\rho+32}^{i}(o)$, and every $v\in T_{x}M$. As a
consequence, the map
$F_{x}:(B^{i}_{16\rho+16}(x),g_{i})\to(\mathbb{R}^{3},g_{\mathrm{eu}})$ is
$(1+3\eta)$-biLipschitz with its image. Notice also that the image of this map
contains the ball $B_{8\rho+8}^{\mathbb{R}^{3}}(0)$.

As a consequence of (4.6), if $\eta$ is small enough, the constants

$C_{1,\mathrm{Sob}}(B^{i}_{4\rho+4}(o)),C_{P}(B^{i}_{4\rho+4}(o)),C_{A}(B^{i}_{4\rho+4}(o))$

appearing in Theorem 3.1 (and defined in (3.2), (3.9), (3.10)), are uniformly
bounded from above by a universal constant multiplied by the value of the
corresponding constants on $\mathbb{R}^{3}$, which are independent on $\rho$.
Moreover, since
$F_{x}:(B^{i}_{16\rho+16}(x),g_{i})\to(\mathbb{R}^{3},g_{\mathrm{eu}})$ is
$(1+3\eta)$-biLipschitz with its image, and contains the ball
$B_{8\rho+8}^{\mathbb{R}^{3}}(0)$, the following holds: for every
$r\leq\rho+1$ we can connect any two points $p,q\in\partial B^{i}_{r}(o)$ with
a continuous curve $\gamma$ in the annulus $A^{i}_{3r/4,5r/4}(o)$. Indeed, it
suffices to connect $F_{x}(p)$ with $F_{x}(q)$ with a curve
$\tilde{\gamma}\subset\mathbb{R}^{3}$ in the annulus
$A^{\mathbb{R}^{3}}_{7r/8,9r/8}(0)$, and take
$\gamma:=F_{x}^{-1}(\tilde{\gamma})$. Moreover, with the same reasoning, the
constant $C_{\mathrm{cov}}(B_{\rho+1}(o))$ defined in (3.11) is bounded above
by a universal constant which only depends on the following universal constant
independent on $\rho$:

$\begin{split}\tilde{C}\vcentcolon=\min\Big{\\{}N\in\mathbb{N}\ :\
&\text{$A^{\mathbb{R}^{3}}_{r/2,3r/2}(0)$ is covered by $N$ open balls of
radius $r/4$}\\\ &\text{with centers in $A^{\mathbb{R}^{3}}_{r/2,3r/2}(o)$ for
any $0<r\leq\rho+1$}\Big{\\}}.\end{split}$ (4.7)

Hence, we can apply Theorem 3.1 on the ball
$B^{i}_{2R}(o):=B^{i}_{4\rho+4}(o)$, where the radius $\rho+1$ in this proof
corresponds to the number $\rho$ in the statement of Theorem 3.1. As a result
of the discussion above, the constant $C$ appearing in Theorem 3.1 when
applied to the ball $B^{i}_{4\rho+4}(o)$ is bounded above by a universal
constant $\xi>1$ independent of $i$, $\rho$, for $i$ large enough.

Now let $w_{i}$ be the weak IMCF issuing from $o$ in $B^{i}_{4\rho+4}(o)$,
given by Theorem 3.1. Denote $\Omega_{t}^{i}:=\\{w_{i}<t\\}$. Let
$T_{\rho,\xi}:=2\log(\rho-1)-\xi-1$, and take $\rho>1$ such that

$4\pi e^{T_{\rho,\xi}}=4\pi e^{2\log(\rho-1)-\xi-1}=\mathscr{P}.$ (4.8)

Notice that the choice of $\rho$ only depends on $\mathscr{P}$ and on the
universal constant $\xi$. By the last assertion of Theorem 3.1, applied with
$r:=\rho-1$, we have that

$\\{w_{i}<T_{\rho,\xi}\\}=\Omega_{T_{\rho,\xi}}^{i}\subset
B_{\rho-1}^{i}(o)\subset\subset B_{\rho}(o).$

Now we aim at applying item (3) of Section 3.2 with the choices
$U:=B_{4\rho+4}^{i}(o)$, $\Omega:=F_{o}^{-1}(B_{2\rho}^{\mathbb{R}^{3}}(0))$,
and $t_{1}:=T_{\rho,\xi}$. We stress that
$\\{w_{i}<T_{\rho,\xi}\\}\subset\subset
F_{o}^{-1}(B_{2\rho}^{\mathbb{R}^{3}}(0))$, and

