# Rigidity and compactness
with constant mean curvature
in warped product manifolds

Francesco Maggi  Department of Mathematics, The University of Texas at Austin,
2515 Speedway, Stop C1200, Austin TX 78712-1202, United States of America
<EMAIL_ADDRESS>and  Mario Santilli  Dipartimento di Ingegneria e
Scienze dell’Informazione e Matematica, Universitá degli Studi dell’Aquila,
67100 L’Aquila, Italy<EMAIL_ADDRESS>
###### Abstract.

We prove the rigidity of rectifiable boundaries with constant distributional
mean curvature in the Brendle class of warped product manifolds (which
includes important models in General Relativity, like the
deSitter–Schwarzschild and Reissner–Nordstrom manifolds). As a corollary we
characterize limits of rectifiable boundaries whose mean curvatures converge,
as distributions, to a constant. The latter result is new, and requires the
full strength of distributional CMC-rigidity, even when one considers smooth
boundaries whose mean curvature oscillations vanish in arbitrarily strong
$C^{k}$-norms. Our method also establishes that rectifiable boundaries of sets
of finite perimeter in the hyperbolic space with constant distributional mean
curvature are finite unions of possibly mutually tangent geodesic spheres.

###### Contents

  1. 1 Introduction
  2. 2 Sets of positive reach in Riemannian manifolds
  3. 3 A Lusin-type property of White’s $(m,\lambda)$-sets
  4. 4 Viscous Heintze–Karcher inequalities
  5. 5 Rigidity and compactness theorem
  6. 6 Proof of Theorem 1.3
  7. A Assumption ${\rm(H3)^{*}}$ and models in General Relativity

##  1\. Introduction

###  1.1. Overview

We move from two recent extensions of the classical Alexandrov theorem
[Ale62]: in the Euclidean space, spheres are the only connected, constant mean
curvature (CMC) boundaries enclosing finite volumes:

(i) In [Bre13], Brendle has proved CMC-rigidity in a class of warped product
Riemannian manifolds which includes important models in General Relativity,
like the deSitter–Schwarzschild and the Reissner–Nordstrom manifolds. In
dimension $3\leq n\leq 7$, when isoperimetric sets are smooth by local
regularity theorems, Brendle’s theorem allows one to solve the “horizon
homologous” isoperimetric problem in this class of warped product manifolds.
In turn, since the works of Eichmair and Metzger [EM12, EM13a, EM13b], the
study of horizon-homologous isoperimetric problems in the large volume regime
has played a prominent role in the analysis of the Huisken–Yau problem
[Hui96]– see, e.g. [Cho16, CCE16, CEV17, CESZ19, CESY21, CE22].

(ii) In [DM19], Delgadino and the first-named author have extended the
Alexandrov theorem to the class of (Borel) sets with finite volume, finite
distributional perimeter, and constant distributional mean curvature. CMC-
rigidity in the distributional setting in turn implies, by basic varifolds
theory, a general compactness theorem for almost CMC-boundaries: in the
Euclidean space, finite unions of disjoint spheres with equal radii are the
only possible limits of sequences of boundaries converging in volume and in
perimeter, and whose mean curvatures converge, as distributions, to a
constant. With stronger controls on the mean curvature oscillation one can
even provide quantitative rates of convergence towards finite unions of balls,
as done, for example, in [CM17, DMMN18, JN20, JMS21]. In addition to their
geometric interest, these results find applications to the resolution of
geometric variational problems (see, e.g., [CM17, DW22] for the
characterization of local minimizers of “free droplet energies”) and to the
long time behavior of the volume-preserving mean curvature flow (see, e.g.,
[JN20, MS22, JMS21]).

The main results of this paper are the distributional version of the
Alexandrov theorem in the Brendle class (Theorem 1.1) and a consequent
compactness theorem for almost-CMC boundaries (Theorem 1.2).

An interesting aspect of the method of proof of Theorem 1.1 is that we avoid
the use of “smoothness intensive” tools, like the Schätzle strong maximum
principle for integer varifolds [Sch04], which was central in [DM19]; or the
Greene–Wu “approximation by convolution” theorem [GW73, GW79], which was
crucially used in [Bre13]. We rather put emphasis on the metric notion of set
of positive reach, and work with one-sided viscous mean-curvature/dimensions
bounds (as formulated by White in [Whi16]; see also [CC93]); and, as a by-
product of these efforts, throughout our analysis, we only need to use the
weak maximum principle (meant as “one-sided inclusion and contact imply mean
curvature ordering”). We take the success of this synthetic approach as a
significant indication that the Alexandrov theorem should hold in some metric
version of the Brendle class.

###  1.2. The Brendle class and the main theorem

Given $n\geq 3$, we denote by $\mathcal{B}_{n}$ the class of the
$n$-dimensional manifolds $(M,g)$ of the form

$M=N\times[0,\bar{r})\,,\qquad g=dr\otimes dr+h(r)^{2}\,g_{N}\,,$ (1.1)

for some $\bar{r}\in(0,\infty]$, compact $(n-1)$-dimensional Riemannian
manifold $(N,g_{N})$, and smooth positive function
$h:[0,\bar{r})\to\mathbb{R}$, such that:

  1. (H0)

for some $\rho>0$, ${\rm Ric}_{N}\geq\rho(n-2)\,g_{N}$ on $N$ ;

  2. (H1)

$h^{\prime}(0)=0$ and $h^{\prime\prime}(0)>0$;

  3. (H2)

$h^{\prime}>0$ on $(0,\bar{r})$;

  4. (H3)

$2\,(h^{\prime\prime}/h)+(n-2)\,[((h^{\prime})^{2}-\rho)/h^{2}]$ is increasing
on $(0,\bar{r})$ ;

  5. (H4)

$(h^{\prime\prime}/h)+[(\rho-(h^{\prime})^{2})/h^{2}]$ is positive on
$(0,\bar{r})$ .

To obtain geometric interpretations of these conditions we denote by

$M^{\circ}=N\times(0,\bar{r})\,,\qquad N_{0}=N\times\\{0\\}\,,\qquad
N_{t}=N\times\\{t\\}\quad(t>0)\,,$

the interior, the horizon, and the slices of $M$, and notice that

$\displaystyle{\rm Ric}_{M}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!{\rm
Ric}_{N}-\big{\\{}h\,h^{\prime\prime}+(n-2)\,(h^{\prime})^{2}\big{\\}}\,g_{N}-(n-1)\,\big{(}h^{\prime\prime}\big{/}h\big{)}\,dr\otimes
dr\,,\,\,\,\,$ (1.2) $\displaystyle{\rm R}_{M}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\big{(}{\rm
R}_{N}\big{/}h^{2}\big{)}-(n-1)\,\Big{(}2\,\big{(}h^{\prime\prime}\big{/}h\big{)}+(n-2)\,\big{(}h^{\prime}\big{/}h\big{)}^{2}\Big{)}\,.$
(1.3)

The scalar mean curvature of $N_{t}$ with respect to its $g$-unit normal
vector field $\partial/\partial r$ is
$H_{N_{t}}=\langle\vec{H}_{N_{t}},\partial/\partial
r\rangle_{g}=(n-1)\,h^{\prime}(t)\big{/}h(t)$, so that the horizon of $M$ is a
minimal surface by ${\rm(H1)}$, and the slices of $M$ have positive CMC
(w.r.t. to $\partial/\partial r$) thanks to ${\rm(H2)}$. We next notice that
${\rm(H3)}$ implies (in combination with (1.3), ${\rm(H0)}$, and ${\rm(H2)}$)
that ${\rm R}_{M}$ is decreasing along $\partial/\partial r$. Finally, while
(1.2) implies that $\partial/\partial r$ is an eigenvector of ${\rm Ric}_{M}$,
${\rm(H4)}$ (combined with ${\rm(H0)}$) adds the information that
$\partial/\partial r$ is a simple eigenvector. We then have:

Brendle’s theorem: [Bre13, Theorem 1.1] If $n\geq 3$,
$(M,g)\in\mathcal{B}_{n}$, and $\Sigma$ is a smooth closed, embedded,
orientable, CMC hypersurface in $M$, then $\Sigma$ is a slice of $M$.

######  Remark 1.1 (Dropping ${\rm(H4)}$).

A simple remark (which seems to have gone uncommented so far) is that
Brendle’s theorem also holds in the class $\mathcal{B}_{n}^{*}$ of those
$(M,g)$ satisfying (1.1), ${\rm(H0)}$, ${\rm(H1)}$, ${\rm(H2)}$, and

  1. (H3)∗

$2\,(h^{\prime\prime}/h)+(n-2)\,[((h^{\prime})^{2}-\rho)/h^{2}]$ is strictly
increasing on $(0,\bar{r})$ .

In other words, condition ${\rm(H4)}$ is not needed to conclude rigidity as
soon as ${\rm R}_{M}$ is strictly decreasing along $\partial/\partial r$. (For
more details on this point, see the discussions in Section 1.4 and Remark 4.1
below.) In terms of applications to General Relativity, it is interesting to
notice that while the Reissner–Nordstrom manifolds belong to the class
$\mathcal{B}_{n}^{*}$, the deSitter–Schwarzschild manifolds do not; in
particular, a stronger stability mechanism for almost-CMC hypersurfaces is at
work in the former class than in the latter; see Appendix A for more
information.

######  Remark 1.2 (Formulation with boundaries).

Brendle’s theorem is, actually, a statement about boundaries in $M$. Indeed,
as noticed also in [Bre13, Section 3], under the assumptions of Brendle’s
theorem on $\Sigma$,

$\begin{split}&\mbox{there is $(a,b)\subset\subset(0,\bar{r})$ and
$\Omega\subset M$ open such that}\\\ &\mbox{$\Sigma\subset N\times(a,b)$ and
either $\partial\Omega=\Sigma$ or $\partial\Omega=\Sigma\cup
N_{0}$}\,.\end{split}$ ($\Sigma\leftrightarrow\Omega$)

Now, as explained in more detail later on, there are two basic geometric
problems – the characterization of horizon-homologous isoperimetric regions
and the study of sequences of (smooth) boundaries with vanishing mean
curvature oscillation – that call for the extension of Brendle’s theorem to
the class of sets of finite perimeter. This extension is the content of our
main theorem, Theorem 1.1 below. Referring to [Mag12] for a complete
discussion of the subject, we just recall here that a Borel set $\Omega$ in
$(M,g)$ is a set of finite perimeter if ${\rm
Per}(\Omega):=\sup\\{\small\int_{\Omega}{\rm
div}\,X\,d\mathcal{H}^{n}:X\in\mathcal{X}(M)\,,|X|_{g}\leq 1\\}$ is finite
(where $\mathcal{X}(M)=\\{\mbox{smooth vector fields on $M$}\\}$). Then one
can define the reduced boundary $\partial^{*}\Omega$ (a locally
$\mathcal{H}^{n-1}$-rectifiable set with ${\rm
Per}(\Omega)=\mathcal{H}^{n-1}(\partial^{*}\Omega)$) and the measure-theoretic
outer normal $\nu_{\Omega}$ (a Borel $g$-unit vector field defined on
$\partial^{*}\Omega$) so that the distributional divergence theorem
$\small{\int}_{\Omega}\,{\rm
div}\,\,X\,d\mathcal{H}^{n}=\small{\int}_{\partial^{*}\Omega}\langle
X,\nu_{\Omega}\rangle_{g}\,d\mathcal{H}^{n-1}$ holds for every
$X\in\mathcal{X}(M)$. Finally, one says that $H$ is the distributional mean
curvature of $\partial^{*}\Omega$ with respect to $\nu_{\Omega}$, if $H$ is
summable in $\mathcal{H}^{n-1}\llcorner\partial^{*}\Omega$, and

$\int_{\partial^{*}\Omega}\,{\rm
div}\,^{\partial^{*}\Omega}X\,d\mathcal{H}^{n-1}=\int_{\partial^{*}\Omega}H\,\langle
X,\nu_{\Omega}\rangle_{g}\,d\mathcal{H}^{n-1}\,,\qquad\forall
X\in\mathcal{X}(M)\,,$

where ${\rm div}\,^{\partial^{*}\Omega}X:={\rm
div}\,X-\langle\nabla_{\nu_{\Omega}}X,\nu_{\Omega}\rangle_{g}$. We thus have
the following distributional version of Brendle’s theorem:

######  Theorem 1.1 (Rigidity).

If $n\geq 3$, $(M,g)\in\mathcal{B}_{n}\cup\mathcal{B}_{n}^{*}$, $\Omega$ is a
set of finite perimeter in $M$ such that
$M^{\circ}\cap\overline{\partial^{\ast}\Omega}$ is compact in $M$ and
$\lambda\in\mathbb{R}$ is such that

$\int_{M^{\circ}\cap\partial^{*}\Omega}\\!\\!{\rm
div}\,^{\partial^{*}\Omega}X\,d\mathcal{H}^{n-1}=\lambda\,\int_{M^{\circ}\cap\partial^{*}\Omega}\langle
X,\nu_{\Omega}\rangle_{g}\,d\mathcal{H}^{n-1}\,,$ (1.4)

for every $X\in\mathcal{X}(M)$, then, for some $t_{0}\in(0,\bar{r})$, either
$\Omega$ is $\mathcal{H}^{n}$-equivalent to $N\times(0,t_{0})$ (and
$\lambda>0$) or $\Omega$ is $\mathcal{H}^{n}$-equivalent to
$N\times(t_{0},\bar{r})$ (and $\lambda<0$).

######  Remark 1.3.

By Allard’s regularity theorem [All72, Sim83, DL18], (1.4) implies that
$\Sigma=\partial^{*}\Omega$ is a smooth, CMC hypersurface in $M$ with
$\mathcal{H}^{n-1}(\overline{\Sigma}\setminus\Sigma)=0$. Since $\Sigma$ is not
necessarily closed, rigidity cannot be deduced by the direct application of
Brendle’s theorem. The difficulty addressed in Theorem 1.1 is expressing how
the distributional CMC condition (1.4) “ties together” the (potentially
countably many) connected components of $\Sigma$, and forces them to align
into a single slice, rather than, say, allowing them to combine through the
singular set $\overline{\Sigma}\setminus\Sigma$ into a non-slice CMC
hypersurface.

######  Remark 1.4.

Condition (1.4) is equivalent to ask that, if $f_{t}$ is a smooth volume-
preserving flow of $\Omega$ ($f_{0}={\rm id}\,$, $(\partial f/\partial
t)|_{t=0}=X\in\mathcal{X}(M)$, and
$\mathcal{H}^{n}(f_{t}(\Omega))=\mathcal{H}^{n}(\Omega)$ for every $|t|$
small), then

$\big{(}d\big{/}dt\big{)}\big{|}_{t=0}\,{\rm Per}(f_{t}(\Omega))=0\,.$

Theorem 1.1 then says that among sets of finite perimeter, slices are the only
volume-constrained critical points of the area functional in
$(M,g)\in\mathcal{B}_{n}\cup\mathcal{B}_{n}^{*}$.

######  Remark 1.5 (Necessity of “$M^{\circ}\cap\overline{\partial^{*}\Omega}$
compact” in Theorem 1.1).

If $v>0$ is small enough and $\Omega_{v}$ is a minimizer of
$\mathcal{H}^{n-1}(M^{\circ}\cap\partial^{*}\Omega)$ among sets $\Omega\subset
M$ with $\mathcal{H}^{n}(\Omega)=v$, then
$M^{\circ}\cap\overline{\partial^{*}\Omega_{v}}$ is a smooth CMC hypersurface,
diffeomorphic to a hemisphere sitting on the horizon (see, e.g., [MM16] for a
detailed analysis of this kind of result in the capillarity setting).
Alternatively, one can first apply the perturbative construction of Pacard and
Xu [PX09] on the horizon of the doubled Schwarzschild manifold, as described
in [BE13]. Either way, one obtains non-slice, CMC hypersurfaces bounding sets
$\Omega$.

###  1.3. Compactness for almost-CMC boundaries

Theorem 1.1 is of course strongly motivated by the following natural
compactness problem for almost-CMC hypersurfaces, which can be formulated, in
very general terms, as follows:

$\begin{split}&\mbox{{\it If $(M,g)\in\mathcal{B}_{n}\cup\mathcal{B}_{n}^{*}$,
does every sequence $\\{\Sigma_{j}\\}_{j}$ of closed,}}\\\ &\mbox{{\it
embedded, orientable smooth hypersurfaces in $M$}}\\\ &\mbox{{\it whose scalar
mean curvatures in $(M,g)$ converge}}\\\ &\mbox{{\it to a constant, have
slices as their only possible limits?}}\end{split}$ (CP)

This basic question appears naturally in several contexts. Two important
examples are: the analysis of the Huisken–Yau problem [Hui96, QT07, NT09,
NT10, Hua10, LMS11, BE14, Ner18, CE20], e.g., an outlying CMC hypersurface in
an asymptotically Schwarzschild manifold can be seen as an almost-CMC
hypersurface in the Schwarzschild manifold; and the study of the long-time
behavior of the volume-preserving mean curvature flow – since the
$L^{2}$-oscillation of the mean curvature is the dissipation of the flow.

In light of ($\Sigma\leftrightarrow\Omega$), for each $\Sigma_{j}$ in (CP)
there are open sets $\Omega_{j}$ in $M$ and intervals
$(a_{j},b_{j})\subset\subset(0,\bar{r})$ such that $\Sigma_{j}\subset
N\times(a_{j},b_{j})$ and (up to extracting subsequences) either
$\Sigma_{j}=\partial\Omega_{j}$ (for every $j$) or $\Sigma_{j}\cup
N_{0}=\partial\Omega_{j}$ (for every $j$). In both cases, under the natural
set of assumptions that

$\displaystyle\mbox{$a=\inf_{j}\,a_{j}>0$}\,,\qquad\mbox{$b=\sup_{j}b_{j}<\bar{r}$}\,,\qquad\mbox{$\sup_{j}\mathcal{H}^{n-1}(\Sigma_{j})<\infty$}\,,$
(1.5)

one finds a set $\Omega$ with finite perimeter in $(M,g)$ such that, and up to
extracting subsequences, it holds

$\lim_{j\to\infty}\mathcal{H}^{n}(\Omega_{j}\Delta\Omega)=0\,,\qquad\liminf_{j\to\infty}\mathcal{H}^{n-1}(\partial\Omega_{j})\geq{\rm
Per}(\Omega)\,.$ (1.6)

In the above basic setting, Theorem 1.1, combined with standard closure
results for integer varifolds, leads to an affirmative answer to (CP). This is
the content of our second main result, where the most general case of
sequences of sets of finite perimeter is directly addressed:

######  Theorem 1.2 (Compactness).

If $n\geq 3$, $(M,g)\in\mathcal{B}_{n}\cup\mathcal{B}_{n}^{*}$, and
$\\{\Omega_{j}\\}_{j}$ is a sequence of sets of finite perimeter in $M$ such
that

  1. (i)

there are $(a,b)\subset\subset(0,\bar{r})$ and a Borel set $\Omega\subset M$
s.t. $M^{\circ}\cap\overline{\partial^{*}\Omega_{j}}\subset N\times(a,b)$ for
every $j$, and $\mathcal{H}^{n}(\Omega_{j}\Delta\Omega)\to 0$ as $j\to\infty$;

  2. (ii)

${\rm Per}(\Omega_{j})\to{\rm Per}(\Omega)$ as $j\to\infty$;

  3. (iii)

there is $\lambda\in\mathbb{R}$ such that, for every $X\in\mathcal{X}(M)$, as
$j\to\infty$,

$\int_{M^{\circ}\cap\partial^{*}\Omega_{j}}{\rm
div}\,^{\partial^{*}\Omega_{j}}X\,d\mathcal{H}^{n-1}-\lambda\,\int_{M^{\circ}\cap\partial^{*}\Omega_{j}}\langle
X,\nu_{\Omega_{j}}\rangle_{g}\,d\mathcal{H}^{n-1}\to 0\,;$ (1.7)

then there is $t_{0}\in(0,\bar{r})$ such that $\Omega$ if
$\mathcal{H}^{n}$-equivalent either to $N\times(0,t_{0})$ (and then
$\lambda>0$) or to $N\times(t_{0},\bar{r})$ (and then $\lambda<0$).

######  Remark 1.6.

The proof of Theorem 1.2 requires the full strength of Theorem 1.1 even if one
is only interested in sequences $\\{\Sigma_{j}\\}_{j}$ of closed, embedded,
orientable smooth hypersurfaces in $M$ whose mean curvature oscillations are
assumed to vanish in every $C^{k}$-norm.

######  Remark 1.7.

In (1.7) the mean curvature oscillation is required to vanish only in
distributional sense. This feature points to the possibility of applying
Theorem 1.2 to minimizing sequences of horizon-homologous isoperimetric
problems that have been suitably selected by means of the Ekeland variational
principle; see, e.g., [CL12, Theorem 3.2(iv)]. Similarly, Theorem 1.2 will be
easily applied to the study of horizon-homologous isoperimetric sets with
fixed volume with respect to metrics $\\{g_{j}\\}_{j}$ on $M$ such that, as
$j\to\infty$, $g_{j}\to g$ with
$(M,g)\in\mathcal{B}_{n}\cup\mathcal{B}_{n}^{*}$. In both these examples, the
perimeter convergence assumption (ii) is trivially checked by energy
comparison.

###  1.4. Strategy of proof and organization of the paper

Section 2 and Section 3 are devoted to establishing in the Riemannian setting
the several tools from GMT that lie at the core of our analysis. In Section
2.1 we collect several properties of normal bundles to closed sets in complete
Riemannian manifolds (Theorem 2.1), extending from the Euclidean case a series
of recent results obtained in [Alb15, MS19, San20a, KS23, HS22]. In Section
2.2 we review some theorems of Kleinjohann [Kle81] and Bangert [Ban82]
concerning sets of positive reach in complete Riemannian manifolds (Theorem
2.2 and Theorem 2.3). In Section 3.1 we recall the viscous notion of “being
$m$-dimensional with mean curvature vector bounded by $\lambda$” introduced on
closed subsets of Riemannian manifolds by White in [Whi16], and recall its
relation to distributional mean curvature (Theorem 3.1). Finally, in Section
3.2, we extend to the Riemannian setting a delicate “Lusin-type property” of
normal bundles of White’s $(m,\lambda)$-sets that, in the Euclidean case, was
proved in [San20b] (Theorem 3.3).

At the basis of the distributional version of the Alexandrov theorem proved in
[DM19] (as well as of the previously cited quantitative versions of it), lies
the approach to CMC-rigidity of Ros [Ros87] and Montiel-Ros [MR91]. Their
method is based on the analysis of equality cases in the (Euclidean) Heintze-
Karcher inequality. Brendle’s theorem, in turn, is based on the analysis of
the equality cases of two different Heintze–Karcher-type inequalities for
subsets $\Omega$ of $(M,g)\in\mathcal{B}_{n}\cup\mathcal{B}_{n}^{*}$
(corresponding to the cases $\partial\Omega=\Sigma$ and
$\partial\Omega=\Sigma\cup N_{0}$ appearing in
($\Sigma\leftrightarrow\Omega$)), and, specifically, to the fact that, when
$\Sigma$ is such an equality case, then, $\Sigma$ is umbilic; the umbilicality
of $\Sigma$ is then combined with ${\rm(H4)}$ to deduce rigidity (i.e.,
$\Sigma$ is a slice).

