# Multiplicity of solutions to the multiphasic Allen–Cahn–Hilliard system with
a small volume constraint on closed parallelizable manifolds

João Henrique Andrade  ,  Jackeline Conrado  ,  Stefano Nardulli  ,  Paolo
Piccione  and  Reinaldo Resende  Department of Mathematics, University of
British Columbia
V6T 1Z2, Vancouver-BC, Canada
and
Institute of Mathematics and Statistics, University of São Paulo
05508-090, São Paulo-SP, Brazil
and
Department of Mathematics, Federal University of Paraíba
58051-900, João Pessoa-PB, Brazil<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Institute of Mathematics and Statistics, University of
São Paulo
05508-090, São Paulo-SP, Brazil<EMAIL_ADDRESS>Department of Mathematics,
Federal University of ABC
09210-580, São Paulo-SP, Brazil<EMAIL_ADDRESS>Institute of
Mathematics and Statistics, University of São Paulo
05508-090, São Paulo-SP, Brazil<EMAIL_ADDRESS>Institute of Mathematics
and Statistics, University of São Paulo
05508-090, São Paulo-SP, Brazil<EMAIL_ADDRESS>
###### Abstract.

We prove the existence of multiple solutions to the Allen–Cahn–Hilliard (ACH)
vectorial equation (with two equations) involving a triple-well (triphasic)
potential with a small volume constraint on a closed parallelizable Riemannian
manifold. More precisely, we find a lower bound for the number of solutions
depending on some topological invariants of the underlying manifold. The phase
transition potential is considered to have a finite set of global minima,
where it also vanishes, and a subcritical growth at infinity. Our strategy is
to employ the Lusternik–Schnirelmann and infinite-dimensional Morse theories
for the vectorial energy functional. To this end, we exploit that the
associated ACH energy $\Gamma$-converges to the weighted multi-perimeter for
clusters, which combined with some deep theorems from isoperimetric theory
yields the suitable setup to apply the photography method. Along the way, the
lack of a closed analytic expression for the multi-isoperimetric function for
clusters imposes a delicate issue. Furthermore, using a transversality
theorem, we also show the genericity of the set of metrics for which solutions
to the ACH system are nondegenerate.

###### Key words and phrases:

Lusternik–Schnirelmann and Morse theories, Allen–Cahn–Hilliard system,
$\Gamma$-convergence, Multiphasic potential, Isoperimetric clusters, Compact
manifolds

###### 2020 Mathematics Subject Classification:

35J20, 35J25, 58E05, 49Q20, 58E99, 53A10, 49Q05, 28A75

*Corresponding author.

###### Contents

  1. 1 Introduction
  2. 2 Abstract photography method
  3. 3 Riemannian isoperimetric theory for weighted clusters
  4. 4 Convergence results for the vectorial energy
  5. 5 Multiplicity
    1. 5.1 Framework
    2. 5.2 Lower boundedness
    3. 5.3 Palais–Smale condition
    4. 5.4 Photography map
    5. 5.5 Quasi-minima sublevel sets
    6. 5.6 Barycenter map
    7. 5.7 Homotopy equivalence
    8. 5.8 Proof of Theorem Theorem 1: Multiplicity
  6. 6 Generic nondegeneracy
    1. 6.1 Framework
    2. 6.2 The case of nonconstant solutions
    3. 6.3 Proof of Theorem Theorem 1: Generic nondegeneracy

##  1\. Introduction

It is well-known that any open bounded domain in the flat Euclidean space
contains an isoperimetric cluster enclosing fixed (sufficiently small) volumes
[4]. Roughly speaking, an isoperimetric cluster is a solution to a minimal
partition problem, or more generally, a minimizing critical point of the
(vectorial) perimeter. A classical problem in differential geometry is to
count the number of isoperimetric clusters enclosing a small volume on a
Riemannian manifold or more generally the number of critical clusters for the
perimeter (of clusters) under a vectorial volume constraint. In the light of
the $\Gamma$-convergence theory, our approach to attack the latter problem
consists of a first step in studying a problem with a relaxation parameter,
which will be described below and as a second step in sending this parameter
to zero. In this paper, we study the relaxed problem. The second step will be
the subject of a forthcoming paper.

Let $n,N,m\in\mathbb{N}$ be such that $n,N\geqslant 2$ and $m\geqslant 1$
(otherwise explicitly mentioned this will always be the case in this paper).
Let $(M^{n},g)$ be a (smooth) closed (compact and without boundary) Riemannian
manifold. We study the existence and multiplicity of vectorial $m$-map
solutions $\mathbf{u}=(\mathrm{u}_{1},\dots,\mathrm{u}_{m})\in
C_{g}^{\infty}(M,\mathbb{R}^{m})$ to the following Allen–Cahn–Hilliard (ACH)
system

$\displaystyle\begin{cases}\mathrm{d}\boldsymbol{\mathcal{E}}_{\varepsilon,\boldsymbol{W}}(\mathbf{u})=\mathbf{\Lambda}\quad{\rm
on}\quad M,\\\
\boldsymbol{\mathcal{V}}_{g}(\mathbf{u})=\mathbf{v},\end{cases}$
($ACH_{\varepsilon,\mathbf{v},m,N,g}$)

where $\mathbf{\Lambda}=(\Lambda_{1},\dots,\Lambda_{m})\in\mathbb{R}^{m}$ is a
Lagrange multiplier, $m\in\mathbb{N}$ is the number of equations,
$\boldsymbol{\mathcal{V}}_{g}:{C}_{g}^{\infty}(M,\mathbb{R}^{m})\rightarrow\mathbb{R}^{m}$
is the volume functional given by
$\boldsymbol{\mathcal{V}}_{g}(\mathbf{u})=(\int_{M}\mathrm{u}_{1}\mathrm{d}\mathcal{L}^{n}_{g},\dots,\int_{M}\mathrm{u}_{m}\mathrm{d}\mathcal{L}^{n}_{g})$,
and
$\mathbf{v}=(\mathrm{v}_{1},\dots,\mathrm{v}_{m})\in\mathbb{R}_{+}^{m}:=\\{x\in\mathbb{R}^{m}:x_{i}\geqslant
0\\}$ with $\mathrm{v}=|\mathbf{v}|:=\sum_{i=1}^{m}\mathrm{v}_{i}\ll 1$. The
functional
$\boldsymbol{\mathcal{E}}_{\varepsilon,\boldsymbol{W}}:L_{g}^{1}(M,\mathbb{R}^{m})\rightarrow\overline{\mathbb{R}}$
is called the vectorial ACH energy