$F_{o}^{-1}(B_{2\rho}^{\mathbb{R}^{3}}(0)),\,\,\text{and}\,\,\partial
F_{o}^{-1}(B_{2\rho}^{\mathbb{R}^{3}}(0))=F_{o}^{-1}(\partial
B_{2\rho}^{\mathbb{R}^{3}}(0))\,\text{are connected}.$

Moreover, for any arbitrary $t_{0}<T_{\rho,\xi}$, one can define
$\Omega^{\prime\prime}$ to be a sufficiently small ball so that all the
hypotheses of item (2) in Section 3.2 are met, since we have
$w_{i}(x)\to-\infty$ as $x\to o$. Finally noticing that
$H_{1}(F_{o}^{-1}(B_{2\rho}^{\mathbb{R}^{3}}(0));\mathbb{Z})=\\{0\\}$, one
gets that all the hypotheses of item (2) and (3) in Section 3.2 are met, and
thus $\partial\\{w_{i}<t\\}$ is connected for every $t<T_{\rho,\xi}$.

Therefore, recalling that $R_{g_{i}}\geq-\varepsilon_{i}$ on
$B^{i}_{4\rho+4}(o)$, by Section 4.1 we get

$|\Omega_{T_{\rho,\xi}}^{i}|_{i}\geq\frac{1}{\sqrt{1+\frac{2}{3}\varepsilon_{i}e^{T_{\rho,\xi}}}}\frac{P_{i}(\Omega_{T_{\rho,\xi}}^{i})^{3/2}}{6\sqrt{\pi}}=\frac{1}{\sqrt{1+\frac{2}{3}\varepsilon_{i}e^{T_{\rho,\xi}}}}\frac{(4\pi
e^{T_{\rho,\xi}})^{3/2}}{6\sqrt{\pi}}=\frac{1}{\sqrt{1+\frac{2}{3}\varepsilon_{i}\frac{\mathscr{P}}{4\pi}}}\frac{\mathscr{P}^{3/2}}{6\sqrt{\pi}},$
(4.9)

where we used that $P_{i}(\Omega_{t}^{i})=4\pi e^{t}$, see Section 3.1, where
$P_{i}(\cdot)$ denotes perimeter computed with respect to the metric $g_{i}$.

Now, since $\Omega^{i}_{T_{\rho,\xi}}\subset\subset B_{\rho}(o)$, and the
perimeters $P_{i}(\Omega^{i}_{T_{\rho,\xi}})$ are equibounded, we can use the
precompactness and lower semicontinuity result in Section 2.3 to get a set
$E_{\rho}\subset B_{\rho}(o)$ such that $\Omega^{i}_{T_{\rho,\xi}}\to
E_{\rho}\subset\subset\Omega\setminus\mathcal{C}$ in $L^{1}$,
$P(E_{\rho})\leq\mathscr{P}$, and, by passing (4.9) to the limit, such that

$|E_{\rho}|\geq\frac{\mathscr{P}^{3/2}}{6\sqrt{\pi}},$ (4.10)

which completes the proof of (4.3).

Finally, notice that $E_{\rho}\subset\subset B_{\rho+1}(o)$. Moreover (4.5)
holds. This implies that, if $\eta<1$ is small enough, on $B_{\rho+1}(o)$
there holds a $(1,1^{*})$-Sobolev inequality (3.2) with $C_{1,{\rm
Sob}}(B_{\rho+1}(o))$ bounded from above by a universal constant only
depending on the constant in the Euclidean $(1,1^{*})$-Sobolev inequality. By
Section 3, this implies that $P(E)\geq\tilde{\vartheta}|E|^{2/3}$ for a
universal $\tilde{\vartheta}$, for every $E\subset\subset B_{\rho+1}(o)$.
Applying the latter inequality on $E_{\rho}$, and using again (4.10), we
finally get $P(E_{\rho})\geq\vartheta\mathscr{P}^{2/3}$ for a universal
$\vartheta$, concluding the proof. ∎

###  4.2. Proof of the main results and consequences

We are now ready to prove the main results of the paper.