A natural strategy for proving Theorem 1.1 thus consists in: (a) establishing
the two Heintze–Karcher-type inequalities of Brendle on sets with finite
perimeter; (b) addressing the analysis of their equality cases in the
distributional setting; and, (c) deducing rigidity from an established set of
necessary conditions for equality.

In Section 4 we implement this strategy. There we work with not necessarily
closed, smooth, embedded, hypersurfaces $\Sigma$ satisfying three assumptions:
first, $\mathcal{H}^{n-1}(\overline{\Sigma}\setminus\Sigma)=0$ with
$\mathcal{H}^{n-1}(\Sigma)<\infty$ ($\overline{\Sigma}\setminus\Sigma$ is
understood as the “singular set” of $\Sigma$); second, either
$\overline{\Sigma}=\partial\Omega$, or
$N_{0}\cup\overline{\Sigma}=\partial\Omega$, for an open set $\Omega$ in $M$;
and, finally, $\overline{\Sigma}$ is compactly contained in $M^{\circ}$ and
has bounded mean curvature in the viscous sense of White, while $\Sigma$ has
positive mean curvature with respect to the outer $g$-unit normal
$\nu_{\Omega}$ to $\Omega$.

In Theorem 4.1 we take care of steps (a) and (b). Implementing step (a) is
particularly delicate not only because, as expected, several passages of
Brendle’s argument make a crucial use of smoothness, and thus require
considerable effort to be repeated or redesigned in a non-smooth framework;
but also because, in anticipation of the non-smooth rigidity discussion of
step (c), we need a more detailed list of necessary conditions for equality
cases in Brendle’s Heintze–Karcher-type inequalities. In this direction, we
notice that we shall establish three such conditions: (E1) $\Sigma$ is
umbilical in $(M,g)$ (which is the condition already pointed out in [Bre13]);
(E2) $M\setminus\Omega$ has positive reach in $(M,g^{*})$ (where $g^{*}$ is a
certain complete metric on $M$, conformal to $g$); (E3) under ${\rm(H3)^{*}}$,
$g_{N}(\nu_{\Omega},\nu_{\Omega})=0$ on $\Sigma$. The identification of
condition (E3) is new even in the smooth case. We also notice that, in the
smooth case, condition (E2) is automatically true; and, indeed, the derivation
of (E2) requires some careful work on Brendle’s argument which is specific to
the viscous setting.

In Theorem 4.2, we further assume that $\overline{\Sigma}$ has constant mean
curvature in distributional sense, and then address step (c). If
${\rm(H3)^{*}}$, and thus condition (E3), holds, then we can infer rigidity
directly from it, without using umbilicality: indeed (E3) implies the very
strong information that $\nu_{\Omega}$ is parallel to $\partial/\partial r$
$\mathcal{H}^{n-1}$-a.e. along $\Sigma$ – an information that gives
$\Sigma=N_{t_{0}}$ by a simple property of sets of finite perimeter (cf. with
[Mag12, Exercise 15.18]). If, otherwise, only ${\rm(H3)}$ holds, then, as in
[Bre13], we need to combine ${\rm(H4)}$ and umbilicality to deduce that,
$\mathcal{H}^{n-1}$-a.e. on $\Sigma$, $\nu_{\Omega}$ is either parallel or
orthogonal to $\partial/\partial r$. The dichotomy parallel/orthogonal
prevents the use of something as simple as [Mag12, Exercise 15.18]. In the
smooth case, one immediately excludes “orthogonality”, and thus conclude
rigidity, by a sliding argument. However, in the non-smooth setting, sliding
arguments are not equally effective (because of multiplicity issues preventing
the use of Allard’s regularity theorem at contact points). This is the passage
where condition (E2) reveals useful, and ultimately allows us to conclude the
proof of Theorem 4.2.

Finally, Section 5 contains the proofs of Theorem 1.1 and Theorem 1.2.

###  1.5. Rigidity and bubbling in the hyperbolic space

It is a classical and well known result that Alexandrov rigidity result can be
generalized to _smooth_ boundaries in the hyperbolic space. In [Bre13, Theorem
1.4] a new proof is obtained based on the Heintze-Karcher inequality. It is a
natural and interesting question to understand if finite unions of possibly
mutually tangent balls are the only examples of sets of finite perimeter in
the hyperbolic space with constant distributional mean curvature. The methods
developed in this paper provides a positive answer to this question.

######  Theorem 1.3.

Suppose $\Omega$ is a set of finite perimeter with compact closure in the
hyperbolic space $\mathbb{H}^{n}$ and $\lambda\in\mathbb{R}$ such that

$\int_{\partial^{\ast}\Omega}{\rm
div}^{\partial^{\ast}\Omega}X\,d\mathcal{H}^{n-1}=\lambda\int_{\partial^{\ast}\Omega}\langle
X,\nu_{\Omega}\rangle_{\mathbb{H}^{n}}\,d\mathcal{H}^{n-1},$ (1.8)

for every $X\in\mathcal{X}(M)$. Then $\Omega$ is a finite union of disjoint
(possibly mutually tangent) open geodesic balls with equal radii.

######  Remark 1.8.

In [HHW23] the authors develop a beautiful moving plane method for a class of
varifolds satisfying a suitable _tameness condition_ and they employ it to
prove several rigidity results for stationary and CMC varifolds. In
particular, cf. [HHW23, Theorem 1.8], they prove that if
$\Sigma\subseteq\mathbb{H}^{n}$ is the support of a $(n-1)$-dimensional _tame_
varifold without boundary and with constant mean curvature111cf. [HHW23, eq.
(8)] for the definition of varifold with bounded mean curvature, and [HHW23,
Definition 1.6] for the definition of tameness for varifolds, and if $\Sigma$
is connected and compact, then it is a geodesic sphere. In this direction we
point out that unions of mutually tangent geodesic spheres cannot be tame
varifolds in the sense of [HHW23] (since the tangent cone at the singular
point between the spheres is a multiplicity two plane). On the other hand,
these configurations naturally arise as limits of sequences of connected and
compact smooth hypersurfaces with mean curvatures converging to a constant
(see [CM17] and references therein). Hence, Theorem 1.3, in addressing
rigidity and compactness in the hyperbolic space under assumptions that do not
prevent bubbling, provides a useful improvement of [HHW23, Theorem 1.8].

###  1.6. Further directions

A natural question is that of obtaining quantitative estimates for almost-CMC
boundaries, both in the Brendle class and on space forms. In this direction we
mention the recent results on space forms [CV20, CRV21], where, based on the
moving planes method, sharp decays are obtained under a “bubbling-preventing”
exterior/interior ball assumption. There are alternative proofs of the
Heintze–Karcher type inequalities behind Brendle’s theorem, based on integral
identities rather than on geodesic flows, that have been developed, for
example, in [LX19, FP22]. Correspondingly, non-sharp quantitative estimate
have been derived in [Sch21, Theorem 1.3] on space forms, and in [SX22,
Theorem 1.4] on a sub-class of Brendle’s class (which still includes the model
manifolds from General Relativity). Both these results require bounds on the
$C^{2,\beta}$-geometry of the considered boundaries (in addition to interior
ball conditions), and the resulting stability constants (together with the
non-sharp stability exponents in the case of [SX22, Theorem 1.4]) depend to
the particular $\beta$ under consideration. For this reason the compactness
result in Theorem 1.2 is entirely new even on smooth boundaries, as it does
not require any uniform control on their geometry.

Acknowledgements: FM wishes to thank Claudio Arezzo for having introduced him
to the framework considered in this work. FM is supported by NSF-DMS RTG
1840314, NSF-DMS FRG 1854344, and NSF-DMS 2000034. MS acknowledges support of
the INDAM-GNSAGA project ”Analisi Geometrica: Equazioni alle Derivate Parziali
e Teoria delle Sottovarietà”.

##  2\. Sets of positive reach in Riemannian manifolds

In this section $(M,g)$ is a complete Riemannian manifold of dimension $n$,
with exponential function $\exp$ and Riemannian distance $d$, and we denote by
$\Psi:TM\times[0,\infty)\to M$ the map

$\Psi(p,\eta,t)=\exp(p,t\,\eta)\,,\qquad\forall(p,\eta)\in TM\,,t\geq 0\,.$
(2.1)

We denote by $\mathrm{d}f$ the differential of $f:M\to\mathbb{R}$, and by
$\nabla f$ and $\mathrm{D}^{2}f$ the gradient and Hessian of $f$ with respect
to $g$. We use $\nabla$ also to denote the metric connection of $(M,g)$. A
segment in $M$ is a unit speed geodesic $\gamma:[a,b]\to M$ such that
$d(\gamma(a),\gamma(b))=b-a$.

###  2.1. Normal bundles of closed sets

Given a closed set $\Gamma\subset M$, the projection map on $\Gamma$ is
defined for at $p\in M$ as the subset of $\Gamma$ given by

$\xi_{\Gamma}(p)=\big{\\{}a\in\Gamma:{\rm dist}(p,\Gamma)=d(p,a)\big{\\}}\,.$

The unit normal bundle $\mathcal{N}^{1}\Gamma$ and the normal bundle
$\mathcal{N}\Gamma$ of $\Gamma$ are defined by setting, for $a\in\Gamma$,

$\displaystyle\mathcal{N}^{1}_{a}\Gamma$ $\displaystyle=$
$\displaystyle\big{\\{}\eta\in T_{a}M:|\eta|=1\,,\mbox{$\exists s>0$ s.t.
$s={\rm dist}\big{(}\exp(a,s\,\eta),\Gamma\big{)}$}\big{\\}}\,,$
$\displaystyle\mathcal{N}_{a}\Gamma$ $\displaystyle=$
$\displaystyle\big{\\{}t\,\eta:t\geq
0\,,\eta\in\mathcal{N}^{1}_{a}\Gamma\big{\\}}\,.$

We define $\rho_{\Gamma}:\mathcal{N}^{1}\Gamma\to(0,\infty]$,
$A_{\Gamma}\subset TM$, and ${\rm Cut}(\Gamma)\subset M$ by setting

$\displaystyle\rho_{\Gamma}(x,\eta)=\sup\big{\\{}s>0:s={\rm
dist}(\exp(x,s\,\eta),\Gamma)\big{\\}}\,,$ (2.4) $\displaystyle
A_{\Gamma}=\big{\\{}(p,\eta,t):(p,\eta)\in\mathcal{N}^{1}\Gamma\,,t\in(0,\rho_{\Gamma}(p,\eta))\big{\\}}\,,$
$\displaystyle{\rm
Cut}(\Gamma)=\big{\\{}\exp(x,s\,\eta):(x,\eta)\in\mathcal{N}^{1}\Gamma\,,s=\rho_{\Gamma}(x,\eta)<\infty\big{\\}}\,,$

so that, when $\Gamma$ is a closed $C^{2}$-hypersurface in $M$,
$\rho_{\Gamma}$ is continuous on $\Gamma$, ${\rm Cut}(\Gamma)$ corresponds to
the usual notion of cut-locus of $\Gamma$ and satisfies $\mathcal{H}^{n}({\rm
Cut}(\Gamma))=0$, and $\Psi|_{A_{\Gamma}}$ is a diffeomorphism between
$A_{\Gamma}$ and

$\mathcal{U}(\Gamma)=\Psi(A_{\Gamma})=\big{\\{}\exp(a,s\,\eta):(a,\eta)\in\mathcal{N}^{1}\Gamma\,,s\in\big{(}0,\rho_{\Gamma}(a,\eta)\big{)}\big{\\}}\,.$

The following theorem relies on a series of recent results [Alb15, MS19,
San20a, KS23, HS22] to adapt/extend the above classical facts to the case when
$\Gamma$ is merely a closed set.

######  Theorem 2.1.

If $(M^{n},g)$ is a complete Riemannian manifold and $\Gamma\subset M$ is
closed, then:

  1. (i)

$\mathcal{N}\,\Gamma$ is a countably $n$-rectifiable Borel subset of $TM$;

  2. (ii)

$\mathcal{N}_{a}\Gamma$ is a convex cone in $T_{a}M$ for every $a\in\Gamma$;

  3. (iii)

for each $m=0,\ldots,n-1$ the set

$\Gamma^{(m)}:=\\{a\in\Gamma:\dim\mathcal{N}_{a}\Gamma=n-m\\}$

is countably $(\mathcal{H}^{m},m)$-rectifiable; in particular222Indeed, if
$\mathcal{H}^{0}(\mathcal{N}^{1}_{a}\Gamma)>2$, then the convexity of
$\mathcal{N}_{a}\Gamma$ implies that $\dim\mathcal{N}_{a}\Gamma\geq 2$, i.e.
$a\in\Gamma^{(m)}$ for some $m\leq n-2$.

$\mathcal{H}^{n-1}(\\{a\in\Gamma:\mathcal{H}^{0}(\mathcal{N}^{1}_{a}\Gamma)>2\\})=0\,;$
(2.5)

  4. (iv)

$\rho_{\Gamma}:\mathcal{N}^{1}(\Gamma)\rightarrow(0,+\infty]$ is an upper-
semicontinuous function;

  5. (v)

$\mathcal{H}^{n}({\rm Cut}(\Gamma))=0$.

  6. (vi)

For each $p\in\mathcal{U}(\Gamma)$ there exists a unique $(a,\eta)\in TM$,
$|\eta|=1$, such that $d(p,a)={\rm dist}(p,\Gamma)$ and
$\exp(a,d(p,a)\eta)=p$;

  7. (vii)

If $\tau_{0}>0$ and $\rho_{\Gamma}(a,\eta)\geq\tau_{0}$ for
$\mathcal{H}^{n-1}$ a.e. $(a,\eta)\in\mathcal{N}^{1}\Gamma$, then

$\big{\\{}x\in M:0<{\rm
dist}(x,\Gamma)<\tau_{0}\big{\\}}\subseteq\mathcal{U}(\Gamma)\,.$

###### Proof.

Let $\phi:U\subset M\rightarrow V\subset\mathbb{R}^{n}$ be a local chart of
$M$. For each $a\in U$ there is a unique symmetric bijective linear map
$S_{a}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ such that

$\langle
u,v\rangle_{g}=\mathrm{d}\phi(a)(v)\,\cdot\,S_{a}\big{[}\mathrm{d}\phi(a)(u)\big{]}\,,\qquad\forall
u,v\in T_{a}M\,.$

A routine argument shows that the map $\Phi:TU\to V\times\mathbb{R}^{n}$
defined by $\Phi(a,v)=(\phi(a),S_{a}[\mathrm{d}\phi(a)(v)])$, $(a,v)\in TU$,
is a diffeomorphism, with

$\Phi\big{(}\mathcal{N}\,\Gamma\,\llcorner
U\big{)}=\big{(}\mathcal{N}\,\overline{\phi(\Gamma\cap U)}\big{)}\,\llcorner
V\,.$

Similarly, for every $a\in\Gamma$ and $m=0,...,n-1$, we have

$[S_{a}\circ\mathrm{d}\phi(a)]\big{(}\mathcal{N}_{a}\Gamma\big{)}=\mathcal{N}_{\phi(a)}\,\overline{\phi(\Gamma\cap
U)}\,,\quad\phi(\Gamma^{(m)}\cap U)=[\overline{\phi(\Gamma\cap U)}]^{(m)}\cap
V\,.$

Hence, conclusions (i), (ii), and (iii) follow from the analogous statements
in the Euclidean case proved in [San20a, Remark 4.3] and [MS19]. Conclusion
(iv) follows by an obvious adaptation of the argument for the Euclidean space
proved in [KS23, Lemma 2.35]. Conclusion (v) is proved in [Alb15, Theorem 1].
Conclusion (vi) follows from the remark that if $s>0$ and $\alpha$ is a unit-
speed geodesic such that $\alpha(0)\in\Gamma$ and ${\rm
dist}(\alpha(s),\Gamma)=s$, then, for every $0\leq t<s$, we have ${\rm
dist}(\alpha(t),\Gamma)=t$, $\xi_{\Gamma}(\alpha(t))=\\{\alpha(0)\\}$, and
$\alpha|_{[0,t]}$ is the unique segment joining $\alpha(0)$ and $\alpha(t)$.
We are left to prove conclusion (vii), which requires modifications to the
proof of [HS22, Lemma 3.19]. With $\Psi$ as in (2.1), we define

$Q^{\ast}=\\{(a,\eta,t):(a,\eta)\in\mathcal{N}^{1}\Gamma,\;0<t<\inf\\{\tau_{0},\rho_{\Gamma}(a,\eta)\\}\\}$

and we notice that $\Psi(Q^{\ast})=\\{x\in M:0<{\rm
dist}(x,\Gamma)<\tau_{0}\\}\setminus{\rm Cut}(\Gamma)$. Let

$Q=Q^{\ast}\cap\\{(a,\eta,t):\rho_{\Gamma}(a,\eta)\geq\tau_{0}\\}$

so that $\mathcal{H}^{n}(Q^{\ast}\setminus Q)=0$ by assumption. Hence
$\mathcal{H}^{n}(\Psi(Q^{\ast})\setminus\Psi(Q))=0$, and, by (v), we conclude
that $\Psi(Q)$ is dense in $\\{x\in M:0<{\rm dist}(x,\Gamma)<\tau_{0}\\}$. Let
$x\in M$ and $t={\rm dist}(x,\Gamma)$ with $0<t<\tau_{0}$. There exists a
sequence $(a_{i},\eta_{i},t_{i})\in Q$ such that
$\Psi(a_{i},\eta_{i},t_{i})\to x$. Since $d(a_{i},x)\leq
d(a_{i},\Psi(a_{i},\eta_{i},t_{i}))+d(\Psi(a_{i},\eta_{i},t_{i}),x)$ for
$i\geq 1$ and $t_{i}={\rm dist}(\Psi(a_{i},\eta_{i},t_{i}),\Gamma)\to t$, we
infer that $\limsup_{i\to\infty}d(a_{i},x)\leq t$. By compactness, there are
$(a,\eta)\in TM$ with $|\eta|=1$ and a subsequence $(a_{i_{j}},\eta_{i_{j}})$
such that $(a_{i_{j}},\eta_{i_{j}})\to(a,\eta)$ as $j\to\infty$. It follows
that

$t=\lim_{j\to\infty}{\rm
dist}(\Psi(a_{i_{j}},\eta_{i_{j}},t_{i_{j}}),\Gamma)={\rm
dist}(\Psi(a,\eta,t),\Gamma)$

and $(a,\eta)\in\mathcal{N}^{1}\Gamma$. By conclusion (iv) and by
$\rho_{\Gamma}(a_{i},\eta_{i})\geq\tau_{0}$ we find
$\rho_{\Gamma}(a,\eta)\geq\tau_{0}$, and thus $x\in\mathcal{U}(\Gamma)$. ∎

###  2.2. Sets of positive reach

Introduced in the Euclidean setting by Federer [Fed59], sets of positive reach
have been studied in the Riemannian setting by Kleinjohann [Kle81] and Bangert
[Ban82]. Given a closet set $\Gamma\subset M$, the set of unique projection
over $\Gamma$ is defined as

$\operatorname{{\rm UP}}(\Gamma)=\big{\\{}x\in
M:\mathcal{H}^{0}(\xi_{\Gamma}(x))=1\big{\\}}\,.$

Given $a\in\Gamma$, we denote by $\operatorname{reach}(\Gamma,a)$ the supremum
of those $r\geq 0$ such that $B(a,r)\subseteq\operatorname{{\rm UP}}(\Gamma)$,
and say that $\Gamma$ is a set of locally positive reach if
$\operatorname{reach}(\Gamma,a)>0$ for each $a\in\Gamma$, and is a set of
positive reach if $\operatorname{reach}(\Gamma,\cdot)$ is uniformly positive
on $\Gamma$.

######  Remark 2.1.

If $\Gamma$ is closed and $\rho_{\Gamma}(x,\eta)\geq\tau_{0}>0$ for
$\mathcal{H}^{n-1}$ a.e. $(x,\eta)\in\mathcal{N}^{1}\Gamma$, then by Theorem
2.1-(vii),

$\\{x\in M:0<{\rm
dist}(x,\Gamma)<\tau_{0}\\}\subset\mathcal{U}(\Gamma)\subseteq\operatorname{{\rm
UP}}(\Gamma)\,.$

In particular, $\Gamma$ is of positive reach, with
$\operatorname{reach}(\Gamma,\cdot)\geq\tau_{0}$ on $\Gamma$.

The following result, contained in [Kle81], is crucial in obtaining (4.50) and
(4.51) in the proof of Theorem 4.1.

######  Theorem 2.2 (Kleinjohann).

If $A\subseteq M$ is a set of positive reach with compact boundary, then there
exists $\varepsilon(A)>0$ such that for every $t\in(0,\varepsilon(A))$ the set

$A_{t}=\\{x\in M:{\rm dist}(A,x)=t\\}$

is a compact $C^{1,1}$-hypersurface contained in $\operatorname{{\rm UP}}(A)$,
and the geodesic-flow map $\Phi_{t}:\mathcal{N}^{1}(A)\rightarrow A_{t}$,
defined by $\Phi_{t}(a,\eta)=\exp(a,t\eta)$ for
$(a,\eta)\in\mathcal{N}^{1}(A)$, is bi-Lipschitz on $\mathcal{N}^{1}(A)$. In
particular, $\mathcal{N}^{1}(A)$ is an $(n-1)$-dimensional compact Lipschitz
submanifold of $TM$.

