# Finite total curvature and soap bubbles with almost constant higher-order
mean curvature

Mario Santilli  Dipartimento di Ingegneria e Scienze dell’Informazione e
Matematica, Universitá degli Studi dell’Aquila, 67100 L’Aquila, Italy
<EMAIL_ADDRESS>
###### Abstract.

Given $n\geq 2$ and $k\in\\{2,\ldots,n\\}$, we study the asymptotic behaviour
of sequences of bounded $C^{2}$-domains of finite total curvature in
$\mathbb{R}^{n+1}$ converging in volume and perimeter, and with the $k$-th
mean curvature functions converging in $L^{1}$ to a constant. Under natural
mean convexity hypothesis, and assuming an $L^{\infty}$-control on the mean
curvature outside a set of vanishing area, we prove that finite unions of
mutually tangent balls are the only possible limits. This is the first result
where such a uniqueness is proved without assuming uniform bounds on the
exterior or interior touching balls.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Legendrian cycles and Curvature measures
  4. 4 Heintze-Karcher inequality for varifolds of bounded mean curvature
  5. 5 Proof of Theorem 1.1
  6. A Soap bubbles with positive reach

##  1\. Introduction

###  1.1. Overview

If $\Omega\subseteq\mathbb{R}^{n+1}$ is an open set whose boundary
$\partial\Omega$ is a closed embedded $C^{2}$-hypersurface and if
$\kappa_{\Omega,1}\leq\ldots\leq\kappa_{\Omega,n}$ are the principal
curvatures of $\partial\Omega$ with respect to the exterior normal of
$\Omega$, then _the $k$-th mean curvature function of $\Omega$_ is given by

$H_{\Omega,k}=\sum_{\lambda\in\Lambda(n,k)}\kappa_{\Omega,\lambda(1)}\cdots\kappa_{\Omega,\lambda(k)},$

where $\Lambda(n,k)$ is the set of all increasing maps from $\\{1,\ldots,k\\}$
to $\\{1,\ldots,n\\}$. The function $H_{\Omega,1}$ is also called mean
curvature of $\Omega$.

The following theorem is a classical and well known result.

Soap bubble theorem. If $k\in\\{1,\ldots,n\\}$ and
$\Omega\subseteq\mathbb{R}^{n+1}$ is a bounded and connected open set with
$C^{2}$-boundary such that $H_{\Omega,k}$ is constant, then $\Omega$ is a
round ball.

For $k=1$ this result is due to Alexandrov and was proved using his celebrated
moving plane method; see [Ale58]. For arbitrary $k$ the result was proved in
[Ros87] and [MR91], using an approach based on an optimal geometric inequality
inspired by the work of Heintze and Karcher in [HK78], and on the Minkowski
identities [Hsi54]. A proof based on the moving plane method for arbitrary $k$
is due to Korevaar, see [Ros88, Appendix].

Motivated by the soap bubble theorem, we are interested in the following
uniqueness problem

$(UP_{k})\begin{split}&\mbox{{\it Does every sequence
$\Omega_{\ell}\subseteq\mathbb{R}^{n+1}$ of bounded, connected, open}}\\\
&\mbox{{\it smooth sets with bounded perimeters, whose $k$-th mean
curvature}}\\\ &\mbox{{\it functions converge to a constant, have finite
unions of}}\\\ &\mbox{{\it mutually tangent balls as their only possible
limits?}}\end{split}$

In general it is not possible to deduce convergence to a single ball from the
sole hypothesis of small oscillation of the mean curvature functions. In fact,
by truncating and smoothly completing unduloids with thin necks one can
construct a sequence of bounded and connected smooth boundaries converging to
an array of mutually tangent balls while the mean curvatures converge to a
constant.

The problem $(UP_{1})$ was thoroughly investigated, even in quantitative ways.
In [DM19], Delgadino and Maggi extended the Alexandrov theorem proving that if
a set of finite perimeter has constant distributional mean curvature, then it
is a finite union of closed balls with disjointed interiors. This result is
proved using a measure-theoretic generalization of the Montiel-Ros argument,
and implies the following uniqueness result (see [DM19, Corollary 2]): _finite
unions of mutually tangent balls are the only possible limits of sequences of
sets of finite perimeter converging in volume and in perimeter, and whose
distributional mean curvatures converge to a constant._ Quantitative rates of
convergence towards finite unions of balls are obtained in [CM17], [DMMN18]
and in [JN23], employing integral-geometric methods inspired by the Montiel-
Ros argument. See also [DRKS20] and [MS23] for extensions of [DM19] to the
anisotropic and Riemannian setting.

The Heintze-Karcher inequality was generalized in the setting of arbitrary
closed sets in [HS22, Theorem 3.20]. This result opens the way to obtain a
measure-theoretic version of the soap bubble theorem for sets of positive
reach in terms of their curvature measures; see [HS22, Theorem A, Theorem
6.15] (see also Theorem A.1 for an extension). As a corollary (see Theorem
A.2) we obtain the following answer to $(UP_{k})$ for arbitrary $k$: _one
single ball is the only possible limit in the sense of Hausdorff converge, if
one assumes that the sets $\Omega_{\ell}$ in $(UP_{k})$ satisfies a uniform
bound on the outer touching balls (i.e. lower uniform bound on the reach) and
their $(k-1)$-th mean curvature functions become asymptotically non-negative._
Optimal quantitative rates of convergence in Hausdorff distance towards one
single balls can be deduced from the results in [CV18] (for $k=1$) and [CRV21]
(for $1\leq k\leq n$), assuming a uniform bound both on the interior and
exterior touching balls. The results in [CV18] and [CRV21] are based on a
quantitative version of the Alexandrov moving plane method. See also [MP20]
for other related quantitative results.

###  1.2. The main theorem

The uniqueness problem $(UP_{k})$, for $k\geq 2$ and without assuming uniform
bounds on the interior or exterior touching balls, is a natural and
interesting problem, which is to author’s knowledge completely open, even in
the $3$-dimensional Euclidean space for sequences with vanishing oscillation
of the Gaussian curvature (in some $L^{p}$-norm). In this paper we study this
problem for sequences of _finite total curvature_ , see Definition 2. As
expected, studying this problem under this new hypothesis requires the
introduction of a substantial novel method of proof, with respect to the
approaches used in Theorem A.2 and in [CRV21], where the problem is treated
under uniform bounds on the touching balls. Before to state the main result of
the paper, firstly we introduce some definitions and notations. For a function
$f$ we write

$f^{+}=\sup\\{f,0\\}\quad\textrm{and}\quad f^{-}=-\inf\\{f,0\\}.$

We denote with $A_{\Omega}$ the norm of the second fundamental form of
$\partial\Omega$:

$A_{\Omega}(x)=\bigg{(}\sum_{i=1}^{n}\kappa_{\Omega,i}(x)^{2}\bigg{)}^{\frac{1}{2}}\quad\textrm{for
$x\in\partial\Omega$.}$

######  Definition 1 (Compactly supported sequences).

We say that a sequence $\Omega_{\ell}$ of subsets of $\mathbb{R}^{n+1}$ is
_compactly supported_ if there exists a ball $B_{R}$ such that
$\Omega_{\ell}\subseteq B_{R}$ for all $\ell\geq 1$.

######  Definition 2 (Sequences of finite total curvature).

We say that a sequence $\Omega_{\ell}\subseteq\mathbb{R}^{n+1}$ of open sets
with $C^{2}$-boundary has _finite total curvature_ if

$\sup_{\ell\geq
1}\int_{\partial\Omega_{\ell}}A_{\Omega_{\ell}}^{n}\,d\mathcal{H}^{n}<\infty.$

######  Definition 3 (Asymptotically $k$-mean convex sequences).

Let $k\in\\{1,\ldots,n\\}$. We say that a sequence
$\Omega_{\ell}\subseteq\mathbb{R}^{n+1}$, $\ell\geq 1$, of open sets with
$C^{2}$-boundary is _asymptotically $k$-mean convex_ if and only if

$\lim_{\ell\to\infty}\int_{\partial\Omega_{\ell}}(H_{\Omega_{\ell},i})^{-}\,d\mathcal{H}^{n}=0\quad\textrm{for
$i=1,\ldots,k$.}$

If $k=1$ we simply say that $\Omega_{\ell}$ is asymptotically mean convex.

######  Remark 1.1.

In relation to Definition 3, we recall a well known fact. If
$k\in\\{1,\ldots,n\\}$ and $\Omega\subseteq\mathbb{R}^{n+1}$ is a bounded and
connected $C^{2}$-domain with constant $k$-th mean curvature function, then

$H_{\Omega,i}(x)>0\quad\textrm{for every $x\in\partial\Omega$ and for $1\leq
i\leq k$.}$ (1.1)

See [Ros87, page 450] for details. The condition (1.1) is usually called
$k$-convexity or $k$-mean convexity. It naturally appears in many problems
involving higher-order mean curvature functions as it guarantees the
ellipticity of the related fully non-linear PDE’s equations; see the
pioneering [CNS85].

This is the main result of the paper.

######  Theorem 1.1.

Let $n\geq 2$, $k\in\\{2,\ldots,n\\}$ and let
$\Omega_{\ell}\subseteq\mathbb{R}^{n+1}$ be a compactly supported and
asymptotically $(k-1)$-mean convex sequence with finite total curvature.
Suppose that the sets $\Omega_{\ell}$ converge in volume and perimeter to a
set $\Omega\subseteq\mathbb{R}^{n+1}$, and there exist $\lambda\in\mathbb{R}$
and $M>0$ such that

$\lim_{\ell\to\infty}\int_{\partial\Omega_{\ell}}|H_{\Omega_{\ell},k}-\lambda|\,d\mathcal{H}^{n}=0$
(1.2)
$\lim_{\ell\to\infty}\mathcal{H}^{n}(\\{x\in\partial\Omega_{\ell}:H_{\Omega_{\ell},1}(x)\geq
M\\})=0.$ (1.3)

Then $\Omega$ is $\mathcal{L}^{n+1}$ almost equal to a finite union of closed
balls of the same radius with disjointed interiors. The radius $\rho$ of the
balls satisfies the relations

$\rho=\frac{(n+1)\mathcal{L}^{n+1}(\Omega)}{\mathcal{H}^{n}(\partial^{\ast}\Omega)}\quad\textrm{and}\quad\lambda={n\choose
k}\rho^{-k}.$ (1.4)

In the classical case of $2$-dimensional surfaces in $\mathbb{R}^{3}$, in
Theorem 1.1 (i.e. $n=2$ and $k=2$) one can equivalently replace the condition
of finite total curvature with a uniform bound on the $L^{2}$-norm of the mean
curvature functions, since
$A_{\Omega_{\ell}}^{2}=H_{\Omega_{\ell},1}^{2}-2H_{\Omega_{\ell},2}$.

######  Corollary 1.2.

Let $\Omega_{\ell}\subseteq\mathbb{R}^{3}$ be a compactly supported and
asymptotically mean convex sequence with

$\sup_{\ell\geq
1}\int_{\partial\Omega_{\ell}}H_{\Omega_{\ell},1}^{2}\,d\mathcal{H}^{2}<\infty.$

Suppose that the sets $\Omega_{\ell}$ converge in volume and perimeter to a
set $\Omega\subseteq\mathbb{R}^{3}$, and there exist $\lambda\in\mathbb{R}$
and $M>0$ such that

$\lim_{\ell\to\infty}\int_{\partial\Omega_{\ell}}|H_{\Omega_{\ell},2}-\lambda|\,d\mathcal{H}^{2}=0$
$\lim_{\ell\to\infty}\mathcal{H}^{2}(\\{x\in\partial\Omega_{\ell}:H_{\Omega_{\ell},1}(x)\geq
M\\})=0.$ (1.5)

Then the conclusion of Theorem 1.1 holds with $n=k=2$.

######  Remark 1.2.

The hypothesis (1.3) in Theorem 1.1 (or (1.5) in Corollary 1.2) amounts to
require that the mean curvatures are uniformly bounded from above outside a
set of vanishing area. This is a technical condition, which is used in section
4 to derive some key fine properties of the varifold associated with the
reduced boundary of the limiting set $\Omega$. This condition is clearly
satisfied whenever one has a sequence such that each set $\Omega_{\ell}$ is
made of several domains of uniformly bounded mean curvature connected by small
necks of vanishing area (in which case we do not need to care about the
behaviour of the mean curvatures on the vanishing necks).