###### Proof of Theorem 1.1.

It is a direct consequence of Section 4.1 and the definition of isoperimetric
mass in Section 1.1. ∎

###### Proof of Theorem 1.2.

Suppose by contradiction that there exists $v_{0}>0$ such that for every
$v\in(v_{0},+\infty)$ there are no isoperimetric sets of volume $v$ in $M$. We
claim that then $I$ is strictly increasing on $(2v_{0},+\infty)$.

Let $v\in(2v_{0},+\infty)$. By Theorem 2.2 there is an isoperimetric set
$E\subset M$ with $|E|\leq v_{0}$ (possibly empty) and a ball
$B\subset\mathbb{R}^{3}$ with $|B|\geq v-v_{0}\geq v/2$ such that

$v=|E|+|B|_{\mathrm{eu}},\qquad I(v)=P(E)+P_{\mathrm{eu}}(B).$

Let now $\varepsilon>0$ be such $B_{\varepsilon}$ is a ball in
$\mathbb{R}^{3}$ concentric to $B$ and with volume
$|B_{\varepsilon}|_{\mathrm{eu}}=|B|_{\mathrm{eu}}-\varepsilon$. Notice that,
by approximating $B_{\varepsilon}\subset\mathbb{R}^{3}$ with sets diverging
along the manifold, arguing as in the proof of the upper bound for the
isoperimetric profile in Section 2.3, we get $I(v-\varepsilon)\leq
P(E)+P_{\mathrm{eu}}(B_{\varepsilon})$. Thus, taking $\varepsilon\to 0$, we
find

$\frac{I(v)-I(v-\varepsilon)}{\varepsilon}\geq\frac{P_{\mathrm{eu}}(B)-P_{\mathrm{eu}}(B_{\varepsilon})}{\varepsilon}\xrightarrow[\varepsilon\to
0]{}2\left(\frac{4\pi}{3}\right)^{1/3}|B|_{\mathrm{eu}}^{-1/3}\geq
2\left(\frac{4\pi}{3}\right)^{1/3}v^{-1/3}.$

Thus we get that the lower left Dini derivative satisfies

$\overline{D^{-}}I(v):=\liminf_{\varepsilon\to
0^{+}}\frac{I(v)-I(v-\varepsilon)}{\varepsilon}\geq
2\left(\frac{4\pi}{3}\right)^{1/3}v^{-1/3}>0,\qquad\forall
v\in(2v_{0},+\infty).$

Since $I$ is continuous by Section 2.3, the latter implies that $I$ is
strictly increasing on $(2v_{0},+\infty)$. Thus the sought claim is proved.

We aim now at showing that there exists an isoperimetric set with volume
strictly greater than $2v_{0}$, thus reaching a contradiction. Fix $v>2v_{0}$.
By Theorem 2.2 there is an isoperimetric set $E\subset M$ with $|E|\leq v_{0}$
and a ball $B\subset\mathbb{R}^{3}$ with $|B|\geq v-v_{0}>0$ such that

$v=|E|+|B|_{\mathrm{eu}},\qquad I(v)=P(E)+P_{\mathrm{eu}}(B).$

By Theorem A.1 we know that $E$ is bounded.

We apply Section 4.1 with $K=\emptyset$ and $K^{\prime}=\overline{B}_{r}(o)$
for some ball $B_{r}(o)$ with $r>1$ such that $E\cup\mathcal{C}\subset
B_{r-1}(o)$. Then there is a set $F\subset M$ such that $F\subset\subset
M\setminus K^{\prime}$, $P(F)\leq P_{\mathrm{eu}}(B)$, and

$|F|\geq\frac{1}{6\sqrt{\pi}}P_{\mathrm{eu}}(B)^{3/2}=|B|_{\mathrm{eu}}.$

Thus the set $E\cup F$ is such that

$|E\cup F|=|E|+|F|\geq|E|+|B|_{\mathrm{eu}}=v.$

Hence, since $I$ is strictly increasing on $(2v_{0},+\infty)$, one gets

$I(|E\cup F|)\geq I(v)=P(E)+P_{\mathrm{eu}}(B),$

but at the same time

$\begin{split}I(|E\cup F|)&\leq P(E\cup F)=P(E)+P(F)\leq
P(E)+P_{\mathrm{eu}}(B).\end{split}$