###### Proof.

Let $\mathcal{U}$ be an open neighborhood of the null section of $TM$ and
$\mathcal{V}$ an open subset of $M\times M$ such that the map
$\mathcal{U}\ni(a,v)\mapsto(a,\exp(a,v))$ is a diffeomorphism of $\mathcal{U}$
onto $\mathcal{V}$. Let $\Phi$ be its inverse, so that $\exp(p,\Phi(p,q))=q$
for every $(p,q)\in\mathcal{V}$. As explained in [Kle81, middle of page 336],
one can choose for each $a\in A$ a number $\zeta(a)>0$333Denoted with
$z^{\prime}(a,\epsilon)$ in [Kle81, page 336]. so that

$B(a,\zeta(a))\times B(a,\zeta(a))\subseteq\mathcal{V}\quad\textrm{for $a\in
A$}$

and, defining $W:=\bigcup_{a\in A}B(a,\zeta(a))$, we have that for every $x\in
W$ there exists a unique minimizing geodesic joining $x$ and $A$ and $\xi_{A}$
is locally lipschitz on $W$ (see [Kle81, Satz (2.5)]). For $x\in W\setminus A$
we set

$\nu(x)=\frac{\Phi(x,\xi_{A}(x))}{|\Phi(x,\xi_{A}(x))|}\quad\textrm{and}\quad\eta(x)=\frac{\Phi(\xi_{A}(x),x)}{|\Phi(\xi_{A}(x),x)|}.$

It follows from [Kle81, Satz (2.1) and Satz (2.3)] that ${\rm dist}(A,\cdot)$
is continuously differentiable on $W\setminus A$, with $\nabla{\rm
dist}(A,x)=-\nu(x)$ for $x\in W\setminus A$. In particular $\nabla{\rm
dist}(\cdot,A)$ is locally Lipschitz on $W\setminus A$. Since ${\rm
dist}(A,x)=|\Phi(\xi_{A}(x),x)|$ for $x\in W$, it follows that

$\Psi_{t}(\xi_{A}(x),\eta(x))=\exp(\xi_{A}(x),\Phi(\xi_{A}(x),x))=x$

for $x\in W\cap A_{t}$. By applying the Lebesgue covering lemma to
$\\{B(a,\zeta(a)):a\in\partial A\\}$, we find a positive number
$\varepsilon(A)>0$ such that $\\{x\in M:{\rm
dist}(A,x)<\varepsilon(A)\\}\subseteq W$, and thus conclude the proof. ∎

Following a standard convention (see, e.g. [Gro93]) we say that $p\in
M\setminus\Gamma$ is a critical point for ${\rm dist}(\cdot,\Gamma)$ if for
every $v\in T_{p}(M)$, $v\neq 0$, there exists $a\in\Gamma$ with $d(p,a)={\rm
dist}(p,\Gamma)$ and a segment $\gamma:[0,d(p,a)]\rightarrow M$ such that
$\gamma(0)=p$, $\gamma(d(p,a))=a$ and ${\rm
angle}(v,\gamma^{\prime}(0))\leq\pi/2$. Correspondingly, $\tau>0$ is a regular
value for ${\rm dist}(\Gamma,\cdot)$ if there are no critical points $p$ of
${\rm dist}(\cdot,\Gamma)$ with ${\rm dist}(p,\Gamma)=\tau$. The following
result is a special case of the main result obtained by Bangert in [Ban82],
and plays an important role in our analysis (see the proof of (4.48)).

######  Theorem 2.3 (Bangert).

If $\Gamma\subseteq M$ is compact and $\tau>0$ is a regular value of ${\rm
dist}(\cdot,\Gamma)$, then $\\{x\in M:{\rm dist}(x,\Gamma)\geq\tau\\}$ is a
set of positive reach.

###### Proof.

By [Man03, Proposition 3.4], $f=-{\rm dist}(\Gamma,\cdot)$ is locally
semiconvex on $M\setminus\Gamma$. In particular, $f$ belongs to the class
$\mathcal{F}(M\setminus\Gamma)$ introduced by Bangert in [Ban82]. Moreover, by
[RZ12, Lemma 5.5], $p\in M\setminus\Gamma$ is a regular point of $f$ if and
only if there exists $v\in T_{p}M$ such that444The function $f$ being
semiconcave, it may fail to have a differential at $p$. However, the limit
$\partial_{p}f(v)$ will exist for every $p$ and $v$. Here are we using the
same notation found in [Ban82]. $\partial_{p}f(v)=\lim_{t\to
0^{+}}(f(p+t\,v)-f(p))/t$ is negative. It follows that all points of
$f^{-1}(-\tau)$ are regular in the sense of [Ban82, Definition (i)], so that
$f^{-1}((-\infty,-\tau])$ is a set of locally positive reach by the main
theorem of [Ban82]. Since $(M,g)$ is complete,
$f^{-1}(-\tau)=\partial[f^{-1}((-\infty,-\tau])]$ is compact, and thus we
conclude by the general fact that, if $A\subseteq M$ is a set of locally
positive reach and $\partial A$ is compact, then $A$ is a set of positive
reach (for example because, by [Kle81, Lemma 1.1],
$\operatorname{reach}(A,\cdot)$ is continuous on $A$). ∎

##  3\. A Lusin-type property of White’s $(m,\lambda)$-sets

In Section 3.1 we recall a viscosity formulation of the notion of “being
$m$-dimensional with mean curvature vector bounded by $\lambda$” for closed
subsets of a Riemannian manifold $(M,g)$, as introduced by White in [Whi16].
Then, in Section 3.2, we extend from the Euclidean to the Riemannian setting a
“Lusin condition” for normal bundles proved in [San20b].

###  3.1. White’s $(m,\lambda)$-sets

Given an integer $m\in\\{1,...,n-1\\}$ and a constant $\lambda\geq 0$, we say
that a closed subset $\Gamma$ of a Riemannian manifold $(M,g)$ is a White
$(m,\lambda)$-set in $(M,g)$ if, for every $f\in C^{2}(M)$ such that
$f|_{\Gamma}$ admits a local maximum at $x\in\Gamma$, it holds that

${\rm trace}_{m}(\mathrm{D}^{2}f(x))\leq\lambda|\nabla f(x)|\,.$ (3.1)

Here ${\rm trace}_{m}(\mathrm{D}^{2}f(x))=\lambda_{1}+\cdots+\lambda_{m}$ if
$\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}$ are the eigenvalues of
$\mathrm{D}^{2}f(x)$ listed in increasing order. A fundamental result
concerning White $(m,\lambda)$-sets relates condition (3.1) to the notion of
distributional mean curvature for a varifold. This theorem plays a key role in
our analysis (specifically, it allows us to use Theorem 4.1 in proving Theorem
4.2, see the next section).

######  Theorem 3.1 (White).

If $\lambda\geq 0$, $V$ is an $m$-dimensional varifold in $(M,g)$, and
$\vec{H}$ is a Borel vector field in $M$ such that
$\|\vec{H}_{V}\|_{L^{\infty}({\rm spt}\|V\|)}\leq\lambda$ and

$\int({\rm
div}\,^{\tau}X)(x)\,dV(x,\tau)=-\int_{M}\,\langle\vec{H},X\rangle_{g}\,d\|V\|\,,$

for every $X\in\mathcal{X}(M)$, then ${\rm spt}\,\|V\|$ is a White
$(m,\lambda)$-set in $(M,g)$.

###### Proof.

This is [Whi16, Corollary 2.8]. ∎

We shall also need the following simple fact:

######  Lemma 3.2.

If $(M,g)$ is a Riemannian manifold, $\Gamma$ is a White $(m,\lambda)$-set in
$(M,g)$, $\varphi\in C^{\infty}(M)$, $g^{*}=e^{2\,\varphi}g$, and

$C(\Gamma,\varphi)=\sup_{\Gamma}|\nabla\varphi|<\infty\,,$

then $\Gamma$ is a White $(m,\lambda+3\,m\,C(\Gamma,\varphi))$-set in
$(M,g^{*})$.

###### Proof.

Denoting with $\nabla^{*}$ and $\mathrm{D}^{2}_{*}$ the Riemannian connection
and the Hessian operator with respect to $g^{*}$, it is enough to prove that

$\mathrm{D}^{2}_{*}f(x)\leq\mathrm{D}^{2}f(x)+3\,|\nabla\varphi(x)|\,|\nabla
f(x)|\,g_{x}\,$ (3.2)

whenever $x\in O$, $O$ is open in $M$, and $f\in C^{2}(O)$. To this end, by
[Sak96, II, Proposition 3.9], we compute

$\displaystyle\mathrm{D}^{2}_{*}f(U,V)=U(V(f))-(\nabla^{*}_{U}V)(f)=\mathrm{D}^{2}f(U,V)-g(\nabla\varphi,U)g(\nabla
f,V)$ $\displaystyle\hskip 142.26378pt-g(\nabla\varphi,V)g(\nabla
f,U)-g(U,V)g(\nabla\varphi,\nabla f)\,,$

whenever $U,V\in\mathcal{X}(M)$. Hence,

$\mathrm{D}^{2}_{*}f(U,U)\leq\mathrm{D}^{2}f(U,U)+3|\nabla
f||\nabla\varphi||U|^{2}\,.$

If now $O$ is a neighborhood of some $x\in\Gamma$ and $f|_{\Gamma}$ has a
local maximum at $x$, then this last inequality, combined with [Whi16, Lemma
12.3] and the fact that $\Gamma$ is a White $(m,\lambda)$-set in $(M,g)$,
implies that

$\displaystyle{\rm trace}_{m}(\mathrm{D}^{2}_{*}f(x))$ $\displaystyle\leq{\rm
trace}_{m}\big{(}\mathrm{D}^{2}f(x)+3\,|\nabla
f(x)|\,|\nabla\varphi(x)|\,g_{x}\big{)}$ $\displaystyle={\rm
trace}_{m}(\mathrm{D}^{2}f(x))+3\,m\,|\nabla f(x)|\,|\nabla\varphi(x)|$
$\displaystyle\leq(\lambda+3\,m\,C(\Gamma,\varphi))\,|\nabla f(x)|\,,$

thus completing the proof. ∎

###  3.2. A Lusin-type property of normal bundles

The following theorem is needed at a crucial step (4.29) in the proof of
Theorem 4.1.

######  Theorem 3.3 (Lusin-type property of normal bundles).

Suppose $\Gamma\subseteq M$ is a White $(m,\lambda)$-set in $(M,g)$ such that
$\Gamma$ is a countable union of sets with finite $\mathcal{H}^{m}$-measure.
Then the following implication holds:

$Z\subseteq\Gamma,\;\mathcal{H}^{m}(Z\cap\Gamma^{(m)})=0\quad\implies\quad\left\\{\begin{split}&\mathcal{H}^{n}(\mathcal{N}(\Gamma)\llcorner
Z)=0\,,\\\ &\mathcal{H}^{n-1}(\mathcal{N}^{1}(\Gamma)\llcorner
Z)=0\,.\end{split}\right.$ (3.3)

###### Proof.

By [San20b, 3.3, 3.8, 3.9], if $W\subseteq\mathbb{R}^{k}$ is open, $\Gamma$ is
a White $(m,\lambda)$-set in $W$, and $\Gamma$ is a countable union of sets
with finite $\mathcal{H}^{m}$-measure, then

$Z\subseteq\Gamma,\;\mathcal{H}^{m}(Z\cap\Gamma^{(m)})=0\quad\implies\quad\left\\{\begin{split}&\mathcal{H}^{k}\big{(}\mathcal{N}(\Gamma)\llcorner
Z\big{)}=0\,,\\\ &\mathcal{H}^{k-1}\big{(}\mathcal{N}(\Gamma)\llcorner
Z\big{)}=0\,.\end{split}\right.$ (3.4)

We now reduce the proof of (3.3) to an application of the Euclidean case
(3.4). To this end, by the Nash embedding theorem, we can directly assume that
$M$ is an $n$-dimensional embedded submanifold of the Euclidean space
$\mathbb{R}^{k}$ (for some large $k$) and $g$ is the Riemannian metric on $M$
induced by the Euclidean metric $\langle\cdot,\cdot\rangle$ of
$\mathbb{R}^{k}$. If $X\subseteq M$ is relatively closed in $M$, then we
denote by $\widetilde{\mathcal{N}}\overline{X}$ the normal bundle of the
closure in $\mathbb{R}^{k}$ of $X$, while we keep the symbol $\mathcal{N}X$
for the normal bundle of $X$ as a subset of $(M,g)$ (notice
$\mathcal{N}X\subseteq TM$). If $a\in M$, then $\pi_{a}$ and $\pi^{\perp}_{a}$
denote the orthogonal projections of $\mathbb{R}^{k}$ onto $T_{a}M$ and
$T^{\perp}_{a}M$ respectively. We now divide the rest of the proof into two
claims and one final argument.

Claim one : If $X\subseteq M$ is relatively closed in $M$, then

$\widetilde{\mathcal{N}}_{a}\overline{X}=\mathcal{N}_{a}X+T^{\perp}_{a}M\,,\qquad\forall
a\in X\,.$

Indeed, let $u\in\widetilde{\mathcal{N}}_{a}\overline{X}$ be such that
$\pi_{a}(u)\neq 0$ and $B\cap\overline{X}=\varnothing$, where, for some $r>0$,
$B$ is the open Euclidean ball in $\mathbb{R}^{k}$ of radius $r|u|$ centered
at $a+ru$. Since $u\in T^{\perp}_{a}(\partial B)$ and
$\langle\pi_{a}(u),u\rangle>0$, we conclude that $\pi_{a}(u)\in
T_{a}M\setminus T_{a}(\partial B)$. Then we choose an open neighborhood $V$ of
$a$ and a continuous function $\eta:V\cap M\rightarrow\mathbf{S}^{k-1}$ such
that $\eta(a)=\pi_{a}(u)/|\pi_{a}(u)|$ and $\eta(b)\in T_{b}M$ for every $b\in
V\cap M$. Since $[b\in V\cap M\cap\partial B]\mapsto{\rm
dist}(\eta(b),T_{b}(\partial B))$ is continuous and ${\rm
dist}(\eta(a),T_{a}(\partial B))>0$, there is an open neighborhood $W\subseteq
V$ of $a$ such that ${\rm dist}(\eta(b),T_{b}(\partial B))>0$ for every $b\in
W\cap\partial B\cap M$. Hence, $\dim\big{[}T_{b}M\cap T_{b}(\partial
B)\big{]}\leq n-1$ and $T_{b}M+T_{b}(\partial B)=\mathbb{R}^{k}$ for every
$b\in W\cap M\cap\partial B$. This means that the submanifolds $W\cap\partial
B$ and $W\cap M$ are transversal and consequently

$\partial B\cap M\cap W\quad\textrm{is an $n-1$-dimensional smooth submanifold
of $M$}$ (3.5)

with $T_{b}M\cap T_{b}(\partial B)=T_{b}(M\cap\partial B)$ for $b\in W\cap
M\cap\partial B$; see [GP74, pp. 29-30]. Now, observing that if $v\in
T_{a}(\partial B\cap M)=T_{a}(\partial B)\cap T_{a}(M)$, then
$\langle\pi_{a}^{\perp}(u),v\rangle=0$ and $\langle\pi_{a}(u),v\rangle=\langle
u,v\rangle=0$, we find that

$\pi_{a}(u)\in T^{\perp}_{a}(\partial B\cap M)\,.$ (3.6)

For $t$ sufficiently small, since $\langle\pi_{a}(u),u\rangle>0$, we notice
that $\exp(a,t\pi_{a}(u))\in B\cap M$ and we conclude from (3.6) that the open
geodesic ball of $(M,g)$ centred at $\exp(a,t\pi_{a}(u))$ and radius
$t|\pi_{a}(u)|$ is contained in $B\cap M$. This means, always for $t$
sufficiently small, that $d(\exp(a,t\pi_{a}(u)),X)=t|\pi_{a}(u)|$ (thanks to
$B\cap X=\varnothing$) and $\pi_{a}(u)\in\mathcal{N}_{a}(X)$.

We have thus proved
$\widetilde{\mathcal{N}}_{a}\overline{X}\subseteq\mathcal{N}_{a}X+T^{\perp}_{a}M$
for $a\in X$. To prove the opposite inclusion, let now $v\in\mathcal{N}_{a}X$
with $v\neq 0$. An open geodesic ball $G$ in $M$ with sufficiently small
radius is such that $\partial G$ is a smooth $n-1$ dimensional submanifold in
$\mathbb{R}^{k}$, $G\cap X=\varnothing$, $a\in\partial G$, and $v$ is an
interior normal of $G$ at $a$. Then there exists $r>0$ such that the open
Euclidean ball $B$ in $\mathbb{R}^{k}$ of radius $r$ centered at $a+rv$
satisfies $B\cap\partial G=\varnothing$ and $B\cap G\neq\varnothing$. Choosing
$r$ smaller if necessary, we can also ensure that $B$ is contained in the
tubular neighbourhood of $M$ where the nearest point projection onto $M$ is
single valued. This means that $B\cap M$ is a connected subset of
$M\setminus\partial G$; since $G$ is a connected component of
$M\setminus\partial G$ and $G\cap B\neq\varnothing$, we infer that $B\cap
M\subseteq G$. The latter inclusion implies that $B\cap X=\varnothing$. It
follows that $B\cap\overline{X}=\varnothing$ and
$\mathcal{N}_{a}X\subseteq\widetilde{\mathcal{N}}_{a}\overline{X}$. Now the
convexity of $\widetilde{\mathcal{N}}_{a}\overline{X}$ and the obvious
inclusion $T_{a}^{\perp}M\subseteq\widetilde{\mathcal{N}}_{a}\overline{X}$
allow to conclude
$\mathcal{N}_{a}X+T^{\perp}_{a}M\subseteq\widetilde{\mathcal{N}}_{a}\overline{X}+\widetilde{\mathcal{N}}_{a}\overline{X}=\widetilde{\mathcal{N}}_{a}\overline{X}$
and to complete the proof of claim one.

Claim two: If $W\subseteq\mathbb{R}^{k}$ is an open set such that
$\overline{W}\cap M$ is compact, then there is $\lambda_{W}\geq 0$ such that
$\Gamma\cap W$ is a White $(m,\lambda_{W})$-set in $W$. Firstly, notice that
$M\cap W$ and $\Gamma\cap W$ are relatively closed in $W$. Now, let
$b\in\Gamma\cap W$ and $f\in C^{2}(W)$ such that $f|_{\Gamma}$ has a local
maximum at $b$. Denoting by $\overline{\nabla}$ both the Euclidean gradient
operator and metric connection, and by $\overline{\mathrm{D}}^{\,2}$ the
Euclidean Hessian operator, we set
$\eta(x)=\pi^{\perp}_{x}(\overline{\nabla}f(x))$ for $x\in W\cap M$, notice
that $\pi_{x}(\overline{\nabla}f(x))=\nabla f(x)$, and compute

$\displaystyle\overline{\mathrm{D}}^{\,2}f(x)(v,w)$
$\displaystyle=\langle\overline{\nabla}_{v}\overline{\nabla}f(x),w\rangle=\langle\overline{\nabla}_{v}\nabla
f(x),w\rangle+\langle\overline{\nabla}_{v}\eta(x),w\rangle$
$\displaystyle=\mathrm{D}^{2}f(x)(v,w)+\langle
A_{\eta(x)}(v),w\rangle=\mathrm{D}^{2}f(x)(v,w)-Q(x)(v,w)$

for every $x\in M\cap W$ and $v,w\in T_{x}(M)$, where
$A_{\eta(x)}:T_{x}M\rightarrow T_{x}$ is the shape operator of $M$ in the
direction $\eta(x)$, $Q(x):T_{x}M\times T_{x}M\rightarrow\mathbb{R}$ is the
symmetric bilinear form defined as $Q(x)(v,w)=\langle S(v,w),\eta(x)\rangle$
and $S$ is the second fundamental form of $M$. If $Q(b)(v,v)\geq 0$ for all
$v\in T_{b}M$ then we obtain from [JT03, Lemma 2.3] and [Whi16, Lemma 12.3]

$\displaystyle{\rm trace}_{m}(\overline{\mathrm{D}}^{\,2}f(b))$
$\displaystyle\leq{\rm
trace}_{m}\big{[}\overline{\mathrm{D}}^{\,2}f(b)|T_{b}M\times T_{b}M\big{]}$
$\displaystyle={\rm trace}_{m}[\mathrm{D}^{2}f(b)-Q(b)]$
$\displaystyle\leq{\rm trace}_{m}(\mathrm{D}^{2}f(b))\leq\lambda|\nabla
f(b)|\leq\lambda|\overline{\nabla}f(b)|.$

Now we assume that $Q(b)(v,v)<0$ for some $v\in T_{b}M$. Then we define

$\mu_{W}=\inf\\{\langle S(u,u),\nu\rangle:x\in\overline{W}\cap M,\,u\in
T_{x}M,\,\nu\in(T_{x}M)^{\perp},\,|u|=|\nu|=1\\}$

and we notice that $-\infty<\mu_{W}<0$ and

$Q(b)(v,v)\geq\mu_{W}|\eta(b)|\langle v,v\rangle\qquad\textrm{for $v\in
T_{b}M$.}$

Therefore, by [JT03, Lemma 2.3] and [Whi16, Lemma 12.3]

$\displaystyle{\rm trace}_{m}(\mathrm{D}^{2}f(b))$ $\displaystyle={\rm
trace}_{m}\big{[}\overline{\mathrm{D}}^{\,2}f(b)|(T_{b}M\times
T_{b}M)+Q(b)\big{]}$ $\displaystyle\geq{\rm
trace}_{m}\big{[}\big{(}\overline{\mathrm{D}}^{\,2}f(b)+\mu_{W}|\eta(b)|\langle\cdot,\cdot\rangle\big{)}\big{|}T_{b}M\times
T_{b}(M)\big{]}$ $\displaystyle\geq{\rm
trace}_{m}\big{[}\overline{\mathrm{D}}^{\,2}f(b)+\mu_{W}|\eta(b)|\langle\cdot,\cdot\rangle\big{]}$
$\displaystyle={\rm
trace}_{m}(\overline{\mathrm{D}}^{\,2}f(b))+m\mu_{W}|\eta(b)|,$

and we deduce that

${\rm trace}_{m}(\overline{\mathrm{D}}^{\,2}f(b))\leq\lambda|\nabla
f(b)|-m\mu_{W}|\eta(b)|\leq(\lambda-m\mu_{W})|\overline{\nabla}f(b)|.$

Conclusion of the proof: Let $Z\subseteq\Gamma$ such that
$\mathcal{H}^{m}(Z\cap\Gamma^{(m)})=0$. Since
$\mathcal{N}_{a}(\Gamma)\subseteq T_{a}M$ it follows from claim one that

$\dim\widetilde{\mathcal{N}}_{a}\overline{\Gamma}=\dim\mathcal{N}_{a}\Gamma+\dim
T^{\perp}_{a}M\quad\textrm{for $a\in\Gamma$,}$

so that

$\mathcal{H}^{m}(Z\cap\\{a:\dim\widetilde{\mathcal{N}}_{a}\overline{\Gamma}=k-m\\})=0.$

It follows from claim two and (3.4) that
$\mathcal{H}^{k}(\widetilde{\mathcal{N}}\,\overline{\Gamma}\llcorner Z)=0$.
Let $P:M\times\mathbb{R}^{k}\rightarrow TM$ be the smooth map defined by

$P(a,u)=(a,\pi_{a}(u))\quad\textrm{for $(a,u)\in M\times\mathbb{R}^{k}$.}$

Since
$P(\widetilde{\mathcal{N}}\,\overline{\Gamma}\llcorner\Gamma)=\mathcal{N}\,\Gamma$
by claim one, noting by Theorem 2.1 that
$\widetilde{\mathcal{N}}\,\overline{\Gamma}\llcorner\Gamma$ is a countably
$k$-rectifiable subset of $M\times\mathbb{R}^{k}$ and $\mathcal{N}\,\Gamma$ is
a countably $n$-rectifiable subset of $TM$. By the coarea formula [Fed69a,
3.2.22],

$0=\int_{\widetilde{\mathcal{N}}\,\overline{\Gamma}\llcorner
Z}\textrm{ap}J_{n}P\,d\mathcal{H}^{k}=\int_{\mathcal{N}\Gamma\llcorner
Z}\mathcal{H}^{k-n}(T^{\perp}_{a}M)\,d\mathcal{H}^{n}_{(a,u)}\,.$

Since $\mathcal{H}^{k-n}(T^{\perp}_{a}M)=+\infty$ for every $a\in M$, we have
$\mathcal{H}^{n}(\mathcal{N}\Gamma\llcorner Z)=0$. ∎

##  4\. Viscous Heintze–Karcher inequalities

In Theorem 4.1 below, we prove the Heintze–Karcher inequalities of Brendle
[Bre13], and address their equality cases, in the viscous setting of White
[Whi16]. Starting from this result, in Theorem 4.2, we extend Brendle’s
rigidity theorem to the distributional setting. Throughout the section, $n\geq
3$ and $(M,g)$ is a Riemannian manifold which satisfies at least
${\rm(H0)}$–${\rm(H3)}$. We consider $\Sigma\subset M$ such that:

  1. (A1)

$\Sigma$ is a smooth embedded hypersurface in $M$ with
$\overline{\Sigma}\subset M^{\circ}$ and

$\mathcal{H}^{n-1}(\overline{\Sigma}\setminus\Sigma)=0\,,\qquad\mathcal{H}^{n-1}(\Sigma)<\infty\,.$

Notice carefully that we do not assume $\Sigma$ to be closed. Thus,
$\overline{\Sigma}\setminus\Sigma$ may be non-empty and may contain singular
points (i.e., $\overline{\Sigma}$ may fail to be an hypersurface at points in
$\overline{\Sigma}\setminus\Sigma$), and $\Sigma$ may consists of countably
many connected components. Our second main assumption is that
$\overline{\Sigma}$ is (topologically) a boundary, namely,

  1. (A2)

there is $\Omega\subset M$ open such that

either $\partial\overline{\Omega}=\overline{\Sigma}$ or
$\partial\overline{\Omega}=\overline{\Sigma}\cup N_{0}$, (4.1)

Now, under (4.1), assumption ${\rm(A1)}$ implies that $\Omega$ is a set of
finite perimeter in $M$ thanks to Federer’s criterion, see [Fed69b, 4.5.12].
In particular, if we denote by $\partial^{*}\Omega$ the reduced boundary of
$\Omega$, and by $\nu_{\Omega}$ its measure theoretic outer $g$-unit normal,
then we observe that $\Sigma\subset\partial^{*}\Omega$ and $\nu_{\Omega}$ is
smooth on $\Sigma$. It thus makes sense to define

$H_{\Sigma}=\vec{H}_{\Sigma}\cdot\nu_{\Omega}\quad\textrm{on $\Sigma$},$

where $\vec{H}_{\Sigma}$ is the mean curvature vector of $\Sigma$ in $(M,g)$.