$\boldsymbol{\mathcal{E}}_{\varepsilon,\boldsymbol{W}}(\mathbf{u}):=\begin{cases}\int_{M}\left(\varepsilon|{\nabla}_{g}\mathbf{u}|^{2}+{\varepsilon^{-1}}\boldsymbol{W}(\mathbf{u})\right)\mathrm{d}\mathcal{L}^{n}_{g},&\
\mbox{if}\ \mathbf{u}\in\boldsymbol{\mathfrak{M}}_{\mathbf{v}}\\\ \infty,&\
\mbox{if}\ \mathbf{u}\in
L_{g}^{1}(M,\mathbb{R}^{m})\setminus\boldsymbol{\mathfrak{M}}_{\mathbf{v}},\end{cases}$
(1)

where
${\nabla}_{g}\mathbf{u}=({\nabla}_{g}\mathrm{u}_{1},\dots,{\nabla}_{g}\mathrm{u}_{m})$
is the gradient acting on $m$-maps, $0<\varepsilon\ll 1$ is the relaxation
(temperature) parameter, $\mathrm{d}\mathcal{L}^{n}_{g}$ is the volume measure
induced by $g$ (also denoted $\mathrm{v}_{g}$), $\boldsymbol{W}\in
C^{\infty}(\mathbb{R}^{m})$ is a multi-well (multiphasic) potential vanishing
at a finite set of (global) minima points
$\mathcal{Z}\subset\mathbb{R}_{+}^{m}$ containing the origin and such that
$\\#\mathcal{Z}=N$ (or $\mathcal{H}_{g}^{0}(\mathcal{Z})=N$)111The notation
$\mathcal{H}_{g}^{d}$ for $d=0,\dots,n$ stands for the $d$-dimensional
Hausdorff measure on $M$. Also, recall that $\mathcal{H}_{g}^{0}=\\#$ (the
counting measure) and $\mathcal{H}_{g}^{n}=\mathcal{L}_{g}^{n}$ (the Lebesgue
measure) (see Definition 1), and

$\boldsymbol{\mathfrak{M}}_{\mathbf{v}}=\left\\{\mathbf{u}\in{H}_{g}^{1}(M,\mathbb{R}^{m}):\boldsymbol{\mathcal{V}}_{g}(\mathbf{u})=\mathbf{v}\right\\}.$
(2)

From now on, we simply denote
$\boldsymbol{\mathcal{E}}_{\varepsilon,\boldsymbol{W}}=\boldsymbol{\mathcal{E}}_{\varepsilon}$.

Here, for any $1\leqslant q<\infty$, let us consider
$L_{g}^{q}(M,\mathbb{R}^{m})$ the Banach space, completion of
$C_{g}^{\infty}(M,\mathbb{R}^{m})$ with respect to the norm
$\|\mathbf{u}\|_{L_{g}^{q}(M,\mathbb{R}^{m})}:=(\int_{M}|\mathbf{u}|^{q}\mathrm{d}\mathcal{L}^{n}_{g})^{1/q}$.
We can also define the Sobolev space
${H}_{g}^{1}(M,\mathbb{R}^{m})=\\{\mathbf{u}=(\mathrm{u}_{1},\dots,\mathrm{u}_{m}):|\mathbf{u}|\in
H_{g}^{1}(M)\\}$ furnished with the inner product
$\langle\mathbf{u},\mathbf{v}\rangle_{{H}_{g}^{1}(M,\mathbb{R}^{m})}=\langle\nabla_{g}|\mathbf{u}|,\nabla_{g}|\mathbf{v}|\rangle$
and the metric
$\|\mathbf{u}\|_{H^{1}_{g}(M,\mathbb{R}^{m})}:=(\int_{M}|\nabla_{g}|\mathbf{u}||^{q}\mathrm{d}\mathcal{L}^{n}_{g})^{1/q}$.
The higher order Sobolev spaces $W^{\ell,q}_{g}(M,\mathbb{R}^{m})$ for
$1\leqslant\ell<\infty$ are defined similarly. We omit subscripts when dealing
with the standard Euclidean metric.

In this language, a simple computation shows that
($ACH_{\varepsilon,\mathbf{v},m,N,g}$) is the Euler–Lagrange equation
associated to the energy (1) with Fréchet derivative
$\mathrm{d}\boldsymbol{\mathcal{E}}_{\varepsilon}:H_{g}^{1}(M,\mathbb{R}^{m})\rightarrow{\mathbb{R}}\cup\\{\infty\\}$
defined by

$\mathrm{d}\boldsymbol{\mathcal{E}}_{\varepsilon}(\mathbf{u})=-\varepsilon{\Delta}_{g}\mathbf{u}+\varepsilon^{-1}\nabla\boldsymbol{W}(\mathbf{u}),$

where
${\Delta}_{g}\mathbf{u}=(\Delta_{g}\mathrm{u}_{1},\dots,\Delta_{g}\mathrm{u}_{m})$
denotes the vectorial Laplace–Beltrami operator.

