# Isoperimetric sets in nonnegative scalar curvature and their role through
various concepts of mass

Luca Benatti  L. Benatti, Università degli Studi di Pisa, Largo Bruno
Pontecorvo 5, 56127 Pisa, Italy<EMAIL_ADDRESS>and  Mattia
Fogagnolo  M. Fogagnolo, Università degli Studi di Padova, via Trieste 63,
35121 Padova, Italy<EMAIL_ADDRESS>
###### Abstract.

We review some recent results about the relations among isoperimetric sets,
Penrose inequalities and related concepts in the analysis of $3$-manifolds of
nonnegative scalar curvature. We also show that if the isoperimetric sets of
big volume have connected boundaries, the equivalence among suitable notions
of mass hold.

MSC (2020): 53C21, 53E10, 83C99, 49J45.

Keywords: isoperimetric sets, Penrose inequality, positive mass theorem,
isoperimetric mass, isocapacitary mass, inverse mean curvature flow,
nonnegative scalar curvature.

##  1\. Introduction

The topics we will deal with will gravitate around isoperimetric properties of
Riemannian manifolds of dimension $3$ with nonnegative scalar curvature that
are Asymptotically Flat in some suitable sense, usually endowed with a closed,
minimal and outermost boundary. With the latter adjective, we indicate that no
other closed, minimal surface exists enclosing $\partial M$. We will
occasionally refer to boundaries with these properties with the word _horizon_
, or _horizon boundary_.

One of the classical results in this class of manifold is the Riemannian
Penrose Inequality. Leaving all the discussion and the details to the main
body of the work, we just point out that Penrose inequalities read as bounds
from above of the area of the minimal boundary of $(M,g)$ in terms of suitable
global geometric invariants, which are interpreted as physical global
_masses_. In [BFM22a] we show an isoperimetric version of this inequality
holding in a very large class of manifolds. We will review on such result
focusing, focusing on its relation with the isoperimetric sets and on the
techniques that led to its existence in this context [CCE16]. We will also
deal with the equivalence of various, apparently very different, notions of
mass. Particular attention will be put in being as sharp as possible in the
decay requirements for the asymptotically flat condition on manifolds
considered.

In Section 2, we review the main properties of the Inverse Mean Curvature Flow
we are going to employ, mainly obtained by Huisken-Ilmanen [HI01]. In Section
3, we review and detail the beautiful proof of the existence of isoperimetric
sets of any volume in nonnegative scalar curved $3$-manifolds, obtained by
Carlotto-Chodosch-Eichmair [CCE16] after the fundamental insight of Shi
[Shi16]. In doing so, we sensibly weaken the decay assumptions on the metric
to $\mathscr{C}^{0}$-Asymptotic Flatness. In Section 4, after having discussed
Huisken’s notion of Isoperimetric mass [Hui06], we review the proof of the
related Isoperimetric Penrose inequality obtained in collaboration with
Mazzieri [BFM22a]. In Section 5, we analyze the relations with other notions
of mass, most notably the classical $\operatorname{ADM}$ mass [ADM61]. This
also gives us the occasion for discussing the physical relevance of the
concepts involved. The other notions of mass that will be taken into account
are the Isocapacitary masses [Jau20, BFM23]. We show, as partly new results,
relations among the connectedness of isoperimetric sets of large volume and
the equivalence of such notions of masses. We conclude with Section 6, where
we gather various directions of research that may by undertaken in connection
with the topics studied in this paper.

### Acknowledgements.

Part of this work has been carried out during the authors’ attendance to the
_Thematic Program on Nonsmooth Riemannian and Lorentzian Geometry_ that took
place at the Fields Institute in Toronto. The authors warmly thank the staff,
the organizers and the colleagues for the wonderful atmosphere and the
excellent working conditions set up there. L.B. is supported by the European
Research Council’s (ERC) project n.853404 ERC VaReg – Variational approach to
the regularity of the free boundaries, financed by the program Horizon 2020,
by PRA_2022_11 and by PRA_2022_14. F.M. has been supported by the European
Union – NextGenerationEU and by the University of Padova under the 2021 STARS
Grants@Unipd programme “QuASAR”. The authors are members of Gruppo Nazionale
per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA),
which is part of the Istituto Nazionale di Alta Matematica (INdAM), and are
partially funded by the GNAMPA project “Problemi al bordo e applicazioni
geometriche”.

The authors are grateful to G. Antonelli, S. Hirsch, L. Mazzieri, F. Oronzio
and M. Pozzetta for countless, precious and pleasureful discussions on topics
related to this work. Moreover, M. F. thanks A. Pluda, V. Franceschi and G.
Saracco for having organized, with the support of INdAM, the wonderful
workshop “Anisotropic Isoperimetric Problems & Related Topics” in Rome. This
paper rises as a conference proceeding of that event.

##  2\. The Inverse Mean Curvature Flow and the Hawking mass

A fundamental tool to understand some intimate geometric properties of
$3$-manifolds with nonnegative scalar curvature is the evolution of surfaces
through the inverse of the mean curvature. It evolves an immersion
$F_{0}:\Sigma\hookrightarrow M$ through

$\frac{\partial}{\partial
t}F_{t}(\Sigma)=\frac{1}{\operatorname{H}_{t}}\nu_{t},$ (2.1)

where $\nu_{t}$ is the normal pointing towards infinity and
$\operatorname{H}_{t}$ the mean curvature. It is clear that, at least at
points where the mean-curvature tends to zero, such flow must develop
singularities111Singularities happen in this case only [HI08, Corollary 2.3].
Representing $F_{t}(\Sigma)$ as a level set $\\{w=t\\}$, it is readily checked
that then

$\operatorname{div}\left(\frac{\mathrm{D}\kern
0.28436ptw}{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\mathrm{D}\kern
0.28436ptw$}}\right\rvert$}}{\lvert\mathrm{D}\kern
0.28436ptw\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\mathrm{D}\kern
0.28436ptw$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\mathrm{D}\kern
0.28436ptw$}}\right\rvert$}}}}\right)={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\mathrm{D}\kern
0.28436ptw$}}\right\rvert$}}{\lvert\mathrm{D}\kern
0.28436ptw\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\mathrm{D}\kern
0.28436ptw$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\mathrm{D}\kern
0.28436ptw$}}\right\rvert$}}}.$ (2.2)

Indeed, the left-hand side coincides with the mean curvature of the level sets
of $w$, while the right-hand side with the inverse of the velocity of the
level set flow.

Building on this, Huisken-Ilmanen [HI01] were able to define and fully
describe a weak notion of Inverse Mean Curvature Flow, which is a weak (in a
suitable, original sense) solution to 2.1 starting at $\Sigma=\partial\Omega$,
and crucially prescribing a procedure to _jump_ through singularities. Namely,
for any closed $\Omega\subset M$ with smooth boundary homologous to $\partial
M$, it consists in a _proper_ function
$w\in\operatorname{Lip}_{\operatorname{loc}}(M\smallsetminus\operatorname{Int}\Omega)$
solving

$\left\\{\kern-3.0pt\begin{array}[]{@{\
}rcl@{\hspace{1cm}}l}\operatorname{div}\left(\dfrac{\mathrm{D}\kern
0.28436ptw}{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\mathrm{D}\kern
0.28436ptw$}}\right\rvert$}}{\lvert\mathrm{D}\kern
0.28436ptw\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\mathrm{D}\kern
0.28436ptw$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\mathrm{D}\kern
0.28436ptw$}}\right\rvert$}}}}\right)&=&{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\mathrm{D}\kern
0.28436ptw$}}\right\rvert$}}{\lvert\mathrm{D}\kern
0.28436ptw\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\mathrm{D}\kern
0.28436ptw$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\mathrm{D}\kern
0.28436ptw$}}\right\rvert$}}}\hfil\hskip 28.45274pt&\text{on
$M\smallsetminus\Omega$,}\\\ w&=&0\hfil\hskip 28.45274pt&\text{on
$\partial\Omega$,}\\\ w&\to&+\infty\hfil\hskip 28.45274pt&\text{as
$\mathrm{d}(x,o)\to+\infty$}.\end{array}\right.$ (2.3)

The set $\Omega$ being closed allows us to consider also the case
$\partial\Omega=\partial M$.

Leaving the rigorous definition of weak solution to 2.3 to the original source
[HI01], we just heuristically describe what happens when a singular time is
next. As long as $\left\\{w=t\right\\}$ does not _fatten_ , meaning that it
does not develop positive full Lebesgue measure, the weak flow consists in a
foliation of $\mathscr{C}^{1,\alpha}$-hypersurfaces dictated by the inverse of
a suitable $L^{2}$-weak version of mean curvature. In this case, it turns out
that
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{t}$}}\right\rvert$}}{\lvert\partial\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{t}$}}\right\rvert$}}}$,
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\Omega_{t}$}}\right\rvert$}}{\lvert\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\Omega_{t}$}}\right\rvert$}}}$
and also $\int_{\partial\Omega_{t}}\operatorname{H}^{2}$ are continuous, where
$\Omega_{t}=\left\\{w\leq t\right\\}$. The fattening of a level set at a time
$\overline{t}$ (jump time) is a jump from $\left\\{w<\overline{t}\right\\}$ to
its _strictly outward minimizing hull_ , whihc is defined as the set
$E_{t}\supseteq\left\\{w<\overline{t}\right\\}$ of maximal volume among those
minimizing the perimeter from the outside. Jump times are exactly those when
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
E_{t}$}}\right\rvert$}}{\lvert
E_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
E_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
E_{t}$}}\right\rvert$}}}-{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\left\\{w<t\right\\}$}}\right\rvert$}}{\lvert\left\\{w<t\right\\}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\left\\{w<t\right\\}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\left\\{w<t\right\\}$}}\right\rvert$}}}>0$
but
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E_{t}$}}\right\rvert$}}{\lvert\partial
E_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E_{t}$}}\right\rvert$}}}={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\left\\{w<t\right\\}$}}\right\rvert$}}{\lvert\partial\left\\{w<t\right\\}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\left\\{w<t\right\\}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\left\\{w<t\right\\}$}}\right\rvert$}}}$.
The (closed) hull $E_{t}$ will be given by $\left\\{w\leq t\right\\}$. A
precise study of minimizing hulls (also in connection with IMCF) is performed
in [FM22], after [BT84].

A first decisive result of Huisken-Ilmanen’s work [HI01] is a characterization
of the existence of the proper, global weak flow with the mere existence of a
global subsolution. We are not describing this result in its full generality,
but we just point out its application to $\mathscr{C}^{0}$-Asymptotically Flat
Riemannian $3$-manifold, together with some basic properties we are going to
explicitly need. The statement will follow the precise definition of
$\mathscr{C}^{k}_{\tau}$-Asymptotic Flatness.

######  Definition 2.1 (Asymptotically Flat Riemannian manifolds).

A Riemannian $3$-manifold $(M,g)$ with (possibly empty) boundary is said to be
_$\mathscr{C}^{k}_{\tau}$ -Asymptotically Flat_, with $k\in\mathbb{N}$ and
$\tau\geq 0$, if the following conditions are satisfied.

  1. (1)

There exists a compact set $K\subseteq M$ such that $M\smallsetminus K$ is
differmorphic to
$\mathbb{R}^{3}\smallsetminus\left\\{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
x$}}\right\rvert$}}{\lvert
x\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
x$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
x$}}\right\rvert$}}}\leq R\right\\}$, through a map $(x^{1},x^{2},x^{3})$
whose component are called _asymptotically flat coordinates_.

  2. (2)

In the chart $(M\smallsetminus K,(x^{1},x^{2},x^{3}))$ the metric tensor is
expressed as

$g=g_{ij}\mathrm{d}x^{i}\otimes\mathrm{d}x^{j}=(\delta_{ij}+\eta_{ij})\mathrm{d}x^{i}\otimes\mathrm{d}x^{j}$
(2.4)

with

$\displaystyle\qquad\sum_{i,j=1}^{3}\sum_{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\beta$}}\right\rvert$}}{\lvert\beta\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\beta$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\beta$}}\right\rvert$}}}=0}^{k}{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
x$}}\right\rvert$}}{\lvert
x\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
x$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
x$}}\right\rvert$}}}^{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\beta$}}\right\rvert$}}{\lvert\beta\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\beta$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\beta$}}\right\rvert$}}}+\tau}{\mathchoice{\raisebox{2.23334pt}{$\displaystyle\left\lvert\raisebox{-2.23334pt}{\hbox{\set@color$\displaystyle\partial_{\beta}\eta_{ij}$}}\right\rvert$}}{\lvert\partial_{\beta}\eta_{ij}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial_{\beta}\eta_{ij}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial_{\beta}\eta_{ij}$}}\right\rvert$}}}=O(1)\text{
($=o(1)$ resp.)}$ $\displaystyle\text{as
}{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
x$}}\right\rvert$}}{\lvert
x\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
x$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
x$}}\right\rvert$}}}\to+\infty.$ (2.5)

We will denote the $\mathscr{C}^{1}_{0}$-Asymptotically Flat condition simply
with $\mathscr{C}^{1}$-Asymptotically Flat.

The following statement substantially gathers [BFM22a, Proposition 3.2] and
[HI01, Connectedness Lemma 4.2], see also [Ago+22, p.9-10] and [Cha+22, Lemma
2.1] for the connectedness part.