######  Theorem 4.1.

If $n\geq 3$, $(M,g)$ satisfies ${\rm(H0)}$-${\rm(H3)}$, and the pair
$(\Sigma,\Omega)$ satisfies assumptions ${\rm(A1)}$, ${\rm(A2)}$, and

  1. (A3)

for some $\lambda\geq 0$, $\overline{\Sigma}$ is a White $(n-1,\lambda)$-set
in $M$,

then, denoting by $r$ the projection of $M=N\times(0,\bar{r})$ over
$(0,\bar{r})$, and setting

$f=h^{\prime}\circ r\,,\qquad g^{*}=f^{-2}\,g\,,$

the following statements hold:

  1. (a)

if $\partial\Omega=\overline{\Sigma}$, then

$(n-1)\int_{\Sigma}\frac{f}{H_{\Sigma}}\,d\mathcal{H}^{n-1}\geq
n\int_{\Omega}f\,d\mathcal{H}^{n}\,;$ (4.2)

  2. (b)

if $\partial\Omega=\overline{\Sigma}\cup N_{0}$, then

$(n-1)\int_{\Sigma}\frac{f}{H_{\Sigma}}\,d\mathcal{H}^{n-1}\geq
n\int_{\Omega}f\,d\mathcal{H}^{n}+h(0)^{n}{\rm vol}\,(N,g_{N})\,;$ (4.3)

  3. (c)

if either $(a)$ or $(b)$ holds with equality, then $\Sigma$ is umbilic in
$(M,g)$, $M\setminus\Omega$ has positive reach in $(M,g^{*})$, and555Here,
given $\nu=(\tau,a)\in T_{(x,t)}M\equiv T_{x}N\times\mathbb{R}$, we have set
$(g_{N})|_{(x,t)}(\nu,\nu)=(g_{N})_{x}(\tau,\tau)$.

$\Big{(}2\,\frac{h^{\prime\prime}}{h}+(n-2)\,\frac{((h^{\prime})^{2}-\rho)}{h^{2}}\Big{)}^{\prime}\,g_{N}(\nu_{\Omega},\nu_{\Omega})=0\,,\qquad\mbox{on
$\Sigma$}\,.$ (4.4)

######  Theorem 4.2.

If $n\geq 3$, $(M,g)\in\mathcal{B}_{n}\cup\mathcal{B}_{n}^{*}$, and the pair
$(\Sigma,\Omega)$ satisfies assumptions ${\rm(A1)}$,

  1. (A2)’

there is $\Omega\subset M$ open such that either
$\partial\overline{\Omega}=\overline{\Sigma}$ or
$\partial\overline{\Omega}=\overline{\Sigma}\cup N_{0}$;

  2. (A3)’

there is $H_{0}\geq 0$ such that, for every $Y\in\mathcal{X}(M)$,

$\int_{\Sigma}{\rm
div}\,^{\Sigma}Y\,d\mathcal{H}^{n-1}=H_{0}\,\int_{\Sigma}\langle\nu_{\Omega},Y\rangle\,d\mathcal{H}^{n-1}\,;$
(4.5)

then, for some $t_{0}\in(0,\bar{r})$, $\Omega=N\times(0,t_{0})$ and
$H_{0}=(n-1)\,h^{\prime}(t_{0})/h(t_{0})>0$.

######  Remark 4.1 (On the relation between Theorem 4.1 and Theorem 4.2).

As detailed in the proof of Theorem 4.2, by testing the constant mean
curvature condition (4.5) with the vector field $h\,(\partial/\partial r)$, we
see that $H_{0}$ appearing in (4.5) is positive (so that ${\rm(A2)}$’ implies
${\rm(A2)}$), and that either (4.2) or (4.3) (depending on whether
$\partial\Omega=\overline{\Sigma}$ or $\partial\Omega=\overline{\Sigma}\cup
N_{0}$) holds as an identity. Moreover, by Theorem 3.1, ${\rm(A3)}$’ implies
the validity of ${\rm(A3)}$. Therefore, under the assumptions of Theorem 4.2,
conclusion (c) of Theorem 4.1 holds too. When ${\rm(H3)^{*}}$ holds, then
(4.4) immediately implies that $\nu_{\Omega}(p)$ is parallel to
$(\partial/\partial r)|_{p}$ at every $p\in\Sigma$: this information, combined
with standard facts on sets of finite perimeter and with the positivity of
$H_{\Sigma}$, immediately implies that $\Omega$ is bounded by a single slice.
When, instead, only ${\rm(H3)}$ is assumed, the information in (4.4) may be
trivial. In this second case, arguing as in [Bre13], we deduce from
umbilicality and the Codazzi equations that $\nu_{\Omega}(p)$ is an
eigenvector of $({\rm Ric}_{M})|_{p}$ at every $p\in\Sigma$. Since ${\rm(H4)}$
implies that $(\partial/\partial r)|_{p}$ is a simple eigenvector of $({\rm
Ric}_{M})|_{p}$, we thus find that, at each $p\in\Sigma$, $\nu_{\Omega}(p)$ is
either parallel or orthogonal to $(\partial/\partial r)|_{p}$. Concluding
rigidity from this weaker information using only standard facts on sets of
finite perimeter does not seem immediate; however, the fact that
$M\setminus\Omega$ has positive reach in $(M,g^{*})$ can be exploited to
quickly reach the desired conclusion.

###### Proof of Theorem 4.1.

Preparation of $M$: The results of Section 2 and Section 3 require the
completeness of the ambient manifold. Notice that $(M,g)$ is not complete. A
first problem is that geodesics in $(M,g)$ may arrive in finite time to the
horizon $N_{0}$: this issue is fixed by passing from $g$ to $g^{*}=f^{-2}\,g$.
A different issue, however, is the behavior of geodesics near the
$\bar{r}$-end of $M$. To fix this second problem we argue as follows. By
assumption $\overline{\Sigma}\subset M^{\circ}$, there is
$(a,b)\subset\subset(0,\bar{r})$ such that

$\overline{\Sigma}\subset
N\times\Big{[}a,\frac{a+b}{2}\Big{]}\,,\qquad\Omega\subset
N\times\Big{(}0,\frac{a+b}{2}\Big{)}\,.$ (4.6)

Correspondingly we can consider a smooth positive function
$h_{b}:[0,+\infty)\to\mathbb{R}$ so that $h_{b}=h$ on $[0,b]$,
$h_{b}^{\prime}>0$ on $(0,\infty)$, and

$\sup_{[0,\infty)}h_{b}^{\prime}<\infty\,,\qquad\int_{b}^{\infty}\frac{dt}{h_{b}^{\prime}(t)}=+\infty\,.$

We introduce the metrics $g_{b}=dr\otimes dr+h_{b}(r)^{2}g_{N}$ and
$g^{*}_{b}=f_{b}^{-2}\,g_{b}$ on $N\times(0,\infty)$, where
$f_{b}(p)=h_{b}^{\prime}(r(p))$. Since ${\rm(H1)}$ and $h_{b}=h$ on $[0,b]$
imply

$\int_{0}^{b}\frac{dt}{h_{b}^{\prime}(t)}=+\infty\,,$

we easily see that $g^{*}_{b}$-geodesic balls centered at points $p\in
N\times(0,+\infty)$ are contained in compact slabs of the form $N\times[s,t]$
($0<s<t<\infty$). In particular, by the Hopf–Rinow theorem,
$(N\times(0,\infty),g^{*}_{b})$ is a complete Riemannian manifold. Notice that
$h_{b}$ satisfies assumption ${\rm(H3)}$ on $(0,b)$, where it coincides with
$h$, but, possibly, not on $(0,\infty)$.

We have thus reduced to the following situation: $M=N\times(0,\infty)$; the
metric $g=dr\otimes dr+h(r)^{2}\,g_{N}$ is such that ${\rm(H0)}$ and
${\rm(H1)}$ hold, ${\rm(H2)}$ holds on $(0,\infty)$ (i.e., $h^{\prime}>0$ on
$(0,\infty)$), ${\rm(H3)}$ holds on $(0,b)$, and

$\sup_{M}f<\infty\,;$ (4.7)

the metric $g^{*}=f^{-2}\,g$ is such that $(M,g^{*})$ is a complete Riemannian
manifold; and, finally, $\Sigma$ and $\Omega$ satisfy (4.6) in addition to
assumptions ${\rm(A1)}$, ${\rm(A2)}$, and ${\rm(A3)}$.

The “vertical” vector field $X$: The shortest path between points $(x,t_{1})$
and $(x,t_{2})$ in $(M,g^{*})$ with $t_{1}\in(0,t_{2})$ is given by
$s\in[t_{1},t_{2}]\mapsto(x,s)$ (while ${\rm
dist}_{g^{*}}((x,0),(x,t))=+\infty$ for every $x\in N$ and $t>0$). “Vertical”
segments are thus length minimizing geodesics in $(M,g^{*})$, and the vector
field $\partial/\partial r$ has a special role in the geometry of $(M,g^{*})$.
It is also convenient to consider, alongside with $\partial/\partial r$, its
rescaled version

$X=h\,\frac{\partial}{\partial r}\,.$

Simple computations show that

$\displaystyle{\rm div}\,(\partial/\partial r)=(n-1)\,\frac{f}{h\circ r}\,,$
(4.8) $\displaystyle{\rm div}\,^{\Gamma}(\partial/\partial
r)=\Big{\\{}(n-1)-\sum_{i=1}^{n}\,\langle\sigma_{i},\partial_{n}\rangle_{g}^{2}\Big{\\}}\,\frac{f}{h\circ
r}\,,\qquad\mbox{on $\Gamma$}\,,$ (4.9) $\displaystyle{\rm
div}\,\,X=n\,f\quad\mbox{on $M$}\,,\qquad{\rm
div}\,^{\Gamma}X=(n-1)\,f\quad\mbox{on $\Gamma$}\,,$ (4.10)

whenever $\Gamma$ is a $C^{1}$-hypersurface and $\\{\sigma_{i}\\}_{i=1}^{n-1}$
denotes a $g$-orthonormal basis of $T_{p}\Gamma$ for some $p\in\Gamma$. An
immediate consequence of (4.9) is that if $\Gamma$ is a closed
$C^{1,1}$-hypersurface in $M^{\circ}$ with $\mathcal{H}^{n-1}(\Gamma)<\infty$
and with mean curvature vector $\vec{H}_{\Gamma}$ in $(M,g)$, and if
$h^{\prime\prime}\circ r\geq 0$ on $\Gamma$, then

$(n-1)\,\mathcal{H}^{n-1}(\Gamma)\geq\int_{\Gamma}\frac{\langle\vec{H}_{\Gamma},X\rangle_{g}}{f}\,d\mathcal{H}^{n-1}\,.$
(4.11)

Indeed, by (4.10) we find

${\rm
div}\,^{\Gamma}\Big{(}\frac{X}{f}\Big{)}=(n-1)-\frac{\langle\nabla^{\Gamma}f,X\rangle_{g}}{f^{2}}\,,$

where $\nabla f=[(h^{\prime\prime}/h)\circ r]\,X$. Denoting by $X^{\Gamma}$
the projection of $X$ along $T\Gamma$, we find $\langle
X,\nabla^{\Gamma}f\rangle_{g}=[(h^{\prime\prime}/h^{\prime})\circ
r]\,|X^{\Gamma}|_{g}^{2}$. Hence, by applying the divergence theorem to $X/f$
on $\Gamma$ and by using $h^{\prime\prime}\circ r\geq 0$ on $\Gamma$, we find
(4.11).

“Immersed” geodesic flow from $\Sigma$: The vector field

$\nu^{*}_{\Omega}=f\,\nu_{\Omega}$

is a smooth $g^{*}$-unit normal to $\Sigma$, pointing out of $\Omega$. Now,
denoting by $\exp^{*}$ the exponential map in the complete Riemannian manifold
$(M,g^{*})$, we can define a smooth map $\Phi:\Sigma\times(0,\infty)\to M$ by
setting

$\Phi(x,t)=\Phi_{t}(x)=\exp^{*}(x,-t\,\nu^{*}_{\Omega}(x))\,,\qquad
x\in\Sigma\,,t>0\,.$

In this way, denoting by $J^{\Sigma}\Phi_{t}$ the tangential Jacobian666Here
and in the following, $\mathcal{H}^{k}$ and $J$ always denote Hausdorff
measures and Jacobians computed with respect to the metric $g$. of $\Phi_{t}$
along $\Sigma$,

$\Phi(x,0)=x\,,\quad\frac{\partial\Phi}{\partial
t}(x,0)=-f(x)\,\nu_{\Omega}(x)\,,\quad J^{\Sigma}\Phi_{0}(x)=1\,,\qquad\forall
x\in\Sigma\,.$ (4.12)

By (4.6) and (4.12), we have that $\Phi_{t}(x)\in N\times(0,b)$ with
$J^{\Sigma}\Phi_{t}(x)>0$ for every $t$ small enough. We can thus define a
lower semicontinuous, positive function $R_{\Sigma}:\Sigma\to(0,\infty]$ by
setting

$\displaystyle R_{\Sigma}(x)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\min\Big{\\{}\inf\big{\\{}t>0:J^{\Sigma}\Phi_{t}(x)=0\big{\\}},$
$\displaystyle\hskip 56.9055pt\inf\big{\\{}t>0:\Phi_{t}(x)\not\in
N\times(0,b)\big{\\}}\Big{\\}}\,,\qquad x\in\Sigma\,,$

so to have

$\displaystyle\Phi_{t}(x)\in N\times(0,b)\quad\mbox{and}\quad
J^{\Sigma}\Phi_{t}(x)>0\,,$ (4.13) $\displaystyle\forall(x,t)\in
A_{\Sigma}:=\Big{\\{}(x,t):x\in\Sigma\,,t\in(0,R_{\Sigma}(x))\Big{\\}}\,.$

By the Gauss lemma (see, e.g., [Sak96, pag. 60]), for every $(x,t)\in
A_{\Sigma}$,

$\frac{\partial\Phi}{\partial
t}(x,t)\in\Big{(}d\Phi_{t}(x)[T_{x}\Sigma]\Big{)}^{\perp}\,,\qquad\Big{|}\frac{\partial\Phi}{\partial
t}(x,t)\Big{|}_{g}=f(\Phi_{t}(x))\,.$ (4.14)

In particular, the tangential Jacobian $J^{A_{\Sigma}}\Phi$ of $\Phi$ along
$A_{\Sigma}$ is related to $J^{\Sigma}\Phi_{t}$ by the identity

$(J^{A_{\Sigma}}\Phi)(x,t)=f(\Phi_{t}(x))\,J^{\Sigma}\Phi_{t}(x)\,,\qquad\forall(x,t)\in
A_{\Sigma}\,.$ (4.15)

We now notice that, for every $t\in(0,\infty)$, $\\{R_{\Sigma}>t\\}$ is an
open subset of $\Sigma$, and

$\Gamma_{t}=\Phi_{t}\big{(}\\{R_{\Sigma}>t\\}\big{)}\,,\qquad t>0\,,$

is a smooth immersed hypersurface in $M$. Indeed, by construction, for every
$(x,t)\in A_{\Sigma}$ there is an open neighborhood $W$ of $x$ in $\Sigma$
such that $(\Phi_{t})|_{W}$ is a smooth embedding. Correspondingly, we
denote777Notice carefully that $\Phi$ may not be injective on the whole
$A_{\Sigma}$, therefore we will not be able to consider $H$ and ${\rm II}$ as
functions on $\Phi(A_{\Sigma})\subset M$. by $H(x,t)$ and ${\rm II}(x,t)$ the
scalar mean curvature and the second fundamental form (in the metric $g$) of
the smooth hypersurface $\Phi_{t}(W)$ at the point $\Phi_{t}(x)$ and with
respect to the normal

$\nu(x,t)=-\frac{1}{f(\Phi_{t}(x))}\frac{\partial\Phi}{\partial t}(x,t),$
(4.16)

(see (4.15)). Notice that $\nu(x,0)=\nu_{\Omega}(x)$ for $x\in\Sigma$. Since,
by (4.13), $\Phi_{t}(x)\in N\times(0,b)$ for every $t\in(0,R_{\Sigma}(x))$,
and since $h$ satisfies ${\rm(H3)}$ on $(0,b)$, the pointwise calculations in
[Bre13, Proposition 3.2] (which are based on ${\rm(H0)}$ and ${\rm(H3)}$ and
the Riccati equation) can be repeated verbatim to show that, everywhere on
$A_{\Sigma}$,

$\displaystyle\frac{\partial}{\partial
t}\Big{(}\frac{H}{f\circ\Phi}\Big{)}\geq|{\rm
II}|^{2}\geq\frac{H^{2}}{n-1}\,,$ (4.17)
$\displaystyle\frac{\partial}{\partial
t}\Big{(}\frac{f\circ\Phi}{H}\Big{)}\leq-|{\rm
II}|^{2}\,\frac{(f\circ\Phi)^{2}}{H^{2}}\,\leq-\frac{(f\circ\Phi)^{2}}{n-1}\,.$
(4.18)

Since $H(\cdot,0)>0$ on $\Sigma$ by assumption ${\rm(A2)}$, we see that (4.17)
implies

$\mbox{$H$ is positive on $A_{\Sigma}$}\,.$ (4.19)

Moreover, we have that

$\frac{\partial}{\partial
t}\,J^{\Sigma}\Phi_{t}(x)=-[(f\circ\Phi)\,H](x,t)\,J^{\Sigma}\Phi_{t}(x)\,,\qquad\forall(x,t)\in
A_{\Sigma}\,.$ (4.20)

Indeed, given $(x,t)\in A_{\Sigma}$ and $W$ as above, if $W^{\prime}$ is an
arbitrary open subset of $W\subset\Sigma$ then, by the area formula,

$\displaystyle\frac{d}{ds}\Big{|}_{s=t}\mathcal{H}^{n-1}(\Phi_{s}(W^{\prime}))=\frac{d}{ds}\Big{|}_{s=t}\,\int_{W^{\prime}}J^{\Sigma}\Phi_{s}\,d\mathcal{H}^{n-1}=\int_{W^{\prime}}\frac{\partial}{\partial
t}J^{\Sigma}\Phi_{t}\,d\mathcal{H}^{n-1}$

while, by the formula for the first variation of the area and by (4.14)

$\displaystyle\frac{d}{ds}\Big{|}_{s=t}\mathcal{H}^{n-1}(\Phi_{s}(W^{\prime}))\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!-\int_{W^{\prime}}[(f\circ\Phi)\,H](y,t)\,J^{\Sigma}\Phi_{t}(y)\,d\mathcal{H}^{n-1}_{y}\,,$

so that (4.20) follows by arbitrariness of $W^{\prime}$. We finally notice
that

$\displaystyle\mbox{if $x\in\Sigma$, $R_{\Sigma}(x)<\infty$, and
$J^{\Sigma}\Phi_{t}(x)\to 0$ as $t\to R_{\Sigma}(x)^{-}$}\,,$
$\displaystyle\mbox{then $H(x,t)\to+\infty$ as $t\to R_{\Sigma}(x)^{-}$}\,.$
(4.21)

Indeed, (4.20) gives

$\log\big{(}J^{\Sigma}\Phi_{t}(x)\big{)}=-\int_{0}^{t}\,[(f\circ\Phi)\,H](x,s)\,ds\,,\qquad\forall
t\in(0,R_{\Sigma}(x))\,.$

A refinement of (4.18): We claim that, everywhere on $A_{\Sigma}$,

$\displaystyle-\Big{\\{}\frac{\partial}{\partial
t}\Big{(}\frac{f\circ\Phi}{H}\Big{)}+\frac{(f\circ\Phi)^{2}}{n-1}\Big{\\}}$
$\displaystyle=$ $\displaystyle(f\circ\Phi)^{2}\,\Big{\\{}\frac{|{\rm
II}|^{2}}{H^{2}}-\frac{1}{n-1}\Big{\\}}$
$\displaystyle+\frac{f\circ\Phi}{H^{2}}\,h^{\prime}\,\big{\\{}{\rm
Ric}_{N}-\rho\,(n-2)\,g_{N}\big{\\}}(\nu,\nu)$
$\displaystyle+\frac{h^{3}}{2}\,\big{(}\mathcal{M}[h]\big{)}^{\prime}\,g_{N}(\nu,\nu)\,,$

where, by definition,

$\mathcal{M}[h]=2\,\frac{h^{\prime\prime}}{h}-(n-2)\,\frac{\rho-(h^{\prime})^{2}}{h^{2}}\,.$
(4.23)

Indeed, setting $T=(\Delta f)\,g-\mathrm{D}^{2}\,f+f\,{\rm Ric}_{M}$, by
[Bre13, Proposition 2.1] we have that

$\displaystyle T=h^{\prime}\,\big{\\{}{\rm
Ric}_{N}-\rho\,(n-2)\,g_{N}\big{\\}}+\frac{h^{3}}{2}\,\big{(}\mathcal{M}[h]\big{)}^{\prime}\,g_{N}\,,$
(4.24)

while the computations in [Bre13, Proposition 3.2] give

$\frac{\partial}{\partial
t}\Big{(}\frac{f\circ\Phi}{H}\Big{)}=(f\circ\Phi)^{2}\,\Big{\\{}\frac{1}{n-1}-\frac{|{\rm
II}|^{2}}{H^{2}}\Big{\\}}-\frac{(f\circ\Phi)}{H^{2}}\,T(\nu,\nu)\,.$ (4.25)

The combination of (4.24) and (4.25) leads immediately to (4).