We restrict ourselves to the suitable class of subcritical multi-well
(multiphasic) potentials

######  Definition 1.

Let $n,m,N\in\mathbb{N}$ with $n,N\geqslant 2$ and $m\geqslant 1$. A
nonnegative real function $\boldsymbol{W}\in C^{2}(\mathbb{R}^{m})$ is denoted
by $\boldsymbol{W}\in\mathcal{W}^{+}$. Also, we denote
$\boldsymbol{W}\in\mathcal{W}^{+}_{N}$, if it has a finite set of vanishing
global minima, denoted by
$\mathcal{Z}=\\{\mathbf{p}_{1},\dots,\mathbf{p}_{N}\\}\subset\mathbb{R}^{m}_{+}$.
In other terms, $\boldsymbol{W}(\mathbf{p}_{i})=0$,
$\nabla\boldsymbol{W}(\mathbf{p}_{i})=0$, and
$\nabla^{2}\boldsymbol{W}(\mathbf{p}_{i})>0$ for all $i=1,\dots,N$. Also, it
satisfies that

  * •

$m=N-1$, $\mathbf{p}_{N}=0$ and $\\{\mathbf{p}_{1},\dots,\mathbf{p}_{N-1}\\}$
is a linearly independent set such that

$\omega_{ij}(\boldsymbol{W})<\omega_{i\ell}(\boldsymbol{W})+\omega_{\ell
j}(\boldsymbol{W})\quad{\rm for\ all}\quad i,j,\ell\in\\{1,\dots,N\\}\quad{\rm
and}\quad\ell\notin\\{i,j\\},$ (${{\rm W}_{0}}$)

where
$\omega_{ij}(\boldsymbol{W})=\mathrm{d}_{\boldsymbol{W}}(\mathbf{p}_{i},\mathbf{p}_{j})$
and the degenerate metric
$\mathrm{d}_{\boldsymbol{W}}:\mathbb{R}^{m}\times\mathbb{R}^{m}\rightarrow\mathbb{R}$
is defined by

$\mathrm{d}_{\boldsymbol{W}}(\mathbf{p}_{i},\mathbf{p}_{j}):=\inf_{\mathbf{c}\in
C^{1}([0,1],\mathbb{R}^{m})}\left\\{\int_{0}^{1}\boldsymbol{W}^{1/2}(\mathbf{c}(t))|\mathbf{c}^{\prime}(t)|\mathrm{d}t:\begin{aligned}
\mathbf{c}(&t)\in\mathbb{R}^{m}_{+}\ {\rm for\ all}\ t\in[0,1],\\\
\mathbf{c}&(0)=\mathbf{p}_{i}\;{\rm
and}\;\mathbf{c}(1)=\mathbf{p}_{j}\end{aligned}\right\\}.$

When $\boldsymbol{W}\in\mathcal{W}_{N}^{+}$ satisfies (${{\rm W}_{0}}$), we
denote $\boldsymbol{W}\in\mathcal{W}_{N,0}^{+}$;

  * •

there exists $k_{1}>0$ such that

$|\nabla\boldsymbol{W}(z)|\leqslant k_{1}(1+|z|^{p-1})\quad{\rm for\ all}\quad
z\in\mathbb{R}^{m}$ (${{\rm W}_{1}}$)

and for any $1<p<2^{*}$ if $n\geqslant 3$ $($or $p<\infty$ if $n=2$$)$ with
$2^{*}:=\frac{2n}{n-2}$ the critical Sobolev exponent of the embedding
$H^{1}(\mathbb{R}^{n})\hookrightarrow L^{q}(\mathbb{R}^{n})$ for $q>1$. When
$\boldsymbol{W}\in\mathcal{W}_{N}^{+}$ satisfies (${{\rm W}_{1}}$), we denote
$\boldsymbol{W}\in\mathcal{W}_{N,1}^{+}$;

  * •

there exists $k_{2}>0$ such that

$|\nabla^{2}\boldsymbol{W}(z)|\leqslant k_{2}(1+|z|^{p-2})\quad{\rm for\
all}\quad z\in\mathbb{R}^{m}$ (${{\rm W}_{2}}$)

and for any $1<p<2^{*}$ if $n\geqslant 3$ $($or $p<\infty$ if $n=2$$)$. When
$\boldsymbol{W}\in\mathcal{W}_{N}^{+}$ satisfies (${{\rm W}_{2}}$), we denote
$\boldsymbol{W}\in\mathcal{W}_{N,2}^{+}$;

  * •

there exist $p_{1},p_{2},k_{3},k_{4},R>0$ such that

$k_{3}|z|^{p_{1}}<\boldsymbol{W}(z)<k_{4}|z|^{p_{2}}\quad{\rm for\
all}\quad|z|\geqslant R,$ (${{\rm W}_{3}}$)

where $2<p_{1}<2^{\\#}$ with $p_{1}\leqslant p_{2}\leqslant 2(p_{1}-1)$ and
$2^{\\#}:=\frac{2n-1}{n-1}$. When $\boldsymbol{W}\in\mathcal{W}_{N}^{+}$
satisfies (${{\rm W}_{3}}$), we denote
$\boldsymbol{W}\in\mathcal{W}_{N,3}^{+}$.

In this fashion, we define the admissible class of potentials
$\widetilde{\mathcal{W}}^{+}_{N}=\cap_{i=0}^{3}\mathcal{W}_{N,i}^{+}$.

######  Remark 2.

It is not hard to check that $\widetilde{\mathcal{W}}^{+}_{N}\neq\emptyset$.
In fact, one can always find a nonnegative polynomial with degree six and
suitably chosen coefficients $\boldsymbol{P}\in C^{\infty}(\mathbb{R}^{N-1})$
such that $\boldsymbol{P}(\mathbf{p}_{i})=0$,
$\nabla\boldsymbol{P}(\mathbf{p}_{i})=0$, and
$\nabla^{2}\boldsymbol{P}(\mathbf{p}_{i})>0$ for all $i=1,\dots,N$. Next, we
fix $R\gg 1$ and $0<\tau\ll 1$ such that $\overline{\boldsymbol{W}}\in
C(\mathbb{R}^{m})$, where

$\overline{\boldsymbol{W}}(z)=\begin{cases}\boldsymbol{P}(z),&\quad{\rm
if}\quad 0<|z|<R,\\\ |z|^{2+\tau},&\quad{\rm if}\quad|z|\geqslant
R.\end{cases}$

Finally, using an approximation theorem there exists ${\boldsymbol{W}}\in
C^{2}(\mathbb{R}^{m})$ such that
${\boldsymbol{W}}\in\widetilde{\mathcal{W}}^{+}_{N}$.