######  Theorem 2.2 (Existence and basic properties of the weak IMCF).

Let $(M,g)$ be a Riemannian $3$-manifold possibly with boundary. Suppose that
$(M,g)$ is $\mathscr{C}^{0}$-Asymptotically Flat. Then, for any closed
$\Omega$ homologous to $\partial M$ with smooth boundary there exists a weak
solution $w$ to 2.3. If $\partial\Omega$ is connected, and $H_{2}(M,\partial
M,\mathbb{Z})=\\{0\\}$, then $\partial\\{w\leq t\\}$ is connected for any
$t\in[0,+\infty)$.

###  2.1. The monotonicity of the Hawking mass

The weak Inverse Mean Curvature Flow is key in the analysis of nonnegative
scalar curvature in three dimensions due to the monotonicity of the _Hawking
mass_ ,

$\mathfrak{m}_{H}(\partial\Omega)=\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega$}}\right\rvert$}}{\lvert\partial\Omega\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega$}}\right\rvert$}}}^{\frac{1}{2}}}{16\pi^{\frac{3}{2}}}\left(4\pi-\int_{\partial\Omega}\frac{\operatorname{H}^{2}}{4}\,\mathrm{d}\sigma\right)$
(2.6)

along this evolution by level sets. This quantity has been substantially
conceived in [Haw68], while Geroch [Ger73] showed it to be nondecreasing along
the smooth IMCF, and devised it as a tool to provide the Positive Mass
Theorem. We are discussing positive mass results below, in connection also
with isoperimetry.

The computation along the smooth classical evolution of _connected_
hypersurfaces is straightforward. It relies on classical evolution equations
and on the Gauss-Bonnet Theorem, which is the reason why connectedness is
needed along the evolution. The monotonicity calculation reads as follows.
Denote $\Sigma_{t}=F_{t}(\Sigma)$ and $\mathrm{d}\sigma_{t}$ the area measure
on $\Sigma_{t}$. We are just employing well-known evolution equations (see
e.g. [HP99, Theorem 3.2]). We have

$\frac{\mathrm{d}}{\mathrm{d}t}{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\Sigma_{t}$}}\right\rvert$}}{\lvert\Sigma_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\Sigma_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\Sigma_{t}$}}\right\rvert$}}}=\int_{\Sigma_{t}}\frac{\partial}{\partial
t}(\mathrm{d}\sigma_{t})=\int_{\Sigma}\,\mathrm{d}\sigma_{t}={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\Sigma_{t}$}}\right\rvert$}}{\lvert\Sigma_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\Sigma_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\Sigma_{t}$}}\right\rvert$}}},$
(2.7)

immediately implying
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
F_{t}(\Sigma)$}}\right\rvert$}}{\lvert
F_{t}(\Sigma)\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
F_{t}(\Sigma)$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
F_{t}(\Sigma)$}}\right\rvert$}}}=\operatorname{e}^{t}{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\Sigma$}}\right\rvert$}}{\lvert\Sigma\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\Sigma$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\Sigma$}}\right\rvert$}}}$.
Hence, we get

$\displaystyle(16\pi)^{\frac{3}{2}}\frac{\mathrm{d}}{\mathrm{d}t}\mathfrak{m}_{H}(\Sigma_{t})$
$\displaystyle={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\Sigma_{t}$}}\right\rvert$}}{\lvert\Sigma_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\Sigma_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\Sigma_{t}$}}\right\rvert$}}}^{\frac{1}{2}}\left(8\pi-\int_{\Sigma_{t}}\frac{\operatorname{H}^{2}_{t}}{2}\,\mathrm{d}\sigma_{t}-\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Sigma_{t}}\operatorname{H}^{2}_{t}\,\mathrm{d}\sigma_{t}\right)$
(2.8)
$\displaystyle={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\Sigma_{t}$}}\right\rvert$}}{\lvert\Sigma_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\Sigma_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\Sigma_{t}$}}\right\rvert$}}}^{\frac{1}{2}}\left(8\pi-\int_{\Sigma_{t}}\frac{3\operatorname{H}^{2}_{t}}{2}\,\mathrm{d}\sigma_{t}-2\int_{\Sigma_{t}}\operatorname{H}_{t}\frac{\partial
H_{t}}{\partial t}\,\mathrm{d}\sigma_{t}\right)$ (2.9)
$\displaystyle={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\Sigma_{t}$}}\right\rvert$}}{\lvert\Sigma_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\Sigma_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\Sigma_{t}$}}\right\rvert$}}}^{\frac{1}{2}}\left(8\pi-\int_{\Sigma_{t}}\frac{3\operatorname{H}^{2}_{t}}{2}\,\mathrm{d}\sigma_{t}+2\int_{\Sigma_{t}}\operatorname{H}_{t}\left(\Delta_{\Sigma_{t}}\frac{1}{\operatorname{H}_{t}}+\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\operatorname{h}_{t}$}}\right\rvert$}}{\lvert\operatorname{h}_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\operatorname{h}_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\operatorname{h}_{t}$}}\right\rvert$}}}^{2}+\operatorname{Ric}(\nu_{t},\nu_{t})}{\operatorname{H}_{t}}\right)\,\mathrm{d}\sigma_{t}\right),$
(2.10) where $\operatorname{h}_{t}$ is the second fundamental form of
$\Sigma_{t}$. Integrating by parts and using the classical (traced) Guass-
Codazzi equation, we are left with
$\displaystyle(16\pi)^{\frac{3}{2}}\frac{\mathrm{d}}{\mathrm{d}t}\mathfrak{m}_{H}(\Sigma_{t})$
$\displaystyle={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\Sigma_{t}$}}\right\rvert$}}{\lvert\Sigma_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\Sigma_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\Sigma_{t}$}}\right\rvert$}}}^{\frac{1}{2}}\left(8\pi-\int_{\Sigma_{t}}\frac{3\operatorname{H}^{2}_{t}}{2}\,\mathrm{d}\sigma_{t}+\int_{\Sigma_{t}}2\frac{{\mathchoice{\raisebox{2.30127pt}{$\displaystyle\left\lvert\raisebox{-2.30127pt}{\hbox{\set@color$\displaystyle\nabla_{\Sigma_{t}}\operatorname{H}_{t}$}}\right\rvert$}}{\lvert\nabla_{\Sigma_{t}}\operatorname{H}_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\nabla_{\Sigma_{t}}\operatorname{H}_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\nabla_{\Sigma_{t}}\operatorname{H}_{t}$}}\right\rvert$}}}^{2}}{\operatorname{H}_{t}^{2}}+\mathrm{R}-\mathrm{R}^{\Sigma_{t}}+{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\operatorname{h}_{t}$}}\right\rvert$}}{\lvert\operatorname{h}_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\operatorname{h}_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\operatorname{h}_{t}$}}\right\rvert$}}}^{2}+\operatorname{H}_{t}^{2}\,\mathrm{d}\sigma_{t}\right)$
(2.11)
$\displaystyle={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\Sigma_{t}$}}\right\rvert$}}{\lvert\Sigma_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\Sigma_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\Sigma_{t}$}}\right\rvert$}}}^{\frac{1}{2}}\left(8\pi-\int_{\Sigma_{t}}\mathrm{R}^{\Sigma_{t}}\,\mathrm{d}\sigma_{t}+\int_{\Sigma_{t}}2\frac{{\mathchoice{\raisebox{2.30127pt}{$\displaystyle\left\lvert\raisebox{-2.30127pt}{\hbox{\set@color$\displaystyle\nabla_{\Sigma_{t}}\operatorname{H}_{t}$}}\right\rvert$}}{\lvert\nabla_{\Sigma_{t}}\operatorname{H}_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\nabla_{\Sigma_{t}}\operatorname{H}_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\nabla_{\Sigma_{t}}\operatorname{H}_{t}$}}\right\rvert$}}}^{2}}{\operatorname{H}_{t}^{2}}+\mathrm{R}+{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\mathring{\operatorname{h}}_{t}$}}\right\rvert$}}{\lvert\mathring{\operatorname{h}}_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\mathring{\operatorname{h}}_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\mathring{\operatorname{h}}_{t}$}}\right\rvert$}}}^{2}\,\mathrm{d}\sigma_{t}\right),$
(2.12)

where $\mathring{\operatorname{h}}_{t}$ is the trace-less second fundamental
form of $\Sigma_{t}$. Coupling $\mathrm{R}\geq 0$ with the Gauss-Bonnet
theorem applied to the connected surface $\Sigma_{t}$, reading

$\int_{\Sigma_{t}}\mathrm{R}^{\Sigma_{t}}\,\mathrm{d}\sigma_{t}=4\pi\raisebox{0.0pt}{$\chi$}(\Sigma_{t})\leq
8\pi,$ (2.13)

ensures that the deriviative of the Hawking mass is nonnegative.

The analysis of the monotonicity in the general, weak formulation constitutes
a central part of Huisken-Ilmanen’s work. However, once accepted the heuristic
description of the level set flow given above, one realizes that, at jump
times, part of the evolving boundary should be replaced by a piece of minimal
surface, so that the $L^{2}$-norm of the mean curvature can only decrease,
while the area is continuous. At jump times, hence, the Hawking mass should
only increase. This is in fact what happens.

######  Theorem 2.3 (Geroch Monotonicity along the weak IMCF [HI01]).

Let $(M,g)$ be a $\mathscr{C}^{0}$-Asymptotically Flat Riemannian $3$-manifold
possibly with boundary, satisfying $H_{2}(M,\partial M,\mathbb{Z})=\\{0\\}$.
Then, the Hawking mass 2.6 is well defined and monotone nonincreasing along
the weak IMCF of any outward minimizing connected closed $\Omega$ homologous
to the boundary $\partial M$, as long as the flow is contained in a region of
nonnegative scalar curvature.

##  3\. Isoperimetry in nonnegative scalar curvature

###  3.1. Nonnegative scalar curvature and the reverse Isoperimetric
inequality

To see plainly the role of the monotonicity of the Hawking mass in
isoperimetric issues, we focus on the case of manifolds with no boundary. it
is not difficult to conceive that, shrinking and shrinking an initial set
$\Omega$ around a point, one can build through a limiting procedure a weak
IMCF originating from a point $o$. This can actually be obtained with a
similar argument as in [HI01, Proposition 7.2]. Moreover, arguing as in [HI01,
Blowdown Lemma 7.1], $w(x)$ can be shown to behave as the Euclidean model
$(n-1)\log(\mathrm{d}(x,o))$ around $o$. In particular, one immediately
observes that $\mathfrak{m}_{H}(\partial\Omega_{t})\to 0$ when approaching
$o$, that is a $t\to-\infty$, where $\Omega_{t}=\left\\{w\leq t\right\\}$.
Consequently, being $\mathfrak{m}_{H}$ nondecreasing along the flow by Theorem
2.3, the level sets of $w$ satisfy _reverse Willmore inequality_ , meaning

$\int_{\partial\Omega_{t}}\operatorname{H}^{2}d\sigma\leq 16\pi.$ (3.1)

This immediately suggests that the isoperimetric quotient of such set is
smaller than the Euclidean, and can be seen, in the smooth flow case, as
follows.

Let
$v(t)={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\Omega_{t}$}}\right\rvert$}}{\lvert\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\Omega_{t}$}}\right\rvert$}}}$.
Then, by coarea formula and equation solved by $w$ 2.3, one has

$v^{\prime}(t)=\int_{\partial\Omega_{t}}\frac{1}{\operatorname{H}}\,\mathrm{d}\sigma.$
(3.2)

On the other hand, by the Hölder’s inequality

${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{t}$}}\right\rvert$}}{\lvert\partial\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{t}$}}\right\rvert$}}}=\int_{\partial\Omega_{t}}\operatorname{H}^{\alpha}\operatorname{H}^{-\alpha}\,\mathrm{d}\sigma\leq\left(\int_{\partial\Omega_{t}}\operatorname{H}^{\alpha
p}\,\mathrm{d}\sigma\right)^{\frac{1}{p}}\left(\int_{\partial\Omega_{t}}\operatorname{H}^{-\alpha\frac{p}{(p-1)}}\,\mathrm{d}\sigma\right)^{\frac{p-1}{p}}$
(3.3)

applied with $\alpha=2/3$ and $p=3$, one finds out that

$\left(\int_{\partial\Omega_{t}}\frac{1}{\operatorname{H}}\,\mathrm{d}\sigma\right)^{-1}\leq\frac{\left(\int_{\partial\Omega_{t}}\operatorname{H}^{2}\,\mathrm{d}\sigma\right)^{\frac{1}{2}}}{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{t}$}}\right\rvert$}}{\lvert\partial\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set<EMAIL_ADDRESS>(3.4)

Now, the numerator in the right-hand side is estimated by 3.1, while, as far
as the denominator is concerned, recall that the evolution forces the area
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{t}$}}\right\rvert$}}{\lvert\partial\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{t}$}}\right\rvert$}}}$
to equal $\mathrm{e}^{t}$ times a constant. This is due to 2.7 (see [HI01,
Lemma 1.6] for the computation in the setting of weak solutions). By the
asymptotic behaviour at the pole pointed out above, the constant is in fact
$4\pi$, the area of the round unit $2$-sphere, so that

${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{t}$}}\right\rvert$}}{\lvert\partial\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set<EMAIL_ADDRESS>(3.5)

Consequently, combining these pieces of information with 3.4 and 3.2 leaves us
with

$v^{\prime}(t)\geq 2\pi\mathrm{e}^{\frac{3}{2}t}.$ (3.6)

Integrating it from $t\to-\infty$, which corresponds to the pole, where both
the volume and the area of the level sets vanishes, and taking into account
again 3.5, yields

${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{t}$}}\right\rvert$}}{\lvert\partial\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{t}$}}\right\rvert$}}}\leq(36\pi)^{\frac{1}{3}}{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\Omega_{t}$}}\right\rvert$}}{\lvert\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set<EMAIL_ADDRESS>(3.7)

In other words, the level sets of $w$ satisfy a reverse sharp Eucliden
Isoperimetric inequality, at least when the evolution is smooth. The following
result of Shi [Shi16], subsumed in particular by [Shi16, (15)], constitutes
the general statement. Its less transparent formulation is ultimately due to
the fact that not any volume is covered along the weak evolution, where jumps
are allowed, and actually occur.