Hence all the inequalities in the previous formula are equalities, and then
$E\cup F$ is an isoperimetric set with volume $>2v_{0}$, which is a
contradiction. ∎

In the proof of Theorem 1.2 we argued that, if for some $v_{0}>0$ no
isoperimetric sets exist for volumes $v\geq v_{0}$, then the isoperimetric
profile is strictly increasing for large volumes. This implied that some
isoperimetric set of large volume must exist, resulting in a contradiction. In
fact, if the isoperimetric profile is strictly increasing, then one can deduce
that isoperimetric sets exist for any volume. This is what happens in a smooth
asymptotically flat $3$-manifold of nonnegative scalar curvature that is
complete with no closed minimal surfaces or endowed with a horizon boundary.
For this reason, the following result can be seen as a generalization of the
existence result for any volume obtained in [23, Proposition K.1] (see also
[10, Theorem 3.6]). Moreover, when in addition the isoperimetric profile
diverges at infinity, such isoperimetric sets realize the isoperimetric mass
in the sense of (4.11). The isoperimetric profile diverges at infinity for
instance when a global Euclidean-like isoperimetric inequality is in force.
The latter happens, for example, when the manifold is $C^{0}$-asymptotic to
$\mathbb{R}^{3}$ according to Section 1.4.

######  Proposition .

Let $(M,g)$ be a $C^{0}$-Riemannian manifold that is
$C^{0}_{\mathrm{loc}}$-asymptotic to $\mathbb{R}^{3}$, and such that
$R_{g}\geq 0$ in the approximate sense on $M\setminus K$, where $K\subset M$
is a compact set. Assume that its isoperimetric profile $I(v)$ is strictly
increasing. Then, for every volume $v$, there exists an isoperimetric set
$E_{v}$ of volume $v$ on $M$. If in addition $\lim_{v\to+\infty}I(v)=+\infty$,
then

$\mathfrak{m}_{\mathrm{iso}}=\limsup_{v\to+\infty}\frac{2}{P(E_{v})}\left(|E_{v}|-\frac{P(E_{v})^{3/2}}{6\sqrt{\pi}}\right).$
(4.11)

###### Proof.

We assume by contradiction that, for some $v>0$, there exists no isoperimetric
sets of volume $v$. By Theorem 2.2, we have $I(v)=P(E)+P_{\rm eu}(B)$ for a
possibly empty $E\subset M$ realizing $P(E)=I(|E|)$ and
$B\subset\mathbb{R}^{3}$ a nonempty ball, such that $|E|+|B|_{\rm eu}=v$. The
proof of Theorem 1.2 shows that we can find a set $F\subset\subset
M\setminus\overline{B}_{r+1}(o)$, for some ball such that $E\subset B_{r}(o)$,
with $P(F)\leq P_{\rm eu}(B)$ and with volume $|F|\geq|B|_{\rm eu}$. If
$|F|=|B|_{\rm eu}$, then $E\cup F$ is an isoperimetric set of volume $v$,
giving a contradiction. If $\left\lvert F\right\rvert>\left\lvert
B\right\rvert_{\rm eu}$, we derive a contradiction with strict monotonicity of
$I$. Indeed, on the one hand we would have

$I(\left\lvert E\right\rvert+\left\lvert F\right\rvert)>I(\left\lvert
E\right\rvert+|B|_{\rm eu})=I(v),$

and on the other hand there holds

$I(\left\lvert E\right\rvert+\left\lvert F\right\rvert)\leq P(E)+P(F)\leq
P(E)+P_{\rm eu}(B)=I(v).$

This proves the existence of isoperimetric sets of any volume.

We are left to prove (4.11). Observe that, since the isoperimetric profile
diverges at infinity, the isoperimetric sets $E_{v}$ have perimeter diverging
to infinity as $v\to+\infty$, and thus they are valid competitors in the
definition (1.1) of $\mathfrak{m}_{\mathrm{iso}}$. Hence

${\mathfrak{m}}_{\mathrm{iso}}\geq\limsup_{v\to+\infty}\frac{2}{P(E_{v})}\left(|E_{v}|-\frac{P(E_{v})^{3/2}}{6\sqrt{\pi}}\right).$