We conjecture that the hypothesis (1.3) can be completely removed from Theorem
1.1.

######  Remark 1.3.

Besides its own interest, the uniqueness problem $(UP_{k})$ for $k\geq 2$
naturally arises in the study of higher-order isoperimetric type inequalities
(see [GL09] and [CW13]) and in the study of the geometric properties of
hypersurfaces with prescribed higher-order mean curvatures (see [GLL12] and
references therein).

###  1.3. Method of proof and organization of the paper

The proof of Theorem 1.1 is based on a novel geometric-measure theoretic
extension of the method pioneered by Montiel and Ros. In [Ros87] and [MR91]
the proof is based on two fundamental steps: (1) proving a sharp Heintze-
Karcher inequality with equality achieved only by balls, (2) checking that a
domain with constant $k$-th mean curvature realizes the equality case. This
last step for $k\geq 2$ is crucially based on the Minkowski identities in
[Hsi54], while it is an immediate consequence of divergence theorem for $k=1$.

The reduced boundary of the limiting set $\Omega$ in Theorem 1.1 is a
$n$-dimensional varifold $V_{\Omega}$ of bounded mean curvature in
$\mathbb{R}^{n+1}$. In section 4, combining results in varifolds theory
([Men13] and [San20b]) with fine properties of the curvature for arbitrary
closed sets ([San20a]), we can quickly deduce from the general Heintze-Karcher
inequality in [HS22, Theorem 3.20] a sharp geometric inequality for sets of
finite perimeter and bounded distributional mean curvature; see Theorem 4.2.
This completes the first step of the proof. In order to complete the proof, we
have to check that the limiting set $\Omega$ satisfies the equality case in
Theorem 4.2, and this is the key new difficulty with respect to other
aforementioned results based on the Montiel-Ros method. In fact, one needs to
use a Minkowski-type formula for the limiting set in order to use the
geometric information given by the vanishing oscillation hypothesis (1.2)
within the geometric inequality from Theorem 4.2. However, Minkowski formulae
for singular geometric sets are known only in very special cases, namely sets
of positive reach and subanalytic sets, and their proof is a quite subtle
issue based on the existence of a normal cycle; see [Fu98, section 3]. In
particular, no Minkowski formulae are known in the varifolds setting. To deal
with this point in Theorem 1.1, we consider the sequence of normal cycles
$N_{\Omega_{\ell}}$ associated with the exterior unit-normal bundles of the
sets $\Omega_{\ell}$. These are $n$-dimensional integral currents in the
product space $\mathbb{R}^{n+1}\times\mathbb{S}^{n}$, which are cycles (i.e.
$\partial N_{\Omega_{\ell}}=0$ in the sense of currents) and satisfy a
Legendrian-type property, see Remark 3.4. It follows from the finite total
curvature assumption that the masses of the integral currents
$N_{\Omega_{\ell}}$ are uniformly bounded; henceforth we can apply Federer-
Fleming compactness theorem to find, up to subsequences, that the currents
$N_{\Omega_{\ell}}$ converge weakly to a Legendrian cycle $T$; see section 3.
Using a representation formula for the curvature measures associated with $T$
(see Lemma 3.2), we pass to limit in the Minkowski formulae for
$\Omega_{\ell}$ to find a Minkowski-type formula for the Legendrian cycle $T$;
see Lemma 3.4. Now it is still not clear how to use the Minkowski formulae for
$T$ within the Heintze-Karcher inequality for $V_{\Omega}$, since $V_{\Omega}$
and $T$ come from two completely different limit procedures: $T$ is the limit
in the sense of currents of the normal cycles of $\Omega_{\ell}$, while
$V_{\Omega}$ is the limit in the sense of varifolds of the boundaries of
$\Omega_{\ell}$. This is a quite subtle point and occupies most of the proof
of Theorem 1.1 in section 5.

Finally we mention that in the appendix, firstly we generalize the soap bubble
theorem [HS22, Theorem A] (allowing the constant $\lambda\in\mathbb{R}$), then
we show how this theorem allows to characterize the limit of sequences of
$C^{2}$-domains with almost constant $k$-th mean curvature function and with a
uniform bound on the exterior touching balls.

Acknowledgements: The author wishes to thank Francesco Maggi for useful
comments on a preliminary version of this work. The author is partially
supported by INdAM-GNSAGA.

##  2\. Preliminaries

Let
$\pi_{0}:\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}\rightarrow\mathbb{R}^{n+1}$
and
$\pi_{1}:\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}\rightarrow\mathbb{R}^{n+1}$ be
defined as

$\pi_{0}(x,u)=x,\qquad\pi_{1}(x,u)=u.$

If $S\subseteq\mathbb{R}^{n+1}$ we define ${\rm dist}(x,S)=\inf\\{|x-a|:a\in
S\\}$ for $x\in\mathbb{R}^{n+1}$. If
$Q\subseteq\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}$ and
$S\subseteq\mathbb{R}^{n+1}$ we define

$Q\operatorname{\llcorner}S=\pi_{0}^{-1}(S)\cap Q.$

We say that a subset $S\subseteq\mathbb{R}^{m}$ is _countably
$\mathcal{H}^{k}$-rectifiable_ if there exists a countable family $F$ of
$k$-dimensional embedded $C^{1}$-submanifolds of $\mathbb{R}^{m}$ such that
$\mathcal{H}^{k}(S\setminus\bigcup F)=0$. We denote with
$\operatorname{Tan}^{k}(\mathcal{H}^{k}\operatorname{\llcorner}S,x)$ the
_$\mathcal{H}^{k}$ -approximate tangent cone_ of $S$ at $x$; see [Fed69,
3.2.16]. If $\mathcal{H}^{k}(S)<\infty$, $f$ is a Lipschitz function on $S$,
then $\operatorname{Tan}^{k}(\mathcal{H}^{k}\operatorname{\llcorner}S,x)$ is a
$k$-dimensional plane at $\mathcal{H}^{k}$ a.e. $x\in S$ and we denote with
$J_{k}^{S}f$ the $k$-dimensional approximate tangential jacobian function of
$f$; see [Fed69, 3.2.19, 3.2.20].

###  2.1. Sets of finite perimeter

We refer to [AFP00, Chapter 3] or [Mag12] for details. We recall that
$\Omega\subseteq\mathbb{R}^{n+1}$ is a set of finite perimeter in
$\mathbb{R}^{n+1}$ if its characteristic function $\bm{1}_{\Omega}$ is a
function of bounded variation in $\mathbb{R}^{n+1}$. The reduced boundary
$\partial^{\ast}\Omega$ of $\Omega$ is the set of points
$x\in\mathbb{R}^{n+1}$ such that the following limit

$\lim_{r\to
0}\frac{\mathrm{D}\bm{1}_{\Omega}(B_{r}(x))}{|\mathrm{D}\bm{1}_{\Omega}(B_{r}(x))|}$

exists and belongs to $\mathbb{S}^{n}$, in which case we denote it by
$\nu_{\Omega}(x)$ (here $\mathrm{D}\bm{1}_{\Omega}$ is the distributional
gradient of $\bm{1}_{\Omega}$). The reduced boundary is countably
$\mathcal{H}^{n}$-rectifiable and $\mathcal{H}^{n}(\partial^{\ast}\Omega)$
equals the total variation of $\mathrm{D}\bm{1}_{\Omega}$; this number is the
perimeter of $\Omega$. The map
$\nu_{\Omega}:\partial^{\ast}\Omega\rightarrow\mathbb{S}^{n}$ is the measure-
theoretic exterior unit-normal of $\Omega$ and we define

$\overline{\nu}_{\Omega}:\partial^{\ast}\Omega\rightarrow\mathbb{R}^{n+1}\times\mathbb{S}^{n},\qquad\overline{\nu}_{\Omega}(x)=(x,\nu_{\Omega}(x)).$

Notice that if $\Omega$ is a $C^{1}$-domain then
$\partial^{\ast}\Omega=\partial\Omega$ and $\nu_{\Omega}$ is the classical
exterior unit-normal vector field of $\Omega$.

###  2.2. Currents

We refer to [Sim83, Chapter 6] for details. The space of compactly supported
$k$-forms on an open subset $U$ of $\mathbb{R}^{p}$ is denoted by the usual
$\mathcal{D}^{k}(U)$ and the space of $k$-currents on $U$ by
$\mathcal{D}_{k}(U)$. We denote with $\bm{M}_{W}(T)$ the mass of $T$ over an
open subset $W$ of $U$. A sequence $T_{\ell}\in\mathcal{D}_{k}(U)$ _weakly
converges_ to $T\in\mathcal{D}_{k}(U)$ if and only if

$T_{\ell}(\phi)\to T(\phi)\qquad\textrm{for all $\phi\in\mathcal{D}^{k}(U)$.}$

We say that a $k$-current $T\in\mathcal{D}_{k}(U)$ is an _interger
multiplicity rectifiable $k$-current of $U$_ provided

$T(\phi)=\int_{M}\langle\eta(x),\phi(x)\rangle\theta(x)\,d\mathcal{H}^{k}(x)\qquad\textrm{for
all $\phi\in\mathcal{D}^{k}(U)$,}$

where $M$ is a countably $\mathcal{H}^{k}$-rectifiable subset of $U$,
$\theta:M\rightarrow\mathbb{Z}_{+}$ is an $\mathcal{H}^{k}$-measurable
function such that $\int_{K\cap M}\theta\,d\mathcal{H}^{k}<\infty$ for every
compact subset $K$ of $U$, and
$\eta(x)=\tau_{1}(x)\wedge\ldots\wedge\tau_{k}(x)$ for $\mathcal{H}^{k}$ a.e.
$x\in M$, where $\tau_{1}(x),\ldots,\tau_{k}(x)$ form an orthonormal basis of
$\operatorname{Tan}^{k}(\mathcal{H}^{k}\operatorname{\llcorner}M,x)$ for
$\mathcal{H}^{k}$ a.e. $x\in M$. The set $M$ is called _carrier_ of $T$ and it
is $\mathcal{H}^{n}$ almost uniquely determined by $T$. Henceforth for each
integer multiplicity rectifiable $k$-current $T$ we introduce the symbol
$W_{T}$ for the carrier of $T$. One can easily check that if $W\subseteq U$ is
an open set then

$\bm{M}_{W}(T)=\int_{W\cap M}\theta\,d\mathcal{H}^{k}.$

###  2.3. Varifolds

We refer to [All72] for details. Let $U\subseteq\mathbb{R}^{n+1}$ be an open
set. The space of $k$-dimensional varifolds on $U$ is denoted with
$\bm{V}_{k}(U)$ and the space of $k$-dimensional integral varifolds on $U$
with the usual $\bm{IV}_{k}(U)$. Associated with $V\in\bm{V}_{k}(U)$ we
consider the weight measure $\|V\|$, which is the Radon measure on $U$ defined
by $\|V\|(S)=V(S\times\bm{G}(n+1,k))$ for every $S\subseteq U$, and we denote
the _first variation of $V$_ by $\delta V$; see [All72, section 4] for
details. A varifold $V\in\bm{V}_{k}(U)$ is a varifold of _bounded mean
curvature_ if there exists a function $h\in
L^{\infty}(\|V\|,\mathbb{R}^{n+1})$ so that

$\delta V(X)=\int h(x)\bullet X(x)\,d\|V\|(x),\quad\textrm{for all
$X\in\mathcal{C}^{\infty}_{c}(U,\mathbb{R}^{n+1})$.}$

The function $h$ is uniquely determined by $V$ and we write
$h=\bm{h}(V,\cdot)$.