######  Theorem 3.1 (Shi’s reverse Isoperimetric Inequality).

Let $(M,g)$ be a $\mathscr{C}^{0}$-Asymptotically Flat Riemannian
$3$-manifold, satisfying $H_{2}(M,\partial M,\mathbb{Z})=\\{0\\}$, and let
$o\in M$. Let, for $o\in M$, $w_{o}$ be the weak IMCF issuing from $o$, and
let, for $v>0$

$t(v)=\inf\left\\{\tau\,\middle|\,{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\left\\{w_{o}\leq
t\right\\}$}}\right\rvert$}}{\lvert\left\\{w_{o}\leq
t\right\\}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\left\\{w_{o}\leq
t\right\\}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\left\\{w_{o}\leq
t\right\\}$}}\right\rvert$}}}\geq v\right\\}.$ (3.8)

Then, as long as $(\\{w_{o}\leq t(v)\\},g)$ has nonnegative scalar curvature,
we have

${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\\{w_{o}\leq
t(v)\\}$}}\right\rvert$}}{\lvert\partial\\{w_{o}\leq
t(v)\\}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\\{w_{o}\leq
t(v)\\}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\\{w_{o}\leq
t(v)\\}$}}\right\rvert$}}}\leq(36\pi)^{\frac{1}{3}}v^{\frac{2}{3}}.$ (3.9)

The scalar curvature will rule the existence of isoperimetric sets
substantially by means of the above result. In fact, in light of the general
principle first devised by [Nar14], the only way a minimizing sequence may
fail to provide an isoperimetric set would consist in losing part of its
volume at infinity. However, by the $\mathscr{C}^{0}$-Asymptotic Flatness,
such volume would be fated to converge to a round ball in $\mathbb{R}^{3}$. By
Theorem 3.1, one can then safely replace this ball with a sublevel set of the
IMCF enclosing the same drifted away volume, obtaining a competitor that is
not worse.

######  Remark 3.2 (Comparison with nonnegative Ricci curvature).

A scheme to isoperimetric existence like the one being illustrated here works
in any dimension if nonnegative scalar curvature is promoted to nonnegative
Ricci curvature, as a consequence of the more basic Laplacian Comparison and
Bishop-Gromov Theorems. Such classical results respectively state that, in a
suitable weak sense, the mean curvature of a geodesic ball of radius $r$ is
bounded from above by $(n-1)/r$, while its area
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
B(r)$}}\right\rvert$}}{\lvert\partial
B(r)\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
B(r)$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
B(r)$}}\right\rvert$}}}$ is controlled from above by
${\mathchoice{\raisebox{3.06888pt}{$\displaystyle\left\lvert\raisebox{-3.06888pt}{\hbox{\set@color$\displaystyle\mathbb{S}^{n-1}$}}\right\rvert$}}{\lvert\mathbb{S}^{n-1}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\mathbb{S}^{n-1}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\mathbb{S}^{n-1}$}}\right\rvert$}}}r^{n-1}$.
We stick now to dimension $3$, in order to facilitate the comparison with 3.7.
Fix a point $o\in M$, a manifold endowed with a metric of nonnegative Ricci
curvature $g$. Letting $r(V)$ be the radius of the ball $B(r(V))$ centered in
$o$ of volume $V$, it is easily computed through classical variation formulas
that

$r^{\prime}(V)=\frac{1}{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
B(r(V))$}}\right\rvert$}}{\lvert\partial
B(r(V))\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
B(r(V))$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
B(r(V))$}}\right\rvert$}}}}\geq\frac{1}{4\pi r(V)^{2}}.$ (3.10)

Integrating such inequality, one gets

$r(V)\geq\left(\frac{3}{4\pi}V\right)^{\frac{1}{3}}.$ (3.11)

On the other hand, we have

$I(r(V))^{\prime}=\frac{1}{I(r(V)}\int_{\partial
B(r(V)}\operatorname{H}\,\mathrm{d}\sigma\leq\frac{2}{r(V)}\leq\frac{2}{V^{\frac{1}{3}}}\left(\frac{4\pi}{3}\right)^{\frac{1}{3}},$
(3.12)

where we plugged 3.11 in. Integrating this other differential inequality, we
conclude

${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
B(r)$}}\right\rvert$}}{\lvert\partial
B(r)\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
B(r)$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
B(r)$}}\right\rvert$}}}\leq(36\pi)^{\frac{1}{3}}{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
B(r)$}}\right\rvert$}}{\lvert
B(r)\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
B(r)$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
B(r)$}}\right\rvert$}}}^{\frac{2}{3}}$ (3.13)

for any radius $r\geq 0$, that fully mirrors 3.7, but for a different
exhaustion. It should be clear that no particular features of the dimension
have been exploited here, contrarily to the Gauss-Bonnet Theorem 2.13 utilized
to infer the monotonicity of the Hawking mass, that in turn led to 3.7. The
reverse isoperimetric inequality in nonnegative Ricci curvature was first
pointed out by Morgan-Johnson [MJ00], and it was first exploited to infer the
existence of isoperimetric sets under suitable asymptotic assumptions by
Mondino-Nardulli [MN16].

###  3.2. Isoperimetric analysis on manifolds with nonnegative scalar
curvature

The following is a version of Nardulli’s Generalized Compactness Principle,
crafted for the Asymptotically Flat framework. It substantially asserts that
the runaway volume in an isoprimetric sequence is fully recovered with a ball
in $\mathbb{R}^{3}$. The sublevel sets of the IMCF satisfying, as just
explained, a reverse Isoperimetric inequality, can thus be exploited to
replace such lost volume and provide an isoperimetric set.

######  Theorem 3.3 (Asymptotic Decomposition of Isoperimetric minimizing
sequences).

Let $(M,g)$ be a smooth, $\mathscr{C}^{0}$-Asymptotically Flat Riemannian
manifold $(M,g)$ with Ricci curvature bounded from below, and let $I$ be its
isoperimetric profile. Then, for any $V>0$, there exists a possibly empty,
_bounded_ $E\subset M$ and a ball $B$ in $\mathbb{R}^{n}$ such that
$V={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
E$}}\right\rvert$}}{\lvert
E\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
E$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
E$}}\right\rvert$}}}+{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
B$}}\right\rvert$}}{\lvert
B\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
B$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
B$}}\right\rvert$}}}$ and

$I(V)={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E$}}\right\rvert$}}{\lvert\partial
E\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E$}}\right\rvert$}}}+{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
B$}}\right\rvert$}}{\lvert\partial
B\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
B$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
B$}}\right\rvert$}}}.$ (3.14)

The version above is actually a consequence of the way more general [AFP22,
Theorem 1.1].

Clearly, the number of drifted away balls being at most one is due to the fact
that a union of balls in $\mathbb{R}^{3}$ is manifestly isoperimetrically less
convenient than one.

######  Remark 3.4.

Nardulli’s earlier work has been recently vastly exploited and empowered in
the context of possibly nonsmooth metric spaces with Ricci lower bounds,
[AFP22, Ant+22, Ant+22a, ANP22, AP23]. It led to far-reaching consequences
regarding the existence of isoperimetric sets, isoperimetric inequalities and
sharp properties of the isoperimetric profile in such regime.

There is a last issue to be taken into account before safely running the
argument sketched above for the existence of Isoperimetric sets. Indeed, as
already pointed out, the weak IMCF can jump, and in particular there could be
some value $V$ such that $\\{w\leq t\\}$ encloses a volume $V$ for no $t$.
This may cause trouble in case such volume is exactly the volume of the ball
at infinity that we would like to replace. This problem is bypassed by the
strict monotonicity of the Isoperimetric profile, a property that is
intimately related to the outermost property of the minimal boundary.

######  Proposition 3.5.

Let $(M,g)$ be a $\mathscr{C}^{0}$-Asymptotically Flat Riemannian $3$-manifold
with Ricci curvature bounded from below and endowed with a closed, outermost
minimal boundary. Then, its isoperimetric profile $I$ is strictly increasing.

  * Proof.

We first of all recall that the isoperimetric profile is continuous in this
case, see [MN19]. Fix a volume $V$. Then, by Theorem 3.3, we know that

$I(V)={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E$}}\right\rvert$}}{\lvert\partial
E\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E$}}\right\rvert$}}}+{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
B$}}\right\rvert$}}{\lvert\partial
B\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
B$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
B$}}\right\rvert$}}}_{\mathbb{R}^{3}}$ (3.15)

for a possibly empty $E\subset M$ and for a Euclidean ball $B$ satisfying
$V={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
E$}}\right\rvert$}}{\lvert
E\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
E$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
E$}}\right\rvert$}}}+{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
B$}}\right\rvert$}}{\lvert
B\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
B$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
B$}}\right\rvert$}}}_{\mathbb{R}^{3}}$. Clearly, $E$ must be isoperimetric for
its own volume, i.e.
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E$}}\right\rvert$}}{\lvert\partial
E\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E$}}\right\rvert$}}}=I({\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
E$}}\right\rvert$}}{\lvert
E\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
E$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
E$}}\right\rvert$}}})$, and in particular it is smooth. Assume $E$ is
nonempty. Then, performing an inward variation of $\partial E$ supported in
$\partial E\setminus\partial M$, giving rise, for any $\varepsilon>0$, to
$E_{\varepsilon}\subset E$ of volume
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
E$}}\right\rvert$}}{\lvert
E\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
E$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
E$}}\right\rvert$}}}-\varepsilon$, we get

$\liminf_{\varepsilon\to
0^{+}}\frac{I(V)-I(V-\varepsilon)}{\varepsilon}\geq\liminf_{\varepsilon\to
0^{+}}\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E$}}\right\rvert$}}{\lvert\partial
E\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E$}}\right\rvert$}}}-{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E_{\varepsilon}$}}\right\rvert$}}{\lvert\partial
E_{\varepsilon}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E_{\varepsilon}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E_{\varepsilon}$}}\right\rvert$}}}}{\varepsilon}=\operatorname{H}_{E},$ (3.16)

where $\operatorname{H}_{E}$ is the constant mean curvature of $\partial
E\setminus\partial M$. In the inequality in 3.16, we used

$I(V-\varepsilon)\leq{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E_{\varepsilon}$}}\right\rvert$}}{\lvert\partial
E_{\varepsilon}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E_{\varepsilon}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E_{\varepsilon}$}}\right\rvert$}}}+{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
B$}}\right\rvert$}}{\lvert\partial
B\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
B$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
B$}}\right\rvert$}}}_{\mathbb{R}^{3}},$ (3.17)

in force, since the sets $E_{\varepsilon}\cup B_{j}$, with $B_{j}\subset M$
approaching $B$ in the $\mathscr{C}^{0}$-topology, form a valid family of
competitors for the isoperimetric problem of volume $V-\varepsilon$. The key
step consists thus in showing that the constant $\operatorname{H}_{E}$ is
strictly positive. To see this, we can argue as done for [BFM22a, Lemma 2.8],
after [HI01, Remark, p. 394]. Namely, one can flow a geodesic ball in the
asymptotic region, that is mean-convex by [BFM22b, Lemma 4.3], through the
Mean Curvature Flow of mean-convex surfaces with surgery in Riemannian
$3$-manifolds [BH18]; as proved in such paper, this flow is fated to smoothly
converge to the outermost minimal boundary $\partial M$. One can then find a
surface in the evolution touching $\partial E$. If $\operatorname{H}_{E}$ were
nonpositive, this would result in a contradiction with the Maximum Principle.
This completes the proof in neighbourhoods of volumes such that $\partial E$
is nonempty.

In case $E$ were empty, deforming $B\subset\mathbb{R}^{3}$ inwardly as above
yields 3.16, this time in terms of the mean curvature of
$B\subset\mathbb{R}^{n}$, that is obviously strictly positive. This concludes
the proof. ∎

With Theorem 3.1, Theorem 3.3 and Proposition 3.5 at hand, we can prove that
isoperimetric sets exist in any $\mathscr{C}^{0}$-Asymptotically Flat
Riemannian $3$-manifold with nonnegative scalar curvature and horizon
boundary, for any volume, provided some lower bound on the Ricci curvature is
in force. This is a refinement of [CCE16, Proposition K.1]. Useful insights
about the strategy employed were actually proposed by Brendle-Chodosh [BC14],
including the key computation leading to Theorem 3.1.

######  Theorem 3.6 (Existence of Isoperimetric sets in nonnegative scalar
curvature).

Let $(M,g)$ be $\mathscr{C}^{0}$-Asymptotically Flat Riemannian $3$-manifold
with nonnegative scalar curvature and with Ricci curvature bounded from below,
endowed with a closed, minimal outermost boundary. Then, for any $V>0$, there
exists an isoperimetric set of volume $V$.

  * Proof.