On the other hand, let $(\Omega_{j})_{j\in\mathbb{N}}$ be any other sequence
of finite perimeter sets such that $P(\Omega_{j})\to+\infty$. Let $E_{j}$ be
an isoperimetric set of volume $V_{j}=\left\lvert\Omega_{j}\right\rvert$.
Then, the sequence $(E_{j})_{j\in\mathbb{N}}$ satisfies

$\frac{2}{P(\Omega_{j})}\left(|\Omega_{j}|-\frac{P(\Omega_{j})^{3/2}}{6\sqrt{\pi}}\right)\leq\frac{2}{P(E_{j})}\left(|E_{j}|-\frac{P(E_{j})^{3/2}}{6\sqrt{\pi}}\right),$

implying (4.11). ∎

##  Appendix A Auxiliary results

In this appendix we collect two technical results we used in the paper.

######  Lemma .

Let $(M,g)$ be an $n$-dimensional complete $C^{0}$-Riemannian manifold. Then
the following hold.

  * •

Let $f\in\operatorname{Lip}_{\rm loc}(M)$. Then ${\rm lip\,}f=|\nabla f|$
almost everywhere.

  * •

Let $\Omega\subset M$ be an open set and suppose that $g_{i}$ is a sequence of
smooth Riemannian metrics converging to $g$ uniformly on $\Omega$. Then for
any $f\in L^{1}_{\rm loc}(\Omega,g)$ there holds

$\lim_{i}|Df|_{i}(\Omega)=|Df|(\Omega),$

where $|Df|_{i}$ denotes the total variation of $f$ as a function in
$L^{1}_{\rm loc}(\Omega,g_{i})$.

  * •

For any $f\in\operatorname{Lip}_{\rm loc}(M)$, there holds $|Df|={\rm
lip\,}f\,\mathcal{H}^{n}=|\nabla f|\,\mathcal{H}^{n}$. In particular

$|D\mathsf{d}_{x_{0}}|=\mathcal{H}^{n},$

for any $x_{0}\in M$, where $\mathsf{d}_{x_{0}}$ denotes distance from
$x_{0}$.

###### Proof.

Let $f\in\operatorname{Lip}_{\rm loc}(M)$. Recall that the identity ${\rm
lip\,}f=|\nabla f|$ almost everywhere readily follows on smooth Riemannian
manifolds exploiting the exponential map. Fix $x\in M$ and let $g_{i}$ be a
sequence of smooth Riemannian metrics converging to $g$ in $C^{0}$-sense on a
neighborhood $A$ of $x$. We can write $(1-\varepsilon_{i})^{2}g(v,v)\leq
g_{i}(v,v)\leq(1+\varepsilon_{i})^{2}g(v,v)$ for any tangent vector $v$ on
$A$, for some $\varepsilon_{i}\to 0$. Denote by $\mathsf{d}_{i}$ the distance
function on $(A,g_{i})$ defined by taking infimum of lengths of curves
contained in $A$. For any $i$, let $y_{j}\in A$ such that
$\lim_{j}\mathsf{d}_{i}(x,y_{j})=0$ and

${\rm lip}_{i}f(x)=\lim_{j}\frac{|f(x)-f(y_{j})|}{\mathsf{d}_{i}(x,y_{j})},$

where ${\rm lip}_{i}f$ denotes the slope of $f$ as a function in $(A,g_{i})$.
For $j$ large, $\mathsf{d}_{i}(x,y_{j})$ is realized by a curve contained in
$\Omega$. Hence
$\mathsf{d}_{i}(x,y_{j})\geq(1-\varepsilon_{i})\mathsf{d}(x,y_{j})$ for any
$j$ large. Thus

${\rm
lip}f(x)\geq\limsup_{j}\frac{|f(x)-f(y_{j})|}{\mathsf{d}(x,y_{j})}\geq(1-\varepsilon_{i})\lim_{j}\frac{|f(x)-f(y_{j})|}{\mathsf{d}_{i}(x,y_{j})}=(1-\varepsilon_{i}){\rm
lip}_{i}f(x).$

A symmetric argument implies that $\lim_{i}{\rm lip\,}_{i}f(x)={\rm
lip\,}f(x)$. Since ${\rm lip\,}_{i}f=|\nabla^{g_{i}}f|_{g_{i}}$ almost
everywhere, and norms of gradients clearly pass to the limit, then ${\rm
lip\,}f(x)=\lim_{i}|\nabla^{g_{i}}f|_{g_{i}}=|\nabla f|$ almost everywhere.