The assumption $m=N-1$ is chosen suitably so that in the sharp interface
limit, we have the phase separation for a mixture of $N-1$ immiscible fluids.
Also, we highlight that since $\mathrm{d}_{\boldsymbol{W}}$ is a metric it is
always true that
$\omega_{ij}(\boldsymbol{W})\leqslant\omega_{i\ell}(\boldsymbol{W})+\omega_{\ell
j}(\boldsymbol{W})$ for all $i,j,\ell\in\\{1,\cdots,N\\}$ and
$\ell\notin\\{i,j\\}$, which implies that (${{\rm W}_{0}}$) is a generic
condition. This can be regarded as a physical hypothesis on the immiscibility
of the fluids. In Section 3, we will explain a more geometric
$($isoperimetric$)$ reason to consider this assumption. We refer to this as
the immiscibility condition.

Now, we introduce the notion of nondegeneracy of solutions

######  Definition 3.

Let $(M^{n},g)$ be a closed Riemannian manifold and
$\boldsymbol{W}\in\mathcal{W}_{N}^{+}$. We say that a solution
$(\mathbf{u},\mathbf{\Lambda})\in
H_{g}^{1}(M,\mathbb{R}^{m})\times\mathbb{R}^{m}$ to
($ACH_{\varepsilon,\mathbf{v},m,N,g}$) is nondegenerate when the only pair
$(\mathbf{w},\mathbf{\Lambda})\in
H_{g}^{1}(M,\mathbb{R}^{m})\times\mathbb{R}^{m}$ solving the linearized
problem

$\displaystyle\begin{cases}\mathrm{d}^{2}\boldsymbol{\mathcal{E}}_{\varepsilon}(\mathbf{u})\mathbf{w}=\mathbf{\Lambda}\quad{\rm
in}\quad M\\\ \boldsymbol{\mathcal{V}}_{g}(\mathbf{u})=\mathbf{v},\end{cases}$
(3)

is the trivial one $(\mathbf{w},\mathbf{\Lambda})=(0,0)$, where
$\mathrm{d}^{2}\boldsymbol{\mathcal{E}}_{\varepsilon}(\mathbf{u})\mathbf{w}=-\varepsilon\Delta_{g}\mathbf{w}+\varepsilon^{-1}\nabla^{2}\boldsymbol{W}(\mathbf{u})\mathbf{w}$.

Our main result proves the existence of multiple pairs
$(\mathbf{u}_{\varepsilon,\mathbf{v}},\mathbf{\Lambda}_{\varepsilon,\mathbf{v}})\in
H_{g}^{1}(M,\mathbb{R}^{m})\times\mathbb{R}^{m}$ satisfying
($ACH_{\varepsilon,\mathbf{v},m,N,g}$) for $0<\varepsilon,|\mathbf{v}|\ll 1$.
The strategy is to search for critical points of the vectorial ACH functional
in (1) restricted to the weakly closed set
$\boldsymbol{\mathfrak{M}}_{\mathbf{v}}$ in (2). Namely, for each pair
$(\varepsilon,\mathbf{v})\in\mathbb{R}^{m+1}_{+}$, let us consider the
associated moduli space,

$\boldsymbol{\mathfrak{N}}_{\varepsilon,\mathbf{v}}=\left\\{(\mathbf{u}_{\varepsilon,\mathbf{v}},\mathbf{\Lambda}_{\varepsilon,\mathbf{v}})\in
H^{1}_{g}(M,\mathbb{R}^{m})\times\mathbb{R}^{m}:(\mathbf{u}_{\varepsilon,\mathbf{v}},\mathbf{\Lambda}_{\varepsilon,\mathbf{v}})\
\mbox{solves \eqref{oursystem} on}\
\boldsymbol{\mathfrak{M}}_{\mathbf{v}}\right\\},$

and the counting function
$\boldsymbol{\eta}(\varepsilon,\mathbf{v}):\boldsymbol{\mathfrak{N}}_{\varepsilon,\mathbf{v}}\rightarrow\mathbb{N}\cup\\{\infty\\}$
defined by
$\boldsymbol{\eta}(\varepsilon,\mathbf{v}):=\\#\boldsymbol{\mathfrak{N}}_{\varepsilon,\mathbf{v}}$.
We show the existence of $\varepsilon_{*},\mathrm{v}_{*}>0$ such that for
$(\varepsilon,\mathbf{v})\in(0,\varepsilon_{*})\times(0,\mathrm{v}_{*})^{m}\subset(0,\varepsilon_{*})\times{\rm
conv}(\mathcal{Z})$ the function $\boldsymbol{\eta}(\varepsilon,\mathbf{v})$
has a lower bound depending on the topology of the manifold, where ${\rm
conv}(\mathcal{Z})$ is the convex hull of the set $\mathcal{Z}$. More
accurately, this bound depends on the Lusternik–Schnirelman category of $M$
denoted by $\operatorname{cat}\left(M\right)$, and on
$\mathscr{P}_{1}\left(M\right):=\sum_{k\in\mathbb{N}}\beta_{k}\left(M\right)$,
where $\beta_{k}\left(M\right)$ is the $k$-th Betti number of $M$.
Additionally, we prove that the set of metrics for which the nondegeneracy
property holds is large (in the topological sense) on the space of smooth
metrics over $M$, denoted by ${\rm Met}^{\infty}(M)$.