Let $V>0$. As above, we have

$I(V)={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E$}}\right\rvert$}}{\lvert\partial
E\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E$}}\right\rvert$}}}+{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
B$}}\right\rvert$}}{\lvert\partial
B\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
B$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
B$}}\right\rvert$}}}_{\mathbb{R}^{3}}$ (3.18)

for a possibly empty $E\subset M$ and for a Euclidean ball $B$ satisfying
$V={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
E$}}\right\rvert$}}{\lvert
E\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
E$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
E$}}\right\rvert$}}}+{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
B$}}\right\rvert$}}{\lvert
B\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
B$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
B$}}\right\rvert$}}}_{\mathbb{R}^{3}}$. We can assume that
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
B$}}\right\rvert$}}{\lvert
B\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
B$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
B$}}\right\rvert$}}}_{\mathbb{R}^{3}}>0$, otherwise $E$ is already the
isoperimetric set of volume $V$ sought for.

Let now $o\in M$ be far away from the boundary. We can assume that there
exists a weak IMCF $w_{o}$ issuing from $o$, although this is not obtained
through flows homologous to $\partial M$, as Theorem 2.2 would require. In
fact, we can attach a handlebody to $\partial M$, and extend smoothly the
metric $g$ to this new complete manifold [PV16]. Then, Theorem 2.2, coupled
with the limiting procedure already mentioned above, yields a weak IMCF
$w_{o}$ issuing from $o$. By the known topological structure of manifolds with
nonnegative scalar curvature and minimal outermost boundary, see e.g.
[BFM22a], we have $H_{2}(M,\partial M,\mathbb{Z})=\left\\{0\right\\}$.
Consequently, this is also fulfilled by the extended manifold without
boundary, and so, again as stated in Theorem 2.2, the surfaces
$\partial\left\\{w_{o}\leq t\right\\}$ remain connected.

By [MRS22, Theorem 1.3], there exist functions
$f_{1}f_{2},:[0,+\infty)\to\mathbb{R}$ that grow to infinity at infinity such
that

$f_{1}(\mathrm{d}(x,o))\leq w_{o}(x)\leq f_{2}(\mathrm{d}(x,o))$ (3.19)

for any $o\in M$. In fact, such result requires a Ricci lower bound and the
validity of a global, possibly weighted Sobolev inequality. In the
$\mathscr{C}^{0}$-Asymptotically Flat case this in force with no weight, as a
direct consequence of the uniform Euclidean-like Isoperimetric inequality that
can be directly obtained outside some sufficiently big compact set, and then,
by [PST14, Theorem 3.2], on the whole manifold.

In light of 3.19, given $v>0$ we can always find $o$ sufficiently far in space
such that $\\{w_{o}\leq t(v)\\}$ is disjoint from the bounded $E$ and
$\partial M$. Since such sublevel set is contained in a nonnegative scalar
curved region, Shi’s reverse Isoperimetric Inequality 3.9 holds for $t(v)\leq
t$. Choose now
$v={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
B$}}\right\rvert$}}{\lvert
B\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
B$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
B$}}\right\rvert$}}}$, and $o$ as above. Let $F=\left\\{w_{o}\leq
t(v)\right\\}$, and observe that, by definition,
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
F$}}\right\rvert$}}{\lvert
F\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
F$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
F$}}\right\rvert$}}}\geq v$. Now, if
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
F$}}\right\rvert$}}{\lvert
F\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
F$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
F$}}\right\rvert$}}}=v$, then by (3.9) the set $E\cup F$ is isoperimetric of
volume $V$. If instead
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
F$}}\right\rvert$}}{\lvert
F\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
F$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
F$}}\right\rvert$}}}>v$, then by the strict monotonicity of $I$ shown in
Proposition 3.5 and (3.18) we have

$I({\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
E$}}\right\rvert$}}{\lvert
E\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
E$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
E$}}\right\rvert$}}}+{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
F$}}\right\rvert$}}{\lvert
F\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
F$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
F$}}\right\rvert$}}})>I({\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
E$}}\right\rvert$}}{\lvert
E\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
E$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
E$}}\right\rvert$}}}+v)=I(V)={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E$}}\right\rvert$}}{\lvert\partial
E\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E$}}\right\rvert$}}}+(36\pi)^{\frac{1}{3}}v_{B}^{\frac{2}{3}}.$ (3.20)

On the other hand, we also have

$I({\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
E$}}\right\rvert$}}{\lvert
E\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
E$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
E$}}\right\rvert$}}}+{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
F$}}\right\rvert$}}{\lvert
F\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
F$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
F$}}\right\rvert$}}})\leq{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E$}}\right\rvert$}}{\lvert\partial
E\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E$}}\right\rvert$}}}+{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
F$}}\right\rvert$}}{\lvert\partial
F\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
F$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
F$}}\right\rvert$}}}\leq{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E$}}\right\rvert$}}{\lvert\partial
E\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E$}}\right\rvert$}}}+(36\pi)^{\frac{1}{3}}v^{\frac{2}{3}},$ (3.21)

where we have used again (3.9). Chaining (3.20) with (3.21) leads to a
contradiction, that ends the proof. ∎

We find worth spending some words on the arguably unnecessary assumption of
some Ricci lower bound.

######  Remark 3.7 (On the Ricci lower bound assumption).

We exploited the Ricci lower bound in two places. The first one is in the
fundamental Generalized Compactness statement of Theorem 3.3; in fact, both in
[AFP22] and in the subsequent, improved [ANP22], some possibly synthetic bound
on the Ricci tensor is required. On the other hand, it does not seem difficult
to realize that, disposing as in our case of a _global_ Asymptotic Flatness
should be enough to guarantee the availability of all that is needed to carry
out the arguments.

A maybe more serious issue arises in connection with 3.19. In fact, at least
the barrier $f_{2}$ constructed in [MRS22] strongly depends on the bound on
the Ricci curvature. However, such bound does not exploit the Asymptotic
Flatness at all, that should allow one to provide suitable barriers modelled
on the Euclidean model solutions. The difficulties seem to lie in ensuring the
validity of the double bounds up to to $o$ and _uniformly_ on $o$. Without
these specifications, a similar goal was achieved for different aims in
[BFM22a, Proposition 3.2].

##  4\. The Isoperimetric mass and the Isoperimetric Penrose inequality

So far, we did not mention any particular example of $3$-manifold with
nonnegative scalar curvature and minimal, outermost boundary. Let us stick now
to the archetypal one, that will actually constitute the model for the
geometric inequalities that we are going to present. The ($3$-dimensional)
Schwarzschild manifold of (positive) mass $\mathfrak{m}$ is the space
$\mathbb{R}^{3}\smallsetminus\left\\{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
x$}}\right\rvert$}}{\lvert
x\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
x$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
x$}}\right\rvert$}}}<2\mathfrak{m}\right\\}$ endowed with the rotationally
symmetric metric

$g=\left(1+\frac{\mathfrak{m}}{2{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
x$}}\right\rvert$}}{\lvert
x\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
x$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
x$}}\right\rvert$}}}}\right)^{4}\delta_{ij}\,\mathrm{d}x^{i}\otimes\mathrm{d}x^{j}.$
(4.1)

This Riemannian manifold is scalar flat, the boundary $\partial
M=\\{{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
x$}}\right\rvert$}}{\lvert
x\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
x$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
x$}}\right\rvert$}}}=2\mathfrak{m}}\\}$ is minimal, and since any other level
set of
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
x$}}\right\rvert$}}{\lvert
x\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
x$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
x$}}\right\rvert$}}}$ is constantly mean-curved, such boundary is also
outermost, since the presence of any other closed minimal surfaces would
result in a contradiction with the Maximum Principle.

It is known from early work of Bray [Bra97], later generalized in various
directions [BM02, Bre13], that the isoperimetric sets in this warped products
are only the Euclidean annuli given by
$E_{R}=\\{2\mathfrak{m}\leq{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
x$}}\right\rvert$}}{\lvert
x\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
x$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
x$}}\right\rvert$}}}\leq R\\}$. Coherently with Shi’s reverse Isoperimetric
inequality, and with the fact that balls form an Inverse Mean Curvature Flow,
one checks that

${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
E_{R}$}}\right\rvert$}}{\lvert
E_{R}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
E_{R}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
E_{R}$}}\right\rvert$}}}-\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E_{R}$}}\right\rvert$}}{\lvert\partial
E_{R}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E_{R}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E_{R}$}}\right\rvert$}}}^{\frac{3}{2}}}{6\sqrt{\pi}}\geq 0.$ (4.2)

Moreover, this bound is seen to scale as $R\to+\infty$ with $R^{2}$.
Interestingly, one computes that the mass parameter $\mathfrak{m}$ is in fact
recovered with the following limit

$\mathfrak{m}=\limsup_{R\to+\infty}\frac{2}{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E_{R}$}}\right\rvert$}}{\lvert\partial
E_{R}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E_{R}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E_{R}$}}\right\rvert$}}}}\left({\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
E_{R}$}}\right\rvert$}}{\lvert
E_{R}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
E_{R}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
E_{R}$}}\right\rvert$}}}-\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E_{R}$}}\right\rvert$}}{\lvert\partial
E_{R}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E_{R}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E_{R}$}}\right\rvert$}}}^{\frac{3}{2}}}{6\sqrt{\pi}}\right).$ (4.3)

The above observations can serve as a motivation for the notion of
Isoperimetric mass, a concept conceived by Huisken [Hui06].

######  Definition 4.1 (Isoperimetric mass).

Let $(M,g)$ be a Riemannian $3$-manifold possibly with boundary, with infinite
volume. Then, its _Isoperimetric mass_ is defined as

$\mathfrak{m}_{\operatorname{iso}}=\sup_{(\Omega_{j})_{j\in\mathbb{N}}}\limsup_{j\to+\infty}\frac{2}{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega$}}\right\rvert$}}{\lvert\partial\Omega\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega$}}\right\rvert$}}}}\left({\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\Omega$}}\right\rvert$}}{\lvert\Omega\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\Omega$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\Omega$}}\right\rvert$}}}-\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega$}}\right\rvert$}}{\lvert\partial\Omega\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega$}}\right\rvert$}}}^{\frac{3}{2}}}{6\sqrt{\pi}}\right)$
(4.4)

where the supremum is taken among all exhaustions
$(\Omega_{j})_{j\in\mathbb{N}}$ consisting of domains with
$\mathscr{C}^{1,\alpha}$-boundary.

It is immediately checked from the above definition that, if an exhaustion
consists of isoperimetric sets, this automatically realizes the required
supremum. In particular, in nonnegative scalar curvature, the following holds.

######  Lemma 4.2.

Let $(M,g)$ be a $3$-dimensional $\mathscr{C}^{0}$-Asymptotically Flat
Riemannian manifold with nonnegative scalar curvature and Ricci curvature
bounded from below, endowed with a closed, outermost minimal boundary. Then,
we have

$\mathfrak{m}_{\operatorname{iso}}=\limsup_{V\to+\infty}\frac{2}{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E_{V}$}}\right\rvert$}}{\lvert\partial
E_{V}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E_{V}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E_{V}$}}\right\rvert$}}}}\left({\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
E_{V}$}}\right\rvert$}}{\lvert
E_{V}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
E_{V}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
E_{V}$}}\right\rvert$}}}-\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E_{V}$}}\right\rvert$}}{\lvert\partial
E_{V}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E_{V}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E_{V}$}}\right\rvert$}}}^{\frac{3}{2}}}{6\sqrt{\pi}}\right),$ (4.5)

where $E_{V}$ is isoperimetric of volume $V>0$.

We should take into account the following _caveat_. The sequences of
isoperimetric sets $(E_{V_{j}})_{j\in\mathbb{N}}$ of increasing volume $V_{j}$
we have at hand do not a priori form an exhaustion. However, the useful [JL19,
Proposition 37] asserts that, in the $\mathscr{C}^{0}$-Asymptotically Flat
regime, one can relax the definition of $\mathfrak{m}_{\operatorname{iso}}$ in
order to replace the requirement for $(\Omega_{j})_{j\in\mathbb{N}}$ to form
an exhaustion with
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{i}$}}\right\rvert$}}{\lvert\partial\Omega_{i}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{i}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{i}$}}\right\rvert$}}}\to+\infty$
as $i\to+\infty$, certifying the validity of the above Lemma.

Turning back to the observation of the Schwarzschild model, as a consequence
of Lemma 4.2 we immediately deduce that
$\mathfrak{m}_{\operatorname{iso}}=\mathfrak{m}$. Moreover, just by computing
the value of
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
M$}}\right\rvert$}}{\lvert\partial
M\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
M$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
M$}}\right\rvert$}}}=\\{{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
x$}}\right\rvert$}}{\lvert
x\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
x$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
x$}}\right\rvert$}}}=2\mathfrak{m}}\\}$ from the expression for $g$ in 4.1,
one sees that

$\sqrt{\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
M$}}\right\rvert$}}{\lvert\partial
M\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
M$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
M$}}\right\rvert$}}}}{16\pi}}=\mathfrak{m}_{\operatorname{iso}}.$ (4.6)

Together with Mazzieri, we proved that, in the general case considered in the
above Section, and actually with no Ricci lower bound required, the left-hand
side of 4.6 is controlled from above by the right-hand side [BFM22a, Theorem
1.3]. Inequalities bounding a suitable notion of mass in terms of the horizon
boundary are usually denominated Penrose inequalities.