Let now $f,g_{i}$ be as in the second item. We can write again that
$(1-\varepsilon_{i})^{2}g(v,v)\leq
g_{i}(v,v)\leq(1+\varepsilon_{i})^{2}g(v,v)$ for any tangent vector $v$ on
$\Omega$, for some $\varepsilon_{i}\to 0$. Denote by $\mathsf{d}_{i}$ the
distance function on $(\Omega,g_{i})$ defined by taking infimum of lengths of
curves contained in $\Omega$ and by $|D(\cdot)|_{i}$ the corresponding total
variation. Let $f_{k}\in\operatorname{Lip}_{\rm loc}(\Omega)$ be a sequence
converging to $f$ in $L^{1}_{\rm loc}$ on $(\Omega,g)$ such that
$|Df|(\Omega)=\lim_{k}\int_{\Omega}{\rm lip}f_{k}$. As before, one estimates

${\rm lip}f_{k}(x)\geq(1-\varepsilon_{i}){\rm lip}_{i}f_{k}(x).$

Therefore

$|Df|(\Omega)=\lim_{k}\int_{\Omega}{\rm
lip}f_{k}\geq(1-\tilde{\varepsilon}_{i})\liminf_{k}\int_{\Omega}{\rm
lip}_{i}f_{k}\,\mathrm{d}\mathcal{H}^{n}_{g_{i}}\geq(1-\tilde{\varepsilon}_{i})|Df|_{i}(\Omega),$

for suitable $\tilde{\varepsilon}_{i}\to 0$. Hence
$|Df|(\Omega)\geq\limsup_{i}|Df|_{i}(\Omega)$. An analogous argument shows
that $|Df|(\Omega)\leq\liminf_{i}|Df|_{i}(\Omega)$.

Now let $f\in\operatorname{Lip}_{\rm loc}(M)$. For any $x_{0}\in M$ there
exist $r_{0}>0$ and a sequence of smooth metrics $g_{i}$ on $B_{r_{0}}(x_{0})$
uniformly converging to $g$ on $B_{r_{0}}(x_{0})$. The above argument also
shows that ${\rm lip}_{i}f\to{\rm lip}f$ pointwise on $B_{r_{0}}(x_{0})$,
hence in $L^{1}_{\rm loc}$ on $(B_{r_{0}}(x_{0}),g)$, being $f$ locally
Lipschitz. Using the second item, and since $|Df|_{i}={\rm
lip}_{i}f\,\mathcal{H}^{n}_{g_{i}}$ we have

$|Df|(B_{r_{0}}(x_{0}))=\lim_{i}\int_{B_{r_{0}}(x_{0})}{\rm
lip}_{i}f\,\mathrm{d}\mathcal{H}^{n}_{g_{i}}=\int_{B_{r_{0}}(x_{0})}{\rm
lip}f\,\mathrm{d}\mathcal{H}^{n}.$

Taking into account the first item, the third one follows as well. ∎

Boundedness of isoperimetric sets usually follows from a suitable deformation
lemma, see e.g. [4, Theorem B.1]. It is not completely clear how to work out
such an argument in the $C^{0}$ setting, yet one can slightly modify the
argument, taking advantage of the asymptotic behavior imposed, to obtain the
following weaker version, that still suffices for our aims.

######  Theorem A.1.

Let $(M,g)$ be an $n$-dimensional complete $C^{0}$-Riemannian manifold that is
$C^{0}_{\mathrm{loc}}$-asymptotic to $\mathbb{R}^{n}$. Assume that for some
$V>0$ there exist $E\subset M$ and a nonempty Euclidean ball
$B\subset\mathbb{R}^{n}$ such that $\left\lvert E\right\rvert+\left\lvert
B\right\rvert_{\mathrm{eu}}=V$ and $I(V)=P(E)+P_{\mathrm{eu}}(B)$. Then $E$
has a bounded representative.