In this paper we consider integral varifolds
$V\in\bm{IV}_{n}(\mathbb{R}^{n+1})$ associated with the reduced boundary of a
set of finite perimeter; namely if $E$ is a set of finite perimeter in
$\mathbb{R}^{n+1}$ of positive volume, we define the $n$-dimensional varifold
$V_{E}\in\bm{IV}_{n}(\mathbb{R}^{n+1})$ as the unique Radon measure $V_{E}$ on
$\mathbb{R}^{n+1}\times\bm{G}(n+1,n)$ such that

$\int\phi(x,S)\,dV_{E}(x,S)=\int_{\partial^{\ast}E}\phi(x,\operatorname{Tan}^{n}(\mathcal{H}^{n}\operatorname{\llcorner}\partial^{\ast}E,x))\,d\mathcal{H}^{n}(x)$

for every $\phi\in C_{c}(\mathbb{R}^{n+1}\times\bm{G}(n+1,n))$. Notice that
$\|V_{E}\|=\mathcal{H}^{n}\operatorname{\llcorner}\partial^{\ast}E$ and

$\delta V_{E}(X)=\int_{\partial^{\ast}E}{\rm
div}_{\mathbb{R}^{n+1}}(X)\,d\mathcal{H}^{n}-\int_{\partial^{\ast}E}\mathrm{D}X(x)(\nu_{E}(x))\bullet\nu_{E}(x)\,d\mathcal{H}^{n}(x)$
(2.1)

for all $X\in\mathcal{C}^{\infty}_{c}(\mathbb{R}^{n+1},\mathbb{R}^{n+1})$.

###  2.4. Normal bundle of closed sets

Suppose $C\subseteq\mathbb{R}^{n+1}$ is a closed set. We define

$\operatorname{nor}(C)=\\{(x,u)\in C\times\mathbb{S}^{n}:{\rm
dist}(x+su,C)=s\;\textrm{for some $s>0$}\\}.$

and we recall that $\operatorname{nor}(C)$ is always countably
$\mathcal{H}^{n}$-rectifiable; see [San20a]111The unit normal bundle of a
closed set $C$ in [San20a] is denoted with $N(C)$.. On the other hand we
remark that there are closed sets $C$ for which $\operatorname{nor}(C)$ has
not locally finite $\mathcal{H}^{n}$-measure; indeed this can happen even when
$C={\rm spt}\|V\|$ and $V\in\bm{IV}_{2}(\mathbb{R}^{3})$ is a varifold of
bounded mean curvature.

It is proved in [San20a] that for $\mathcal{H}^{n}$ a.e.
$(x,u)\in\operatorname{nor}(C)$ there exist a linear subspace $T_{C}(x,u)$ of
$\mathbb{R}^{n+1}$ and a symmetric bilinear form $Q_{C}(x,u):T_{C}(x,u)\times
T_{C}(x,u)\rightarrow\mathbb{R}$, whose eigenvalues can be used to provide an
explicit representation of the approximate tangent space of
$\operatorname{nor}(C)$ at $\mathcal{H}^{n}$ almost all points. To this end,
for $\mathcal{H}^{n}$ a.e. $(x,u)\in\operatorname{nor}(C)$, we define

$-\infty<\kappa_{C,1}(x,u)\leq\ldots\leq\kappa_{C,n}(x,u)\leq\infty$

in the following way: $\kappa_{C,1}(x,u),\ldots,\kappa_{C,m}(x,u)$ are the
eigenvalues of $Q_{C}(x,u)$, where $m=\dim T_{C}(x,u)$, and
$\kappa_{C,i}(x,u)=+\infty$ for $i=m+1,\ldots,n$. Given this definition we
recall the following lemma.

######  Lemma 2.1 (cf. [San20a, Lemma 4.11]).

Let $Q\subseteq\operatorname{nor}(C)$ be an $\mathcal{H}^{n}$-measurable set
with finite $\mathcal{H}^{n}$ measure. For $\mathcal{H}^{n}$ a.e. $(x,u)\in Q$
there exist $\tau_{1}(x,u),\ldots,\tau_{n}(x,u)\in\mathbb{R}^{n+1}$ such that
$\\{\tau_{1}(x,u),\ldots,\tau_{n}(x,u),u\\}$ is an orthonormal basis of
$\mathbb{R}^{n+1}$ and
$\zeta_{1}(x,u),\ldots,\zeta_{n}(x,u)\in\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}$,
defined as

$\zeta_{i}(x,u)=(\tau_{i}(x,u),\kappa_{i}(x,u)\tau_{i}(x,u))\quad\textrm{if
$\kappa_{C,i}(x,u)<\infty$}$ $\zeta_{i}(x,u)=(0,\tau_{i}(x,u))\quad\textrm{if
$\kappa_{C,i}(x,u)=\infty$},$

form an orthogonal basis of
$\operatorname{Tan}^{n}(\mathcal{H}^{n}\operatorname{\llcorner}Q,(x,u))$.

We also need the following result, relating the numbers $\kappa_{C,i}$ with
the principal curvature of a $C^{2}$-hypersurface that intersects $C$.

######  Lemma 2.2 (cf. [San20a, Lemma 6.1]).

If $\Sigma$ is an embedded $C^{2}$-hypersurface then there exists
$R\subseteq\Sigma\cap\partial C$ such that

$\mathcal{H}^{n}((\Sigma\cap\partial C)\setminus
R)=0,\qquad\operatorname{nor}(C)\operatorname{\llcorner}R\subseteq\operatorname{nor}(\Sigma)$

and

$\kappa_{C,i}(x,u)=\kappa_{\Sigma,i}(x,u)\quad\textrm{for $\mathcal{H}^{n}$
a.e.\ $(x,u)\in\operatorname{nor}(C)\operatorname{\llcorner}R$}$

for $1\leq i\leq n$. Here $\operatorname{nor}(\Sigma)$ is the classical unit-
normal bundle of $\Sigma$ and

$\kappa_{\Sigma,1}(x,u)\leq\ldots\leq\kappa_{\Sigma,n}(x,u)$

are the principal curvatures of $\Sigma$ at $x$ in the direction $u$.

We recall now the general Heintze-Karcher type inequality for arbitrary closed
sets proved in [HS22], which is used in the proof of Theorem 4.2.

######  Theorem 2.3 (cf. [HS22, Theorem 3.20]).

Let $C\subseteq\mathbb{R}^{n+1}$ be a bounded closed set with non empty
interior. Let $K=\mathbb{R}^{n+1}\setminus{\rm interior}(C)$ and assume that

$\sum_{i=1}^{n}\kappa_{K,i}(x,u)\leq 0\quad\textrm{for $\mathcal{H}^{n}$ a.e.\
$(x,u)\in\operatorname{nor}(K)$.}$

Then

$(n+1)\mathcal{L}^{n+1}({\rm
interior}(C))\leq\int_{\operatorname{nor}(K)}J_{n}^{\operatorname{nor}(K)}\pi_{0}(x,u)\,\frac{n}{|\sum_{i=1}^{n}\kappa_{K,i}(x,u)|}\,d\mathcal{H}^{n}(x,u).$

It the equality holds and there exists $q<\infty$ such that
$|\sum_{i=1}^{n}\kappa_{K,i}(x,u)|\leq q$ for $\mathcal{H}^{n}$ a.e.
$(x,u)\in\operatorname{nor}(K)$, then $C$ equals the union of finitely many
closed balls with disjointed interiors.

##  3\. Legendrian cycles and Curvature measures

We denote with $E_{n+1}\in\mathcal{D}^{n+1}(\mathbb{R}^{n+1})$ the _canonical
volume form_ of $\mathbb{R}^{n+1}$, and with

$\alpha_{\mathbb{R}^{n+1}}:\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}\rightarrow{\textstyle\bigwedge^{1}}(\mathbb{R}^{n+1}\times\mathbb{R}^{n+1})$

the _contact $1$-form_; i.e.

$\langle(y,v),\alpha_{\mathbb{R}^{n+1}}(x,u)\rangle=y\bullet
u\qquad\textrm{for $(x,u),(y,v)\in\mathbb{R}^{n+1}$.}$

A _Legendrian cycle of $\mathbb{R}^{n+1}$_ is an integer-multiplicity
rectifiable $n$-current $T$ of $\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}$ such
that

$\textrm{${\rm spt}(T)$ is compact subset of
$\mathbb{R}^{n+1}\times\mathbb{S}^{n}$,}\quad\partial T=0,\quad
T\operatorname{\llcorner}\alpha_{\mathbb{R}^{n+1}}=0.$

The following result describes the approximate tangent space of the carrier of
a Legendrian cycle.

######  Theorem 3.1 (cf. [RZ19, Theorem 9.2]).

Let $T$ be a Legendrian cycle of $\mathbb{R}^{n+1}$ with carrier $W_{T}$. For
$\mathcal{H}^{n}$ a.e. $(x,u)\in W_{T}$ there exist numbers
$-\infty<\kappa_{1}(x,u)\leq\ldots\leq\kappa_{n}(x,u)\leq\infty$ and vectors
$\tau_{1}(x,u),\ldots,\tau_{n}(x,u)$ such that
$\\{\tau_{1}(x,u),\ldots,\tau_{n}(x,u),u\\}$ is a positively oriented basis of
$\mathbb{R}^{n+1}$ (i.e.
$\langle\tau_{1}(x,u)\wedge\ldots\wedge\tau_{n}(x,u)\wedge
u,E_{n+1}\rangle=1$) and the vectors
$\zeta_{1}(x,u),\ldots,\zeta_{n}(x,u)\in\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}$,
defined as

$\zeta_{i}(x,u)=(\tau_{i}(x,u),\kappa_{i}(x,u)\tau_{i}(x,u))\quad\textrm{if
$\kappa_{i}(x,u)<\infty$}$ $\zeta_{i}(x,u)=(0,\tau_{i}(x,u))\quad\textrm{if
$\kappa_{i}(x,u)=\infty$},$

form an orthogonal basis of
$\operatorname{Tan}^{n}(\mathcal{H}^{n}\operatorname{\llcorner}W_{T},(x,u))$.
The numbers $\kappa_{i}(x,u)$ for $i=1,\ldots,n$ are uniquely determined, as
well as the subspaces of $\mathbb{R}^{n+1}$ spanned by
$\\{\tau_{i}(x,u):\pi_{1}(\zeta_{i}(x,u))=\kappa_{k}(x,u)\tau_{i}(x,u)\\}$ for
each $k$.

######  Definition 4.

Given $T$, $\tau_{1},\ldots,\tau_{n}$ and $\zeta_{1},\ldots,\zeta_{n}$ as in
Theorem 3.1, we define

$\mathcal{\mathcal{J}}_{T}(x,u)=\frac{1}{|\zeta_{1}(x,u)\wedge\ldots\wedge\zeta_{n}(x,u)|}$

and

$\zeta_{T}(x,u)=\frac{\zeta_{1}(x,u)\wedge\ldots\wedge\zeta_{n}(x,u)}{|\zeta_{1}(x,u)\wedge\ldots\wedge\zeta_{n}(x,u)|}$

for $\mathcal{H}^{n}$ a.e. $(x,u)\in W_{T}$.

######  Remark 3.1.

This definition is well-posed, in the sense that $\mathcal{J}_{T}$ and
$\zeta_{T}$ do not depend, up to a set of $\mathcal{H}^{n}$-measure zero, on
the choice of the vectors $\tau_{1},\ldots,\tau_{n}$.

######  Remark 3.2.

There exists a $\mathcal{H}^{n}\operatorname{\llcorner}W_{T}$ almost unique
integer-valued map $i_{T}$ such that $i_{T}(x,u)=0$ for $\mathcal{H}^{n}$ a.e.
$(x,u)\in\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}\setminus W_{T}$,
$i_{T}(x,u)>0$ for $\mathcal{H}^{n}$ a.e. $(x,u)\in W_{T}$ and

$T=(\mathcal{H}^{n}\operatorname{\llcorner}W_{T})i_{T}\wedge\zeta_{T}.$

######  Definition 5 (Lipschitz-Killing differential forms).

For each $k\in\\{0,\ldots,n\\}$ we define

$\Sigma_{n,k}=\bigg{\\{}\sigma:\\{1,\ldots,n\\}\rightarrow\\{0,1\\}:\sum_{i=1}^{n}\sigma(i)=n-k\bigg{\\}}.$

The _$k$ -th Lipschitz-Killing differential form_
$\varphi_{k}:\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}\rightarrow\bigwedge^{n}(\mathbb{R}^{n+1}\times\mathbb{R}^{n+1})$
is defined by

$\displaystyle\langle\xi_{1}\wedge\cdots\wedge\xi_{n},\varphi_{k}(x,u)\rangle$
$\displaystyle\qquad=\sum_{\sigma\in\Sigma_{n,k}}\langle\pi_{\sigma(1)}(\xi_{1})\wedge\cdots\wedge\pi_{\sigma(n)}(\xi_{n})\wedge
u,E_{n+1}\rangle,$

for every $\xi_{1},\ldots,\xi_{n}\in\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}$.