Geodesic flow from $\Sigma$: We now consider the (embedded) geodesic flow of
$\Sigma$ (which is the main structure used in Brendle’s argument), i.e. we
relate $\Phi$ to the distance function from $\Sigma$ in $(M,g^{*})$. Let us
define, for the sake of brevity,
$u^{*}_{\Sigma}:\overline{\Omega}\to[0,\infty)$ and
$R_{\Sigma}^{*}:\Sigma\to(0,\infty)$ by setting

$\displaystyle u^{*}_{\Sigma}(p)$ $\displaystyle=$ $\displaystyle{\rm
dist}_{g^{*}}(p,\Sigma)\,,\qquad\qquad\hskip
62.59596ptp\in\overline{\Omega}\,,$ $\displaystyle R^{*}_{\Sigma}(x)$
$\displaystyle=$
$\displaystyle\rho^{*}_{\overline{\Sigma}}(x,-\nu^{*}_{\Omega}(x))$
$\displaystyle=$
$\displaystyle\sup\big{\\{}s>0:s=u_{\Sigma}^{*}(\Phi_{s}(x))\big{\\}}\,,\qquad
x\in\Sigma\,,$

(where, for $\Gamma$ closed in $M$, $\rho^{*}_{\Gamma}$ is defined as in (2.4)
with respect to the metric $g^{*}$) and then consider the sets

$\displaystyle\Omega_{t}$ $\displaystyle=$
$\displaystyle\big{\\{}p\in\Omega:u_{\Sigma}^{*}(p)>t\big{\\}}\,,$
$\displaystyle\Sigma_{t}$ $\displaystyle=$
$\displaystyle\big{\\{}p\in\Omega:u^{*}_{\Sigma}(p)=t\big{\\}}=M^{\circ}\cap\partial\Omega_{t}\,,$
$\displaystyle\Sigma^{\ast}_{t}$ $\displaystyle=$
$\displaystyle\Phi_{t}(\\{R^{*}_{\Sigma}>t\\})\subset\Sigma_{t}\,,$
$\displaystyle A_{\Sigma}^{*}$ $\displaystyle=$
$\displaystyle\big{\\{}(x,t):x\in\Sigma,\,t\in(0,R^{*}_{\Sigma}(x))\big{\\}}\,.$

(Notice that if $\partial\Omega=\overline{\Sigma}$, then
$\partial\Omega_{t}=\Sigma_{t}$; if, otherwise,
$\partial\Omega=\overline{\Sigma}\cup N_{0}$, then
$\partial\Omega_{t}=\Sigma_{t}\cup N_{0}$.) It is easily seen that
$R_{\Sigma}^{*}$ is continuous on $\Sigma$, so that $\\{R^{*}_{\Sigma}>t\\}$
is an open subset of $\Sigma$ for every $t>0$, and $A_{\Sigma}^{*}$ is open.
The fact that $\Phi$ is a diffeomorphism on $A_{\Sigma}^{*}$ with values in
$\Omega$ is standard (since $\Sigma$ is smooth), so that

$\Phi(A_{\Sigma}^{*})\subset\Omega\,,\qquad R_{\Sigma}(x)\geq
R_{\Sigma}^{*}(x)\quad\forall x\in\Sigma\,,\qquad A_{\Sigma}^{*}\subset
A_{\Sigma}\,,$ (4.26)

and $\Phi_{t}$ is a smooth embedding of $\\{R_{\Sigma}^{*}>t\\}$ into $M$. In
particular, for each $t>0$, $\Sigma^{\ast}_{t}$ is a (possibly empty,
embedded) hypersurface in $M$. (Notice that $\Gamma_{t}$ is, in general,
larger than $\Sigma_{t}^{*}$, immersed but not embedded, and unrelated to
$u_{\Sigma}^{*}$.) The vector field (see (4.16))

$\nu_{t}(y)=\nu(\Phi_{t}^{-1}(y),t)\,,\qquad y\in\Sigma_{t}^{*}\,,$ (4.27)

is a unit normal vector field to $\Sigma_{t}^{*}$ in $(M,g)$ with the property
that

$H(\Phi_{t}^{-1}(y),t)=H_{\Sigma_{t}^{*}}(y)\qquad\mbox{$y\in\Sigma_{t}^{*}$}\,,$
(4.28)

where $H_{\Sigma_{t}^{*}}$ is the scalar mean curvature of $\Sigma_{t}^{*}$
with respect to $\nu_{t}$. We now prove three important geometric properties
of the family $\\{\Sigma_{t}^{*}\\}_{t}$, namely, we show that

$\displaystyle\mathcal{H}^{n}(\Omega\setminus\Phi(A_{\Sigma}^{*}))=0\,,$
(4.29)
$\displaystyle\mathcal{H}^{n-1}(\Sigma_{t}\setminus\Sigma^{\ast}_{t})=0\,,\qquad\mbox{for
$\mathcal{L}^{1}$-a.e.\ $t>0$}\,,$ (4.30)

and that, when $\partial\Omega=\Sigma\cup N_{0}$,

$\displaystyle\mathcal{H}^{n-1}(\Sigma_{t}^{*})\geq h(0)^{n-1}\,{\rm
vol}\,(N)\,,\qquad\mbox{for $\mathcal{L}^{1}$-a.e. $t>0$}\,.$ (4.31)

We begin noticing that (4.29) is immediate to prove when $\Sigma$ is a
_closed_ smooth hypersurface, since, in that case, we trivially see that
$\Omega\setminus\Phi(A_{\Sigma}^{*})\subseteq{\rm Cut}^{\ast}(\Sigma)$, where
${\rm Cut}^{*}$ denotes the cut-locus in $(M,g^{*})$, and
$\mathcal{H}^{n}({\rm Cut}^{\ast}(\Sigma))=0$ by Theorem 2.1-(v). In our non-
smooth setting, we begin noticing that, by assumption ${\rm(A3)}$ and Lemma
3.2, there is $\lambda^{*}>0$ such that $\overline{\Sigma}$ is a White
$(n-1,\lambda^{*})$-subset of $(M,g^{*})$. Now, by construction,

$\Omega\setminus\Phi(A_{\Sigma}^{*})\subseteq{\rm
Cut}^{*}(\overline{\Sigma})\cup\exp\big{[}\mathcal{N}(\overline{\Sigma})\llcorner(\overline{\Sigma}\setminus\Sigma)\big{]}\,,$

Since $\overline{\Sigma}$ is a White $(n-1,\lambda^{*})$-set in $(M,g^{*})$
and $\mathcal{H}^{n-1}(\overline{\Sigma}\setminus\Sigma)=0$ we conclude from
Theorem 3.3 that

$\mathcal{H}^{n}(\mathcal{N}(\overline{\Sigma})\llcorner(\overline{\Sigma}\setminus\Sigma))=0\quad\textrm{and}\quad\mathcal{H}^{n}\big{(}\exp^{*}\big{[}\mathcal{N}(\overline{\Sigma})\llcorner(\overline{\Sigma}\setminus\Sigma)\big{]}\big{)}=0\,.$

Since $\mathcal{H}^{n}({\rm Cut}^{\ast}(\overline{\Sigma}))=0$ , we conclude
the proof of (4.29). By (4.29) and the coarea formula we have

$0=\int_{\Omega\setminus\Phi(A_{\Sigma}^{*})}|\nabla
u^{*}_{\Sigma}|_{g}\,d\mathcal{H}^{n}=\int_{0}^{\infty}\mathcal{H}^{n-1}(\Sigma_{s}\setminus\Phi(A_{\Sigma}^{*}))\,ds\,,$

which immediately implies (4.30) by
$\Sigma_{s}^{*}=\Sigma_{s}\cap\Phi(A_{\Sigma}^{*})$. Finally, to prove (4.31),
we notice that, since $u_{\Sigma}^{*}$ is a Lipschitz function,
$\Omega_{t}=\\{u_{\Sigma}^{*}>t\\}$ is a set of finite perimeter in $M$ for
$\mathcal{L}^{1}$-a.e. $t>0$. Since $\partial\Omega=\Sigma\cup N_{0}$ implies
$\partial\Omega_{t}=N_{0}\cup\Sigma_{t}$, by (4.30) the reduced boundary
$\partial^{*}\Omega_{t}$ of $\Omega_{t}$ is $\mathcal{H}^{n-1}$-equivalent to
the union of $N_{0}$ and $\Sigma_{t}^{*}$, with measure theoretic outer
$g$-unit normal $\nu_{\Omega_{t}}$ such that
$\nu_{\Omega_{t}}=-\partial/\partial r$ on $N_{0}$ and
$\nu_{\Omega_{t}}=\nu_{t}$ on $\Sigma_{t}^{*}$; in particular,

$\int_{\Omega_{t}}{\rm div}\,(\partial/\partial
r)=\int_{\Sigma_{t}^{*}}\langle\nu_{t},\partial/\partial
r\rangle_{g}\,d\mathcal{H}^{n-1}-\mathcal{H}^{n-1}(N_{0})\,.$

By (4.8), and since both $\partial/\partial r$ and $\nu_{t}$ have unit length
in $g$, we deduce
$\mathcal{H}^{n-1}(\Sigma_{t}^{*})\geq\mathcal{H}^{n-1}(N_{0})$, which is
(4.31).

A general Heintze–Karcher inequality: We now prove a general Heintze–Karcher
inequality, see (4.33) below, which implies both (4.2) and (4.3), and which
allows one to deduce the crucial positive reach information contained in
conclusion (c) when equality holds in either (4.2) or (4.3). We start noticing
that, by (4.18) and (4.20),

$\displaystyle\int_{\Sigma}\,d\mathcal{H}^{n-1}_{x}\int_{0}^{R_{\Sigma}(x)}(f\circ\Phi)^{2}(x,t)\,J^{\Sigma}\Phi_{t}(x)\,dt$
(4.32) $\displaystyle\hskip
28.45274pt\leq-(n-1)\,\int_{\Sigma}\,d\mathcal{H}^{n-1}_{x}\int_{0}^{R_{\Sigma}(x)}\frac{\partial}{\partial
t}\Big{(}\frac{f\circ\Phi}{H}\Big{)}\,(x,t)\,J^{\Sigma}\Phi_{t}(x)\,dt$
$\displaystyle\hskip
28.45274pt=-(n-1)\int_{\Sigma}\,d\mathcal{H}^{n-1}_{x}\int_{0}^{R_{\Sigma}(x)}(f\circ\Phi)^{2}\,(x,t)\,J^{\Sigma}\Phi_{t}(x)\,dt$
$\displaystyle\hskip
56.9055pt-(n-1)\,\int_{\Sigma}\Big{[}\Big{(}\frac{f\circ\Phi}{H}\Big{)}\,(x,t)\,J^{\Sigma}\Phi_{t}(x)\Big{]}\Big{|}_{t=0}^{t=R_{\Sigma}(x)}d\mathcal{H}^{n-1}_{x}\,.$

By (4.15), the area formula, and (4.29) we obtain

$\displaystyle\int_{\Sigma}\,d\mathcal{H}^{n-1}_{x}\int_{0}^{R_{\Sigma}^{*}(x)}(f\circ\Phi)^{2}\,(x,t)\,J^{\Sigma}\Phi_{t}(x)\,dt$
$\displaystyle=\int_{A_{\Sigma}^{*}}(f\circ\Phi)\,J^{A_{\Sigma}}\Phi\,d\mathcal{H}^{n}=\int_{\Phi(A_{\Sigma}^{*})}f\,d\mathcal{H}^{n}=\int_{\Omega}f\,d\mathcal{H}^{n}\,,$

while by (4.12),

$\Big{[}\int_{\Sigma}\Big{(}\frac{f\circ\Phi}{H}\Big{)}\,(x,t)\,J^{\Sigma}\Phi_{t}(x)d\mathcal{H}^{n-1}_{x}\Big{]}\Big{|}_{t=0}=\int_{\Sigma}\,\frac{f}{H_{\Sigma}}\,d\mathcal{H}^{n-1}\,,$

so that (4.32) gives

$\displaystyle
n\,\int_{\Omega}f\,d\mathcal{H}^{n}+\int_{\Sigma}\,d\mathcal{H}^{n-1}_{x}\int_{R_{\Sigma}^{*}(x)}^{R_{\Sigma}(x)}(f\circ\Phi)^{2}\,(x,t)\,J^{\Sigma}\Phi_{t}(x)\,dt+\mathcal{L}(\Sigma)$
$\displaystyle\leq(n-1)\,\int_{\Sigma}\frac{f}{H_{\Sigma}}\,d\mathcal{H}^{n-1}\,,$
(4.33)

where

$\mathcal{L}(\Sigma)=(n-1)\,\int_{\Sigma}\,\Big{[}\lim_{t\to
R_{\Sigma}(x)^{-}}\,\Big{(}\frac{f\circ\Phi}{H}\Big{)}\,(x,t)\,J^{\Sigma}\Phi_{t}(x)\Big{]}\,d\mathcal{H}^{n-1}_{x}\,.$

Notice that, for every $x\in\Sigma$, $t\mapsto
J^{\Sigma}\Phi_{t}(x)\,[(f\circ\Phi)/H](x,t)$ is decreasing on
$(0,R_{\Sigma}(x))$ thanks to (4.18) and (4.20): in particular, the integrand
in the definition of $\mathcal{L}(\Sigma)$ is a well-defined non-negative
function, (4.2) follows immediately from (4.33), and conclusion (a) is proved.

Conditional proof of conclusions (b) and (c): We now prove conclusions (b) and
(c) assuming the validity of the following inequality:

$\mathcal{L}(\Sigma)\geq h(0)^{n}\,{\rm vol}(N)\,,\qquad\mbox{when
$\partial\Omega=\Sigma\cup N_{0}$}\,.$ (4.34)

Indeed, if (4.34) holds, then (4.33) definitely implies (4.3), that is is
conclusion (b). Moreover, if equality holds in either (4.2) or (4.3), then
inequality (4.32) (appearing in the derivation of (4.33)) must hold as an
identity. Therefore, since (4.18) was used in proving (4.32), we find that if
equality holds in either (4.2) or (4.3), then

$\displaystyle\frac{\partial}{\partial
t}\Big{(}\frac{f\circ\Phi}{H}\Big{)}=-\frac{(f\circ\Phi)^{2}}{n-1}\,,\qquad\mbox{on
$A_{\Sigma}$}\,,$ (4.35)
$\displaystyle\int_{\Sigma}\,d\mathcal{H}^{n-1}_{x}\int_{R_{\Sigma}^{*}(x)}^{R_{\Sigma}(x)}(f\circ\Phi)^{2}\,(x,t)\,J^{\Sigma}\Phi_{t}(x)\,dt=0\,.$
(4.36)

By (4), we see that (4.35) gives

$|{\rm II}|^{2}=\frac{H^{2}}{n-1}\,,\qquad\mbox{on $A_{\Sigma}$}\,,$ (4.37)

which, tested with $t=0$, implies that $\Sigma$ is umbilical in $(M,g)$ (the
first part of conclusion (c)), as well as

$\frac{h^{3}}{2}\,\big{(}\mathcal{M}[h]\big{)}^{\prime}\,g_{N}(\nu,\nu)=0\,,\qquad\mbox{on
$A_{\Sigma}$}\,,$

which, tested with $t=0$, implies the validity of (4.4). A more delicate
argument is needed to deduce from (4.36) that $M\setminus\Omega$ has positive
reach in $(M,g^{*})$ (the second part of conclusion (c)), and it goes as
follows: Since $f>0$ on $M$ (by assumption (${\rm H2}$)) and
$J^{\Sigma}\Phi_{t}(x)>0$ for every $t\in(0,R_{\Sigma}(x))$ (by definition of
$R_{\Sigma}(x)$), (4.36) implies that

$R_{\Sigma}(x)=R_{\Sigma}^{*}(x)\,,\qquad\mbox{for $\mathcal{H}^{n-1}$-a.e.
$x\in\Sigma$}\,,$

whence we infer from the lower semicontinuity of $R_{\Sigma}$ and the
continuity of $R^{\ast}_{\Sigma}$ that

$R_{\Sigma}(x)=R_{\Sigma}^{*}(x)\,,\qquad\mbox{for every $x\in\Sigma$}\,.$
(4.38)

Let $x\in\Sigma$ be such that $R_{\Sigma}(x)<\infty$. Should it be that

${\rm dist}_{g^{*}}(\Phi_{t}(x),N_{0}\cup N_{b})\to 0^{+}\qquad\mbox{as $t\to
R_{\Sigma}(x)^{-}$}\,,$ (4.39)

the facts that $R_{\Sigma}(x)<\infty$ and ${\rm
dist}_{g^{*}}(N_{0},\Sigma)\geq{\rm dist}_{g^{*}}(N_{0},N_{a})=+\infty$ would
imply

${\rm dist}_{g^{*}}(\Phi_{t}(x),N_{b})\to 0^{+}\qquad\mbox{as $t\to
R_{\Sigma}(x)^{-}$}\,,$

and thus, by (4.38), $\Phi(A_{\Sigma}^{*})\subset\Omega$, and (4.6), that

$\lim_{t\to R_{\Sigma}(x)^{-}}\Phi_{t}(x)\in\overline{\Omega}\cap
N_{b}=\varnothing\,,$

a contradiction. Since (4.39) cannot occur, $R_{\Sigma}(x)<\infty$ must then
imply $J^{\Sigma}\Phi_{t}(x)\to 0^{+}$ as $t\to R_{\Sigma}(x)^{-}$: in this
case, (4.21) holds, and we can integrate (4.35) over $t\in(0,R_{\Sigma}(x))$
and take advantage of (4.12) so to find

$-\frac{f(x)}{H_{\Sigma}(x)}=-\int_{0}^{R_{\Sigma}(x)}\frac{f(\Phi_{t}(x))^{2}}{n-1}\,dt\geq-\frac{R_{\Sigma}(x)}{n-1}\,\sup_{M}\,f^{2}\,,$

where $\sup_{M}f^{2}<\infty$ thanks to (4.7). Since $0<H_{\Sigma}\leq\lambda$
on $\Sigma$ by assumption ${\rm(A3)}$, we conclude (using again (4.6)) that

$R_{\Sigma}(x)\geq\frac{(n-1)}{\lambda\,\sup_{M}f^{2}}\,\inf_{\Sigma}f\geq\frac{(n-1)}{\lambda\,\sup_{M}f^{2}}\,\inf_{[a,b]}h^{\prime}\,.$

We have thus proved the existence of a positive constant $c(\Sigma)$ such
that888Since it is false that every
$(x,\eta)\in\mathcal{N}^{1}\overline{\Sigma}$ is an accumulation point of
$\\{(x,-\nu_{\Omega}^{*}(x)):x\in\Sigma\\}$, we cannot deduce a lower bound
for $\rho_{\overline{\Sigma}}$ on $\mathcal{N}^{1}\overline{\Sigma}$ by simply
combining (4.40),
$R_{\Sigma}^{*}(x)=\rho^{*}_{\overline{\Sigma}}(x,-\nu_{\Omega}^{*}(x))$, and
the upper semicontinuity of $\rho^{*}_{\overline{\Sigma}}$ on
$\mathcal{N}^{1}\overline{\Sigma}$. In general,
$\mathcal{N}^{1}\overline{\Sigma}$ minus the closure of
$\\{(x,-\nu_{\Omega}^{*}(x)):x\in\Sigma\\})$ may even be of positive
$\mathcal{H}^{n-1}$-measure.  $R_{\Sigma}^{*}(x)=R_{\Sigma}(x)\geq
c(\Sigma)\,,\qquad\forall x\in\Sigma\,.$ (4.40) By assumption ${\rm(A2)}$,
$\partial(M\setminus\Omega)=\overline{\Sigma}$, therefore

$\displaystyle\mathcal{N}^{1}(M\setminus\Omega)\\!\\!$ $\displaystyle\subset$
$\displaystyle\\!\\!\mathcal{N}^{1}\overline{\Sigma}\,,$ (4.41)
$\displaystyle\mathcal{N}^{1}(M\setminus\Omega)\cap\big{(}\mathcal{N}^{1}\overline{\Sigma}\,\,\llcorner\Sigma\big{)}\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\big{\\{}(x,-\nu_{\Omega}^{*}(x)):x\in\Sigma\big{\\}}\,.$
(4.42)

At the same time, by applying Theorem 3.3 with $\Gamma=\overline{\Sigma}$
(which is admissible by assumption ${\rm(A3)}$), $m=n-1$, and
$Z=\overline{\Sigma}\setminus\Sigma$ (which is admissible since, by assumption
${\rm(A1)}$, $\mathcal{H}^{n-1}(\overline{\Sigma}\setminus\Sigma)=0$), we find
that

$\mathcal{H}^{n-1}\big{(}\mathcal{N}^{1}\overline{\Sigma}\,\,\llcorner\,\big{(}\overline{\Sigma}\setminus\Sigma\big{)}\big{)}=0\,.$
(4.43)

By combining (4.41), (4.42) and (4.43) we thus find

$(x,\eta)=(x,-\nu_{\Omega}^{*}(x))\qquad\mbox{for $\mathcal{H}^{n-1}$-a.e.
$(x,\eta)\in\mathcal{N}^{1}(M\setminus\Omega)$}\,,$

so that the function, for $\mathcal{H}^{n-1}$-a.e.
$(x,\eta)\in\mathcal{N}^{1}(M\setminus\Omega)$ we have

$\displaystyle\rho^{*}_{M\setminus\Omega}(x,\eta)$ $\displaystyle=$
$\displaystyle\rho^{*}_{M\setminus\Omega}(x,-\nu_{\Omega}^{*}(x))$
$\displaystyle=$ $\displaystyle\sup\big{\\{}s>0:s={\rm
dist}_{g^{*}}\big{(}\exp^{*}\big{(}x,-s\,\nu_{\Omega}^{*}(x)\big{)},M\setminus\Omega\big{)}\big{\\}}\,.$

Since ${\rm dist}_{g^{*}}(p,M\setminus\Omega)={\rm
dist}_{g^{*}}(p,\overline{\Sigma})$ for $p\in\Omega$ and since

$\exp^{*}(x,-s\,\nu_{\Omega}^{*}(x))\in\Omega\,,\qquad\forall
s\in(0,\rho^{*}_{M\setminus\Omega}(x,-\nu_{\Omega}^{*}(x)))\,,$

by (4.40) we conclude that, for $\mathcal{H}^{n-1}$-a.e.
$(x,\eta)\in\mathcal{N}^{1}(M\setminus\Omega)$,

$\rho^{*}_{M\setminus\Omega}(x,\eta)=\rho^{*}_{\overline{\Sigma}}(x,-\nu_{\Omega}^{*}(x))=R_{\Sigma}^{*}(x)\geq
c(\Sigma)>0\,.$

We can thus apply Theorem 2.1-(vii) and Remark 2.1 to finally conclude that
$M\setminus\Omega$ is a set of positive reach in $(M,g^{*})$. We are thus left
to prove (4.34) to complete the proof of conclusions (b) and (c).