###### Proof.

Without loss of generality, we can assume that $E$ has positive volume. Take
$p\in M$. Let $V(r)=|{E\setminus B_{r}(p)}|$. Call $A(r)=P(E,M\setminus
B_{r}(p))$. By coarea (see Theorem 2.1), and since by Appendix A we have
$|D\mathsf{d}_{p}|=\mathcal{H}^{n}$, notice that
$V^{\prime}(r)=-P(B_{r}(p),E)$. Moreover, a Euclidean-like isoperimetric
inequality holds for small volumes. Indeed, the proof of Section 2.3 (compare
also with item (1) in Section 2.3) shows that there is a constant $\zeta$ such
that $|B_{r}(p)|\geq\zeta r^{3}$ for every $r\in(0,1]$, and every $p\in M$.
Thus [6, Proposition 3.20] gives that $I(v)\geq\xi v^{2/3}$ for every
$v\in(0,\eta)$ for some $\eta,\xi>0$. Now arguing verbatim as after [4,
Equation (B.1)], we can reduce ourselves to show that, for some constant $C$,
there holds

$A(r)\leq-V^{\prime}(r)+CV(r),$ (A.1)

for almost all large enough $r$. Assuming by contradiction that $V(r)>0$ for
any $r>0$, the latter would result in a contradiction by ODE comparison on
$V(r)$, see [4, Theorem B.1]. To this aim, let $B^{r}$ be a smooth deformation
of $B\subset\mathbb{R}^{n}$ of volume $\left\lvert B\right\rvert+V(r)$ such
that

$P_{\mathrm{eu}}(B^{r})\leq P_{\mathrm{eu}}(B)+CV(r),$ (A.2)

where $C$ only depends on $B$. Notice it is enough to take $B^{r}$ to be a
ball containing $B$. Then, $\left\lvert E\cap B_{r}(p)\right\rvert+\left\lvert
B^{r}\right\rvert_{\mathrm{eu}}=V$ and thus, by using the analogue of [4,
Proposition 3.2], which can be proved analogously as in [4] (again exploiting
that, arguing as in [4, Lemma 2.17], the isoperimetric profile is achieved by
bounded sets), we get

$P(E)+P_{\mathrm{eu}}(B)=I(V)\leq I(|E|-V(r))+I_{\mathbb{R}^{n}}(|B|+V(r))\leq
P(E\cap B_{r}(p))+P_{\mathrm{eu}}(B^{r}).$ (A.3)

On the other hand, for almost every radius, we have (see [7, Proposition 2.6],
which can be applied thanks to Section 2.3) that

$P(E\cap B_{r}(p))\leq P(E)-P(E,M\setminus B_{r}(p))+P(B_{r}(p),E).$ (A.4)

Plugging (A.2) and (A.4) into (A.3), we are left with (A.1). So the proof is
concluded. ∎