######  Theorem 4.3 (Isoperimetric Penrose inequality).

Let $(M,g)$ be a $\mathscr{C}^{0}$-Asymptotically Flat Riemannian $3$-manifold
with nonnegative scalar curvature and closed, minimal outermost _connected_
boundary. Then,

$\sqrt{\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
M$}}\right\rvert$}}{\lvert\partial
M\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
M$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
M$}}\right\rvert$}}}}{16\pi}}\leq\mathfrak{m}_{\operatorname{iso}}.$ (4.7)

Moreover, equality holds if and only if $(M,g)$ is a Schwarzschild
$3$-manifold with $\mathfrak{m}_{\operatorname{iso}}=\mathfrak{m}$.

Additional discussion on Penrose inequalities, including physical motivation
behind them will be clearer in the next Section, where the Isoperimetric mass
will be compared with other concepts of mass, most notably with the classical
$\operatorname{ADM}$ mass. For the time being, we just point out that the
latter is well defined only under stronger asymptotic assumptions on the
metric decaying towards the flat one, and that these two notions of mass do
actually coincide in these cases.

The general strategy for the proof of 4.7 follows Huisken-Ilmanen’s [HI01],
where the inequality was proved in terms of the $\operatorname{ADM}$ mass. It
consists in exploiting again the monotonicity of the Hawking mass, this time
along the weak Inverse Mean Curvature Flow of the minimal boundary. Since the
mean curvature $\operatorname{H}$ vanishes on $\partial M$, the initial value
of this quantity becomes

$\mathfrak{m}_{H}(\partial
M)=\sqrt{\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
M$}}\right\rvert$}}{\lvert\partial
M\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
M$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
M$}}\right\rvert$}}}}{16\pi}}.$ (4.8)

Hence, since $\mathfrak{m}_{H}$ is nondecreasing along the flow by Theorem
2.3, to prove 4.7 it suffices to show that

$\lim_{t\to+\infty}\mathfrak{m}_{H}{(\partial\Omega_{t})}\leq\liminf_{t\to+\infty}\frac{2}{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{t}$}}\right\rvert$}}{\lvert\partial\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{t}$}}\right\rvert$}}}}\left({\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\Omega_{t}$}}\right\rvert$}}{\lvert\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\Omega_{t}$}}\right\rvert$}}}-\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{t}$}}\right\rvert$}}{\lvert\partial\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{t}$}}\right\rvert$}}}^{\frac{3}{2}}}{6\sqrt{\pi}}\right)\leq\mathfrak{m}_{\mathrm{iso}}.$
(4.9)

This last step is completely different than Huisken-Ilmanen’s asymptotic
comparison with $\mathfrak{m}_{\operatorname{ADM}}$. Notably, it is by far
computationally easier, and way more general in the asymptotic assumptions
needed.

######  Remark 4.4.

In fact in 4.7 we do not even require any kind of Asymptotic Flatness in order
to reach for 4.7; we just have to assume the existence of proper weak IMCF’s.
The issue of the existence of weak IMCF’s is a very interesting topic _per se_
, related to deep metric properties of the underlying manifold, see [KN09,
MRS22].

We provide a (sketch of) the proof of the crucial asymptotic comparison 4.9,
in order to highlight the similarities with the computations leading to 3.4
above.

  * Proof of 4.9.

We directly assume, for simplicity, that, as $t\to+\infty$,

$\int_{\partial\Omega_{t}}\operatorname{H}^{2}\,\mathrm{d}\sigma\to 16\pi$
(4.10)

as $t\to+\infty$. We actually proved in [BFM22a] that this always can be
assumed along the subsequence we will be interested in, otherwise
$\mathfrak{m}_{\mathrm{iso}}$ would be infinite, making 4.9 trivial. We can
apply a version of de L’Hôpital rule, to get

$\liminf_{t\to+\infty}\frac{2}{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{t}$}}\right\rvert$}}{\lvert\partial\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{t}$}}\right\rvert$}}}}\left({\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\Omega_{t}$}}\right\rvert$}}{\lvert\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\Omega_{t}$}}\right\rvert$}}}-\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{t}$}}\right\rvert$}}{\lvert\partial\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{t}$}}\right\rvert$}}}^{\frac{3}{2}}}{6\sqrt{\pi}}\right)\geq\liminf_{t\to+\infty}\frac{2}{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{t}$}}\right\rvert$}}{\lvert\partial\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{t}$}}\right\rvert$}}}}\left(\kern
2.84544pt\int_{\partial\Omega_{t}}\frac{1}{\operatorname{H}}\,\mathrm{d}\sigma-\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{t}$}}\right\rvert$}}{\lvert\partial\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{t}$}}\right\rvert$}}}^{\frac{3}{2}}}{4\sqrt{\pi}}\right).$
(4.11)

Then, the same application of the Hölder inequality 3.4 then gives

$\begin{split}\liminf_{t\to+\infty}\frac{2}{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{t}$}}\right\rvert$}}{\lvert\partial\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{t}$}}\right\rvert$}}}}\left({\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\Omega_{t}$}}\right\rvert$}}{\lvert\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\Omega_{t}$}}\right\rvert$}}}-\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{t}$}}\right\rvert$}}{\lvert\partial\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{t}$}}\right\rvert$}}}^{\frac{3}{2}}}{6\sqrt{\pi}}\right)&\geq\liminf_{t\to+\infty}2\left(\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{t}$}}\right\rvert$}}{\lvert\partial\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{t}$}}\right\rvert$}}}}{\int_{\partial\Omega_{t}}\operatorname{H}^{2}_{t}\,\mathrm{d}\sigma}\right)^{\frac{1}{2}}\left(1-\frac{1}{4\sqrt{\pi}}\left(\kern
2.84544pt\int_{\partial\Omega_{t}}\operatorname{H}^{2}\,\mathrm{d}\sigma\right)^{\frac{1}{2}}\right)\\\
&=\liminf_{t\to+\infty}2\left(\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega_{t}$}}\right\rvert$}}{\lvert\partial\Omega_{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{t}$}}\right\rvert$}}}}{\int_{\partial\Omega_{t}}\operatorname{H}^{2}_{t}\,\mathrm{d}\sigma}\right)^{\frac{1}{2}}\frac{1-\frac{1}{16\pi}\int_{\partial\Omega_{t}}\operatorname{H}^{2}\,\mathrm{d}\sigma}{1+(\frac{1}{16\pi}\int_{\partial\Omega_{t}}\operatorname{H}^{2}\,\mathrm{d}\sigma)^{\frac{1}{2}}}\\\
&=\liminf_{t\to+\infty}\frac{2\mathfrak{m}_{H}(\partial\Omega_{t})}{(\frac{1}{16\pi}\int_{\partial\Omega_{t}}\operatorname{H}^{2}\,\mathrm{d}\sigma)^{\frac{1}{2}}+\frac{1}{16\pi}\int_{\partial\Omega_{t}}\operatorname{H}^{2}\,\mathrm{d}\sigma}\,.\end{split}$
(4.12)

The proof is complete resorting to 4.10 and the monotonicity of
$\mathfrak{m}_{H}$ along the weak IMCF. ∎

The inequality 4.7 states that nonnegatively scalar curved manifolds with
minimal outermost boundary always allow confining more volume in a region of a
prescribed area than the Schwarzchild model would do, in the scaling invariant
limit given by the isoperimetric mass. Once the notion of $\operatorname{ADM}$
mass is explained, this will give a direct link between isoperimetry (for
large volumes) in nonnegative scalar curvature and energy/matter concepts in
general relativity.

##  5\. The $\operatorname{ADM}$ mass and equivalence between masses

In order to better understand the heuristics behind the definition of
$\mathfrak{m}_{\operatorname{ADM}}$ below, we start by recalling why manifolds
fulfilling the assumptions considered above are so natural in the context of
General Relativity. This popular, groundbreaking physical theory is formulated
in the context of Lorentzian Geometry, and specifically in a $4$-manifold
$(L,\mathfrak{g})$ satisfying the _vacuum_ Einstein Field Equations

$\operatorname{Ric}_{\mathfrak{g}}-\frac{1}{2}\mathrm{R}_{\mathfrak{g}}=\mathrm{T},$
(5.1)

where $T$ is named _stress-energy_ tensor and should be thought of as a datum.
A largely accepted assumption on $\mathrm{T}$ is the _dominant energy
condition_ , amounting simply to $\mathrm{T}(V,V)\geq 0$ for any _timelike_
$V$, that is $\mathfrak{g}(V,V)<0$. Consider now a _spacelike_
$3$-hypersurface $(M,g)$, with $g$ induced $\mathfrak{g}$. With spacelike, it
is just meant $g(X,X)>0$ for any $X\in T_{p}M$, for any $p\in M$. We also
consider the simplest case of so-called _time simmetry_ , consisting in the
vanishing of the second fundamental form of the immersion $M\hookrightarrow
L$. By means of the Gauss-Codazzi equations, 5.1 coupled with the dominant
energy condition $\mathrm{T}\geq 0$ induces on $(M,g)$ the identity

$\mathrm{R}_{g}=16\pi\rho\geq 0,$ (5.2)

where $\rho=\mathrm{T}(V,V)$ for some timelike $V$, so that the inequality
follows from the dominant energy condition. We address the reader to [Car21,
Section 1.1] for a detailed derivation of 5.2.

The _event horizon_ of a _black hole_ in $(L,\mathfrak{g})$ manifests in
$(M,g)$ as a minimal outermost boundary. Finally, when modeling an _isolated
gravitational system_ , it becomes natural to assume some kind of Asymptotic
Flatness, that in this note was only at $\mathrm{\mathscr{C}^{0}}$-level, at
least so far. Physically speaking, this condition says that the gravitational
field is not influenced by the presence of some mass “at infinity”. We address
the interested reader to [HE73, Lee19] for a comprehensive treatment of the
relativistic concepts involved.

###  5.1. The $\operatorname{ADM}$ mass

To get an idea of the $\operatorname{ADM}$ mass, we briefly stick with the
more familiar Newtonian framework. Suppose to have a mass density $\rho$ in a
$3$D-time snapshot represented by an isolated gravitational system $(M,g)$.
Newton’s formulation models the gravitational field by a function $V$ called
_gravitational potential_ , which satisfies the relation $\Delta V=4\pi\rho$.
Hence, using the divergence theorem, one can reconstruct the total mass of the
system by observing the effects of the gravitational potential at infinity,
since

$\mathfrak{m}=\int_{M}\rho\,\mathrm{d}\mu=\int_{M}\frac{\Delta
V}{4\pi}\,\mathrm{d}\mu=\lim_{r\to+\infty}\int_{\partial B_{r}}\frac{\partial
V}{\partial r}\,\mathrm{d}\sigma.$ (5.3)

On the other hand, we just recalled that, in Einstein’s formulation, the
gravitational field is modelled by a metric $g$ on $M$ which is only
constrained (in the time-symmetric case) by the equation
$\mathrm{R}_{g}=16\pi\rho$. One would like to compute the system’s total mass
simply by knowing the metric in this case as well. A first idea may be
integrating the quantity $\mathrm{R}_{g}/16\pi$ as we did in the Newtonian
case. However, this approach has at least two main issues. The first one is
that, even in a special paradigmatic case such as Schwarzschild’s, the total
mass would be zero, which does not correspond to what we expect. The second
problem is that the superposition principle does not hold, since the scalar
curvature is not a linear operator of $g$. In the case $(M,g)$ is close to the
Euclidean flat space, one could try to circumvent this by replacing the scalar
curvature with its first-order linearization, that is

$\mathrm{R}_{g}\approx\mathrm{R}_{\delta}+\mathrm{D}\kern
0.28436pt\mathrm{R}|_{\delta}(g-\delta)=\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\mathrm{R}(\delta+\varepsilon(g-\delta))\big{|}_{\varepsilon=0}$
(5.4)

since $\mathrm{R}_{\delta}=0$. Observe that, in the above formula,
$h=\delta+\varepsilon(g-\delta)$ is still a metric for
${\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\varepsilon$}}\right\rvert$}}{\lvert\varepsilon\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\varepsilon$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\varepsilon$}}\right\rvert$}}}$
small enough, and thus it makes sense to compute the scalar curvature of $h$,
which is

$\displaystyle\mathrm{R}_{h}$
$\displaystyle=h^{ij}(\partial_{k}H^{k}_{ij}-\partial_{i}H^{k}_{kj}+H^{k}_{ij}H^{s}_{ks}-H^{k}_{is}H^{s}_{jk})$
(5.5)
$\displaystyle=\frac{\varepsilon}{2}\delta^{ij}\delta^{ks}\partial_{k}(-\partial_{s}g_{ij}+\partial_{j}g_{is}+\partial_{i}g_{js})-\frac{\varepsilon}{2}\delta^{ij}\delta^{ks}\partial_{i}(-\partial_{s}g_{kj}+\partial_{j}g_{ks}+\partial_{k}g_{js})+O_{1}(\varepsilon^{2})$
(5.6)
$\displaystyle=\frac{\varepsilon}{2}\delta^{ij}\delta^{ks}\partial_{k}(-\partial_{s}g_{ij}+2\partial_{j}g_{is})-\frac{\varepsilon}{2}\delta^{ij}\delta^{ks}\partial_{i}\partial_{j}g_{ks}+O_{1}(\varepsilon^{2})$
(5.7)
$\displaystyle=\varepsilon\delta^{ij}\delta^{ks}(\partial_{k}\partial_{i}g_{js}-\partial_{s}\partial_{k}g_{ij})+O_{1}(\varepsilon^{2})$
(5.8)

where $H$ are the Christoffel symbols of $h$. Hence,

$\mathrm{R}_{g}\approx\delta^{ij}\delta^{ks}(\partial_{k}\partial_{i}g_{js}-\partial_{k}\partial_{s}g_{ij}).$
(5.9)