######  Theorem 1.

Let $(M^{n},g)$ be a closed parallelizable Riemannian manifold and
$\boldsymbol{W}\in\widetilde{\mathcal{W}}^{+}_{N}$ with $N=3$. There exists
$\mathrm{v}_{*}=\mathrm{v}_{*}(M,g)>0$ such that for any
$\mathbf{v}\in(0,\mathrm{v}_{*})^{2}\subset{\rm conv}(\mathcal{Z})$, one can
find $\varepsilon_{*}(M,g,\boldsymbol{W},\mathrm{v})>0$ satisfying for every
$\varepsilon\in(0,\varepsilon_{*})$,

  * (i)

$\boldsymbol{\eta}({\varepsilon,\mathbf{v}})\geqslant\operatorname{cat}\left(M\right)+1$,
if the solutions are counted without multiplicity;

  * (ii)

$\boldsymbol{\eta}({\varepsilon,\mathbf{v}})\geqslant
2\mathscr{P}_{1}\left(M\right)-1$, if such solutions are nondegenerate.

Moreover, for any fixed $g_{0}\in{\rm Met}^{\infty}(M)$ the set

$\boldsymbol{\mathcal{G}}_{\boldsymbol{W},\mathbf{v}}=\left\\{(\varepsilon,g)\in(0,\infty)\times{\rm
Met}^{\infty}(M):\begin{aligned} &{\rm any\ solution}\
(\mathbf{u},\mathbf{\Lambda})\in
H^{1}_{g_{0}}(M,\mathbb{R}^{m})\times\mathbb{R}^{m}\ {\rm to}\\\
&\quad\eqref{oursystem}\ {\rm is}\ {\rm nondegenerate}\end{aligned}\right\\}$

is Baire generic with respect to the Gromov–Hausdorff topology.

######  Remark 4.

When $n=3$, the condition of being parallelizable can be interchanged by
orientability, by Stiefel’s theorem $($see [48]$)$. Also, $\mathbb{S}^{n}$ is
parallelizable if, and only if, $n=1,3,7$. Also, for $N=3$, thanks to the
solution of the _weighted_ double bubble conjecture in $\mathbb{R}^{n}$ $($see
[38]$)$ this hypothesis can be weakened to the existence of a smooth global
section of the unit tangent bundle $UTM$.

######  Remark 5.

About the assumption $N=3$. It is worth to observe that the techniques that we
use to prove the preceding theorem only needs the following property:
isoperimetric weighted cluster with small volumes have small diameter in the
tangent space. In fact, we use this property to prove that the photography map
is well-defined, continuous, and to prove the $\Gamma$-convergence in an easy
way. This property holds in the Euclidean space for $N=3$ thanks to [38],
therefore, if one assume this property for $N>3$, the preceding theorem holds
true. Even if, it is not known whether this property is always valid for $N>3$
or not, we expect that it indeed holds for every $N\in\mathbb{N}$, due to
several special cases where it holds (e.g. [47, 38]). However, the proof of
this property for $N\in\mathbb{N}$ goes beyond of the scope of this manuscript
and it will be the subject of a forthcoming paper. Furthermore, we emphasize
that based on the recent classification results of [47], our techniques apply
for the case $2\leqslant N\leqslant\min(5,n+1)$ when the potential is such
that $\omega_{ij}(\boldsymbol{W})=1$ for $i\neq j$ and
$\omega_{ij}(\boldsymbol{W})=0$ for $i\neq j$, where $i,j\in\\{1,\dots,N\\}$.
In particular, Theorem Theorem 1 holds in this context.

Let us now compare our main result with its scalar counterpart. More
precisely, when $m=1$ and $N=2$, System ($ACH_{\varepsilon,\mathbf{v},m,N,g}$)
becomes the classical ACH equation with a small volume constraint,

$\displaystyle\begin{cases}-\varepsilon\Delta_{g}\mathrm{u}+\varepsilon^{-1}W^{\prime}(\mathrm{u})=\Lambda,\quad{\rm
on}\quad M,\\\
\int_{M}\mathrm{u}\mathrm{d}\mathcal{L}_{g}^{n}=\mathrm{v},\end{cases}$
($ACH_{\varepsilon,\mathrm{v},g}$)

where $0<\varepsilon,\mathrm{v}\ll 1$, $\Lambda\in\mathbb{R}$ is a Lagrange
multiplier, and $W\in\widetilde{\mathcal{W}}^{+}_{2}$ is a symmetric double-
well potential. The study of qualitative properties for
($ACH_{\varepsilon,\mathrm{v},g}$) was addressed in [16, 59] (see also [17]
for the Euclidean case). We consider
${\mathfrak{M}}_{\mathrm{v}}=\\{\mathrm{u}\in{H}_{g}^{1}(M):\int_{M}\mathrm{u}\mathrm{d}\mathcal{L}_{g}^{n}=\mathrm{v}\\}$,
${\mathfrak{N}}_{\varepsilon,\mathrm{v}}=\\{(\mathrm{u}_{\varepsilon,\mathrm{v}},{\Lambda}_{\varepsilon,\mathrm{v}})\in
H^{1}_{g}(M)\times\mathbb{R}:(\mathrm{u}_{\varepsilon,\mathrm{v}},{\Lambda}_{\varepsilon,\mathrm{v}})\
\mbox{solves \eqref{ourequation} on}\ \mathfrak{M}_{{\mathrm{v}}}\\}$, and the
counting function
${\eta}(\varepsilon,\mathrm{v}):=\\#\mathfrak{N}_{\varepsilon,\mathrm{v}}$. In
the same spirit of our main theorem, the following result is proved

######  Theorem A ([16, 59]).

Let $N=2$ and let $(M^{n},g)$ be a closed Riemannian manifold and
$\boldsymbol{W}\in\widetilde{\mathcal{W}}^{+}_{2}$. There exists
$\mathrm{v}_{*}=\mathrm{v}_{*}(M,g)>0$ such that for every
$\mathrm{v}\in(0,\mathrm{v}_{*})$, one can find
$\varepsilon_{*}(M,g,W,\mathrm{v})>0$ satisfying for every
$\varepsilon\in(0,\varepsilon_{*})$:

  * (i)

$\eta({\varepsilon,\mathrm{v}})\geqslant\operatorname{cat}\left(M\right)+1$,
if the solutions are counted without multiplicity;

  * (ii)

$\eta({\varepsilon,\mathrm{v}})\geqslant 2\mathscr{P}_{1}\left(M\right)-1$, if
such solutions are nondegenerate.