Proof of (4.34): We start by setting, for every
$(x,t)\in\Sigma\times[0,\infty)$,

$g(x,t)=\left\\{\begin{split}&\Big{(}\frac{f\circ\Phi}{H}\Big{)}\,(x,t)\,J^{\Sigma}\Phi_{t}(x)\,,\qquad
x\in\Sigma\,,t\in(0,R_{\Sigma}(x))\,,\\\ &0\,,\qquad\hskip
101.29173ptx\in\Sigma\,,t\geq R_{\Sigma}(x)\,.\end{split}\right.$

By (4.18), (4.20), and the non-negativity of $f$, $H$ and
$J^{\Sigma}\Phi_{t}$, we see that $t\in[0,\infty)\mapsto g(x,t)$ is
decreasing, and thus provides a non-negative extension of
$J^{\Sigma}\Phi\,(f\circ\Phi)/H$ from $A_{\Sigma}$ to the whole
$\Sigma\times[0,\infty)$ such that

$\mathcal{L}(\Sigma)=(n-1)\,\lim_{t\to\infty}\int_{\Sigma}\,g(x,t)\,d\mathcal{H}^{n-1}_{x}\,.$

Now, since $\\{R_{\Sigma}^{*}>t\\}\subset\Sigma$ and since
$t<R_{\Sigma}^{*}(x)$ implies $t<R_{\Sigma}(x)$ we have

$\int_{\Sigma}\,g(x,t)\,d\mathcal{H}^{n-1}_{x}\geq\int_{\\{R_{\Sigma}^{*}>t\\}}\Big{(}\frac{f\circ\Phi}{H}\Big{)}\,J^{\Sigma}\Phi_{t}\,d\mathcal{H}^{n-1}_{x}=\int_{\Sigma_{t}^{*}}\frac{f}{H_{\Sigma_{t}^{*}}}\,d\mathcal{H}^{n-1}\,,$

where we have used (4.28). Now, by the Cauchy–Schwartz inequality,

$\displaystyle\int_{\Sigma_{t}^{*}}\frac{f}{H_{\Sigma_{t}^{*}}}\,d\mathcal{H}^{n-1}\geq\mathcal{H}^{n-1}(\Sigma_{t}^{*})^{2}\,\Big{(}\int_{\Sigma_{t}^{*}}\frac{H_{\Sigma_{t}^{*}}}{f}\,d\mathcal{H}^{n-1}\Big{)}^{-1}$

so that, in summary,

$\mathcal{L}(\Sigma)\geq(n-1)\,\limsup_{t\to\infty}\mathcal{H}^{n-1}(\Sigma_{t}^{*})^{2}\,\Big{(}\int_{\Sigma_{t}^{*}}\frac{H_{\Sigma_{t}^{*}}}{f}\,d\mathcal{H}^{n-1}\Big{)}^{-1}\,.$
(4.44)

Claim: for every $\lambda\in(0,1)$ there is $t_{0}=t_{0}(\lambda)$ so that, if
$t>t_{0}$, then

$\displaystyle\inf_{\Sigma_{t}^{*}}\langle X,\nu_{t}\rangle_{g}\\!\\!$
$\displaystyle\geq$ $\displaystyle\\!\\!\lambda\,h(0)\,,$ (4.45)
$\displaystyle(n-1)\,\mathcal{H}^{n-1}(\Sigma_{t}^{*})\\!\\!$
$\displaystyle\geq$
$\displaystyle\\!\\!\int_{\Sigma_{t}^{*}}H_{\Sigma_{t}^{*}}\,\frac{\langle
X,\nu_{t}\rangle_{g}}{f}\,d\mathcal{H}^{n-1}\,,$ (4.46)

with $\nu_{t}$ as in (4.27). Notice that by combining (4.44), (4.45) and
(4.46) we obtain indeed that

$\displaystyle\mathcal{L}(\Sigma)$ $\displaystyle\geq$
$\displaystyle(n-1)\,\limsup_{t\to\infty}\inf_{\Sigma_{t}^{*}}\langle
X,\nu_{t}\rangle_{g}\,\mathcal{H}^{n-1}(\Sigma_{t}^{*})^{2}\,\Big{(}\int_{\Sigma_{t}^{*}}\frac{H_{\Sigma_{t}^{*}}}{f}\,\langle
X,\nu_{t}\rangle_{g}\,\Big{)}^{-1}$ $\displaystyle\geq$
$\displaystyle\lambda\,h(0)\,\limsup_{t\to\infty}\,\mathcal{H}^{n-1}(\Sigma_{t}^{*})\geq\lambda
h(0)^{n}\,{\rm vol}\,(N)\,,$

where in the last step we have used (4.31). By letting $\lambda\to 1^{-}$ we
deduce (4.34). We are thus left to prove (4.45) and (4.46) to complete the
proof of conclusions (b) and (c).

Proof of (4.45): Recalling that $\Omega_{t}=\\{x\in\Omega:{\rm
dist}_{g^{*}}(x,\overline{\Sigma})>t\\}$, we now consider

$\displaystyle u_{\Omega_{t}}^{*}={\rm
dist}_{g^{*}}(\cdot,\Omega_{t})\,,\qquad t>0\,,$

and notice that, with the same argument used in the proof of [Bre13, Lemma
3.6], for every $\lambda\in(0,1)$ there is $t_{0}=t_{0}(\lambda)$ such that,
if $p\in\Omega_{t_{0}}$ and $\alpha$ is a $g^{*}$-unit speed geodesic with
$\alpha(0)=p$ and $\alpha(u^{*}_{\Sigma}(p))\in\overline{\Sigma}$, then
$|\alpha^{\prime}(0)|_{g}=f(p)$ and

$\langle\alpha^{\prime}(0),\partial/\partial r\rangle_{g}\geq\lambda\,f(p)\,.$
(4.47)

(In more geometric terms, every $g^{*}$-unit speed geodesic that ends up in
$\overline{\Sigma}$ after originating in $\Omega$ from a point at a
sufficiently large distance from $\overline{\Sigma}$, must have an initial
velocity with “almost vertical” direction). If $t>t_{0}$ we can apply this
statement to any $p\in\Sigma_{t}^{*}\subset\Sigma_{t}\subset\Omega_{t_{0}}$
and with $\alpha^{\prime}(0)=f(p)\,\nu_{t}(p)$, so to find

$\langle\nu_{t}(p),\partial/\partial r\rangle_{g}\geq\lambda\qquad\forall
p\in\Sigma_{t}^{*}\,,$

from which (4.45) follows since $X=h\,\partial/\partial r$ and $h\geq h(0)$ on
$M$.

As an additional consequence of (4.47), setting from now on
$t_{0}=t_{0}(\lambda)|_{\lambda=1/2}$, we also notice that, for every $t\geq
t_{0}$,

$\mbox{$\Omega_{t}$ has positive reach in $(M,g^{*})$}\,.$ (4.48)

To prove this, thanks to Theorem 2.3, we only need to show that
$u_{\Sigma}^{*}$ has no critical points in $\Omega_{t_{0}}$
($t_{0}=t_{0}(\lambda)|_{\lambda=1/2}$), that is, that there cannot be
$p\in\Omega_{t_{0}}$ such that for every $v\in T_{p}M$ with $|v|_{g^{*}}=1$
one can find a $g^{*}$-unit speed geodesic $\alpha$ with $\alpha(0)=p$ and
$\alpha(u_{\Sigma}^{*}(p))\in\overline{\Sigma}$ such that $\langle
v,\alpha^{\prime}(0)\rangle_{g}\geq 0$; and, indeed, any such $\alpha$ would
satisfy $\langle\alpha^{\prime}(0),\partial/\partial r\rangle_{g^{*}}\geq
f(p)/2$ by (4.47), so that, taking $v=-f(p)\,(\partial/\partial r)$, we would
obtain a contradiction.

Proof of (4.46): The proof is based on an approximation argument. Precisely,
for $t>0$ and $\varepsilon<\min\\{1,t\\}$, we consider the sets

$W_{t,\varepsilon}=\big{\\{}x\in
M:u_{\Omega_{t}}^{*}(x)=\varepsilon\big{\\}}\,,$

so that

$\Sigma_{t-\varepsilon}^{*}\subset
W_{t,\varepsilon}\subset\Omega_{t-1}\setminus\overline{\Omega}_{t}\,,\qquad\forall
t>0\,,\varepsilon<\min\\{1,t\\}\,,$ (4.49)

and reduce the proof of (4.46) to showing that

$\displaystyle\lim_{\varepsilon\to
0^{+}}\mathcal{H}^{n-1}(W_{t,\varepsilon})=\mathcal{H}^{n-1}(\Sigma_{t})\,,$
(4.50) $\displaystyle\lim_{\varepsilon\to
0^{+}}\int_{\Sigma^{\ast}_{t-\varepsilon}}\\!\\!\\!H_{\Sigma^{\ast}_{t-\varepsilon}}\frac{\langle
X,\nu_{t-\varepsilon}\rangle_{g}}{f}\,d\mathcal{H}^{n-1}=\int_{\Sigma^{\ast}_{t}}H_{\Sigma^{\ast}_{t}}\,\frac{\langle
X,\nu_{t}\rangle_{g}}{f}\,d\mathcal{H}^{n-1}\,,\hskip 28.45274pt$ (4.51)
$\displaystyle\liminf_{\varepsilon\to
0^{+}}\int_{W_{t,\varepsilon}\setminus\Sigma^{*}_{t-\varepsilon}}\,\frac{\langle\vec{H}_{W_{t,\varepsilon}},X\rangle_{g}}{f}\,d\mathcal{H}^{n-1}\geq
0\,.$ (4.52)

Indeed, thanks to ${\rm(H1)}$, there is $r_{1}>0$ such that
$h^{\prime\prime}>0$ on $(0,2\,r_{1})$. Up to further increase the value of
$t_{0}$, we can ensure $\Omega_{t-1}\subseteq N\times[0,r_{1}]$ for every
$t\geq t_{0}$. In particular, by $W_{t,\varepsilon}\subset\Omega_{t-1}$, we
conclude that $h^{\prime\prime}>0$ on $W_{t,\varepsilon}$. Since
$W_{t,\varepsilon}$ is a closed $C^{1,1}$-hypersurface in $M^{\circ}$ with
$\mathcal{H}^{n-1}(W_{t,\varepsilon})<\infty$, by (4.11) we find

$\displaystyle(n-1)\,\mathcal{H}^{n-1}(W_{t,\varepsilon})\geq\int_{W_{t,\varepsilon}}\,\frac{\langle\vec{H}_{W_{t,\varepsilon}},X\rangle_{g}}{f}\,d\mathcal{H}^{n-1}$
(4.53)
$\displaystyle=\int_{W_{t,\varepsilon}\setminus\Sigma^{*}_{t-\varepsilon}}\,\frac{\langle\vec{H}_{W_{t,\varepsilon}},X\rangle_{g}}{f}\,d\mathcal{H}^{n-1}+\int_{\Sigma^{*}_{t-\varepsilon}}\,H_{\Sigma^{*}_{t-\varepsilon}}\,\frac{\langle\nu_{t-\varepsilon},X\rangle_{g}}{f}\,d\mathcal{H}^{n-1}$

where we have used $\Sigma^{*}_{t-\varepsilon}\subset W_{t,\varepsilon}$ to
deduce

$\vec{H}_{W_{t,\varepsilon}}=\vec{H}_{\Sigma^{*}_{t-\varepsilon}}=H_{\Sigma^{*}_{t-\varepsilon}}\,\nu_{t-\varepsilon}\quad\mbox{on
$\Sigma^{*}_{t-\varepsilon}$}\,.$

By using (4.50), (4.51), (4.52), and (4.53) we deduce immediately (4.46). We
now turn to the proof of (4.50), (4.51), and (4.52). We shall use the
following preliminary step:

Proof of (4.50) and (4.51): Thanks to (4.48) we can apply Theorem 2.2 with
$A=\overline{\Omega}_{t}$, $t\geq t_{0}$ and find
$\varepsilon_{t}\in(0,\min\\{1,t\\})$ such that, if
$\varepsilon\in(0,\varepsilon_{t})$, then
$W_{t,\varepsilon}=\\{u_{\Omega_{t}}^{*}=\varepsilon\\}$ is a compact
$C^{1,1}$-hypersurface contained in $\operatorname{{\rm UP}}(\Omega_{t})$ and
the $g^{*}$-geodesic flow
$\Psi^{t}:\mathcal{N}^{1}(\overline{\Omega}_{t})\times[0,\varepsilon_{t})\to\\{0\leq
u_{\Omega_{t}}^{*}<\varepsilon_{t}\\}$ defined by

$\Psi^{t}(p,\eta,\varepsilon)=\Psi_{\varepsilon}^{t}(p,\eta)=\exp^{*}(p,\varepsilon\,\eta)\,,\qquad(p,\eta,\varepsilon)\in\mathcal{N}^{1}(\overline{\Omega}_{t})\times[0,\varepsilon_{t})\,,$

is such that $\Psi^{t}_{\varepsilon}$ is a locally bi-Lipschitz map
$\mathcal{N}^{1}(\overline{\Omega}_{t})$ to $W_{t,\varepsilon}$ when
$\varepsilon>0$, with $\Psi^{t}_{0}(p,\eta)=p$. In particular,

$\Theta^{t}:=\mathcal{N}^{1}(\overline{\Omega}_{t})$

is a $(n-1)$-dimensional compact Lipschitz submanifold of $TM$. On noticing
that $\mathcal{N}^{1}_{p}(\overline{\Omega}_{t})=\varnothing$ for every
$p\in\Omega_{t}\cup N_{0}$, since $\partial\Omega_{t}=\Sigma_{t}\cup N_{0}$ we
find that

$\mathcal{N}^{1}(\overline{\Omega}_{t})=\mathcal{N}^{1}(\overline{\Omega}_{t})\,\llcorner\,\Sigma_{t}\,,\qquad\mbox{i.e.
$(p,\eta)\in\Theta^{t}$ implies $p\in\Sigma_{t}$}\,.$ (4.54)

Moreover, by Theorem 2.1-(iii) (applied to $\overline{\Omega}_{t}$, see, in
particular, (2.5)),

$\mathcal{H}^{0}(\mathcal{N}^{1}_{p}(\overline{\Omega}_{t}))=1\,,\qquad\mbox{for
$\mathcal{H}^{n-1}$-a.e. $p\in\Sigma_{t}$}\,.$ (4.55)

Finally, the smoothness of $(p,\eta)\in
TM\mapsto\exp^{*}(p,\varepsilon\,\eta)\in M$ ensures that

$\lim_{\varepsilon\to
0^{+}}J^{\Theta^{t}}\Psi^{t}_{\varepsilon}(p,\eta)=J^{\Theta^{t}}\Psi^{t}_{0}(p,\eta)\,,$
(4.56)

for $\mathcal{H}^{n-1}$-a.e. $(p,\eta)\in\Theta^{t}$ (i.e., at every
$(p,\eta)$ such that $T_{(p,\eta)}\Theta^{t}$ exists). Since
$\Psi^{t}_{\varepsilon}$ and its differential are locally bounded in $(M,g)$,
we can apply the area formula (to $\Psi^{t}_{\varepsilon}$), the dominated
convergence theorem (in combination with (4.56)), and the area formula again
(to $\Psi^{t}_{0}$) to find

$\displaystyle\lim_{\varepsilon\to
0^{+}}\mathcal{H}^{n-1}(W_{t,\varepsilon})=\lim_{\varepsilon\to
0^{+}}\int_{\Theta^{t}}J^{\Theta^{t}}\Psi^{t}_{\varepsilon}\,d\mathcal{H}^{n-1}=\int_{\Theta^{t}}J^{\Theta^{t}}\Psi^{t}_{0}\,d\mathcal{H}^{n-1}$
$\displaystyle=\int_{\Psi^{t}_{0}(\Theta^{t})}\\!\\!\mathcal{H}^{0}\big{(}(\Psi^{t}_{0})^{-1}(x)\big{)}\,d\mathcal{H}^{n-1}_{x}=\int_{\Psi^{t}_{0}(\Theta^{t})}\\!\\!\mathcal{H}^{0}\big{(}\mathcal{N}_{x}^{1}(\overline{\Omega}_{t})\big{)}\,d\mathcal{H}^{n-1}_{x}\,,$

which, combined with (4.55) and $\Psi^{t}_{0}(\Theta^{t})\subset\Sigma_{t}$
(i.e. (4.54)), gives

$\lim_{\varepsilon\to
0^{+}}\mathcal{H}^{n-1}(W_{t,\varepsilon})=\int_{\Psi^{t}_{0}(\Theta^{t})}\\!\\!\mathcal{H}^{0}\big{(}\mathcal{N}_{x}^{1}(\overline{\Omega}_{t})\big{)}\,d\mathcal{H}^{n-1}_{x}\leq\mathcal{H}^{n-1}(\Sigma_{t})\,.$
(4.57)

Now, to prove (4.50), let us consider the diffeomorphisms
$\phi_{\varepsilon}:\Sigma^{\ast}_{t}\rightarrow\Sigma^{\ast}_{t-\varepsilon}$
defined by

$\phi_{\varepsilon}(x)=\Phi_{t-\varepsilon}\big{(}\Phi_{t}^{-1}(x)\big{)}=\exp^{*}(x,\varepsilon\,f(x)\,\nu_{t}(x))\,,\qquad\forall
x\in\Sigma_{t}^{*}\,.$

By the area formula,

$\mathcal{H}^{n-1}(\phi_{\varepsilon}(\Sigma^{\ast}_{t}))=\int_{\Sigma^{\ast}_{t}}J^{\Sigma^{\ast}_{t}}\phi_{\varepsilon}\,d\mathcal{H}^{n-1}\,.$

Since $\phi_{\varepsilon}\to{\rm Id}$ and
$J^{\Sigma^{\ast}_{t}}\phi_{\varepsilon}\to 1$ on $\Sigma^{\ast}_{t}$ as
$\varepsilon\to 0$, we conclude by dominated convergence that

$\lim_{\varepsilon\to
0^{+}}\mathcal{H}^{n-1}(\Sigma^{\ast}_{t-\varepsilon})=\mathcal{H}^{n-1}(\Sigma^{\ast}_{t}).$
(4.58)

Thus, by combining (4.57) and (4.58) with the facts that
$\Sigma^{\ast}_{t-\varepsilon}\subset W_{t,\varepsilon}$ and $\Sigma_{t}^{*}$
is $\mathcal{H}^{n-1}$-equivalent to $\Sigma_{t}$ (recall (4.30)), we deduce
(4.50) and

$\lim_{\varepsilon\to
0^{+}}\mathcal{H}^{n-1}(W_{t,\varepsilon}\setminus\Sigma^{\ast}_{t-\varepsilon})=0\,.$
(4.59)

To prove (4.51) we notice that, since
$(\Phi^{-1})|_{\Sigma_{s}^{*}}=(\Phi_{s})^{-1}$ on $\Sigma_{s}^{*}$, we have

$(\Phi^{-1})|_{\Sigma^{*}_{t-\varepsilon}}\circ\phi_{\varepsilon}=(\Phi_{t})^{-1}=(\Phi^{-1})|_{\Sigma^{*}_{t}}\qquad\mbox{on
$\Sigma_{t}^{*}$}\,,$

so that, the area formula gives

$\displaystyle\int_{\Sigma^{\ast}_{t-\varepsilon}}\\!\\!\\!H_{\Sigma_{t-\varepsilon}^{*}}\frac{\langle
X,\nu_{t-\varepsilon}\rangle_{g}}{f}\,d\mathcal{H}^{n-1}\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\int_{\Sigma^{\ast}_{t-\varepsilon}}\\!\\!\\!(H\circ(\Phi_{t-\varepsilon})^{-1})\,\frac{\langle
X,\nu_{\Omega}\circ(\Phi_{t-\varepsilon})^{-1}\rangle_{g}}{f}\,$
$\displaystyle=$
$\displaystyle\\!\\!\\!\int_{\Sigma^{\ast}_{t}}H_{\Sigma^{*}_{t}}\,\frac{\langle
X\circ\phi_{\varepsilon},\nu_{t}\rangle_{g}}{f\circ\phi_{\varepsilon}}\,J\phi_{\varepsilon}\,d\mathcal{H}^{n-1}\,,$

for every $\varepsilon>0$. Then (4.51) follows by dominated convergence.

Proof of (4.52): We finally prove (4.52), that is,

$\liminf_{\varepsilon\to
0^{+}}\int_{W_{t,\varepsilon}\setminus\Sigma^{*}_{t-\varepsilon}}\,\frac{\langle\vec{H}_{W_{t,\varepsilon}},X\rangle_{g}}{f}\,d\mathcal{H}^{n-1}\geq
0\,.$ (4.60)

Setting, for the sake of brevity,

$\displaystyle\nu_{t,\varepsilon}=\frac{\nabla u^{*}_{\Omega_{t}}}{|\nabla
u^{*}_{\Omega_{t}}|_{g}}\,,\qquad
H_{t,\varepsilon}=\vec{H}_{W_{t,\varepsilon}}\cdot\nu_{t,\varepsilon}\,,\qquad\mbox{on
$W_{t,\varepsilon}$}\,,$ (4.61)

we notice that $\nu_{t,\varepsilon}$ defines a Lipschitz continuous $g$-unit
normal to $W_{t,\varepsilon}$, and that $H_{t,\varepsilon}$ is the scalar mean
curvature (as usual, with respect to $g$) of $W_{t,\varepsilon}$ relative to
$\nu_{t}$. With this notation, and thanks to (4.59), (4.60) follows by showing
that

$\displaystyle\inf_{W_{t,\varepsilon}}\langle\nu_{t,\varepsilon},\partial/\partial
r\rangle_{g}\geq 0\,,$ (4.62) $\displaystyle
H_{t,\varepsilon}(x)\geq-\Lambda(t)\,,\qquad\mbox{for $\mathcal{H}^{n-1}$-a.e.
$x\in W_{t,\varepsilon}$}\,,$ (4.63)

for a (positive) constant $\Lambda(t)$ independent of $\varepsilon$.