######  Definition 6 (Curvature measures).

If $T$ is a Legendrian cycle in $\mathbb{R}^{n+1}$ and $k\in\\{0,\ldots,n\\}$,
then _$k$ -th curvature measure_ of $T$ is the $0$-current

$T\operatorname{\llcorner}\varphi_{k}\in\mathcal{D}_{0}(\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}).$

######  Definition 7 (Principal curvatures).

Given a Legendrian cycle $T$ of $\mathbb{R}^{n+1}$ we define
$\kappa_{T,i}=\kappa_{i}$, where $\kappa_{1}\leq\ldots\leq\kappa_{n}$ are the
functions defined $\mathcal{H}^{n}$ a.e. on $W_{T}$ given by Theorem 3.1.

The numbers $\kappa_{T,1}\leq\ldots\leq\kappa_{T,n}$ are the principal
curvatures of $T$.

######  Definition 8 (Mean curvature functions).

Given a Legendrian cycle $T$ of $\mathbb{R}^{n+1}$ we define

$W_{T}^{(i)}=\\{(x,u)\in
W_{T}:\kappa_{T,i}(x,u)<\infty,\;\kappa_{T,i+1}(x,u)=+\infty\\}\quad\textrm{for
$i=1,\ldots,n-1$}$ $W_{T}^{(0)}=\\{(x,u)\in
W_{T}:\kappa_{T,1}(x,u)=+\infty\\}$ $W_{T}^{(n)}=\\{(x,u)\in
W_{T}:\kappa_{T,n}(x,u)<+\infty\\}.$

Then we define the _$r$ -th mean curvature function of $T$_ as

$H_{T,r}=\bm{1}_{W_{T}^{(n-r)}}+\sum_{i=1}^{r}\sum_{\lambda\in\Lambda_{n-r+i,i}}\kappa_{T,\lambda(1)}\cdots\kappa_{T,\lambda(i)}\bm{1}_{W_{T}^{(n-r+i)}}\qquad\textrm{for
$r=1,\ldots,n$}$

where $\Lambda_{n,k}$, for $k\leq n$, is the set of all increasing functions
from $\\{1,\ldots,k\\}$ to $\\{1,\ldots,n\\}$. Additionally,

$H_{T,0}=\bm{1}_{W_{T}^{(n)}}.$

######  Remark 3.3.

It follows from Theorem 3.1 that

$J_{n}^{W_{T}}\pi_{0}(x,u)=\mathcal{J}_{T}(x,u)>0\quad\textrm{for
$\mathcal{H}^{n}$ a.e.\ $(x,u)\in W_{T}^{(n)}$}$

and

$J_{n}^{W_{T}}\pi_{0}(x,u)=0\quad\textrm{for $\mathcal{H}^{n}$ a.e.\ $(x,u)\in
W_{T}\setminus W_{T}^{(n)}$}.$

The definition of mean curvature functions is clearly motivated by the
following result.

######  Lemma 3.2.

If $T$ is a Legendrian cycle of $\mathbb{R}^{n+1}$ and $k\in\\{0,\ldots,n\\}$
then

$(T\operatorname{\llcorner}\varphi_{k})(\phi)=\int\phi(x,u)\,i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\,H_{T,n-k}(x,u)\,d\mathcal{H}^{n}(x,u)$

for every $\phi\in\mathcal{D}^{0}(\mathbb{R}^{n+1}\times\mathbb{R}^{n+1})$.

###### Proof.

For $j\in\\{0,\ldots,n\\}$ we define

$\Sigma_{n,k}^{(j)}=\\{\sigma\in\Sigma_{n,k}:\sigma(i)=1\;\textrm{for every
$i>j$}\\}.$

Notice that $\Sigma_{n,k}^{(j)}=\varnothing$ if $j<k$. Moreover,

$\langle\pi_{\sigma(1)}(\zeta_{1}(x,u))\wedge\ldots\wedge\pi_{\sigma(n)}(\zeta_{n}(x,u))\wedge
u,E_{n+1}\rangle=0$

for $(x,u)\in W_{T}^{(j)}$ and
$\sigma\in\Sigma_{n,k}\setminus\Sigma_{n,k}^{(j)}$, where
$\tau_{1},\ldots,\tau_{n}$ and $\zeta_{1},\ldots,\zeta_{n}$ are given as in
Theorem 3.1. Therefore we infer that

$\displaystyle(T\operatorname{\llcorner}\varphi_{k})(\phi)$
$\displaystyle\quad=\sum_{j=k}^{n}\sum_{\sigma\in\Sigma_{n,k}^{(j)}}\int_{W_{T}^{(j)}}\phi(x,u)\,i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\,\langle\pi_{\sigma(1)}(\zeta_{1}(x,u))\wedge\ldots$
$\displaystyle\qquad\qquad\qquad\qquad\ldots\wedge\pi_{\sigma(n)}(\zeta_{n}(x,u))\wedge
u,E_{n+1}\rangle\,d\mathcal{H}^{n}(x,u).$

Noting that $\langle\tau_{1}(x,u)\wedge\ldots\wedge\tau_{n}(x,u)\wedge
u,E_{n+1}\rangle=1$, we infer for $j>k$ that

$\displaystyle\sum_{\sigma\in\Sigma_{n,k}^{(j)}}\int_{W_{T}^{(j)}}\phi(x,u)\,i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\,\langle\pi_{\sigma(1)}(\zeta_{1}(x,u))\wedge\ldots$
$\displaystyle\qquad\qquad\qquad\qquad\ldots\wedge\pi_{\sigma(n)}(\zeta_{n}(x,u))\wedge
u,E_{n+1}\rangle\,d\mathcal{H}^{n}(x,u)$
$\displaystyle\quad\quad=\sum_{\lambda\in\Lambda_{j,j-k}}\int_{W_{T}^{(j)}}\phi(x,u)\,i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\prod_{\ell=1}^{j-k}\kappa_{T,\lambda(\ell)}(x,u)\,d\mathcal{H}^{n}(x,u),$

while for $j=k$ we have that

$\displaystyle\sum_{\sigma\in\Sigma_{n,k}^{(k)}}\int_{W_{T}^{(k)}}\phi(x,u)\,i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\,\langle\pi_{\sigma(1)}(\zeta_{1}(x,u))\wedge\ldots$
$\displaystyle\qquad\qquad\qquad\qquad\ldots\wedge\pi_{\sigma(n)}(\zeta_{n}(x,u))\wedge
u,E_{n+1}\rangle\,d\mathcal{H}^{n}(x,u)$
$\displaystyle\quad\quad=\int_{W_{T}^{(k)}}\phi(x,u)\,i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\,d\mathcal{H}^{n}(x,u).$

We conclude

$\displaystyle(T\operatorname{\llcorner}\varphi_{k})(\phi)$
$\displaystyle\quad=\sum_{i=1}^{n-k}\sum_{\lambda\in\Lambda_{k+i,i}}\int_{W_{T}^{(k+i)}}\phi(x,u)\,i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\prod_{\ell=1}^{i}\kappa_{T,\lambda(\ell)}(x,u)\,d\mathcal{H}^{n}(x,u)$
$\displaystyle\qquad\qquad+\int_{W_{T}^{(k)}}\phi(x,u)\,i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\,d\mathcal{H}^{n}(x,u),$

which is the desired conclusion. ∎

We conclude this section by introducing the normal cycle of a $C^{2}$-domain.

######  Definition 9 (Normal cycle of a $C^{2}$-domain).

Suppose $\Omega\subseteq\mathbb{R}^{n+1}$ is an open set with
$C^{2}$-boundary. Then we define

$N_{\Omega}=(\overline{\nu}_{\Omega})_{\\#}((\mathcal{H}^{n}\operatorname{\llcorner}\partial\Omega)\wedge\star\,\nu_{\Omega})\in\mathcal{D}_{n}(\mathbb{R}^{n+1}\times\mathbb{S}^{n}).$

######  Remark 3.4 (Classical).

Since
$(\mathcal{H}^{n}\operatorname{\llcorner}\partial\Omega)\wedge\star\,\nu_{\Omega}$
is a $n$-dimensional cycle of $\mathbb{R}^{n+1}$, it follows that $N_{\Omega}$
is a $n$-dimensional cycle in $\mathbb{R}^{n+1}\times\mathbb{S}^{n}$.
Moreover, area formula for rectifiable currents (see [Fed69, 4.1.30]) allows
to conclude

$\displaystyle N_{\Omega}(\psi)$
$\displaystyle=\int_{\partial\Omega}\langle{\textstyle\Lambda_{n}}\mathrm{D}\overline{\nu}_{\Omega}(x)(\star\nu_{\Omega}(x)),\psi(\overline{\nu}_{\Omega}(x))\rangle\,d\mathcal{H}^{n}(x)$
$\displaystyle=\big{[}\big{(}\mathcal{H}^{n}\operatorname{\llcorner}\overline{\nu}_{\Omega}(\partial\Omega)\big{)}\wedge\eta\big{]}(\psi),$
(3.1)

where

$\eta(x,u)=\frac{{\textstyle\Lambda_{n}}\mathrm{D}\overline{\nu}_{\Omega}(x)(\star\nu_{\Omega}(x))}{|{\textstyle\Lambda_{n}}\mathrm{D}\overline{\nu}_{\Omega}(x)(\star\nu_{\Omega}(x))|}\quad\textrm{for
$(x,u)\in\overline{\nu}_{\Omega}(\partial\Omega)$}.$

It follows that
$N_{\Omega}\in\bm{I}_{n}(\mathbb{R}^{n+1}\times\mathbb{R}^{n+1})$. We notice
that

$\displaystyle\bm{M}(N_{\Omega})=\mathcal{H}^{n}(\overline{\nu}_{\Omega}(\partial\Omega))=\int_{\partial\Omega}J_{n}^{\partial\Omega}\overline{\nu}_{\Omega}\,d\mathcal{H}^{n}=\int_{\partial\Omega}\prod_{i=1}^{n}(1+\kappa_{\Omega,i}^{2})^{\frac{1}{2}}\,d\mathcal{H}^{n},$

and by arithmetic-geometric mean inequality we have that

$\displaystyle\bm{M}(N_{\Omega})\leq
2^{\frac{n}{2}}\bigg{(}\mathcal{H}^{n}(\partial\Omega)+n^{-\frac{n}{2}}\int_{\Omega}A_{\Omega}(x)^{n}\,d\mathcal{H}^{n}\bigg{)}.$
(3.2)

We check now that
$N_{\Omega}\operatorname{\llcorner}\alpha_{\mathbb{R}^{n+1}}=0$: if
$x\in\partial\Omega$ and $\tau_{1},\ldots,\tau_{n}$ is an orthonormal basis of
$\operatorname{Tan}(\partial\Omega,x)$ such that
$\star\nu_{\Omega}=\tau_{1}\wedge\ldots\wedge\tau_{n}$ and
$\mathrm{D}\nu_{\Omega}(x)(\tau_{i}(x))=\kappa_{\Omega,i}(x)\tau_{i}(x)$ for
$x\in\partial\Omega$, then we use shuffle formula (see [Fed69, pag. 18]) to
compute

$\displaystyle\langle{\textstyle\Lambda_{n}}\mathrm{D}\overline{\nu}_{\Omega}(x)(\star\nu_{\Omega}(x)),(\psi\wedge\alpha_{\mathbb{R}^{n+1}})(\overline{\nu}_{\Omega}(x))\rangle$
$\displaystyle\qquad=\langle(\tau_{1},\kappa_{\Omega,1}(x)\tau_{1})\wedge\ldots\wedge(\tau_{n},\kappa_{\Omega,n}(x)\tau_{n}),(\psi\wedge\alpha_{\mathbb{R}^{n+1}})(\overline{\nu}_{\Omega}(x))\rangle=0$

for every $\psi\in\mathcal{D}^{n-1}(\mathbb{R}^{n+1}\times\mathbb{R}^{n+1})$.
We conclude that $N_{\Omega}$ is a Legendrian cycle of $\mathbb{R}^{n+1}$.
Additionally one easily infer that
$W_{N_{\Omega}}^{(n)}=W_{N_{\Omega}}=\overline{\nu}_{\Omega}(\partial\Omega)$
and

$\kappa_{N_{\Omega},i}(x,u)=\kappa_{\Omega,i}(x),\quad\mathcal{J}_{N_{\Omega}}(x,u)=\frac{1}{|{\textstyle\Lambda_{n}}\mathrm{D}\overline{\nu}_{\Omega}(x)(\star\nu_{\Omega}(x))|}$

for every $(x,u)\in W_{N_{\Omega}}$. In particular,
$H_{N_{\Omega},k}(x,u)=H_{\Omega,k}(x)$ for
$(x,u)\in\overline{\nu}_{\Omega}(\partial\Omega)$ and by Remark 3.3

$(N_{\Omega}\operatorname{\llcorner}\varphi_{n-k})(\phi)=\int_{\partial\Omega}H_{\Omega,k}(x)\,\phi(x,\nu_{\Omega}(x))\,d\mathcal{H}^{n}(x)$
(3.3)

for all $\phi\in\mathcal{D}^{0}(\mathbb{R}^{n+1}\times\mathbb{R}^{n+1})$ and
$k\in\\{0,\ldots,n\\}$.