Moreover, for any fixed $g_{0}\in{\rm Met}^{\infty}(M)$, the moduli space of
metrics

$\mathcal{G}_{{W},\mathrm{v}}=\left\\{(\varepsilon,g)\in(0,\infty)\times{\rm
Met}^{\infty}(M):\begin{aligned} &{\rm any\ solution}\
(\mathrm{u},{\Lambda})\in H^{1}_{g_{0}}(M)\times\mathbb{R}\ {\rm to}\
\eqref{ourequation}\\\ &\quad\quad\quad\quad\quad{\rm is}\ {\rm
nondegenerate}\end{aligned}\right\\}$

is Baire generic.

To explain the geometric idea behind the proof of our main theorem, we need to
establish some standard terminology.

######  Definition 6.

Let $u\in L_{g}^{1}(M)$. We define its distributional derivative
$\nabla_{g}u\in\mathcal{M}_{g}(M,\mathbb{R}^{n})$ as a vector Radon measure
with total variation measure given by

$\|\nabla_{g}u\|(A)=\sup_{x\in\mathfrak{X}_{c}(A)}\left\\{\int_{U}u{\rm
div}_{g}(X)\mathrm{d}\mathcal{L}^{n}_{g}:\|X\|\leqslant 1\right\\},$

whenever $A\subset M$ is an open subset. We also consider the Borel measure
extending $\|\nabla_{g}u\|$ and we denote it with the same symbol. Here
$\mathfrak{X}_{c}(U)$ is the set of smooth vector fields with compact support
and $\|X\|:=\sup_{x\in M}|X_{x}|$, where $|X_{x}|$ is the norm of $X_{x}\in
T_{x}M$. Let us define $BV_{g}(M)=\\{u\in
L_{g}^{1}(M):\|\nabla_{g}u\|<\infty\\}$. In this case, we say that $u$ is a
function of bounded variation. We define the perimeter
$\mathcal{P}_{g}:\mathcal{C}_{g}(M)\rightarrow[0,\infty)$, up to constant, by
$\mathcal{P}_{g}(\Omega)=\|\nabla_{g}\chi_{\Omega}\|$, where
$\mathcal{C}_{g}(M)=\\{\Omega:\mathcal{P}_{g}(\Omega)<\infty\\}$ are the
Caccioppoli $($or the finite perimeter$)$ sets, that is, measurable and finite
perimeter subsets of $M$. The reduced boundary of a Caccioppoli set
$\Omega\subset M$ is denoted by $\partial^{*}\Omega$ and is the set whose a
notion of measure-theoretic normal vector exists and has a length equal to
one. A deep theorem of E. De Giorgi states that
$\mathcal{P}_{g}(\Omega)=\mathcal{H}_{g}^{n-1}(\partial^{*}\Omega)$, where
$\mathcal{H}_{g}^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure induced
by $g$ of the boundary. Here we fix the convention that all the Caccioppoli
sets $\Omega\in\mathcal{C}_{g}(M)$ satisfies $\partial\Omega={\rm
clos}(\partial^{*}\Omega)$, where $\partial\Omega$ is the topological boundary
of $\Omega$.

A symmetric matrix $\alpha\in{\rm Sym}(\mathbb{R}^{N}_{+})$ is said to be
immiscible when

${\rm tr}(\alpha)=0\quad{\rm and}\quad\alpha_{ij}<\alpha_{i\ell}+\alpha_{\ell
j}\quad{\rm for\ all}\quad i,j,\ell\in\\{1,\dots,N\\}\quad{\rm
and}\quad\ell\notin\\{i,j\\}.$ ($\alpha_{0}$)

Along this line, let us introduce the suitable notions of weighted multi-
perimeter and weighted clusters.

######  Definition 7.

Let $(M^{n},g)$ be a Riemannian manifold and $\alpha\in{\rm
Sym}(\mathbb{R}^{N}_{+})$ be a nonnegative symmetric matrix. We define the set
of $\alpha$-weighted $N$-clusters $($or simply weighted clusters$)$ by

$\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{N})=\left\\{\mathbf{\Omega}=(\Omega_{1},\dots,\Omega_{N}):\begin{aligned}
M&=\mathring{\cup}_{i=1}^{N}\Omega_{i},\ \mathrm{v}_{g}(\Omega_{i})<\infty,\
\mathrm{v}_{g}(\Omega_{i}\cap\Omega_{j})=0,\\\ &\ {\rm for\ all}\
i,j=1,\dots,N\ {\rm and}\
\boldsymbol{\mathcal{P}}^{\alpha}_{g}(\mathbf{\Omega})<\infty\end{aligned}\right\\},$

where the $\alpha$-multi-perimeter $($or vectorial $\alpha$-perimeter$)$
functional
$\boldsymbol{\mathcal{P}}^{\alpha}_{g}:\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{N})\rightarrow[0,\infty)$
is

$\boldsymbol{\mathcal{P}}^{\alpha}_{g}(\mathbf{\Omega})=\sum_{i,j=1}^{N}\alpha_{ij}\mathcal{H}^{n-1}_{g}(\partial^{*}\Omega_{i}\cap\partial^{*}\Omega_{j}).$

Also, a weighted cluster
$\mathbf{\Omega}\in\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{N})$ and the
associated perimeter functional $\boldsymbol{\mathcal{P}}^{\alpha}_{g}$ are
said to be immiscible if $\alpha\in{\rm Sym}(\mathbb{R}_{+}^{N})$ is so. For
any weighted cluster
$\mathbf{\Omega}\in\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{N})$, we fix the
terminology