Proof of (4.62): We start by proving the existence of a positive constant $c$
such that

$\big{\langle}\eta,X(p)\big{\rangle}_{g^{*}}\geq
c\,,\quad\forall(p,\eta)\in\Theta_{t}\,,t>t_{0}\,.$ (4.64)

Since $\langle\eta,v\rangle_{g^{*}}\leq 0$ whenever
$\eta\in\mathcal{N}_{p}(\overline{\Omega}_{t})$ and $v\in T_{p}(\Omega_{t})$
(where $T_{p}(\Omega_{t})$ is tangent cone to $\Omega_{t}$ at $p$), and
recalling (4.55), the validity of (4.64) (for an explicitly computable
constant $\sigma_{0}$) can be easily deduced by showing that, for every
$t>t_{0}$ and $p\in\Sigma_{t}\subset\Omega_{t_{0}}$,

$\left\\{\begin{split}&v\in T_{p}M\,,\quad|v|_{g}=1\,,\\\
&\big{\langle}v,-(\partial/\partial
r)|_{p}\big{\rangle}_{g}>\frac{15}{16}\,,\end{split}\right.\qquad\Rightarrow\qquad
v\in T_{p}(\Omega_{t})\,.$ (4.65)

(In geometric terms: leaving $p\in\Sigma_{t}=(\partial\Omega_{t})\cap
M^{\circ}$ along a sufficiently “vertical and downward” direction, we stay
inside $\Omega_{t}$.) The proof of (4.65) follows closely that of [Bre13,
Lemma 3.7], but since the two statements are not immediate to compare, we
include the details. We need to consider an arbitrary $g$-unit speed curve
$\gamma:[0,1]\to M$ with $\gamma(0)=p$ and $\gamma^{\prime}(0)=v$, and prove
the existence of $\sigma\in(0,1)$ such that $u^{*}_{\Sigma}(\gamma(s))>t$ for
every $s\in(0,\sigma)$. To begin with, we can definitely chose $\sigma$ so
that

$\gamma(s)\in\Omega_{t_{0}}\,,\qquad\big{\langle}\gamma^{\prime}(s),-(\partial/\partial
r)|_{\gamma(s)}\big{\rangle}_{g}\geq\frac{15}{16}\,,\quad\forall
s\in[0,\sigma]\,.$

For every $s\in(0,\sigma)$, the fact that $\gamma(s)\in\Omega_{t_{0}}$ implies
$u^{*}_{\Sigma}(\gamma(s))>0$, and thus the existence of a $g^{*}$-unit speed
geodesic $\alpha_{s}$ with $\alpha_{s}(0)=\gamma(s)$ and
$\alpha_{s}(u^{*}_{\Sigma}(\gamma(s)))\in\overline{\Sigma}$. By (4.47) (with
$\lambda=1/2$), since $w(s)=\alpha_{s}^{\prime}(0)/f(\gamma(s))$ is a $g$-unit
vector, we find

$\displaystyle\frac{1}{2}\\!\\!$ $\displaystyle\leq$
$\displaystyle\\!\\!\big{\langle}w(s),(\partial/\partial
r)|_{\gamma(s)}\big{\rangle}_{g}=\big{\langle}w(s),\gamma^{\prime}(s)+(\partial/\partial
r)|_{\gamma(s)}\big{\rangle}_{g}-\langle w(s),\gamma^{\prime}(s)\rangle_{g}$
$\displaystyle\leq$
$\displaystyle\\!\\!\big{|}\gamma^{\prime}(s)+(\partial/\partial
r)|_{\gamma(s)}\big{|}_{g}-\langle w(s),\gamma^{\prime}(s)\rangle_{g}$
$\displaystyle=$
$\displaystyle\\!\\!\sqrt{2-2\,\langle\gamma^{\prime}(s),-(\partial/\partial
r)|_{\gamma(s)}\rangle_{g}}-\langle w(s),\gamma^{\prime}(s)\rangle_{g}$
$\displaystyle\leq$ $\displaystyle\\!\\!\frac{1}{4}-\langle
w(s),\gamma^{\prime}(s)\rangle_{g}\,,\qquad\mbox{i.e.}\quad\langle-w(s),\gamma^{\prime}(s)\rangle_{g}\geq\frac{1}{4}\quad\forall
s\in(0,\sigma)\,.$

Using the facts that $u^{*}_{\Sigma}(p)=t$ and that, for a.e.
$s\in(0,\sigma)$, $u_{\Sigma}^{*}\circ\gamma$ is differentiable at $\gamma(s)$
with

$(u_{\Sigma}^{*}\circ\gamma)^{\prime}(s)=\langle\nabla^{*}u_{\Sigma}^{*}(\gamma(s)),\gamma^{\prime}(s)\rangle_{g^{*}}=\langle-\alpha^{\prime}_{s}(0),\gamma^{\prime}(s)\rangle_{g^{*}}=\frac{\langle\gamma^{\prime}(s),-w(s)\rangle_{g}}{f(\gamma(s))}\,,$

we thus find that, for every $s^{\prime}\in(0,\sigma)$,

$u^{*}_{\Sigma}(\gamma(s^{\prime}))=t+\int_{0}^{s^{\prime}}\frac{ds}{4\,f(\gamma(s))}>t$

as desired. This proves (4.65), and thus, as explained, (4.64).

We are now ready to deduce (4.62) from (4.64). First, by (4.64) there are
positive constants $c$ and $\delta$ such that

$\langle\eta^{\prime},\partial/\partial r\rangle_{g}\geq c\,,$ (4.66)

whenever $(q,\eta^{\prime})$ lies in the $\delta$-neighborhood
$\mathcal{A}_{\delta}(\Theta^{t})$ of
$\Theta^{t}=\mathcal{N}^{1}(\overline{\Omega}_{t})$ in $TM$. Second, by
smoothness of $(p,\eta,\varepsilon)\mapsto\exp^{*}(p,\varepsilon\,\eta)$, we
can find $\varepsilon^{\prime}<\varepsilon_{t}$ depending on $\delta$ such
that

$\Big{\\{}\Big{(}\Psi^{t}(p,\eta,\varepsilon),\frac{\partial\Psi^{t}}{\partial\varepsilon}(p,\eta,\varepsilon)\Big{)}:(p,\eta)\in\Theta^{t}\,,0<\varepsilon<\varepsilon^{\prime}\Big{\\}}\subset\mathcal{A}_{\delta}(\Theta^{t})\,.$

Third, since $\Psi^{t}_{\varepsilon}$ is a bijection from $\Theta^{t}$ to
$W_{t,\varepsilon}$, we see that for each $x\in W_{t,\varepsilon}$ there is a
unique pair $(p,\eta)$ with $p\in\Sigma_{t}$ and
$\eta\in\mathcal{N}^{1}_{p}(\overline{\Omega}_{t})$ such that
$x=\Psi^{t}_{\varepsilon}(p,\eta)$. Thus, taking also into account that (in
general) $|\nabla^{*}v|_{g^{*}}=f\,|\nabla v|_{g}$ (by
$\nabla^{*}v=f^{2}\,\nabla v$), that
$|\nabla^{*}u^{*}_{\Omega_{t}}|_{g^{*}}=1$ (wherever $u^{*}_{\Omega_{t}}$ is
differentiable), and that $u_{\Omega_{t}}^{*}$ is differentiable along
$\varepsilon\mapsto\Psi^{t}_{\varepsilon}(p,\eta)$ with $g^{*}$-gradient given
by $\partial\Psi^{t}_{\varepsilon}/\partial\varepsilon$, we conclude that

$\displaystyle\langle\nu_{t,\varepsilon}(x),\partial/\partial r\rangle_{g}$
$\displaystyle=$ $\displaystyle\Big{\langle}\frac{\nabla
u_{\Omega_{t}}^{*}(\Psi^{t}_{\varepsilon}(p,\eta))}{|\nabla
u_{\Omega_{t}}^{*}(\Psi^{t}_{\varepsilon}(p,\eta))|_{g}},\partial/\partial
r\Big{\rangle}_{g}$ $\displaystyle=$ $\displaystyle
f\big{(}\Psi^{t}_{\varepsilon}(p,\eta)\big{)}\,\langle\nabla
u_{\Omega_{t}}^{*}(\Psi^{t}_{\varepsilon}(p,\eta)),\partial/\partial
r\rangle_{g}$ $\displaystyle=$ $\displaystyle
f\big{(}\Psi^{t}_{\varepsilon}(p,\eta)\big{)}\,\big{\langle}\frac{\partial\Psi^{t}}{\partial\varepsilon}(p,\eta,\varepsilon),\partial/\partial
r\big{\rangle}_{g}\geq c\,\inf_{[a,b]}h^{\prime}>0\,,$

provided $\varepsilon<\varepsilon^{\prime}$. This proves (4.62).

Proof of (4.63): Let us consider the open set

$A_{t,\varepsilon}=\Omega_{t}\cup\big{\\{}p\in\Omega:u_{\Omega_{t}}^{\ast}(p)<\varepsilon\big{\\}}\,,$

that has $C^{1,1}$-boundary and $g$-unit outer unit normal given by
$\nu_{A_{t,\varepsilon}}=\nu_{t,\varepsilon}$ on $\partial
A_{t,\varepsilon}=W_{t,\varepsilon}$, with $\nu_{t,\varepsilon}$ as in (4.61).
Given $x\in W_{t,\varepsilon}$ such that $\nu_{t,\varepsilon}$ is
differentiable at $x$ (this holds at $\mathcal{H}^{n-1}$-a.e. $x\in
W_{t,\varepsilon}$), let $B^{*}_{x}$ denote the $g^{*}$-geodesic ball centered
at $\exp^{*}(x,(\mu_{t}/2)\,\nabla^{*}u_{\Omega_{t}}^{*}(x))$, where
$\mu_{t}<\min\\{1,\varepsilon_{t}\\}$ is smaller than the injectivity radius
of $M$ in $\overline{\Omega}_{t-2}\setminus\Omega_{t+1}$, so to entail that
$B^{*}_{x}$ has smooth boundary. Since, by construction,

$x\in\partial A_{t,\varepsilon}\cap\partial B^{*}_{x}\,,\qquad
A_{t,\varepsilon}\subset M\setminus B^{*}_{x}\,,$

by the weak maximum principle for $C^{1,1}$-vs-smooth hypersurfaces we find

$H_{W_{t,\varepsilon}}(x)=\vec{H}_{A_{t,\varepsilon}}(x)\cdot\nu_{t,\varepsilon}(x)\geq\vec{H}_{M\setminus
B^{*}_{x}}\cdot\nu_{t,\varepsilon}(x)=-H_{B^{*}_{x}}(x)\,,$

where $H_{B^{*}_{x}}(x)$ denotes the scalar mean curvature of $B^{*}_{x}$ in
$(M,g)$, computed with respect to the outer $g$-unit normal to $B^{*}_{x}$ at
$x$ (i.e., with respect to $-\nu_{t,\varepsilon}(x)$). Now, if
$H^{*}_{B^{*}_{x}}$ denotes the scalar mean curvature of $B^{*}_{x}$ in
$(M,g^{*})$ computed with respect to the outer $g^{*}$-unit normal to
$B^{*}_{x}$ at $x$, then by (3.2) we find

$\displaystyle H_{B^{*}_{x}}(x)\\!\\!$ $\displaystyle\leq$
$\displaystyle\\!\\!H^{*}_{B^{*}_{x}}(x)+3\,(n-1)\,|\nabla\log(f)(x)|_{g}$
$\displaystyle\leq$ $\displaystyle\\!\\!(n-1)\,\sqrt{\kappa_{t}}\,{\rm
coth}(\sqrt{\kappa_{t}}\,\mu_{t}/2)+3\,(n-1)\,|\nabla\log(f)(x)|_{g}\,,$

where in the last inequality we have denoted by $-\kappa_{t}$ a negative lower
bound for the sectional curvatures of $(M,g^{*})$ in
$\overline{\Omega}_{t-2}\setminus\Omega_{t+1}$, and have used [Kar89, pag.
184] (comparison with the mean curvature of geodesic balls in an hyperbolic
model space). Since the right-hand side can be bounded by a positive constant
$\Lambda(t)$, we have concluded the proof of (4.63), and thus of conclusions
(b) and (c) of the theorem. ∎

###### Proof of Theorem 4.2.

Preparation: As in the proof of Theorem 4.1, we reduce the case when
$M=N\times(0,\infty)$, ${\rm(H0)}$ and ${\rm(H1)}$ hold, ${\rm(H2)}$ holds on
$(0,\infty)$ (i.e., $h^{\prime}>0$ on $(0,\infty)$), ${\rm(H3)}$ and
${\rm(H4)}$ hold on $(0,b)$ (if $(M,g)\in\mathcal{B}_{n}$) or ${\rm(H3)^{*}}$
holds on $(0,b)$ (if $(M,g)\in\mathcal{B}_{n}^{*}$); $f=h^{\prime}\circ r$ is
bounded on $M$ and the metric $g^{*}=f^{-2}\,g$ is such that $(M,g^{*})$ is a
complete Riemannian manifold; and, finally, $\Sigma$ and $\Omega$ satisfy
(4.6) in addition to assumptions ${\rm(A1)}$, ${\rm(A2)}$’, and ${\rm(A3)}$’.
Next, by testing (4.5) with vector fields compactly supported in
$M\setminus(\overline{\Sigma}\setminus\Sigma)$ we see that

$H_{\Sigma}\equiv H_{0}\qquad\mbox{on $\Sigma$}\,,$

while testing (4.5) with $X=h\,\partial/\partial r$ and taking into account
that ${\rm div}\,^{\Sigma}X=(n-1)\,f$ on $\Sigma$ and ${\rm div}\,(X)=n\,f$ on
$M$ by (4.10), we find, in the case $\partial\Omega=\overline{\Sigma}$,

$\displaystyle(n-1)\,\int_{\Sigma}f\,d\mathcal{H}^{n-1}$ $\displaystyle=$
$\displaystyle\int_{\Sigma}{\rm
div}\,^{\Sigma}X\,d\mathcal{H}^{n-1}=H_{0}\,\int_{\Sigma}\langle
X,\nu_{\Omega}\rangle_{g}\,d\mathcal{H}^{n-1}$ $\displaystyle=$ $\displaystyle
H_{0}\,\int_{\Omega}{\rm div}\,X=H_{0}\,n\,\int_{\Omega}f\,,$

i.e., $H_{\Sigma}\equiv H_{0}>0$ (so that ${\rm(A2)}$ holds) and (4.2) holds
as an equality; and, in the case $\partial\Omega=\overline{\Sigma}\cup N_{0}$,

$\displaystyle(n-1)\,\int_{\Sigma}f\,d\mathcal{H}^{n-1}$ $\displaystyle=$
$\displaystyle\int_{\Sigma}{\rm
div}\,^{\Sigma}X\,d\mathcal{H}^{n-1}=H_{0}\,\int_{\Sigma}\langle
X,\nu_{\Omega}\rangle_{g}\,d\mathcal{H}^{n-1}$ $\displaystyle=$ $\displaystyle
H_{0}\,\int_{\Omega}{\rm div}\,X\,d\mathcal{H}^{n}-\int_{N_{0}}\langle
X,\partial/\partial r\rangle_{g}\,d\mathcal{H}^{n-1}$ $\displaystyle=$
$\displaystyle H_{0}\,n\,\int_{\Omega}f-h(0)^{n}\,{\rm vol}\,(N)\,,$

i.e, $H_{\Sigma}\equiv H_{0}>0$ (so that ${\rm(A2)}$ holds) and (4.3) holds as
an equality; finally, Theorem 3.1 and (4.5) imply the validity of assumption
${\rm(A3)}$. We can thus apply conclusion (c) of Theorem 4.1 to conclude that

$\displaystyle\mbox{$\Sigma$ is umbilical and has constant mean curvature in
$(M,g)$}\,,$ (4.67) $\displaystyle\mbox{$M\setminus\Omega$ has positive reach
in $(M,g^{*})$}\,,$ (4.68)
$\displaystyle\mbox{$(\mathcal{M}[h])^{\prime}\,g_{N}(\nu_{\Omega},\nu_{\Omega})=0$
on $\Sigma$}\,,$ (4.69)

with $\mathcal{M}[h]$ defined as in (4.23).

Conclusion of the proof if $(M,g)\in\mathcal{B}_{n}^{*}$: In this case, (4.6)
and the validity of ${\rm(H3)^{*}}$ on $(0,b)$ implies that
$(\mathcal{M}[h])^{\prime}>0$ on $\Sigma$, so that (4.69) gives

$\mbox{if $p\in\Sigma$, then $\nu_{\Omega}(p)$ is parallel to
$(\partial/\partial r)|_{p}$}\,.$ (4.70)

Now, if $\phi$ is a local chart of $N$, defined on a ball $B$ in
$\mathbb{R}^{n-1}$, then $\psi(x,t)=(\phi(x),t)$ defines a local chart of $M$
defined on the open set $V=B\times(0,\bar{r})$; clearly, $\psi^{-1}(\Omega)$
is a set of finite perimeter in $V\subset\mathbb{R}^{n}$, with
$\nu_{\psi^{-1}(\Omega)}$ parallel to $e_{n}$ $\mathcal{H}^{n-1}$-a.e. on
$\partial^{*}[\psi^{-1}(\Omega)]$; by [Mag12, Exercise 15.18],
$\psi^{-1}(\Omega)\cap V$ is $\mathcal{H}^{n}$-equivalent to $B\times J$,
where $J$ is a finite union of open intervals compactly contained in
$(0,\bar{r})$; covering $N$ by such charts $\phi$, and going back to $M$, we
conclude that $\Omega$ is $\mathcal{H}^{n}$-equivalent to a $N_{0}\times J$.
Then, the fact that $H_{\Sigma}$ is constant implies that $J$ is either equal
to $(0,t_{0})$ or to $(t_{0},\bar{r})$ for some $t_{0}\in(0,\bar{r})$, and the
theorem is proved.

Conclusion of the proof if $(M,g)\in\mathcal{B}_{n}$: Condition (4.67)
combined with the Codazzi equations implies that

$({\rm Ric}_{M})_{p}(\nu_{\Omega}(p),\sigma_{i}(p))=0\,,\qquad\forall
p\in\Sigma\,,i=1,...,n-1\,,$ (4.71)

provided $\\{\sigma_{i}(p)\\}_{i=1}^{n-1}$ is an $g$-orthonormal basis of
$T_{p}\Sigma$; in particular,

$\mbox{if $p\in\Sigma$, then $\nu_{\Omega}(p)$ is an eigenvector of ${\rm
Ric}_{p}$}\,.$ (4.72)

Since ${\rm(H4)}$ implies that $(\partial/\partial r)|_{p}$ is a simple
eigenvector of $({\rm Ric}_{M})$ (with eigenvalue
$-(n-1)\,(h^{\prime\prime}/h)(r(p))$), it follows from (4.72) that

$\mbox{if $p\in\Sigma$, then $\nu_{\Omega}(p)$ is either orthogonal or
parallel to $(\partial/\partial r)|_{p}$}\,.$ (4.73)

Now, by (4.6), there is $t_{0}>0$ and $p_{0}\in\overline{\Sigma}$ such that

$N\times(0,t_{0})\subset\Omega\,,\qquad p_{0}\in\overline{\Sigma}\cap
N_{t_{0}}\subset\partial\Omega\,.$ (4.74)

From here, in the smooth case when $\overline{\Sigma}=\Sigma$ (i.e., in the
case considered in [Bre13]), (4.73) and (4.74) immediately imply, first, that
$\nu_{\Omega}(p_{0})=(\partial/\partial r)|_{p_{0}}$, and, second, that
$N_{t_{0}}\subset\Sigma$; from which a sliding argument (also required and
detailed below in the non-smooth case) proves the theorem.