Using the divergence theorem one has

$\displaystyle\frac{1}{16\pi}\int_{M}\mathrm{R}_{g}\,\mathrm{d}\mu_{g}$
$\displaystyle\approx\frac{1}{16\pi}\int_{M}\delta^{ij}\delta^{ks}(\partial_{k}\partial_{i}g_{js}-\partial_{k}\partial_{s}g_{ij})\,\mathrm{d}\mu_{\delta}$
(5.10) $\displaystyle=\lim_{r\to+\infty}\frac{1}{16\pi}\int_{\partial
B_{r}}\delta^{ij}(\partial_{i}g_{jk}-\partial_{k}g_{ij})\frac{x^{k}}{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
x$}}\right\rvert$}}{\lvert
x\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
x$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
x$}}\right\rvert$}}}}\,\mathrm{d}\sigma_{\delta}$ (5.11)
$\displaystyle=\lim_{r\to+\infty}\frac{1}{16\pi}\int_{\partial
B_{r}}g^{ij}(\partial_{i}g_{jk}-\partial_{k}g_{ij})\nu^{k}\,\mathrm{d}\sigma_{g}.$
(5.12)

The quantity appearing on the right-hand side is what will be defined as
Arnowitt-Deser-Miser’s mass $\mathfrak{m}_{\operatorname{ADM}}$ [ADM61]. If
the metric $g$ is sufficiently close to the flat metric, the above computation
shows that the integral of the scalar curvature can be approximated by such
mass. Clearly, the two quantities are not, in general, the same, since the
first one depends on the global behaviour of the metric, while the second one
depends only on its behaviour at large distances. Now one can take the
definition of the $\operatorname{ADM}$ mass and compare it to the integral of
the scalar curvature. Consider then the vector field

$Y^{k}=g^{kl}g^{ij}(\partial_{i}g_{jl}-\partial_{l}g_{ij})=g^{ij}\Gamma^{k}_{ij}-g^{ik}\Gamma^{j}_{ij}$
(5.13)

and observe that its divergence computed with respect to $g$ is

$\displaystyle\partial_{k}Y^{k}+\Gamma^{l}_{lk}Y^{k}$
$\displaystyle=\partial_{k}g^{ij}\Gamma^{k}_{ij}+g^{ij}\partial_{k}\Gamma^{k}_{ij}-\partial_{k}g^{ik}\Gamma^{j}_{ij}-g^{ik}\partial_{k}\Gamma^{j}_{ij}+g^{ij}\Gamma^{l}_{lk}\Gamma^{k}_{ij}-g^{ik}\Gamma^{l}_{lk}\Gamma^{j}_{ij}$
(5.14)
$\displaystyle=\mathrm{R}_{g}+\partial_{k}g^{ij}\Gamma^{k}_{ij}-\partial_{k}g^{ik}\Gamma^{j}_{ij}.$
(5.15)

Assume now that for some $R$ there exists $\mathrm{C}>0$ such that

$\displaystyle\mathrm{C}^{-1}\delta\leq g\leq\mathrm{C}\delta$ in
$M\smallsetminus B_{R}$ $\displaystyle\int_{M\smallsetminus
B_{R}}{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
g$}}\right\rvert$}}{\lvert\partial
g\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
g$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
g$}}\right\rvert$}}}^{2}\,\mathrm{d}\mu_{\delta}<+\infty.$ (5.16)

We want to show that the limit defining the $\operatorname{ADM}$ mass exists.
Appealing again to the divergence theorem we have

$\displaystyle\int_{\partial
B_{s}}{\mathchoice{\left\langle{Y\,}\middle|{\,\nu}\right\rangle}{\langle{Y\,}|{\,\nu}\rangle}{\scriptstyle\langle{Y\,}|{\,\nu}\rangle}{\scriptscriptstyle\langle{Y\,}|{\,\nu}\rangle}}\,\mathrm{d}\sigma_{g}-\int_{\partial
B_{r}}{\mathchoice{\left\langle{Y\,}\middle|{\,\nu}\right\rangle}{\langle{Y\,}|{\,\nu}\rangle}{\scriptstyle\langle{Y\,}|{\,\nu}\rangle}{\scriptscriptstyle\langle{Y\,}|{\,\nu}\rangle}}\,\mathrm{d}\sigma_{g}=\int_{B_{s}\smallsetminus
B_{r}}\operatorname{div}Y\,\mathrm{d}\mu_{g}\geq\mathrm{C}\int_{B_{s}\smallsetminus
B_{r}}{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
g$}}\right\rvert$}}{\lvert\partial
g\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
g$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
g$}}\right\rvert$}}}^{2}\,\mathrm{d}\mu_{\delta}$ (5.17)

for every $R\leq r<s$. Hence taking the inferior limit as $s\to+\infty$ we
have

$\liminf_{s\to+\infty}\int_{\partial
B_{s}}{\mathchoice{\left\langle{Y\,}\middle|{\,\nu}\right\rangle}{\langle{Y\,}|{\,\nu}\rangle}{\scriptstyle\langle{Y\,}|{\,\nu}\rangle}{\scriptscriptstyle\langle{Y\,}|{\,\nu}\rangle}}\,\mathrm{d}\sigma_{g}\geq\int_{M\smallsetminus
B_{r}}{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
g$}}\right\rvert$}}{\lvert\partial
g\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
g$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
g$}}\right\rvert$}}}^{2}\,\mathrm{d}\mu_{\delta}+\int_{\partial
B_{r}}{\mathchoice{\left\langle{Y\,}\middle|{\,\nu}\right\rangle}{\langle{Y\,}|{\,\nu}\rangle}{\scriptstyle\langle{Y\,}|{\,\nu}\rangle}{\scriptscriptstyle\langle{Y\,}|{\,\nu}\rangle}}\,\mathrm{d}\sigma_{g}.$
(5.18)

Taking now the superior limit as $r\to+\infty$, in view of our assumptions we
get

$\liminf_{s\to+\infty}\int_{\partial
B_{s}}{\mathchoice{\left\langle{Y\,}\middle|{\,\nu}\right\rangle}{\langle{Y\,}|{\,\nu}\rangle}{\scriptstyle\langle{Y\,}|{\,\nu}\rangle}{\scriptscriptstyle\langle{Y\,}|{\,\nu}\rangle}}\,\mathrm{d}\sigma_{g}\geq\limsup_{r\to+\infty}\int_{\partial
B_{r}}{\mathchoice{\left\langle{Y\,}\middle|{\,\nu}\right\rangle}{\langle{Y\,}|{\,\nu}\rangle}{\scriptstyle\langle{Y\,}|{\,\nu}\rangle}{\scriptscriptstyle\langle{Y\,}|{\,\nu}\rangle}}\,\mathrm{d}\sigma_{g},$
(5.19)

proving the existence of the limit. Moreover, since

${\mathchoice{\raisebox{2.24667pt}{$\displaystyle\left\lvert\raisebox{-2.24667pt}{\hbox{\set@color$\displaystyle\mathfrak{m}_{\operatorname{ADM}}-\int_{M\smallsetminus
B_{R}}\mathrm{R}_{g}\,\mathrm{d}\mu_{g}$}}\right\rvert$}}{\lvert\mathfrak{m}_{\operatorname{ADM}}-\int_{M\smallsetminus
B_{R}}\mathrm{R}_{g}\,\mathrm{d}\mu_{g}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\mathfrak{m}_{\operatorname{ADM}}-\int_{M\smallsetminus
B_{R}}\mathrm{R}_{g}\,\mathrm{d}\mu_{g}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\mathfrak{m}_{\operatorname{ADM}}-\int_{M\smallsetminus
B_{R}}\mathrm{R}_{g}\,\mathrm{d}\mu_{g}$}}\right\rvert$}}}\leq\int_{\partial
B_{R}}{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
Y$}}\right\rvert$}}{\lvert
Y\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
Y$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
Y$}}\right\rvert$}}}\,\mathrm{d}\sigma_{g}+\mathrm{C}\int_{M\smallsetminus
B_{R}}{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
g$}}\right\rvert$}}{\lvert\partial
g\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
g$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
g$}}\right\rvert$}}}^{2}\,\mathrm{d}\mu_{\delta}$ (5.20)

the $\operatorname{ADM}$ mass is finite if and only the scalar curvature is
integrable on $M$.

In order to complete the presentation of the $\operatorname{ADM}$ mass, we
should ensure its independence on the asymptotic chart at infinity chosen, so
that the rightmost-hand side of 5.10 provides a well-posed definition. This
has been proved slightly strengthening the conditions in 5.16, by Bartnik
[Bar86] and Chruściel [Chr86], that is assuming $g$ to be
$\mathscr{C}^{1}_{\tau}$-Asymptotically Flat, $\tau>1/2$.

######  Definition 5.1.

Let $(M,g)$ be a $\mathscr{C}^{1,\tau}$-Asymptotically Flat Riemannian
manifold, for $\tau>1/2$. Then, the $\mathfrak{m}_{\operatorname{ADM}}$ mass
is defined as

$\mathfrak{m}_{\operatorname{ADM}}=\lim_{r\to+\infty}\frac{1}{16\pi}\int_{\partial
B_{r}}g^{ij}(\partial_{i}g_{jk}-\partial_{k}g_{ij})\nu^{k}\,\mathrm{d}\sigma_{g}.$
(5.21)

It has actually been shown [Chr86] that the requirement $\tau>1/2$ is sharp,
since one can provide $\mathscr{C}^{1}_{1/2}$-Asymptotically Flat metrics such
that the right-hand side in 5.21 gives arbitrary values according to the
selected chart at infinity.

###  5.2. Equivalence between masses

A large amount of literature dealt, so far, with geometric inequalities
involving the $\operatorname{ADM}$ mass. Most notable for us, are Schoen-Yau’s
Positive Mass Theorem [SY79, SY81] ensuring under suitable decay conditions on
the metric that $\mathfrak{m}_{\operatorname{ADM}}\geq 0$ in nonnegatively
scalar curved complete manifolds, and that it vanishes only on flat
$\mathbb{R}^{3}$. Solving a special case of conjecture of Penrose [Pen73],
Huisken-Ilmanen [HI01] sharpened such result in the connected horizon boundary
case, obtaining the (Riemannian) Penrose Inequality

$\sqrt{\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
M$}}\right\rvert$}}{\lvert\partial
M\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
M$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
M$}}\right\rvert$}}}}{16\pi}}\leq\mathfrak{m}_{\operatorname{ADM}}.$ (5.22)

With the description given above of $\mathfrak{m}_{\operatorname{ADM}}$ as an
actual (candidate) physical mass, the validity of the (Riemannian) Penrose
inequality should appear more natural. In fact, the Schwarzschild metric,
where the inequality is checked to hold as an equality, represents a space
where the whole matter is shielded by the horizon $\partial M$. Consequently,
in a suitably restricted class, any other space should only increase the
global mass, in relation with the area of the boundary, as 5.22 actually
states.

Huisken-Ilmanen’s proof of 5.22 exploits their Theorem 2.3 coupled with an
involved asymptotic analysis resulting in
$\lim_{t\to+\infty}\mathfrak{m}_{H}(\partial\Omega_{t})\leq\mathfrak{m}_{\operatorname{ADM}}$
along the weak IMCF of the horizon boundary. Bray [Bra01], with an amazing
completely different proof based on a _ad hoc_ developed conformal flow of
metrics, removed the assumption of connected boundary. In both approaches,
decay assumptions strictly stronger than those required by Definition 5.1 to
give a well-posed definition of $\mathfrak{m}_{\operatorname{ADM}}$ are
imposed, namely $\mathscr{C}^{1}_{1}$-Asymptotically Flat coupled with
$\operatorname{Ric}\geq-\mathrm{C}{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
x$}}\right\rvert$}}{\lvert
x\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
x$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
x$}}\right\rvert$}}}^{-2}$ in [HI01] and $\mathscr{C}^{2}_{1}$-Asymptotically
Flat in [Bra01].

The question about the validity of 5.22 in the optimal
$\mathscr{C}^{1}_{\tau}$-Asymptotically Flat regime arises then naturally, and
has been answered in collaboration with Mazzieri [BFM22a, Theorem 1.1]. Having
then at hand the Isoperimetric Penrose Inequality 4.7 and 5.22, another even
more natural (and more general) question regards the relation between
$\mathfrak{m}_{\operatorname{iso}}$ and $\mathfrak{m}_{\operatorname{ADM}}$.
Sharpening the earlier result envisioned by Huisken [Hui06] and rigorously
obtained (and strengthened) by Jauregui-Lee [JL19], we do obtain also [BFM22a,
Theorem 4.13] the stronger

$\mathfrak{m}_{\operatorname{iso}}=\mathfrak{m}_{\operatorname{ADM}}$ (5.23)

in $\mathscr{C}^{1}_{\tau}$-Asymptotically Flat Riemannian manifolds with
nonnegative scalar curvature and horizon boundary, for $\tau>1/2$. Observe
that such identity provides also a completely different proof of the well-
posedness of $\mathfrak{m}_{\operatorname{ADM}}$ in this asymptotic regime,
first due to Bartnik and Chrùsciel [Bar86, Chr86]. In fact, the Isoperimetric
mass is manifestly independent on the choice of the coordinate chart.