  * •

$\alpha_{ij}$ are the weights;

  * •

$\\{\Omega_{i}\\}_{i\in\\{1,\dots,N\\}}$ $($or
$\\{\mathbf{\Omega}(i)\\}_{i\in\\{1,\dots,N\\}}$$)$ are the chambers with
${\Sigma}_{i}=\partial^{*}\Omega_{i}$;

  * •

$\Sigma_{ij}=\Sigma_{i}\cap\Sigma_{j}$ are the interfaces;

  * •

$\widetilde{{\Omega}}=\cup_{i=1}^{N-1}\Omega_{i}$ are the interior chambers
and ${\Omega}_{N}=M\setminus\cup_{i=1}^{N-1}\Omega_{i}$ is the exterior
chamber;

  * •

$\mathbf{v}_{g}(\mathbf{\Omega})=(\mathrm{v}_{g}(\Omega_{1}),\dots,\mathrm{v}_{g}(\Omega_{N}))$
is the vectorial volume;

  * •

${\rm
diam}_{g}(\widetilde{{\Omega}})=\sup_{x,y\in\widetilde{{\Omega}}}\mathrm{d}_{g}(x,y)$
is the diameter.

In this fashion, we can rewrite
$\boldsymbol{\mathcal{P}}^{\alpha}_{g}(\mathbf{\Omega})=\sum_{i,j=1}^{N}\alpha_{ij}\mathcal{H}^{n-1}_{g}(\Sigma_{ij})$.

######  Remark 8.

Notice that when $\boldsymbol{W}\in{\mathcal{W}}_{N,0}^{+}$ the assumption
(${{\rm W}_{0}}$) implies that $\omega\in{\rm Sym}(\mathbb{R}_{+}^{N})$ given
by
$\omega_{ij}(\boldsymbol{W})=\mathrm{d}_{\boldsymbol{W}}(\mathbf{p}_{i},\mathbf{p}_{j})$
for $i,j\in\\{1,\dots,N\\}$ satisfies ($\alpha_{0}$). Hence, the multi-
perimeter
$\boldsymbol{\mathcal{P}}^{\omega}_{g}:\mathcal{C}^{\omega}_{g}(M,\mathbb{R}^{N})\rightarrow[0,\infty)$
$($weighted by the potential$)$ given by

$\boldsymbol{\mathcal{P}}^{\omega}_{g}(\mathbf{\Omega})=\sum_{i,j=1}^{N}\omega_{ij}(\boldsymbol{W})\mathcal{H}^{n-1}_{g}(\partial^{*}\Omega_{i}\cap\partial^{*}\Omega_{j})$
(4)

is immiscible. In this case, we simply denote
$\mathcal{C}^{\omega}_{g}(M,\mathbb{R}^{N})=\mathcal{C}_{g}(M,\mathbb{R}^{N})$
and $\boldsymbol{\mathcal{P}}^{\omega}_{g}=\boldsymbol{\mathcal{P}}_{g}$.

When the coefficient matrix is given by $\alpha=(\alpha_{ij})\in{\rm
Sym}(\mathbb{R}^{N}_{+})$, where $\alpha_{ij}=1-\delta_{ij}$ for
$i,j\in\\{1,\dots,N\\}$ with $\delta_{ij}$ the Kronecker’s delta, we recover
the standard perimeter, which we denote by $\boldsymbol{\mathcal{P}}^{*}_{g}$.
In this fashion, it is proved in _[60, Proposition 1]_ that we can write the
multi-perimeter

$\boldsymbol{\mathcal{P}}^{*}_{g}(\mathbf{\Omega})=\sum_{\begin{subarray}{c}i,j=1\end{subarray}}^{N}\mathcal{H}^{n-1}_{g}(\Sigma_{ij})=\frac{1}{2}\sum_{i=1}^{N}\mathcal{P}_{g}(\Omega_{i}).$

Such a formula allows us to prove existence and compactness results simply by
relying on the theory for Caccioppoli sets, i.e., $N=2$. Nevertheless, when we
are in the weighted case, the behavior of the multi-perimeter may change. This
was first noticed by F. Almgren [4], which considered a more general perimeter
functional with an extra assumption on the weights, the so-called partitioning
regular condition to proceed with this theory. After that, B. White [64]
showed that the strict triangular inequality ($\alpha_{0}$) is enough to prove
existence and regularity results, which is an assumption weaker than Almgren’s
partitioning regular condition.

A generalization with a complete rigorous proof of White’s results was given
by G. P. Leonardi [40]. Another weaker condition can be found in [20, 5] which
considers existence and regularity issues and says that these coefficients
shall satisfy the BV-ellipticity condition. All these hypotheses on the
weights $\alpha_{ij}$ are used to guarantee that the weighted multi-perimeter
is lower semi-continuous with respect to the flat convergence and enjoys nice
regularity properties; this generally leads to the existence of minimizers by
the direct method of the calculus of variations.

The isoperimetric problem asks to minimize the perimeter among all Caccioppoli
sets subject to a volume constraint. The regions attaining this minimal
configuration are called isoperimetric regions. It is well-known that
isoperimetric regions in the Euclidean space are balls. In the vectorial case,
the analog problem of classifying isoperimetric weighted clusters is still not
solved in its full generality, even if we consider the standard (non-weighted)
multi-perimeter. Nevertheless, one has some partial answers, namely, for non-
weighted double bubbles [37, 26] and triple-bubbles [39], for weighted double-
bubble [38], for multi-bubbles in the Gaussian measure case [46], and lastly
we mention the recent work [47], where the multi-bubble conjecture in
$\mathbb{R}^{n}$ and $\mathbb{S}^{n}$ is proved for $N=2,3,4$. Roughly
speaking, isoperimetric weighted clusters are minimal multi-perimeter
configurations for an enclosing region with a finite number of fixed volumes.

######  Definition 9.

Let $(M^{n},g)$ be a Riemannian manifold.