######  Lemma 3.3.

Let $\Omega_{\ell}$ be an asymptotically $k$-mean convex sequence such that
$N_{\Omega_{\ell}}$ weakly converges to a Legendrian cycle $T$ of
$\mathbb{R}^{n+1}$. Then

$H_{T,i}(x,u)\geq 0\quad\textrm{for $\mathcal{H}^{n}$ a.e.\ $(x,u)\in W_{T}$
and for $i\in\\{1,\ldots,k\\}$.}$

###### Proof.

Let $i\in\\{1,\ldots,k\\}$. By (3.3) we have that

$\displaystyle\big{(}N_{\Omega_{\ell}}\operatorname{\llcorner}\varphi_{n-i}\big{)}(\phi)$
$\displaystyle\qquad=-\int_{\partial\Omega_{\ell}}H_{\Omega_{\ell},i}^{-}(x)\,\phi(x,\nu_{\Omega_{\ell}}(x))\,d\mathcal{H}^{n}(x)+\int_{\partial\Omega_{\ell}}H_{\Omega_{\ell},i}^{+}(x)\,\phi(x,\nu_{\Omega_{\ell}}(x))\,d\mathcal{H}^{n}(x)$

for $\phi\in\mathcal{D}^{0}(\mathbb{R}^{n+1}\times\mathbb{R}^{n+1})$, whence
we infer

$(T\operatorname{\llcorner}\varphi_{n-i})(\phi)=\lim_{i\to\infty}\int_{\partial\Omega_{\ell}}H_{\Omega_{\ell},i}^{+}(x)\,\phi(x,\nu_{\Omega_{\ell}}(x))\,d\mathcal{H}^{n}(x)\geq
0$

for $\phi\in\mathcal{D}^{0}(\mathbb{R}^{n+1}\times\mathbb{R}^{n+1})$ with
$\phi\geq 0$. Now we obtain the desired conclusion from Lemma 3.2. ∎

######  Lemma 3.4 (Minkowski-Hsiung identities).

Suppose $\Omega_{\ell}\subseteq\mathbb{R}^{n+1}$ is a compactly supported
sequence of open sets with $C^{2}$-boundary and $T$ is a Legendrian cycle of
$\mathbb{R}^{n+1}$ such that $N_{\Omega_{\ell}}\rightarrow T$ weakly.

Then the Minkowski identity holds for $T$:

$\displaystyle(n-k+1)\int
i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\,H_{T,k-1}(x,u)\,d\mathcal{H}^{n}(x,u)$
$\displaystyle\qquad\qquad=k\int(x\bullet
u)i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\,H_{T,k}(x,u)\,d\mathcal{H}^{n}(x,u)$ (3.4)

for $k=1,\ldots,n$. Additionally, if $\Omega_{\ell}$ converges in measure to a
set $\Omega$, then

$(n+1)\mathcal{L}^{n+1}(\Omega)=\int(x\bullet
u)i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\,H_{T,0}(x,u)\,d\mathcal{H}^{n}(x,u).$
(3.5)

###### Proof.

Let $R>0$ such that $\Omega_{\ell}\subseteq B_{R}$ for all $\ell\geq 1$.
Define $\sigma:\mathbb{R}^{n+1}\times\mathbb{S}^{n}\rightarrow\mathbb{R}$ as
$\sigma(x,u)=x\bullet u$ for $(x,u)\in\mathbb{R}^{n+1}\times\mathbb{S}^{n}$
and choose $\phi\in\mathcal{D}^{0}(\mathbb{R}^{n+1}\times\mathbb{R}^{n+1})$
such that $\phi(x,u)=1$ for all $(x,u)\in B_{2R}$. The classical Minkowski-
Hsiung identities for $C^{2}$-domains and (3.3) imply

$\displaystyle(n-k+1)N_{\Omega_{\ell}}(\phi\,\varphi_{n-k+1})$
$\displaystyle=(n-k+1)\int_{\partial\Omega_{\ell}}H_{\Omega_{\ell},k-1}(x)\,d\mathcal{H}^{n}(x)$
$\displaystyle=k\int_{\partial\Omega_{\ell}}(x\bullet\nu_{\Omega_{\ell}}(x))H_{\Omega_{\ell},k}(x)\,d\mathcal{H}^{n}(x)$
$\displaystyle=kN_{\Omega_{\ell}}(\sigma\,\phi\,\varphi_{n-k}).$

If $\ell\to\infty$ this yields that

$(n-k+1)T(\phi\,\varphi_{n-k+1})=kT(\sigma\,\phi\,\varphi_{n-k})$

and we obtain (3.4) from Lemma 3.2. Analogously, if $\Omega_{\ell}\to\Omega$
in measure, then divergence theorem and area formula imply

$(n+1)\mathcal{L}^{n+1}(\Omega_{\ell})=\int_{\partial\Omega_{\ell}}(a\bullet\nu_{\Omega_{\ell}}(a))\,d\mathcal{H}^{n}(a)=N_{\Omega_{\ell}}(\sigma\,\phi\,\varphi_{n})$

and we pass to limit to obtain (3.5). ∎

##  4\. Heintze-Karcher inequality for varifolds of bounded mean curvature

The next lemma provides fundamental structural properties of a set of finite
perimeter whose reduced boundary is a varifold of bounded mean curvature.

######  Lemma 4.1.

Let $E\subseteq\mathbb{R}^{n+1}$ be a set of finite perimeter in
$\mathbb{R}^{n+1}$ and positive volume such that $V_{E}$ is a varifold of
bounded mean curvature.

Then there exists a closed set $C\subseteq\mathbb{R}^{n+1}$ with non-empty
interior such that

${\rm spt}\|V_{C}\|=\partial C,\qquad\mathcal{L}^{n+1}((C\setminus
E)\cup(E\setminus C))=0,$ (4.1) $\mathcal{H}^{n}\big{[}\partial
C\setminus\big{(}\partial^{\ast}C\cap\pi_{0}(\operatorname{nor}(C))\big{)}\big{]}=0,$
(4.2) $\mathcal{H}^{n}\big{(}\operatorname{nor}(\partial
C)\operatorname{\llcorner}Z\big{)}=0\quad\textrm{whenever $Z\subseteq\partial
C$ with $\mathcal{H}^{n}(Z)=0$,}$ (4.3)
$\operatorname{nor}(C)\operatorname{\llcorner}\partial^{\ast}C=\\{(x,\nu_{C}(x)):x\in\partial^{\ast}C\cap\pi_{0}(\operatorname{nor}(C))\\},$
(4.4)
$\mathcal{H}^{n}(\operatorname{nor}(C)\setminus\overline{\nu}_{C}(\partial^{\ast}C))=0,$
(4.5)

and

$\sum_{i=1}^{n}\kappa_{C,i}(x,u)=\bm{h}(V_{C},x)\bullet u\quad\textrm{for
$\mathcal{H}^{n}$ a.e.\ $(x,u)\in\operatorname{nor}(C)$.}$ (4.6)

###### Proof.

Since $V_{E}$ is a varifold of bounded mean curvature, it follows from [All72,
Theorem 8.6] that
$\Theta^{n}(\mathcal{H}^{n}\operatorname{\llcorner}\partial^{\ast}E,\cdot)$ is
an upper-semicontinuous function on $\mathbb{R}^{n+1}$. We conclude that
$\Theta^{n}(\mathcal{H}^{n}\operatorname{\llcorner}\partial^{\ast}E,x)\geq 1$
for every $x\in\overline{\partial^{\ast}E}$ and
$\mathcal{H}^{n}(\overline{\partial^{\ast}E}\setminus\partial^{\ast}E)=0$ by
standard density results. We infer from [DRKS20, Lemma 6.2] that there exists
a closed set $C\subseteq\mathbb{R}^{n+1}$ with non-empty interior such that
(4.1) holds and

$\mathcal{H}^{n}(\partial C\setminus\partial^{\ast}C)=0.$ (4.7)

For every $x\in\partial^{\ast}C$, by Allard-Duggan regularity theorem [Dug86],
there exist an open set $U\subseteq\mathbb{R}^{n+1}$ with $x\in U$, an open
set $V\subseteq\mathbb{R}^{n}$ and a function $f\in W^{2,p}(V,\mathbb{R})$ for
every $p<\infty$, such that $U\cap\partial^{\ast}C$ coincides with the graph
of $f$, up to a rotation in $\mathbb{R}^{n+1}$. We now recall Reshetnyak
differentiability theorem, see [Res68, Theorem 2]: if $f\in
W^{2,p}(V,\mathbb{R})$ with $p>n$, then for $\mathcal{L}^{n}$ a.e. $a\in V$
there exists a polynomial function $P_{a}$ of degree at most $2$ such that
$P_{a}(a)=f(a)$ and

$\lim_{b\to a}\frac{|f(b)-P_{a}(b)|}{|b-a|^{2}}=0.$ (4.8)

Denoting with $G$ the graph of $f$ and with $\nu_{G}$ a continuous unit-normal
on $G$, we easily conclude from (4.8) that for $\mathcal{H}^{n}$ a.e. $x\in G$
there exists $s>0$ such that

${\rm dist}(x+s\nu_{G}(x),G)=s\quad\textrm{and}\quad{\rm
dist}(x-s\nu_{G}(x),G).$

Henceforth, it follows that

$\mathcal{H}^{n}(\partial^{\ast}C\setminus\pi_{0}(\operatorname{nor}(C)))=0,$

and we obtain (4.2) from (4.7).

The assertion in (4.3) is proved in [San20b, Theorem 3.8]222In [San20b,
Theorem 3.8] the desired conclusion is actually achieved in the more general
class of $(m,h)$-sets.. To prove (4.4) we recall from De Giorgi theorem
[AFP00, Theorem 3.59] that if $x\in\partial^{\ast}C$ then the sets
$\frac{C-x}{r}$ converge in measure as $r\to 0$ to an halfspace perpendicular
to $\nu_{C}(x)$. Consequently if $(x,u)\in\operatorname{nor}(C)$ and
$x\in\partial^{\ast}C$, we readily conclude $u=\nu_{C}(x)$. This proves (4.4),
whence we infer that

$\operatorname{nor}(C)\operatorname{\llcorner}(\partial
C\setminus\partial^{\ast}C)=\operatorname{nor}(C)\setminus\overline{\nu}_{C}(\partial^{\ast}C).$
(4.9)

Then (4.5) follows from (4.9).