The proofs of both the Penrose Inequality 5.22 and of 5.23 under optimal decay
assumptions in [BFM22a] are based on new insights about the asymptotic
behaviour of the weak IMCF. The sharpening of the decay assumptions in the
Penrose inequality substantially stems from the asymptotic comparison 4.9 and
on its relation with the asymptotic behaviour of harmonic functions that takes
advantage of the potential-theoretic version of the Hawking mass introduced in
[AMO21]. The identification of the two masses 5.23 fundamentally relies also
on a combination with Huisken-Jauregui-Lee argument [Hui06, JL19] yielding,
through a monotonicity formula along a (modified) Mean Curvature Flow, a
direct link among upper bounds on Hawking masses and upper bounds on the
Isoperimetric mass.

In what follows, we outline an alternative strategy devised by Chodosh-
Eichmair-Shi-Yu [Cho+21] that can be useful to prove 5.23, relating with the
isoperimetric sets discussed before.

Before going on, we point out that the inequality

$\mathfrak{m}_{\operatorname{ADM}}\leq\mathfrak{m}_{\operatorname{iso}}$
(5.24)

can be obtained in the optimal asymptotic regime through a direct yet not
trivial computation, that has been carried out in [FST09], and so we will deal
with the reverse one only. A key observation is the following, deduced by the
proof of [Cho+21, Theorem C.1]. When isoperimetric sets are known to exist, as
in the setting of Theorem 3.6, they do realize, in the limit of infinite
volume, the isoperimetric mass Lemma 4.2, and since each one has obviously
constant mean curvature, the asymptotic comparison argument carried out in
[BFM22a] can be reversed in some sense, allowing to estimate the isoperimetric
mass _from above_ with the Hawking mass. This inspiring idea is expressed with
the following computation.

$\displaystyle\begin{split}\mathfrak{m}_{\operatorname{iso}}&=\limsup_{V\to+\infty}\mathfrak{m}_{\operatorname{iso}}(E_{V})=\limsup_{V\to+\infty}\frac{2}{I(V)}\left(V-\frac{I(V)^{\frac{3}{2}}}{6\sqrt{\pi}}\right)\\\
&\leq\limsup_{V\to+\infty}\frac{2}{I^{\prime}(V)}\left(1-\frac{\sqrt{I(V)}I^{\prime}(V)}{4\sqrt{\pi}}\right)\\\
&=\limsup_{V\to+\infty}\frac{2\sqrt{I(V)}\left(1-\frac{I(V)I^{\prime}(V)^{2}}{16\pi}\right)}{\sqrt{I(V)}I^{\prime}(V)\left(1+\frac{\sqrt{I(V)}I^{\prime}(V)}{4\sqrt{\pi}}\right)}\\\
&=\limsup_{V\to+\infty}\frac{32\pi\,\mathfrak{m}_{H}(\partial
E_{V})}{4\sqrt{\pi}I^{\prime}(V)\sqrt{I(V)}+I^{\prime}(V)^{2}I(V)},\end{split}$
(5.25)

where
$I(V)={\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial
E_{V}$}}\right\rvert$}}{\lvert\partial
E_{V}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E_{V}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E_{V}$}}\right\rvert$}}}$. If the Hawking mass of the isoperimetric sets of
large volumes does satisfy the sharp bound $\mathfrak{m}_{H}(\partial
E_{V})\leq\mathfrak{m}_{\operatorname{ADM}}$, 5.25 in fact allows to conclude
the desired bound
$\mathfrak{m}_{\operatorname{iso}}\leq\mathfrak{m}_{\operatorname{ADM}}$. By
our work with Mazzieri [BFM22a], in the optimal decay case this is fulfilled
if the isoperimetric sets of large volume have connected boundaries.

######  Remark 5.2.

Starting from the next proposition and for all results up to the end of
Section 5, we will drop the outermost condition on the boundary. This is due
to known properties of Asymptotically Flat $3$-manifolds of nonnegative scalar
curvature, see the recapitulatory [BFM22a, Lemma 2.8], asserting in particular
that, if the manifold contains minimal surfaces, then one can find a minimal,
outermost $\Sigma$ enclosing $\partial M$ and the analysis is applied on the
new manifold with minimal outermost boundary $\Sigma$. The masses are
obviously the same as those of the original manifold. If $(M,g)$ possesses no
closed minimal surfaces, and is in particular boundaryless, then one can
safely work in such complete manifold.

######  Proposition 5.3.

Let $(M,g)$ be a complete $\mathscr{C}^{1}_{\tau}$-Asymptotically Flat
Riemannian $3$-manifold, $\tau>1/2$, with nonnegative scalar curvature and
(possibly empty) closed, minimal boundary. Assume that there exists $V_{0}>0$
such that, for any $V\geq V_{0}$, there exists an isoperimetric set $E_{V}$ of
volume $V$ with connected boundary. Then,

$\mathfrak{m}_{\operatorname{iso}}=\mathfrak{m}_{\operatorname{ADM}}.$ (5.26)

  * Proof.

As already pointed out, we just have to justify
$\mathfrak{m}_{\operatorname{iso}}\leq\mathfrak{m}_{\operatorname{ADM}}$.
Assume $\mathfrak{m}_{\operatorname{ADM}}<+\infty$ otherwise the proposition
trivially follows. Assume also that $\mathfrak{m}_{\operatorname{iso}}>0$,
otherwise [BFM22a, Theorem 1.3] implies that $(M,g)$ is isometric to
$\mathbb{R}^{3}$ and the result trivially holds. Let
$(V_{k})_{k\in\mathbb{N}}$ a sequence realising the superior limit in 5.25.
Since $\mathfrak{m}_{\operatorname{iso}}>0$, $\mathfrak{m}_{H}(\partial
E_{V_{k}})\geq 0$ for large $k$. In particular,
$I^{\prime}(V_{k})^{2}I(V_{k})\leq 16\pi$. Assume by contradiction that there
exists a not relabeled subsequence such that
$I^{\prime}(V_{k})^{2}I(V_{k})\leq 16\pi-\varepsilon$ for some positive
$\varepsilon>0$. Then, we would have

$\frac{\varepsilon}{32\pi}\sqrt{I(V_{k})}\leq\mathfrak{m}_{H}(\partial
E_{V_{k}})\leq\mathfrak{m}_{\operatorname{ADM}},$ (5.27)

where the last inequality follows as in [BFM22a, Theorem 1.1], giving the
contradiction. In particular, $I^{\prime}(V_{k})^{2}I(V_{k})\to 16\pi$ as
$k\to+\infty$. Plugging this piece of information into 5.25 we get

$\mathfrak{m}_{\operatorname{iso}}\leq\limsup_{k\to+\infty}\mathfrak{m}_{H}(\partial
E_{V_{k}})\leq\mathfrak{m}_{\operatorname{ADM}},$ (5.28)

proving 5.26. We stress the fact that the connectedness of $\partial
E_{V_{k}}$ is required in order to infer the rightmost bound in 5.27 and in
5.28. ∎

The authors [Cho+21] can in fact count on connectedness, as they are working
in $\mathscr{C}^{2}_{1}$-Asymptotically Flat manifolds, where isoperimetric
sets of large volume are close to coordinate balls, thanks to the work of
[Ner15] (actually valid in $\mathscr{C}^{2}_{\tau}$-Asymptotically Flat, for
$\tau>1/2$), that importantly weakens the assumptions of the earlier works of
Eichmair-Metzger [EM13, EM13a], where the manifolds were assumed to be
asymptotic to Schwarzschild. Moreover, in [Cho+21] the authors do also rely on
Huisken-Ilmanen’s bound on the Hawking mass, and thus their analysis could be
pushed to $\mathscr{C}^{2}_{\tau}$-Asymptotically Flat manifolds for
$\tau>1/2$ coupled with
$\operatorname{Ric}\geq-\mathrm{C}{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle
x$}}\right\rvert$}}{\lvert
x\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
x$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
x$}}\right\rvert$}}}^{-2}$. Through the $\operatorname{ADM}$-Penrose
inequality in optimal decay assumptions [BFM22a, Theorem 1.1] the assumption
on the Ricci curvature can be dropped. Actually, in [Cho+21], the authors also
take advantage of an _a priori_ knowledge of $I(V)$ and $I^{\prime}(V)$ as
$T\to+\infty$. The proof of Proposition 5.3 actually shows that this is not
needed.

###  5.3. Nonlinear masses

In this last section, we briefly introduce and discuss the nonlinear potential
theoretic counterpart of the isoperimetric mass. To this end, we first
introduce the following $p$-capacity of a compact set $K\in M$,

$\mathfrak{c}_{p}(K)=\inf\left\\{\frac{1}{4\pi}\left(\frac{p-1}{3-p}\right)^{p-1}\int_{M\smallsetminus
K}{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\mathrm{D}\kern
0.28436ptv$}}\right\rvert$}}{\lvert\mathrm{D}\kern
0.28436ptv\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\mathrm{D}\kern
0.28436ptv$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\mathrm{D}\kern
0.28436ptv$}}\right\rvert$}}}^{p}\,\mathrm{d}\mu\,\middle|\,v\in\mathscr{C}_{c}^{\infty}(M),\,v\geq
1\text{ on }K\right\\}.$ (5.29)

As a consequence of the remarkable [MRS22, Theorem 3.6],
$\mathscr{C}^{0}$-Asymptotically Flat manifolds are $p$-nonparabolic, in
particular implying that the above quantity is positive for any
$K=\partial\Omega\in\mathscr{C}^{1,\alpha}$, if $1<p<3$. Inspired by the
Isoperimetric mass, one can define the $p$-Isocapacitary mass as follows.

######  Definition 5.4.

Let $(M,g)$ be a Riemannian $3$-manifold possibly with boundary, with infinite
volume. Then, its $p$-_Isocapacitary mass_ is defined as

$\mathfrak{m}^{{\text{\tiny$(p)$}}}_{\operatorname{iso}}=\sup_{(\Omega_{j})_{j\in\mathbb{N}}}\limsup_{j\to+\infty}\frac{1}{2p\pi\mathfrak{c}_{p}(\partial\Omega_{j})^{\frac{2}{3-p}}}\left({\mathchoice{\raisebox{2.28223pt}{$\displaystyle\left\lvert\raisebox{-2.28223pt}{\hbox{\set@color$\displaystyle\Omega_{j}$}}\right\rvert$}}{\lvert\Omega_{j}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\Omega_{j}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\Omega_{j}$}}\right\rvert$}}}-\frac{4\pi}{3}\mathfrak{c}_{p}(\partial\Omega_{j})^{\frac{3}{3-p}}\right).$
(5.30)

where the supremum is taken among all exhaustions
$(\Omega_{j})_{j\in\mathbb{N}}$ consisting of domains with
$\mathscr{C}^{1,\alpha}$-boundary.

The above capacitary notion of mass has recently been considered for $p=2$ by
Jauregui [Jau20], and together with Mazzieri [BFM23] for $1<p<3$. After the
discussion that took place in the last sections, one could naturally wonder
about the relation of
$\mathfrak{m}^{{\text{\tiny$(p)$}}}_{\operatorname{iso}}$ with the
Isoperimetric mass and the $\mathfrak{m}_{\operatorname{ADM}}$. A first answer
in this direction is the following, holding the generality of
$\mathscr{C}^{0}$-Asymptotically Flat $3$-manifolds.

######  Proposition 5.5.

Let $(M,g)$ be a $\mathscr{C}^{0}$-Asymptotically Flat Riemannian
$3$-manifold. Then,

$\mathfrak{m}^{{\text{\tiny$(p)$}}}_{\operatorname{iso}}\leq\mathfrak{m}_{\operatorname{iso}}.$
(5.31)

This is [BFM23, Proposition 5.6]. Its proof is classical in nature and can be
compared to the classical derivation of Isocapacitary inequalities from the
Isoperimetric (see [Maz11] or, more directly, the arguments in [Jau12] and in
the proof of [BFM22, Theorem 4.1]). More precisely, it builds on the Pólya-
Szegö inequality to be applied to the sharp asymptotic isoperimetric
inequality

$(6\sqrt{\pi}{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\Omega$}}\right\rvert$}}{\lvert\Omega\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\Omega$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\Omega$}}\right\rvert$}}})^{\frac{2p}{3}}\leq{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega$}}\right\rvert$}}{\lvert\partial\Omega\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega$}}\right\rvert$}}}^{p}+2p\sqrt{\pi}(\mathfrak{m}_{\operatorname{iso}}+o(1)){\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega$}}\right\rvert$}}{\lvert\partial\Omega\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega$}}\right\rvert$}}}^{\frac{2p-1}{2}},$
(5.32)

that is actually a direct consequence of the definition of the Isoperimetric
mass.