However, in the non-smooth case, we cannot immediately conclude the
containment of $N_{t_{0}}$ into $\Sigma$, and, actually, it is not even clear
if $\overline{\Sigma}$ is regular at the contact point $p_{0}$ defined in
(4.74): indeed, the blow-up of (the multiplicity one varifold associated to)
$\overline{\Sigma}$ at $p_{0}$ is an hyperplane with multiplicity possibly
higher than one – thus preventing the use of Allard’s regularity theorem to
infer $p_{0}\in\Sigma$. To exit this impasse we make crucial use of the
positive reach property (4.68), which we use to prove the following
approximation property: for every
$(p,\eta)\in\mathcal{N}^{1}(M\setminus\Omega)$ there are a connected component
$\Sigma^{\prime}$ of $\Sigma$ and a sequence
$\\{p_{j}\\}_{j}\subset\Sigma^{\prime}$ such that

$\big{(}p_{j},-f(p_{j})\,\nu_{\Omega}(p_{j})\big{)}\to(p,\eta)\quad\mbox{in
$\mathcal{N}^{1}(M\setminus\Omega)$}\,.$ (4.75)

Indeed, by (4.68) and Theorem 2.2, there is $s_{0}>0$ such that, for every
$s\in(0,s_{0})$, the sets

$Z_{s}=\big{\\{}x\in M:{\rm dist}^{*}(x,M\setminus\Omega)=s\big{\\}}\,,$

are $C^{1,1}$-hypersurfaces, and the map
$\Psi_{s}(p,\eta)=\exp^{*}(p,s\,\eta)$ is bi-Lipschitz from
$\mathcal{N}^{1}(M\setminus\Omega)$ to $Z_{s}$. We notice that

$\mathcal{N}^{1}(M\setminus\Omega)\lfloor\Sigma=\big{\\{}(p,-f(p)\,\nu_{\Omega}(p)):p\in\Sigma\big{\\}}\,,$

and set

$Z_{s}^{*}=\Phi_{s}\big{(}\mathcal{N}^{1}(M\setminus\Omega)\lfloor\Sigma\big{)}\subset
Z_{s}\,.$

By arguing as in the proof of (4.30), with the aid of Theorem 3.3 we find
that, for $\mathcal{L}^{1}$-a.e. $s\in(0,s_{0})$, $Z_{s}^{*}$ is
$\mathcal{H}^{n-1}$-equivalent to $Z_{s}$. For any such $s$, $Z^{\ast}_{s}$ is
an open dense subset of $Z_{s}$, and we can find a sequence $\\{q_{j}\\}_{j}$,
contained in same connected component $Z_{s}^{**}$ of $Z_{s}^{*}$, such that
$q_{j}\to\Phi_{s}(p,\eta)\in Z_{s}$. Setting $\pi(q,\tau)=q$ for every
$(q,\tau)\in TM$, we see that $p_{j}=\pi[(\Phi_{s})^{-1}(q_{j})]$ defines a
sequence contained in a same connected subset
$\pi[(\Phi_{s})^{-1}(Z_{s}^{**})]$ of $\Sigma$, and such that

$\big{(}p_{j},-f(p_{j})\,\nu_{\Omega}(p_{j})\big{)}=(\Phi_{s})^{-1}(q_{j})\to(p,\eta)\,,$

in $TM$, thus proving (4.75). We now combine (4.75) with the fact that, by
definition of $p_{0}$ (recall (4.74)) it holds

$-f(p_{0})\,\frac{\partial}{\partial
r}\Big{|}_{p_{0}}\in\mathcal{N}^{1}_{p_{0}}(M\setminus\Omega)\,,$ (4.76)

to find a sequence $\\{p_{j}\\}_{j}$, contained in a connected component
$\Sigma^{\prime}$ of $\Sigma$, such that
$(p_{j},-f(p_{j})\,\nu_{\Omega}(p_{j}))\to(p_{0},-f(p_{0})(\partial/\partial
r)|_{p_{0}})$ as $j\to\infty$. By (4.73), up to extracting subsequences, there
are two alternatives: either

$\mbox{$-f(p_{j})\,\nu_{\Omega}(p_{j})$ is parallel to $(\partial/\partial
r)|_{p_{j}}$ for every $j$}\,,$ (4.77)

or $g_{p_{j}}(\nu_{\Omega}(p_{j}),(\partial/\partial r)|_{p_{j}})=0$ for every
$j$, where the latter is clearly contradictory, since $|\partial/\partial
r|_{g}=1$. By smoothness and connectedness of $\Sigma^{\prime}$, by (4.73),
and since $\Sigma^{\prime}$ contains points $p_{j}$ as in (4.77) with
$p_{j}\to p_{0}$ as $j\to\infty$, we conclude that $\Sigma^{\prime}$ is an
open connected subset of $N_{t_{0}}$. In fact, it must be
$\Sigma^{\prime}=N_{t_{0}}$, because the above argument, with $p_{0}$ replaced
by a possible point $p_{0}^{\prime}$ in the boundary of $\Sigma^{\prime}$
relative to $N_{t_{0}}$, would lead to the contradiction that an open
neighborhood of $p_{0}^{\prime}$ in $N_{t_{0}}$ would be contained in
$\Sigma^{\prime}$ itself. We have thus proved that

$N_{t_{0}}\subset\Sigma\,.$ (4.78)

The same argument also shows that

$N_{t_{0}}\cap\overline{\Sigma\cap[N\times(t_{0},\infty)]}=\varnothing\,.$
(4.79)

By (4.79) we could then start sliding $N_{t}$ upwards to prove that either
$\Omega=N\times(0,t_{0})$ with
$M^{\circ}\cap\partial\Omega=\overline{\Sigma}=\Sigma=N_{t_{0}}$, thus
concluding the proof of the theorem, or we could find $t_{1}>t_{0}$ such that

$(t_{0},t_{1})\setminus M\setminus\overline{\Omega}\,,\qquad p_{1}\in
N_{t_{1}}\cap\overline{\Sigma}\subset M\setminus\Omega\,.$ (4.80)

By construction, $-f(p_{1})\,(\partial/\partial
r)|_{p_{1}}\in\mathcal{N}_{p_{1}}^{1}(M\setminus\Omega)$, and, by arguing as
in the proof of (4.78), we would find $N_{t_{1}}\subset\Sigma$, with
$\nu_{\Omega}=-(\partial/\partial r)|_{p_{1}}$ along $N_{t_{1}}$. In turn,
this would give that $H_{\Sigma}$ is negative along $N_{t_{1}}$, a
contradiction. This finally proves the theorem. ∎

##  5\. Rigidity and compactness theorem

###### Proof of Theorem 1.1.

Up to change $\Omega$ with $M\setminus\Omega$, since
$\nu_{\Omega}=-\nu_{M\setminus\Omega}$ on
$\partial^{*}\Omega=\partial^{*}(M\setminus\Omega)$, we can assume that
$\lambda\geq 0$. Let $V_{\Omega}$ be the multiplicity one rectifiable varifold
associated to $M^{\circ}\cap\partial^{\ast}\Omega$. The distributional
constant mean curvature condition (1.4) imply lower density bounds on
$\|V_{\Omega}\|$, which in turn imply

$\mathcal{H}^{n-1}(\overline{\partial^{\ast}\Omega}\setminus\partial^{\ast}\Omega)=0.$

Therefore, it is not restrictive to assume that $\Omega$ is an open set such
that $M^{\circ}\cap\partial\Omega$ is compact,
$\overline{\partial^{\ast}\Omega}=\partial\Omega$ and

$\mathcal{H}^{n-1}(\partial\Omega\setminus\partial^{\ast}\Omega)=0.$ (5.1)

(see for instance the construction in the proof of [DRKS20, Lemma 6.2]).
Notice that
$\partial^{\ast}\Omega=\partial^{\ast}\overline{\Omega}\subseteq\partial\overline{\Omega}\subseteq\partial\Omega$,
hence taking the closure we find that
$\partial\Omega=\partial\overline{\Omega}$. By Allard’s regularity theorem
[All72], if we set

$\Sigma=\Big{\\{}x\in{\rm spt}\|V_{\Omega}\|:\lim_{\rho\to
0^{+}}\frac{\|V_{\Omega}\|(B_{\rho}(x))}{\omega_{n-1}\,\rho^{n-1}}=1\Big{\\}}\,,$

then $\Sigma$ is a smooth, embedded hypersurface and

$\Sigma=M^{\circ}\cap\partial^{*}\Omega\,.$ (5.2)

We now check that the pair $(\Sigma,\Omega)$ satisfies the assumptions
${\rm(A1)}$, ${\rm(A2)}$’ and ${\rm(A3)}$’ of Theorem 4.2, thus concluding the
proof of the theorem. Clearly, ${\rm(A3)}$’ is equivalent to (1.4). Since
$M^{\circ}\cap\partial\Omega$ is compact, we infer that
$\overline{\Sigma}\subseteq M^{\circ}$; moreover (5.1) means that
$\mathcal{H}^{n-1}(\overline{\Sigma}\setminus\Sigma)=0$. Henceforth
${\rm(A1)}$ holds. Concerning ${\rm(A2)}$’, since
$M^{\circ}\cap\partial\Omega$ is compact, we notice that $1_{\Omega}$ is
constant in a neighborhood $A$ of $N_{0}$ in $M$; if $1_{\Omega}=0$ on $A$
then $\partial\Omega=\overline{\Sigma}$; if, otherwise $1_{\Omega}=1$ on $A$,
then $N_{0}\subset\partial\Omega$, and thus
$\partial\Omega=N_{0}\cup\overline{\Sigma}$. ∎

###### Proof of Theorem 1.2.

From $\mathcal{H}^{n}(\Omega_{j}\Delta\Omega)\to 0$ as $j\to\infty$ we easily
deduce that for every $x_{0}\in M^{\circ}\cap\overline{\partial^{*}\Omega}$
there is $x_{j}\in\partial^{*}\Omega_{j}$ such that $x_{j}\to x_{0}$ in $M$;
for, otherwise, there would be $\rho>0$, with
$\overline{B}_{\rho}(x_{0})\cap\overline{\partial^{*}\Omega_{j}}=\varnothing$
for every $j$, and $X\in C^{\infty}_{c}(B_{\rho}(x_{0}))$ such that

$\displaystyle 1\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int_{B_{\rho}(x_{0})\cap\partial^{*}\Omega}\\!\\!\langle
X,\nu_{\Omega}\rangle_{g}\,d\mathcal{H}^{n-1}=\int_{\Omega}\,{\rm
div}\,\,X\,d\mathcal{H}^{n}$ $\displaystyle=$
$\displaystyle\\!\\!\lim_{j\to\infty}\int_{\Omega_{j}}\,{\rm
div}\,\,X\,d\mathcal{H}^{n}=\lim_{j\to\infty}\int_{B_{\rho}(x_{0})\cap\partial^{*}\Omega_{j}}\\!\\!\langle
X,\nu_{\Omega_{j}}\rangle_{g}\,d\mathcal{H}^{n-1}=0\,.$

We thus conclude that

$M^{\circ}\cap\overline{\partial^{*}\Omega}\subset N\times[a,b]\subset\subset
M^{\circ}\,.$ (5.3)

By $\mathcal{H}^{n}(\Omega_{j}\Delta\Omega)\to 0$ and, crucially, by ${\rm
Per}(\Omega_{j})\to{\rm Per}(\Omega)$, as $j\to\infty$, we see that the
multiplicity one rectifiable varifolds $V_{j}$ associated to
$\partial^{*}\Omega_{j}$ converge, in the sense of varifolds on $M$, to the
multiplicity one rectifiable varifold $V$ associated to $\partial^{*}\Omega$:
in particular, for every $X\in\mathcal{X}(M)$,

$\lim_{j\to\infty}\int_{M^{\circ}\cap\partial^{*}\Omega_{j}}{\rm
div}\,^{\partial^{*}\Omega_{j}}X\,d\mathcal{H}^{n-1}=\int_{M^{\circ}\cap\partial^{*}\Omega}{\rm
div}\,^{\partial^{*}\Omega}X\,d\mathcal{H}^{n-1}\,.$

Again by $\mathcal{H}^{n}(\Omega_{j}\Delta\Omega)\to 0$ as $j\to\infty$ and
thanks to the divergence theorem
$\small{\int}_{M^{\circ}\cap\partial^{*}\Omega_{j}}\langle
X,\nu_{\Omega_{j}}\rangle_{g}\,d\mathcal{H}^{n-1}\to\small{\int}_{M^{\circ}\cap\partial^{*}\Omega}\langle
X,\nu_{\Omega}\rangle_{g}\,d\mathcal{H}^{n-1}$ as $j\to\infty$. Therefore,
(1.7) implies

$\int_{M^{\circ}\cap\partial^{*}\Omega}{\rm
div}\,^{\partial^{*}\Omega}X\,d\mathcal{H}^{n-1}=\lambda\,\int_{M^{\circ}\cap\partial^{*}\Omega}\langle
X,\nu_{\Omega}\rangle_{g}\,d\mathcal{H}^{n-1}\,.$ (5.4)

By (5.3) and (5.4) we can apply Theorem 1.1 to $\Omega$ and conclude the proof
of the theorem. ∎

##  6\. Proof of Theorem 1.3

The key observation to prove Theorem 1.3 is contained in the following result,
that can be proved employing the same method of Theorem 4.1.

######  Theorem 6.1.

Suppose $(M,g)$ is a $n$-dimensional Riemannian manifold (notice carefully
that we do not assume this space to be geodesically complete) and $f$ is a
smooth positive function on $M$ such that

$f{\rm Ric}-\mathrm{D}^{2}f+(\Delta f)g\geq 0\quad\textrm{on $M$.}$ (6.1)

Suppose $\Omega\subseteq M$ is an open set with finite perimeter, with
exterior unit-normal $\nu_{\Omega}$, such that $\overline{\Omega}$ is compact,
and suppose $\Sigma\subseteq\partial\overline{\Omega}$ is a smooth embedded
hypersurface such that $\overline{\Sigma}=\partial\overline{\Omega}$ is a
White $(n-1,\lambda)$-set of $(M,g)$,
$\mathcal{H}^{n-1}(\overline{\Sigma}\setminus\Sigma)=0$, and
$H_{\Sigma}=\overrightarrow{H}_{\Sigma}\cdot\nu_{\Omega}$ is positive on
$\Sigma$.

Then

$n\int_{\Omega}f\,d\mathcal{H}^{n}\leq(n-1)\int_{\Sigma}\frac{f}{H_{\Sigma}}\,d\mathcal{H}^{n-1}.$
(6.2)

If the equality holds, then there exists $t_{0}>0$ such that the sets

$\Sigma_{t}=\\{p\in\Omega:{\rm
dist}_{g^{\ast}}(p,\overline{\Sigma})=t\\},\quad\textrm{for $0<t<t_{0}$,}$

where $g^{\ast}=\frac{g}{f^{2}}$, are closed embedded
$\mathcal{C}^{1,1}$-hypersurfaces, and for $\mathcal{L}^{1}$ a.e.
$t\in(0,t_{0})$ there exists a smooth embedded umbilical hypersurface
$\Sigma^{\ast}_{t}\subseteq\Sigma_{t}$ such that

$\mathcal{H}^{n-1}(\Sigma_{t}\setminus\Sigma^{\ast}_{t})=0.$

###### Proof.

Let $N\subseteq M$ be a compact set with smooth boundary such that
$\overline{\Omega}\subseteq{\rm int}(N)$. By [PV20, Corollary B], there exists
a geodesically complete Riemannian extension $(M^{\ast},g^{\ast})$ of
$(N,g/f^{2})$ with $\partial M^{\ast}=\varnothing$. We denote by $\exp^{\ast}$
the exponential map of $(M^{\ast},g^{\ast})$ and define

$\Phi:\Sigma\times[0,+\infty)\rightarrow
M^{\ast},\quad\Phi(x,t)=\Phi_{t}(x)=\exp^{\ast}(x,-tf(x)\nu_{\Omega}(x))$

and, for $x\in\Sigma$,

$R_{\Sigma}(x)=\min\\{\inf\\{t>0:J^{\Sigma}\Phi_{t}(x)=0\\},\,\inf\\{t>0:\Phi_{t}(x)\notin{\rm
int}(N)\\}\\}.$

The conclusion now can be obtained by tracing the argument of Theorem 4.1 that
gives the Heintze-Karcher inequality (4.2). We omit to repeat these details
here, and we point out a couple of remarks. First, one needs to employ (6.1)
in order to obtain (4.17) and (4.18) in the present setting: in fact, one can
repeat verbatim the pointwise computations of [Bre13, Proposition 3.2], where
only (6.1) is used. Second, to analyze the equality case, firstly we observe,
exactly in the same way as in the proof of Theorem 4.1, that
$M^{\ast}\setminus\Omega$ is a set of positive reach; henceforth, by Theorem
2.2, there exists $t_{0}>0$ such that $\Sigma_{t}$ is a compact embedded
$\mathcal{C}^{1,1}$-hypersurface for every $0<t<t_{0}$; then, combining (4.30)
and (4.37) we infer that $\Sigma^{\ast}_{t}$ is a smooth umbilical embedded
hypersurface relatively open in $\Sigma_{t}$ for every $t>0$, and
$\mathcal{H}^{n-1}(\Sigma_{t}\setminus\Sigma^{\ast}_{t})=0$ for
$\mathcal{L}^{1}$ a.e. $t>0$. ∎

We consider the upper-half space model of the hyperbolic space
$\mathbb{H}^{n}$: namely
$\mathbb{H}^{n}=\mathbb{R}^{n}_{+}=\\{x\in\mathbb{R}^{n}:x_{n}>0\\}$ and

$g_{\mathbb{H}^{n}}=\frac{1}{x_{n}^{2}}(dx_{1}^{2}+\ldots+dx_{n}^{2}).$

###### Proof of Theorem 1.3.

Choose $p\in\mathbb{H}^{n}\setminus\overline{\Omega}$ and define

$r(x)={\rm dist}_{\mathbb{H}^{n}}(x,p),\quad f(x)=\cosh(r(x)),$

for $x\in\mathbb{H}^{n}$. Recalling that the geodesic spheres in
$\mathbb{H}^{n}$ of radius $\rho$ are smooth embedded umbilical hypersurface
with all principal curvatures equal to $\coth(\rho)$, a straightforward
computation gives that

$\mathrm{D}^{2}f=\sinh(r)\,\mathrm{D}^{2}r+\cosh(r)\,dr\otimes dr,$
$\mathrm{D}^{2}f(x)(v,v)=\cosh(r(x)),$ (6.3) $\Delta f(x)=n\cosh(r(x)),$ (6.4)
$\displaystyle f(x)\,{\rm Ric}_{\mathbb{H}^{n}}(v,v)$
$\displaystyle-\mathrm{D}^{2}f(x)(v,v)+\Delta f(x)$
$\displaystyle=-(n-1)\cosh(r(x))-\cosh(r(x))+n\cosh(r(x))=0,$

for every $v\in T_{x}(\mathbb{H}^{n})$ with $|v|=1$. Henceforth, the
Riemannian manifold $(\mathbb{H}^{n}\setminus\\{p\\},g_{\mathbb{H}^{n}})$
endowed with the function $f$ satisfies the hypothesis of Theorem 6.1. Arguing
as in the proof of Theorem 1.1, we notice that it is not restrictive to assume
that $\Omega$ is an open set such that
$\partial\overline{\Omega}=\partial\Omega=\overline{\partial^{\ast}\Omega}$
and $\mathcal{H}^{n-1}(\partial\Omega\setminus\partial^{\ast}\Omega)=0$; by
Allard regularity theorem we also have that $\partial^{\ast}\Omega$ is a
smooth embedded hypersurface. We set $\Sigma=\partial^{\ast}\Omega$. By
Theorem 3.1, $\overline{\Sigma}$ is a White $(n-1,\lambda)$-set of
$\mathbb{H}^{n}$. Finally, we need to check that $H_{\Sigma}$ is positive and
the couple $(\Omega,\Sigma)$ fulfills the equality in (6.2). The equality
(1.8) clearly implies that $H_{\Sigma}(x)=\lambda$ for $x\in\Sigma$. Moreover,
since by (6.3) and (6.4) we have that

${\rm div}^{\Sigma}(\nabla f)(x)=\Delta
f(x)-\mathrm{D}^{2}f(\nu_{\Omega}(x),\nu_{\Omega}(x))=(n-1)f(x)\quad\textrm{for
$x\in\Sigma$,}$

we infer from (1.8) that

$\displaystyle(n-1)\int_{\Sigma}f\,d\mathcal{H}^{n-1}=\int_{\Sigma}{\rm
div}^{\Sigma}(\nabla f)\,d\mathcal{H}^{n-1}$
$\displaystyle=\lambda\int_{\Sigma}\langle\nu_{\Omega},\nabla
f\rangle_{\mathbb{H}^{n}}\,d\mathcal{H}^{n-1}$
$\displaystyle=\lambda\int_{\Omega}\Delta
f\,d\mathcal{H}^{n}=n\lambda\int_{\Omega}f\,d\mathcal{H}^{n}.$

This implies that $H_{\Sigma}\equiv\lambda>0$ and $(\Omega,\Sigma)$ fulfills
the equality case in (6.2). We conclude that there exists $t_{0}>0$ such that
the sets

$\Sigma_{t}=\\{p\in\Omega:{\rm
dist}_{g^{\ast}}(p,\overline{\Sigma})=t\\},\quad\textrm{for $\mathcal{L}^{1}$
a.e.\ $0<t<t_{0}$,}$

where $g^{\ast}=\frac{g_{\mathbb{H}^{n}}}{f^{2}}$, are closed embedded
$\mathcal{C}^{1,1}$-hypersurfaces with respect to the hyperbolic metric. Since
the hyperbolic metric is conformally equivalent to the Euclidean metric, and
since changing conformally the metric of the ambient space preserves
umbilicity, we infer that $\Sigma_{t}$ is also umbilical with respect to the
Euclidean metric. Henceforth, by [DRKS20, Lemma 3.2], we conclude that each
$\Sigma_{t}$ is a finite disjoint union of Euclidean spheres. Now the
conclusion follows letting $t\to 0$.

##  Appendix A Assumption ${\rm(H3)^{*}}$ and models in General Relativity

In this appendix we check that the Reissner–Nordstrom manifolds satisfy
assumption ${\rm(H3)^{*}}$, while the deSitter–Schwarzschild manifolds do not.
This simple fact, combined with the analysis of equality cases in Brendle’s
Heintze–Karcher type inequalities, shows that a stronger stability mechanism
for almost-CMC hypersurfaces is at a play in the R–N manifolds. Indeed, when
${\rm(H3)^{*}}$ holds, Brendle’s argument also provides, in addition to
almost-umbilicality, a direct control on the oscillation of the normals with
respect to the radial directions as measured by
$g_{N}(\nu_{\Omega},\nu_{\Omega})$; see, for example, condition (4.4).

Let us set $G=S^{n-1}\times(s_{1},s_{2})$ for some $0<s_{1}<s_{2}\leq+\infty$.
The dS–S manifold is then $(G,g_{{\rm dSS}})$, where

$g_{{\rm dSS}}=\frac{ds\otimes ds}{1-m\,s^{2-n}-\kappa
s^{2}}+s^{2}\,g_{S^{n-1}}\,,$

with

$m>0\,,\qquad-\infty<\kappa<(n-2)\,\Big{(}\frac{4}{n^{n}\,m^{2}}\Big{)}^{1/(n-2)}\,.$

When $\kappa>0$ the upper bound on $\kappa$ guarantees that
$1-m\,s^{2-n}-\kappa s^{2}$ has exactly two zeros $s_{1}<s_{2}$ on
$(0,\infty)$, while if $\kappa\leq 0$ we set $s_{2}=+\infty$, while $s_{1}$ is
the unique zero of $1-m\,s^{2-n}-\kappa s^{2}$ on $(0,\infty)$. The R–N
manifold is defined instead as $(G,g_{{\rm RN}})$, where

$g_{{\rm RN}}=\frac{ds\otimes
ds}{1-m\,s^{2-n}+q^{2}\,s^{4-2n}}+s^{2}\,g_{S^{n-1}}\,,\qquad m>2\,q>0\,.$

In this case $s_{1}$ is the largest of the two solutions of
$1-m\,s^{2-n}+q^{2}\,s^{4-2n}=0$ on $(0,\infty)$, while we set
$s_{2}=+\infty$. Both examples can be modeled as

$g_{\omega}=(1/\omega(s))\,ds\otimes ds+s^{2}\,g_{S^{n-1}}$

for a smooth function $\omega:(s_{1},s_{2})\to(0,\infty)$. We then define

$F(s)=\int_{s_{1}}^{s}\frac{1}{\sqrt{\omega}}\quad\forall
s\in(s_{1},s_{2})\,,\qquad h(t)=F^{-1}(t)\quad\forall
t\in(0,\bar{r})\,,\bar{r}:=F(s_{2})\,,$

so that $h(F(s))=s$ for every $s\in(s_{1},s_{2})$, and

$\displaystyle h^{\prime}(F(s))=\sqrt{\omega(s)}\,,\qquad
h^{\prime\prime}(F(s))=\omega^{\prime}(s)\big{/}2\,,\qquad\forall
t\in(0,\bar{r})\,.$ (A.1)