Finally we prove (4.6). By the $C^{2}$-rectifiability of Menne [Men13] we
obtain a countable family $F$ of embedded $C^{2}$-hypersurfaces such that

$\mathcal{H}^{n}\Big{(}\partial C\setminus\bigcup F\Big{)}=0$ (4.10)

and for each $\Sigma\in F$ we have that

$\bm{h}(\Sigma,x)=\bm{h}(V_{C},x)\qquad\textrm{for $\mathcal{H}^{n}$ a.e.\
$x\in\Sigma\cap\partial C$.}$ (4.11)

(Here $\bm{h}(\Sigma,x)$ is the mean curvature vector of $\Sigma$.) For each
$\Sigma\in F$, if $R_{\Sigma}\subseteq\Sigma\cap\partial C$ is a set as in
Lemma 2.2 and $R^{\prime}_{\Sigma}$ is the set of $x\in\Sigma\cap\partial C$
such that (4.11) holds, we have that
$\mathcal{H}^{n}\big{(}(\Sigma\cap\partial C)\setminus(R_{\Sigma}\cap
R^{\prime}_{\Sigma})\big{)}=0$ and it follows from (4.3) that

$\mathcal{H}^{n}\big{[}\operatorname{nor}(C)\operatorname{\llcorner}\big{(}(\Sigma\cap\partial
C)\setminus(R_{\Sigma}\cap R^{\prime}_{\Sigma})\big{)}\big{]}=0;$

therefore we can conclude that

$\sum_{i=1}^{n}\kappa_{C,i}(x,u)=\sum_{i=1}^{n}\kappa_{\Sigma,i}(x,u)=\bm{h}(\Sigma,x)\bullet
u=\bm{h}(V_{C},x)\bullet u$

for $\mathcal{H}^{n}$ a.e.
$(x,u)\in\operatorname{nor}(C)\operatorname{\llcorner}(\Sigma\cap\partial C)$
and for $1\leq i\leq n$. Now (4.6) follows from (4.10). ∎

We can now state and prove the Heintze-Karcher inequality for sets of finite
perimeter and bounded distributional mean curvature.

######  Theorem 4.2.

Let $E\subseteq\mathbb{R}^{n+1}$ be a set of finite perimeter and finite
volume such that $V_{E}$ is a varifold of bounded mean curvature and

$\bm{h}(V_{E},x)\bullet\nu_{E}(x)\geq 0\quad\textrm{for $\mathcal{H}^{n}$
a.e.\ $x\in\partial^{\ast}E$.}$

Then

$\mathcal{L}^{n+1}(E)\leq\frac{n}{n+1}\int_{\partial^{\ast}E}\frac{1}{\bm{h}(V_{E},x)\bullet\nu_{E}(x)}\,d\mathcal{H}^{n}(x),$
(4.12)

and the equality is achieved if and only if $E$ is $\mathcal{L}^{n+1}$-almost
equal to a finite union of closed ball with disjointed interiors.

###### Proof.

We assume $\mathcal{L}^{n+1}(E)>0$. Define $F=\mathbb{R}^{n+1}\setminus E$ and
notice that

$\partial^{\ast}F=\partial^{\ast}E\quad\textrm{and}\quad\nu_{F}=-\nu_{E}.$

Since $V_{F}=V_{E}$ is a varifold of bounded mean curvature, we can apply
Lemma 4.1 to find a closed set $K\subseteq\mathbb{R}^{n+1}$ with non empty
interior such that (4.1)-(4.6) hold with $E$ and $C$ replaced by $F$ and $K$,
respectively. In particular, (since $V_{E}=V_{K}$ and $\nu_{K}=-\nu_{E}$)

$\sum_{i=1}^{n}\kappa_{K,i}(x,u)=\bm{h}(V_{E},x)\bullet u\quad\textrm{for
$\mathcal{H}^{n}$ a.e.\ $(x,u)\in\operatorname{nor}(K)$},$
$\operatorname{nor}(K)\operatorname{\llcorner}\partial^{\ast}K=\\{(x,-\nu_{E}(x)):x\in\partial^{\ast}K\cap\pi_{0}(\operatorname{nor}(K))\\}$

and

$\mathcal{H}^{n}(\operatorname{nor}(K)\setminus(\operatorname{nor}(K)\operatorname{\llcorner}\partial^{\ast}K))=0.$

Henceforth,

$\sum_{i=1}^{n}\kappa_{K,i}(x,u)=-\bm{h}(V_{E},x)\bullet\nu_{E}(x)\leq
0\quad\textrm{for $\mathcal{H}^{n}$ a.e.\ $(x,u)\in\operatorname{nor}(K)$}$

and it follows from Theorem 2.3 and area formula that

$\displaystyle(n+1)\mathcal{L}^{n+1}(E)$
$\displaystyle\leq\int_{\operatorname{nor}(K)}J_{n}^{\operatorname{nor}(K)}\pi_{0}(x,u)\,\frac{n}{\bm{h}(V_{E},x)\bullet\nu_{E}(x)}\,d\mathcal{H}^{n}(x,u)$
$\displaystyle=\int_{\partial^{\ast}E}\frac{n}{\bm{h}(V_{E},x)\bullet\nu_{E}(x)}\,d\mathcal{H}^{n}(x).$

If the equality is achieved in (4.12), since $V_{E}$ is a varifold of bounded
mean curvature, we obtain the conclusion directly from the rigidity statement
in Theorem 2.3. ∎

######  Corollary 4.3.

Let $E\subseteq\mathbb{R}^{n+1}$ be a set of finite perimeter with positive
and finite volume such that $V_{E}$ is a varifold of bounded mean curvature
and

$\bm{h}(V_{E},x)\bullet\nu_{E}(x)\geq\frac{n\mathcal{H}^{n}(\partial^{\ast}E)}{(n+1)\mathcal{L}^{n+1}(E)}\quad\textrm{for
$\mathcal{H}^{n}$ a.e.\ $x\in\partial^{\ast}E$.}$

Then $E$ is $\mathcal{L}^{n+1}$ almost equal to a finite union of disjoint
open balls of the same radius
$\rho=\frac{(n+1)\mathcal{L}^{n+1}(E)}{\mathcal{H}^{n}(\partial^{\ast}E)}$.

###### Proof.

The result can be deduced from Theorem 4.2 using the same argument as in
[HS22, Corollary 5.16]. We leave the details to the reader. ∎

##  5\. Proof of Theorem 1.1

Let $\Omega\subseteq\mathbb{R}^{n+1}$ be a set of finite perimeter such that
the sequence $\Omega_{\ell}$ converges in measure to $\Omega$ and
$\mathcal{H}^{n}(\partial\Omega_{\ell})\to\mathcal{H}^{n}(\partial^{\ast}\Omega)$.
We can assume that $\mathcal{L}^{n+1}(\Omega)>0$, otherwise there is nothing
to prove. It follows from [AFP00, Propositions 1.80 and 3.13]) that

$\mathcal{H}^{n}\operatorname{\llcorner}\partial\Omega_{\ell}\to\mathcal{H}^{n}\operatorname{\llcorner}\partial^{\ast}\Omega\quad\textrm{and}\quad\nu_{\Omega_{\ell}}\mathcal{H}^{n}\operatorname{\llcorner}\partial\Omega_{\ell}\to\nu_{\Omega}\mathcal{H}^{n}\operatorname{\llcorner}\partial^{\ast}\Omega$
(5.1)

weakly as Radon measures. Employing Reshetnyak theorem [AFP00, Theorem 2.39]
and recalling (2.1) we conclude that

$\delta V_{\Omega_{\ell}}(g)\to\delta V_{\Omega}(g)\quad\textrm{for all $g\in
C^{\infty}_{c}(\mathbb{R}^{n+1},\mathbb{R}^{n+1})$. }$

We estimate

$\displaystyle|\delta
V_{\Omega_{\ell}}(g)|\leq\int_{\partial\Omega_{\ell}}(H_{\Omega_{\ell},1})^{+}|g|\,d\mathcal{H}^{n}+\int_{\partial\Omega_{\ell}}(H_{\Omega_{\ell},1})^{-}|g|\,d\mathcal{H}^{n}$
$\displaystyle\leq M\,\int_{\\{0\leq H_{\Omega_{\ell},1}\leq
M\\}}|g|\,d\mathcal{H}^{n}+\int_{\\{H_{\Omega_{\ell},1}>M\\}}H_{\Omega_{\ell},1}|g|\,d\mathcal{H}^{n}+\int_{\partial\Omega_{\ell}}(H_{\Omega_{\ell},1})^{-}|g|\,d\mathcal{H}^{n}$
$\displaystyle\leq
M\,\int_{\partial\Omega_{\ell}}|g|\,d\mathcal{H}^{n}+\bigg{(}\int_{\partial\Omega_{\ell}}|H_{\Omega_{\ell},1}|^{n}\,d\mathcal{H}^{n}\bigg{)}^{\frac{1}{n}}\bigg{(}\int_{\\{H_{\Omega_{\ell},1}>M\\}}|g|^{\frac{n}{n-1}}\,d\mathcal{H}^{n}\bigg{)}^{\frac{n-1}{n}}$
$\displaystyle\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\int_{\partial\Omega_{\ell}}(H_{\Omega_{\ell},1})^{-}|g|\,d\mathcal{H}^{n}.$

Since the sequence has finite total curvature we have that

$\sup_{\ell\geq
1}\int_{\partial\Omega_{\ell}}|H_{\Omega_{\ell},1}|^{n}\,d\mathcal{H}^{n}<\infty,$

and the second integral on the right-side converges to zero as $\ell\to\infty$
thanks to (1.3). Noting that also the third integral on the right-side
converges to zero since the sequence is asymptotically $(k-1)$-mean convex, we
infer from (5.1) that

$|\delta V_{\Omega}(g)|\leq
M\int_{\partial^{\ast}\Omega}|g|\,d\mathcal{H}^{n}\quad\textrm{for $g\in
C^{\infty}_{c}(\mathbb{R}^{n+1},\mathbb{R}^{n+1})$.}$

Henceforth, $V_{\Omega}$ is a varifold of bounded mean curvature. Let
$C\subseteq\mathbb{R}^{n+1}$ be the closed set given by Lemma 4.1 such that
$\mathcal{L}^{n+1}\big{(}C\triangle\Omega\big{)}=0$. In applying the
conclusion (4.3) of Lemma 4.1 it is useful to notice the trivial inclusion
$\operatorname{nor}(C)\subseteq\operatorname{nor}(\partial C)$. Additionally,
it follows from (3.2) that $\sup_{\ell\geq 1}\bm{M}(N_{\Omega_{\ell}})<\infty$
and we deduce from Federer-Fleming compactness theorem, cf. [Sim83] that, up
to subsequences,

$N_{\Omega_{\ell}}\to T\quad\textrm{weakly in the sense of currents},$

with $T$ being an $n$-dimensional integer-rectifiable current compactly
supported in $\mathbb{R}^{n+1}\times\mathbb{S}^{n}$. Since $\partial
N_{\Omega_{\ell}}=0$ and
$N_{\Omega_{\ell}}\operatorname{\llcorner}\alpha_{\mathbb{R}^{n+1}}=0$, we
readily infer that $T$ is a Legendrian cycle of $\mathbb{R}^{n+1}$.