## References

  * [1] V. Agostiniani, L. Mazzieri and F. Oronzio  “A Green’s Function Proof of the Positive Mass Theorem”  In _Comm. Math. Phys._ 405.2, 2024, pp. Paper no. 54  DOI: 10.1007/s00220-024-04941-8
  * [2] L. Ambrosio  “Fine properties of sets of finite perimeter in doubling metric measure spaces” Calculus of variations, nonsmooth analysis and related topics  In _Set-Valued Anal._ 10.2-3, 2002, pp. 111–128  DOI: 10.1023/A:1016548402502
  * [3] L. Ambrosio and S. Di Marino  “Equivalent definitions of $BV$ space and of total variation on metric measure spaces”  In _J. Funct. Anal._ 266.7, 2014, pp. 4150–4188  DOI: 10.1016/j.jfa.2014.02.002
  * [4] G. Antonelli, M. Fogagnolo and M. Pozzetta  “The isoperimetric problem on Riemannian manifolds via Gromov–Hausdorff asymptotic analysis”  In _Commun. Contemp. Math._ 26.1, 2024, pp. Paper no. 2250068  DOI: 10.1142/S0219199722500687
  * [5] G. Antonelli, S. Nardulli and M. Pozzetta  “The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds”  In _ESAIM Control Optim. Calc. Var._ 28, 2022, pp. Paper No. 5732  DOI: 10.1051/cocv/2022052
  * [6] G. Antonelli, E. Pasqualetto and M. Pozzetta  “Isoperimetric sets in spaces with lower bounds on the Ricci curvature”  In _Nonlinear Anal._ 220, 2022, pp. Paper No. 11283959  DOI: 10.1016/j.na.2022.112839
  * [7] G. Antonelli, E. Pasqualetto, M. Pozzetta and I.. Violo  “Topological regularity of isoperimetric sets in PI spaces having a deformation property”  In _Proceedings of the Royal Society of Edinburgh: Section A Mathematics_ , 2023, pp. 1–23  DOI: 10.1017/prm.2023.105
  * [8] R.. Bamler  “A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature”  In _Math. Res. Lett._ 23.2, 2016, pp. 325–337  DOI: 10.4310/MRL.2016.v23.n2.a2
  * [9] R. Bartnik  “The mass of an asymptotically flat manifold”  In _Comm. Pure Appl. Math._ 39.5, 1986, pp. 661–693  DOI: 10.1002/cpa.3160390505
  * [10] L. Benatti and M. Fogagnolo  “Isoperimetric sets in nonnegative scalar curvature and their role through various concepts of mass” Accepted in Springer INdAM series: Anisotropic Isoperimetric Problems & Related Topics, 2023  arXiv:2305.03643
  * [11] L. Benatti, M. Fogagnolo and L. Mazzieri  “Minkowski Inequality on complete Riemannian manifolds with nonnegative Ricci curvature” To appear in _An. & PDEs_, 2022  arXiv:2101.06063
  * [12] L. Benatti, M. Fogagnolo and L. Mazzieri  “On the Isoperimetric Riemannian Penrose Inequality”, 2023  arXiv:2212.10215
  * [13] L. Benatti, A. Pluda and M. Pozzetta  “Fine properties of nonlinear potentials and Geroch-type monotonicities” In preparation, 2024
  * [14] H.. Bray  “The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature” Thesis (Ph.D.)–Stanford University  ProQuest LLC, Ann Arbor, MI, 1997, pp. 103  URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9810085
  * [15] H.. Bray  “Proof of the Riemannian Penrose inequality using the positive mass theorem”  In _J. Differential Geom._ 59.2, 2001, pp. 177–267  URL: http://projecteuclid.org/euclid.jdg/1090349428
  * [16] H.. Bray, D.. Kazaras, M.. Khuri and D.. Stern  “Harmonic functions and the mass of 3-dimensional asymptotically flat Riemannian manifolds”  In _J. Geom. Anal._ 32.6, 2022, pp. Paper No. 18429  DOI: 10.1007/s12220-022-00924-0
  * [17] S. Brendle  “Scalar curvature rigidity of convex polytopes”  In _Invent. Math._ 235.2, 2024, pp. 669–708  DOI: 10.1007/s00222-023-01229-x
  * [18] S. Brendle and O. Chodosh  “A volume comparison theorem for asymptotically hyperbolic manifolds”  In _Comm. Math. Phys._ 332.2, 2014, pp. 839–846  DOI: 10.1007/s00220-014-2074-1
  * [19] P. Burkhardt-Guim  “Pointwise lower scalar curvature bounds for $C^{0}$ metrics via regularizing Ricci flow”  In _Geom. Funct. Anal._ 29.6, 2019, pp. 1703–1772  DOI: 10.1007/s00039-019-00514-3
  * [20] P. Burkhardt-Guim  “ADM mass for $C^{0}$ metrics and distortion under Ricci-DeTurck flow”  In _J. Reine Angew. Math._ 806, 2024, pp. 187–245  DOI: 10.1515/crelle-2023-0085
  * [21] A.. Burtscher  “Length structures on manifolds with continuous Riemannian metrics”  In _New York J. Math._ 21, 2015, pp. 273–296  URL: http://nyjm.albany.edu:8000/j/2015/21_273.html
  * [22] P. Buser  “A note on the isoperimetric constant”  In _Ann. Sci. École Norm. Sup. (4)_ 15.2, 1982, pp. 213–230  URL: http://www.numdam.org/item?id=ASENS_1982_4_15_2_213_0
  * [23] A. Carlotto, O. Chodosh and M. Eichmair  “Effective versions of the positive mass theorem”  In _Invent. Math._ 206.3, 2016, pp. 975–1016  DOI: 10.1007/s00222-016-0667-3
  * [24] J. Cheeger  “Differentiability of Lipschitz functions on metric measure spaces”  In _Geom. Funct. Anal._ 9.3, 1999, pp. 428–517