The reverse inequality is obtained in [BFM23] in the setting of
$\mathscr{C}^{1}_{\tau}$-Asymptotically Flat Riemannian $3$-manifolds,
$\tau>1/2$, with nonnegative scalar curvature. As these assumptions should
suggest, we strongly rely on the well-posedness of the $\operatorname{ADM}$
mass, appearing in an asymptotic computation along geodesic spheres as that in
[FST09]. The overall argument consists in a strengthening of the one proposed
for [Jau20, Theorem 5] and in the application of a suitable, new
generalization of a capacitary estimate of Bray-Miao [BM08]. This last
estimate reads

$\mathfrak{c}_{p}(\partial\Omega)\leq\left(\frac{{\mathchoice{\raisebox{2.5pt}{$\displaystyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\displaystyle\partial\Omega$}}\right\rvert$}}{\lvert\partial\Omega\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega$}}\right\rvert$}}}}{4\pi}\right)^{\mathrlap{\frac{3-p}{2}}}\kern
5.69046pt\leftidx{{}_{2}}{F}{{}_{1}}\left(\frac{1}{2},\frac{3-p}{p-1},\frac{2}{p-1};1-\frac{1}{16\pi}\int_{\partial\Omega}\operatorname{H}^{2}\,\mathrm{d}\sigma\right)^{\mathrlap{-(p-1)}}\kern
19.91684pt,$ (5.33)

where $\kern 5.69046pt\leftidx{{}_{2}}{F}{{}_{1}}$ denotes the hypergeometric
function, and it has been provided for connected $\partial\Omega$ homologous
to $\partial M$ in [BFM23, Proposition 2.14]. The only feature of the implicit
function $\kern 5.69046pt\leftidx{{}_{2}}{F}{{}_{1}}$ we will need will be
recalled below. Resuming, and crucially exploiting 5.23, we conclude in
[BFM23, Theorem 1.3] that in $\mathscr{C}^{1}_{\tau}$-Asymptotically Flat
$3$-manifolds with nonnegative scalar curvature and closed minimal boundary

$\mathfrak{m}^{{\text{\tiny$(p)$}}}_{\operatorname{iso}}=\mathfrak{m}_{\operatorname{iso}}=\mathfrak{m}_{\operatorname{ADM}}.$
(5.34)

for any $1<p\leq 2$. This was already known for $p=2$ only under the stronger
assumption of harmonic flatness at infinity [Jau20, Corollary 8], a locution
standing for $g=u^{4}\delta$ outside a suitable compact set, where $u$ is
harmonic with respect to the Euclidean Laplacian.

Here, we observe that, the same assumption of connectedness of isoperimetric
sets considered in Proposition 5.3 allows to get the identification
$\mathfrak{m}^{{\text{\tiny$(p)$}}}_{\operatorname{iso}}=\mathfrak{m}_{\operatorname{iso}}$
for $1<p\leq 2$, in $\mathscr{C}^{0}$-Asymptotically Flat manifolds with
nonnegative scalar curvature and minimal boundary. This is particularly
interesting due to the fact that, in this asymptotic regime, the notion of
$\mathfrak{m}_{\operatorname{ADM}}$ is not a priori available, and so one
cannot a priori go through computations involving its expression.

######  Proposition 5.6.

Let $(M,g)$ be a complete $\mathscr{C}^{0}$-Asymptotically Flat Riemannian
$3$-manifold with nonnegative scalar curvature and (possibly empty) closed,
minimal boundary. Assume that there exists $V_{0}>0$ such that, for any $V\geq
V_{0}$, there exists an isoperimetric set $E_{V}$ of volume $V$ with connected
boundary. Then,

$\mathfrak{m}_{\operatorname{iso}}=\mathfrak{m}^{{\text{\tiny$(p)$}}}_{\operatorname{iso}}$
(5.35)

for every $1<p\leq 2$.

  * Proof.

The inequality
$\mathfrak{m}_{\operatorname{iso}}\geq\mathfrak{m}^{{\text{\tiny$(p)$}}}_{\operatorname{iso}}$
has already been pointed out to hold more in general, so we focus on the
reverse
$\mathfrak{m}_{\operatorname{iso}}\leq\mathfrak{m}^{{\text{\tiny$(p)$}}}_{\operatorname{iso}}$.

For every $V>0$ large enough, denote $E_{V}$ the isoperimetric set of volume
$V$. Assume that
$\mathfrak{m}^{{\text{\tiny$(p)$}}}_{\operatorname{iso}}<+\infty$. Assume also
that $\mathfrak{m}_{\operatorname{iso}}>0$, otherwise [BFM22a, Thoerem 1.3]
implies that $(M,g)$ is isometric to $\mathbb{R}^{3}$ and the theorem
trivially holds. Along a sequence $(V_{k})_{k\in\mathbb{N}}$ realising the
superior limit in 5.25, $\mathfrak{m}_{H}(\partial E_{V_{k}})\geq 0$. Evolve
$E_{V_{k}}$ using the weak IMCF and denote $E_{V_{k}}^{t}$ its sublevels. We
briefly point out that, since the isoperimetric sets are not known to be
homologous to the boundary, the evolving hypersurfaces may in principle touch
the boundary. If this happens, one should consider the weak IMCF _with jumps_
, that is the modification described in [HI01, Section 6]. By 4.9 and the
Geroch monotonicity formula Theorem 2.3, we have

$\mathfrak{m}_{H}(\partial
E_{V_{k}})\leq\lim_{t\to+\infty}\mathfrak{m}_{H}(\partial
E_{V_{k}}^{t})\leq\mathfrak{m}_{\operatorname{iso}}.$ (5.36)

Observe now that, $\mathfrak{m}_{H}(\partial E_{V_{k}}^{t})\geq 0$ and for
large $t$ the $E_{V_{k}}^{t}$ has a connected boundary which is homologous to
$\partial M$. Applying 5.33, we have

$\displaystyle\mathfrak{c}_{p}(\partial E^{t}_{V_{k}})$
$\displaystyle\leq\left(\frac{{\mathchoice{\raisebox{2.76555pt}{$\displaystyle\left\lvert\raisebox{-2.76555pt}{\hbox{\set@color$\displaystyle\partial
E^{t}_{V_{k}}$}}\right\rvert$}}{\lvert\partial
E^{t}_{V_{k}}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E^{t}_{V_{k}}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E^{t}_{V_{k}}$}}\right\rvert$}}}}{4\pi}\right)^{\mathrlap{\frac{3-p}{2}}}\kern
5.69046pt\leftidx{{}_{2}}{F}{{}_{1}}\left(\frac{1}{2},\frac{3-p}{p-1},\frac{2}{p-1};1-\frac{1}{16\pi}\int_{\partial
E^{t}_{V_{k}}}\operatorname{H}^{2}\,\mathrm{d}\sigma\right)^{\mathrlap{-(p-1)}}\kern
19.91684pt\leq\left(\frac{{\mathchoice{\raisebox{2.76555pt}{$\displaystyle\left\lvert\raisebox{-2.76555pt}{\hbox{\set@color$\displaystyle\partial
E^{t}_{V_{k}}$}}\right\rvert$}}{\lvert\partial
E^{t}_{V_{k}}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E^{t}_{V_{k}}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E^{t}_{V_{k}}$}}\right\rvert$}}}}{4\pi}\right)^{\mathrlap{\frac{3-p}{2}}},$
(5.37)

since
$\leftidx{{}_{2}}{F}{{}_{1}}\left(\frac{1}{2},\frac{3-p}{p-1},\frac{2}{p-1};t\right)\geq
1$ if $0\leq t\leq 1$. In particular, this implies

$\liminf_{t\to+\infty}\mathfrak{m}_{\operatorname{iso}}(E_{V_{k}}^{t})\leq\liminf_{t\to+\infty}p\frac{1}{2p\pi\mathfrak{c}_{p}(\partial
E_{V_{k}}^{t})^{\frac{2}{3-p}}}\left({\mathchoice{\raisebox{2.76555pt}{$\displaystyle\left\lvert\raisebox{-2.76555pt}{\hbox{\set@color$\displaystyle
E_{V_{k}}^{t}$}}\right\rvert$}}{\lvert
E_{V_{k}}^{t}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
E_{V_{k}}^{t}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
E_{V_{k}}^{t}$}}\right\rvert$}}}-\frac{4\pi}{3}\mathfrak{c}_{p}(\partial
E_{V_{k}}^{t})^{\frac{3}{3-p}}\right)\leq
p\mathfrak{m}^{{\text{\tiny$(p)$}}}_{\operatorname{iso}}.$ (5.38)

Combining 5.36 and 5.38 we get that $\mathfrak{m}_{H}(\partial
E_{V_{k}})\leq\mathfrak{m}_{\operatorname{iso}}^{{\text{\tiny$(p)$}}}<+\infty$.

Suppose now that, up to a subsequence, $I^{\prime}(V_{k})^{2}I(V_{k})\leq
16\pi-\varepsilon$ for some positive $\varepsilon>0$ and every
$k\in\mathbb{N}$. Then we would have

$\mathfrak{m}_{H}(\partial
E_{V_{k}})\geq\frac{\varepsilon}{32\pi}\sqrt{I(V_{k})},$ (5.39)

that diverges, since the isoperimetric constant is positive, contradicting
$\mathfrak{m}^{{\text{\tiny$(p)$}}}_{\operatorname{iso}}<+\infty$. Then

$\mathfrak{m}_{\operatorname{iso}}\leq\limsup_{k\to+\infty}\frac{32\pi\,\mathfrak{m}_{H}(\partial
E_{V_{k}})}{4\sqrt{\pi}I^{\prime}(V_{k})\sqrt{I(V_{k})}+I^{\prime}(V_{k})^{2}I(V_{k})}=\limsup_{k\to+\infty}\mathfrak{m}_{H}(\partial
E_{V_{k}})\leq\mathfrak{m}_{\operatorname{iso}}.$ (5.40)

For every $j\geq 0$ there exists $E_{V_{k_{j}}}$ such that

$\mathfrak{m}_{\operatorname{iso}}-\frac{1}{j}\leq\lim_{t\to+\infty}\mathfrak{m}_{H}(\partial
E_{V_{k_{j}}}^{t})\leq\liminf_{t\to+\infty}\frac{2}{{\mathchoice{\raisebox{2.76555pt}{$\displaystyle\left\lvert\raisebox{-2.76555pt}{\hbox{\set@color$\displaystyle
E^{t}_{V_{k}}$}}\right\rvert$}}{\lvert
E^{t}_{V_{k}}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
E^{t}_{V_{k}}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
E^{t}_{V_{k}}$}}\right\rvert$}}}}\left({\mathchoice{\raisebox{2.76555pt}{$\displaystyle\left\lvert\raisebox{-2.76555pt}{\hbox{\set@color$\displaystyle
E^{t}_{V_{k}}$}}\right\rvert$}}{\lvert
E^{t}_{V_{k}}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle
E^{t}_{V_{k}}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle
E^{t}_{V_{k}}$}}\right\rvert$}}}-\frac{{\mathchoice{\raisebox{2.76555pt}{$\displaystyle\left\lvert\raisebox{-2.76555pt}{\hbox{\set@color$\displaystyle\partial
E^{t}_{V_{k}}$}}\right\rvert$}}{\lvert\partial
E^{t}_{V_{k}}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial
E^{t}_{V_{k}}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial
E^{t}_{V_{k}}$}}\right\rvert$}}}^{\frac{3}{2}}}{6\sqrt{\pi}}\right)\leq\mathfrak{m}_{\operatorname{iso}}.$
(5.41)

Therefore, there exists an exhaustion $\Omega_{j}=E_{V_{k_{j}}}^{t_{j}}$ of
$M$ that are bounded subsets with connected boundary homologous to $\partial
M$ and such that

$\displaystyle\mathfrak{m}_{\operatorname{iso}}-\frac{1}{j}\leq\mathfrak{m}_{H}(\partial\Omega_{j})\leq\mathfrak{m}_{\operatorname{iso}}$
$\displaystyle\mathfrak{m}_{\operatorname{iso}}-\frac{1}{j}\leq\mathfrak{m}_{\operatorname{iso}}(\Omega_{j})\leq\mathfrak{m}_{\operatorname{iso}}.$
(5.42)

In particular, we have

$1-\frac{1}{16\pi}\int_{\partial\Omega_{j}}\operatorname{H}^{2}\,\mathrm{d}\sigma=\frac{4\mathfrak{m}_{\operatorname{iso}}\sqrt{\pi}}{\sqrt{{\mathchoice{\raisebox{2.28223pt}{$\displaystyle\left\lvert\raisebox{-2.28223pt}{\hbox{\set@color$\displaystyle\partial\Omega_{j}$}}\right\rvert$}}{\lvert\partial\Omega_{j}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{j}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{j}$}}\right\rvert$}}}}}(1+o(1)).$
(5.43)

Appealing again to 5.33 and using Taylor’s expansion of
$\leftidx{{}_{2}}{F}{{}_{1}}$ around $0$, we get

$\displaystyle\mathfrak{c}_{p}(\partial\Omega_{j})$
$\displaystyle\leq\left(\frac{{\mathchoice{\raisebox{2.28223pt}{$\displaystyle\left\lvert\raisebox{-2.28223pt}{\hbox{\set@color$\displaystyle\partial\Omega_{j}$}}\right\rvert$}}{\lvert\partial\Omega_{j}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{j}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{j}$}}\right\rvert$}}}}{4\pi}\right)^{\frac{3-p}{2}}\leftidx{{}_{2}}{F}{{}_{1}}\left(\frac{1}{2},\frac{3-p}{p-1},\frac{2}{p-1};\frac{4\mathfrak{m}_{\operatorname{iso}}\sqrt{\pi}}{\sqrt{{\mathchoice{\raisebox{2.28223pt}{$\displaystyle\left\lvert\raisebox{-2.28223pt}{\hbox{\set@color$\displaystyle\partial\Omega_{j}$}}\right\rvert$}}{\lvert\partial\Omega_{j}\rvert}{\raisebox{2.5pt}{$\scriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptstyle\partial\Omega_{j}$}}\right\rvert$}}{\raisebox{2.5pt}{$\scriptscriptstyle\left\lvert\raisebox{-2.5pt}{\hbox{\set@color$\scriptscriptstyle\partial\Omega_{j}$}}\right\rvert$}}}}}(1+o(1))\right)^{-(p-1)}$
(5.44)