We have just seen that two different kind of limits are at play here: the
varifolds $V_{\Omega_{\ell}}$ converging to $V_{C}$ and the normal cycles
$N_{\Omega_{\ell}}$ converging as currents to $T$. To prove the theorem we
need to understand how these two limits are related. We start proving the
following two statements:

$\mathcal{H}^{n}\big{(}\operatorname{nor}(C)\setminus W_{T}^{(n)}\big{)}=0$
(5.2) $H_{T,1}(x,u)=\bm{h}(V_{C},x)\bullet u\qquad\textrm{for
$\mathcal{H}^{n}$ a.e.\ $(x,u)\in\operatorname{nor}(C)$.}$ (5.3)

By Reshetnyak continuity theorem [AFP00, Theorem 2.39], we have that

$(N_{\Omega_{\ell}}\operatorname{\llcorner}\varphi_{n})(\phi)=\int_{\partial\Omega_{\ell}}\phi(a,\nu_{\Omega_{\ell}}(a))\,d\mathcal{H}^{n}(a)\rightarrow\int_{\partial^{\ast}C}\phi(a,\nu_{C}(a))\,d\mathcal{H}^{n}(a)$

for every $\phi\in\mathcal{D}^{0}(\mathbb{R}^{n+1}\times\mathbb{R}^{n+1})$.
Since $N_{\Omega_{\ell}}\operatorname{\llcorner}\varphi_{n}\to
T\operatorname{\llcorner}\varphi_{n}$ weakly, it follows from Lemma 3.2 and
Remark 3.3 that

$\displaystyle\int_{\partial^{\ast}C}\phi(a,\nu_{C}(a))\,d\mathcal{H}^{n}(a)$
$\displaystyle=(T\operatorname{\llcorner}\varphi_{n})(\phi)$
$\displaystyle=\int_{W_{T}^{(n)}}J_{n}^{W_{T}}\pi_{0}(a,u)i_{T}(a,u)\phi(a,u)\,d\mathcal{H}^{n}(a,u)$
(5.4)

for $\phi\in\mathcal{D}^{0}(\mathbb{R}^{n+1}\times\mathbb{R}^{n+1})$. Let
$R=\partial^{\ast}C\cap\pi_{0}(\operatorname{nor}(C))$ and choose a
$\mathcal{H}^{n}$-measurable set $Q\subseteq\operatorname{nor}(C)$ with finite
$\mathcal{H}^{n}$-measure. We notice from (4.2) and (4.3) of Lemma 4.1 that
$\mathcal{H}^{n}(Q\setminus(Q\operatorname{\llcorner}R))=0$. The latter in
combination with area formula, (4.4) of Lemma 4.1 and (5) allows to conclude

$\displaystyle\int_{Q}J_{n}^{Q}\pi_{0}(x,u)\,\phi(x,u)\,d\mathcal{H}^{n}(x,u)$
$\displaystyle\qquad=\int_{Q\operatorname{\llcorner}R}J_{n}^{Q}\pi_{0}(x,u)\,\phi(x,u)\,d\mathcal{H}^{n}(x,u)$
$\displaystyle\qquad=\int_{\pi_{0}(Q\operatorname{\llcorner}R)}\phi(x,\nu_{C}(x))\,d\mathcal{H}^{n}(x)$
$\displaystyle\qquad\leq\int_{W_{T}^{(n)}}J_{n}^{W_{T}}\pi_{0}(x,u)i_{T}(x,u)\phi(x,u)\,d\mathcal{H}^{n}(x,u)$

for every $\phi\in\mathcal{D}^{0}(\mathbb{R}^{n+1}\times\mathbb{R}^{n+1})$
with $\phi\geq 0$. It follows that

$\displaystyle\int_{Q\cap S}J_{n}^{Q}\pi_{0}(x,u)\,d\mathcal{H}^{n}(x,u)$
$\displaystyle\qquad\qquad\leq\int_{W_{T}^{(n)}\cap
S}J_{n}^{W_{T}}\pi_{0}(x,u)i_{T}(x,u)\,d\mathcal{H}^{n}(x,u)$ (5.5)

for every $\mathcal{H}^{n}$-measurable set
$S\subseteq\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}$. We prove now that

$J_{n}^{Q}\pi_{0}(x,u)>0\quad\textrm{for $\mathcal{H}^{n}$ a.e.\ $(x,u)\in
Q$.}$ (5.6)

If $P=\\{(x,u)\in Q:J_{n}^{Q}\pi_{0}(x,u)>0\\}$, then

$0=\int_{Q\setminus
P}J_{n}^{Q}\pi_{0}(x,u)\,d\mathcal{H}^{n}(x,u)=\int_{\pi_{0}(Q\setminus
P)}\mathcal{H}^{0}(\pi_{0}^{-1}(x)\cap(Q\setminus P))\,d\mathcal{H}^{n}(x),$

whence we infer, since $\mathcal{H}^{0}(\pi_{0}^{-1}(x)\cap(Q\setminus P))\geq
1$ for $\mathcal{H}^{n}$ a.e. $x\in\pi_{0}(Q\setminus P)$, that

$\mathcal{H}^{n}(\pi_{0}(Q\setminus P))=0.$

It follows from (4.3) of Lemma 4.1 that $\mathcal{H}^{n}(Q\setminus P)=0$ and
(5.6) is proved. Thanks to (5.6), we can choose
$S=(\mathbb{R}^{n+1}\times\mathbb{R}^{n+1})\setminus W_{T}^{(n)}$ in (5) and
deduce that

$\mathcal{H}^{n}(Q\setminus W_{T}^{(n)})=0.$

Since the latter holds for every $Q\subseteq\operatorname{nor}(C)$ with finite
$\mathcal{H}^{n}$-measure, we obtain (5.2). Notice that (5.2) implies that
$\mathcal{H}^{n}(\operatorname{nor}(C))<\infty$. It follows that

$\operatorname{Tan}^{n}(\mathcal{H}^{n}\operatorname{\llcorner}\operatorname{nor}(C),(x,u))=\operatorname{Tan}^{n}(\mathcal{H}^{n}\operatorname{\llcorner}W_{T},(x,u))$

for $\mathcal{H}^{n}$ a.e. $(x,u)\in\operatorname{nor}(C)$ and, combining
Theorem 3.1 with Lemma 2.1 we conclude that

$\kappa_{C,i}(x,u)=\kappa_{T,i}(x,u)\quad\textrm{for $\mathcal{H}^{n}$ a.e.\
$(x,u)\in\operatorname{nor}(C)$}$ (5.7)

for $1\leq i\leq n$. Now (5.3) follows from Lemma 4.1.

We notice that it follows from (3.3) that

$(N_{\Omega_{\ell}}\operatorname{\llcorner}\varphi_{n-k})(\phi)=\lambda(N_{\Omega_{\ell}}\operatorname{\llcorner}\varphi_{n})(\phi)+\int_{\partial\Omega_{\ell}}(H_{\Omega_{\ell},k}(x)-\lambda)\,\phi(x,\nu_{\Omega_{\ell}}(x))\,d\mathcal{H}^{n}(x)$

for $\phi\in\mathcal{D}^{0}(\mathbb{R}^{n+1}\times\mathbb{R}^{n+1})$. Since
the second integral on the right side goes to $0$ as $\ell\to\infty$, we
obtain

$T\operatorname{\llcorner}\varphi_{n-k}=\lambda(T\operatorname{\llcorner}\varphi_{n}).$

Henceforth it follows from Lemma 3.2

$\displaystyle\int_{W_{T}^{(n)}}\phi(x,u)\,i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\,(H_{T,k}(x,u)-\lambda)\,d\mathcal{H}^{n}(x,u)$
$\displaystyle\qquad=\int_{W_{T}\setminus
W_{T}^{(n)}}\phi(x,u)\,i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\,H_{T,k}(x,u)\,d\mathcal{H}^{n}(x,u)$

for every $\phi\in\mathcal{D}^{0}(\mathbb{R}^{n+1}\times\mathbb{R}^{n+1})$.
Since $i_{T}(x,u)\mathcal{J}_{T}(x,u)>0$ for $\mathcal{H}^{n}$ a.e. $(x,u)\in
W_{T}$, we infer

$H_{T,k}(x,u)=0\quad\textrm{for $\mathcal{H}^{n}$ a.e.\ $(x,u)\in
W_{T}\setminus W_{T}^{(n)}$}$ (5.8)

and

$H_{T,k}(x,u)=\lambda\quad\textrm{for $\mathcal{H}^{n}$ a.e.\ $(x,u)\in
W_{T}^{(n)}$.}$ (5.9)

Employing Lemma 3.4 we obtain

$\displaystyle(n-k+1)\int
i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\,H_{T,k-1}(x,u)\,d\mathcal{H}^{n}(x,u)$
(5.10) $\displaystyle\qquad=k\int(x\bullet
u)\,i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\,H_{T,k}(x,u)\,d\mathcal{H}^{n}(x,u)$
$\displaystyle\qquad=k\lambda\int_{W_{T}^{(n)}}(x\bullet
u)\,i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\,d\mathcal{H}^{n}(x,u)$
$\displaystyle\qquad=k\lambda(n+1)\,\mathcal{L}^{n+1}(C)$

and we conclude that (notice that $0<\mathcal{L}^{n+1}(C)<\infty$)

$H_{T,k}(x,u)=\lambda=\frac{(n-k+1)T(\varphi_{n-k+1})}{k(n+1)\,\mathcal{L}^{n+1}(C)}$
(5.11)

for $\mathcal{H}^{n}$ a.e. $(x,u)\in W_{T}^{(n)}$. Since, by Lemma 3.3,
$H_{T,k-1}(x,u)\geq 0$ for $\mathcal{H}^{n}$ a.e. $(x,u)\in W_{T}$, we
conclude from (5.10) that $\lambda\geq 0$.

We claim now that $\lambda>0$ and we prove it by contradiction. Henceforth we
assume $\lambda=0$ and we prove by induction that

$H_{T,j}(x,u)=0\quad\textrm{for $\mathcal{H}^{n}$ a.e.\ $(x,u)\in W_{T}$}$
(5.12)

for $j=0,\ldots,k$. If $j=k$ then (5.12) follows from (5.8) and (5.9). If
$i\in\\{1,\ldots,k\\}$ and $H_{T,k-i+1}(x,u)=0$ for $\mathcal{H}^{n}$ a.e.
$(x,u)\in W_{T}$, we infer from Minkowski formula in Lemma 3.4 that

$\displaystyle 0=(k-i+1)\int(x\bullet
u)i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\,H_{T,k-i+1}(x,u)\,d\mathcal{H}^{n}(x,u)$
$\displaystyle\qquad=(n-k+i)\int
i_{T}(x,u)\,\mathcal{J}_{T}(x,u)\,H_{T,k-i}(x,u)\,d\mathcal{H}^{n}(x,u).$

Since $H_{T,k-i}(x,u)\geq 0$ (by Lemma 3.3) and
$i_{T}(x,u)\mathcal{J}_{T}(x,u)>0$ for $\mathcal{H}^{n}$ a.e. $(x,u)\in
W_{T}$, we conclude that

$H_{T,k-i}(x,u)=0\quad\textrm{for $\mathcal{H}^{n}$ a.e.\ $(x,u)\in W_{T}$.}$

Henceforth, (5.12) is proved for all $j\in\\{0,\ldots,k\\}$. It follows that
$H_{T,0}(x,u)=0$ for $\mathcal{H}^{n}$ a.e. $(x,u)\in W_{T}$ and
$\mathcal{L}^{n+1}(\Omega)=0$ by formula (3.5) of Lemma 3.4. This a
contradiction, and consequently we conclude that $\lambda$ is positive.

Since $H_{T,i}(x,u)\geq 0$ for $\mathcal{H}^{n}$ a.e. $(x,u)\in W_{T}$ and for
every $1\leq i\leq k$, we can apply [HS22, Lemma 2.2] to obtain

$\frac{H_{T,1}(x,u)}{{n\choose 1}}\geq\Bigg{(}\frac{H_{T,2}(x,u)}{{n\choose
2}}\Bigg{)}^{\frac{1}{2}}\geq\ldots\geq\Bigg{(}\frac{H_{T,k}(x,u)}{{n\choose
k}}\Bigg{)}^{\frac{1}{k}}\geq\Bigg{(}\frac{\lambda}{{n\choose
k}}\Bigg{)}^{\frac{1}{k}}$ (5.13)

for $\mathcal{H}^{n}$ a.e. $(x,u)\in W_{T}^{(n)}$. Noting by (5.2) and Remark
3.3 that

$\mathcal{J}_{T}(x,u)=J_{n}^{W_{T}}\pi_{0}(x,u)=J_{n}^{\operatorname{nor}(C)}\pi_{0}(x,u)\quad\textrm{for
$\mathcal{H}^{n}$ a.e.\ $(x,u)\in\operatorname{nor}(C)$}$

and $H_{T,k-1}(x,u)\geq 0$ for $\mathcal{H}^{n}$ a.e. $(x,u)\in W_{T}$ by
Lemma 3.3, we employ Lemma 3.4, (5.13), (5.2) and area formula to conclude

$\displaystyle(n-k+1)\int\mathcal{J}_{T}(x,u)\,i_{T}(x,u)\,H_{T,k-1}(x,u)\,d\mathcal{H}^{n}(x,u)$
(5.14)
$\displaystyle\qquad\geq(n-k+1)\int_{W_{T}^{(n)}}\mathcal{J}_{T}(x,u)\,i_{T}(x,u)\,H_{T,k-1}(x,u)\,d\mathcal{H}^{n}(x,u)$
$\displaystyle\qquad\geq(n-k+1)\,\lambda^{\frac{k-1}{k}}\,{n\choose
k}^{\frac{1-k}{k}}{n\choose
k-1}\,\int_{W_{T}^{(n)}}\mathcal{J}_{T}(x,u)\,i_{T}(x,u)\,d\mathcal{H}^{n}(x,u)$
$\displaystyle\qquad\geq(n-k+1)\,\lambda^{\frac{k-1}{k}}\,{n\choose
k}^{\frac{1-k}{k}}{n\choose
k-1}\,\int_{\operatorname{nor}(C)}J_{n}^{\operatorname{nor}(C)}\pi_{0}(x,u)\,i_{T}(x,u)\,d\mathcal{H}^{n}(x,u)$
$\displaystyle\qquad\geq(n-k+1)\,\lambda^{\frac{k-1}{k}}\,{n\choose
k}^{\frac{1-k}{k}}{n\choose k-1}\,\mathcal{H}^{n}(\partial^{\ast}C).$

Combining (5.10) and (5.14) we conclude

$\Bigg{(}\frac{\lambda}{{n\choose
k}}\Bigg{)}^{\frac{1}{k}}\geq\frac{\mathcal{H}^{n}(\partial^{\ast}C)}{(n+1)\mathcal{L}^{n+1}(C)},$

whence we infer in combination with (5.13) and (5.3) that

$\bm{h}(V_{C},x)\bullet
u\geq\frac{n\mathcal{H}^{n}(\partial^{\ast}C)}{(n+1)\mathcal{L}^{n+1}(C)}\quad\textrm{for
$\mathcal{H}^{n}$ a.e.\ $(x,u)\in\operatorname{nor}(C)$.}$ (5.15)

We conclude from Corollary 4.3 that $C$ is the union of finitely many closed
balls of the same radius
$\rho=\frac{(n+1)\mathcal{L}^{n+1}(E)}{\mathcal{H}^{n}(\partial^{\ast}E)}$
with disjointed interiors.

We now use Lemma 2.2 with $\Sigma$ replaced by a sphere of radius $\rho$:
noting that $\operatorname{nor}(C)\subseteq\operatorname{nor}(\partial C)$,
this lemma allows to conclude in combination with (4.3) of Lemma 4.1, that

$\kappa_{C,1}(x,u)=\ldots=\kappa_{C,n}(x,u)=\frac{1}{\rho}\quad\textrm{for
$\mathcal{H}^{n}$ a.e.\ $(x,u)\in\operatorname{nor}(C)$}.$

Noting that $\kappa_{T,i}(x,u)=\kappa_{C,i}(x,u)$ for $\mathcal{H}^{n}$ a.e.
$(x,u)\in\operatorname{nor}(C)$ and for $i=1,\ldots,n$, as proved in (5.7), we
conclude from (5.9) that

$\lambda=H_{T,k}(x,u)=H_{C,k}(x,u)={n\choose k}\rho^{-k}$

for $\mathcal{H}^{n}$ a.e. $(x,u)\in\operatorname{nor}(C)$. Since
$\mathcal{H}^{n}(\operatorname{nor}(C))>0$ we finally conclude that
$\lambda={n\choose k}\rho^{-k}$.

##  Appendix A Soap bubbles with positive reach

In this appendix we first generalize [HS22, Theorem A], removing the
restriction $\lambda\in\mathbb{R}\setminus\\{0\\}$. Notice that in [HS22,
Theorem A] a stronger hypothesis is at play than in [HS22, Theorem 6.15] and I
do not know if the restriction $\lambda\in\mathbb{R}\setminus\\{0\\}$ can also
be removed in [HS22, Theorem 6.15]. We use the notation and the terminology of
[HS22], for which we refer to [HS22, Section 2.2].

######  Theorem A.1.

Let $\phi$ be a uniformly convex $C^{2}$-norm, $k\in\\{1,\ldots,n\\}$ and let
$C\subseteq\mathbb{R}^{n+1}$ be a set of positive reach with positive and
finite volume. Assume that

$\Theta^{\phi}_{n-i}(C,\cdot)$ is a non-negative measure for $i=1,\ldots,k-1$
(A.1)

and

$\Theta^{\phi}_{n-k}(C,\cdot)=\lambda\Theta^{\phi}_{n}(C,\cdot)\quad\textrm{for
some $\lambda\in\mathbb{R}$.}$ (A.2)

Then $C$ is a finite union of disjoint union of rescaled and translated Wulff
shapes of radius
$\rho=\frac{(n+1)\mathcal{L}^{n+1}(C)}{\mathcal{P}^{\phi}(C)}$ and
$\lambda={n\choose r}\frac{\rho}{r+1}$.

###### Proof.

The main point is to prove that $\lambda>0$, then the conclusion can be
deduced from [HS22, Theorem A]. The proof of the positivity of $\lambda$
follows closely the idea used in the proof of Theorem 1.1.

We notice from [HS22, Lemma 6.14] that

$\bm{H}^{\phi}_{C,r}(a,\eta)=0\quad\textrm{for $\mathcal{H}^{n}$ a.e.\
$(a,\eta)\in N^{\phi}(C)\setminus\widetilde{N}^{\phi}_{n}(C)$}$ (A.3)

and

$\bm{H}^{\phi}_{C,r}(a,\eta)=(r+1)\lambda\quad\textrm{for $\mathcal{H}^{n}$
a.e.\ $(a,\eta)\in\widetilde{N}^{\phi}_{n}(C)$.}$ (A.4)

Assume by contradiction that $\lambda=0$. Now we prove by induction that

$\bm{H}^{\phi}_{C,j}(a,\eta)=0\quad\textrm{for $\mathcal{H}^{n}$ a.e.\
$(x,u)\in N^{\phi}(C)$}$ (A.5)

for $j=0,\ldots,r$. If $j=r$ then (A.5) follows from (A.3) and (A.4). If
$i\in\\{1,\ldots,r\\}$ and $\bm{H}^{\phi}_{C,r-i+1}(a,\eta)=0$ for
$\mathcal{H}^{n}$ a.e. $(a,\eta)\in N^{\phi}(C)$, we infer from Minkowski
formula in [HS22, Theorem 6.8]

$\displaystyle
0=(r-i+1)\int_{N^{\phi}(C)}(a\bullet\bm{n}^{\phi}(\eta))\,J^{\phi}_{C}(a,\eta)\,\bm{H}^{\phi}_{C,r-i+1}(a,\eta)\,d\mathcal{H}^{n}(a,\eta)$
$\displaystyle\qquad=(n-r+i)\int_{N^{\phi}(C)}\phi(\bm{n}^{\phi}(\eta))\,J^{\phi}_{C}(a,\eta)\,\bm{H}^{\phi}_{C,r-i}(a,\eta)\,d\mathcal{H}^{n}(a,\eta).$

Since it follows from (A.1) and [HS22, Definition 6.2] that

$\bm{H}^{\phi}_{C,r-i}(a,\eta)\geq
0\quad\textrm{and}\quad\phi(\bm{n}^{\phi}(\eta))J^{\phi}_{C}(a,\eta)>0\quad\textrm{for
$\mathcal{H}^{n}$ a.e.\ $(a,\eta)\in N^{\phi}(C)$,}$

we conclude that

$\bm{H}^{\phi}_{C,r-i}(a,\eta)=0\quad\textrm{for $\mathcal{H}^{n}$ a.e.\
$(x,u)\in N^{\phi}(C)$.}$

Henceforth, (A.5) is proved for all $j\in\\{0,\ldots,r\\}$. It follows that
$\bm{H}^{\phi}_{C,0}(a,\eta)=0$ for $\mathcal{H}^{n}$ a.e. $(a,\eta)\in
N^{\phi}(C)$ and $\mathcal{L}^{n+1}(C)=0$ by the second formula in [HS22,
Theorem 6.8]. This a contradiction, and consequently we conclude that
$\lambda$ is positive.

∎

Combining Theorem A.1 with some standard facts on sets of positive reach we
can prove the following uniqueness results for $C^{2}$-boundaries with $k$-th
mean curvature functions converging in $L^{1}$ to a constant. A completely
analogous statement can be obtained in the anisotropic geometry induced by a
uniformly convex $C^{2}$-norm. We choose to state Theorem A.2 in the Euclidean
setting to facilitate the comparison with Theorem 1.1.

######  Theorem A.2.

Let $k\in\\{1,\ldots,n\\}$, let $\Omega_{\ell}$ be a compactly supported and
asymptotically $(k-1)$-mean convex sequence and let $C_{\ell}$ be the closure
of $\Omega_{\ell}$. Suppose there exists $\lambda\in\mathbb{R}$ such that

$\lim_{\ell\to\infty}\int_{\partial\Omega_{\ell}}|H_{\Omega_{\ell},k}-\lambda|\,d\mathcal{H}^{n}=0$
(A.6)

and there exists $\epsilon>0$ so that
$\operatorname{reach}(C_{\ell})\geq\epsilon$ for all $\ell\geq 1$.

If $C\subseteq\mathbb{R}^{n+1}$ is an accumulation point with respect to
Hausdorff convergence of $C_{\ell}$ with positive volume, then $C$ is a closed
ball.

###### Proof.

It follows from [HS22, Lemma 6.12] that $\operatorname{reach}(C)\geq\epsilon$.
Moreover, the $j$-th curvature measures $\Theta_{j}(C_{\ell},\cdot)$
associated with $C_{\ell}$ converge weakly to $\Theta_{j}(C,\cdot)$ for every
$j\in\\{0,\ldots,n\\}$. Recalling that $H_{\Omega_{\ell},m}$ is the $m$-th
mean curvature function of $\partial\Omega_{\ell}$ with respect to the
outward-pointing normal direction, and $H_{\Omega_{\ell},0}\equiv 1$, we
notice from [HS22, Lemma 6.4] that

$\displaystyle(n-j+1)\Theta_{j}(C_{\ell},B)=\int_{\partial\Omega_{\ell}}\bm{1}_{B}(a,\nu_{\Omega_{\ell}}(a))\,H_{\Omega_{\ell},n-j}(a)\,d\mathcal{H}^{n}(a,\eta)$

for every Borel set $B\subseteq\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}$ and
$j=0,\ldots,n$. We consider the negative and positive part of
$\Theta_{j}(C_{\ell},\cdot)$

$\Theta^{\pm}_{j}(C_{\ell},\cdot)=\int_{\partial\Omega_{\ell}}\bm{1}_{(\cdot)}(a,\nu_{\Omega_{\ell}}(a))\,H_{\Omega_{\ell},n-j}(a)^{\pm}\,d\mathcal{H}^{n}(a,\eta)\quad\textrm{for
$j=1,\ldots,n$},$

which are non-negative Radon measures. Noting that
$\Theta^{-}_{n-i}(C_{\ell},\cdot)$ weakly converge to $0$ for
$i\in\\{1,\ldots,k-1\\}$, we conclude that
$\Theta^{+}_{n-i}(C_{\ell},\cdot)=\Theta_{n-i}(C_{\ell},\cdot)+\Theta^{-}_{n-i}(C_{\ell},\cdot)$
weakly converge to $\Theta_{n-i}(C,\cdot)$. Henceforth,
$\Theta_{n-i}(C,\cdot)$ is a non-negative Radon measure for
$i\in\\{1,\ldots,k-1\\}$. Moreover, noting that

$(k+1)\Theta_{n-k}(C_{\ell},B)=\int_{\partial\Omega_{\ell}}\bm{1}_{B}(a,\nu_{\Omega_{\ell}}(a))\,(H_{\Omega_{\ell},k}-\lambda)\,d\mathcal{H}^{n}(a)+\lambda\Theta_{n}(C_{\ell},B)$

for every Borel set $B\subseteq\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}$, we
conclude from (A.6) that $\Theta_{n-k}(C,\cdot)=\lambda\Theta_{n}(C,\cdot)$.
Now the conclusion follows from Theorem A.1. ∎

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