  * (i)

The isoperimetric profile is a function
${{\mathcal{I}}}_{(M,g)}:\left(0,\mathrm{v}_{g}(M)\right)\rightarrow(0,\infty)$
given by

${\mathcal{I}}_{(M,g)}(\mathrm{v}):=\inf\left\\{\mathcal{P}_{g}({\Omega}):{\Omega}\in\mathcal{C}_{g}(M)\
\mbox{and}\ \mathrm{v}_{g}({\Omega})=\mathrm{v}\right\\}.$

The Caccioppoli set $\Omega\in\mathcal{C}_{g}(M)$ that attains this infimum is
called an isoperimetric region.

  * (ii)

The $\alpha$-weighted multi-isoperimetric profile is a function
${\boldsymbol{\mathcal{I}}}^{\alpha}_{(M,g)}:\left(0,\mathrm{v}_{g}(M)\right)^{N}\rightarrow(0,\infty)$
given by

${\boldsymbol{\mathcal{I}}}^{\alpha}_{(M,g)}(\mathbf{v}):=\inf\left\\{\boldsymbol{\mathcal{P}}^{\alpha}_{g}({\mathbf{\Omega}}):{\mathbf{\Omega}}\in\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{N})\
\mbox{and}\ \mathbf{v}_{g}(\mathbf{\Omega})=\mathbf{v}\right\\}.$

The vectorial Caccioppoli set
$\mathbf{\Omega}\in\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{N})$ that attains
this infimum is called an isoperimetric $\alpha$-weighted $N$-cluster. As
before, we simply denote
${\boldsymbol{\mathcal{I}}}^{\omega}_{(M,g)}={\boldsymbol{\mathcal{I}}}_{(M,g)}$.

Next, we present the two abstract results that we will apply to prove our main
theorem.

To prove the multiplicity part, we use Lusternik–Schnirelmann, and Morse’s
theories. This relates to the topology of the finite-dimensional ambient
manifold and the number of critical points of an energy functional defined on
an infinite-dimensional Hilbert manifold. We are based on the abstract theorem
below

######  Theorem B ([12, 14, 15]).

Let $X$ be a topological space, $\mathfrak{M}$ be a $C^{2}$-Hilbert manifold,
$\mathcal{E}:\mathfrak{M}\rightarrow\mathbb{R}$ be a $C^{1}$-functional, and
$\mathcal{E}^{c}:=\\{u\in\mathfrak{M}:\mathcal{E}(u)\leqslant c\\}$ be a
sublevel set for some $c\in\mathbb{R}$. Assume that

  1. ($E_{1}$)

$\inf_{u\in\mathfrak{M}}\mathcal{E}(u)>-\infty$;

  2. ($E_{2}$)

$\mathcal{E}$ satisfies the Palais–Smale condition;

  3. ($E_{3}$)

There exist $c\in\mathbb{R}$ and two continuous maps
$\Psi_{R}:X\rightarrow\mathcal{E}^{c}$ and
$\Psi_{L}:\mathcal{E}^{c}\rightarrow X$ such that $\Psi_{L}\circ\Psi_{R}$ is
homotopic to the identity map of $X$.

Then, the number of critical points in $\mathcal{E}^{c}$ satisfies
$\\#\mathcal{E}^{c}\geqslant\operatorname{cat}(X)$, and if $\mathfrak{M}$ is
contractible and $\operatorname{cat}(X)>1$, there is at least other critical
point of $\mathcal{E}$ outside $\mathcal{E}^{c}$. Moreover, there exists
$c_{0}\in(c,\infty)$ such that one of the two following conditions hold:

  * (i)

$\mathcal{E}^{c_{0}}$ contains infinitely many critical points;

  * (ii)

$\mathcal{E}^{c}$ contains $\mathscr{P}_{1}(X)$ critical points and
$\mathcal{E}^{c_{0}}\setminus\mathcal{E}^{c}$, contains $\mathscr{P}_{1}(X)-1$
critical points if counted with their multiplicity. More exactly, we have the
following relation

$\sum_{u\in{\rm
Crit}(\mathcal{E})}i_{t}(u)=\mathscr{P}_{t}(X)+t[\mathscr{P}_{t}(X)-1]+(1+t)\mathscr{Q}(t),$
(5)

where $\mathscr{Q}(t)$ is a polynomial with nonnegative integer coefficients,
and ${\rm Crit}(\mathcal{E}^{c_{0}})$ denotes the set of critical points of
$\mathcal{E}$ on $\mathcal{E}^{c_{0}}$. In particular, if all the critical
points are nondegenerate, there are at least $\mathscr{P}_{1}(X)$ critical
points with energy less or equal than $c$ and $\mathscr{P}_{1}(X)-1$ with
energy between $c$ and $c_{0}$.

For the proof of the generic nondegeneracy part, we apply an abstract
transversality theorem

######  Theorem C ([36]).

Let $\mathfrak{X}$, $\mathfrak{Y}$, and $\mathfrak{Z}$ be real Banach spaces,
$\mathfrak{U}\subset\mathfrak{X}$, $\mathfrak{V}\subset\mathfrak{Y}$ be open
subsets, $\mathfrak{F}:\mathfrak{V}\times\mathfrak{U}\rightarrow\mathfrak{Z}$
be a map of class $C^{1}$ and $z_{0}\in{\rm im}(\mathfrak{F})$. Suppose that

  1. ($F_{1}$)

Given $y\in\mathfrak{V}$, it follows that
$\mathfrak{F}(y,\cdot):x\mapsto\mathfrak{F}(y,x)$ is a Fredholm map of index
$\ell<1$, i.e.,
$\mathrm{d}\mathfrak{F}(y,\cdot)_{x}:\mathfrak{X}\rightarrow\mathfrak{Z}$ is a
Fredholm operator of index $\ell$ for any $x\in\mathfrak{U}$;

  2. ($F_{2}$)

$z_{0}$ is a regular value of $\mathfrak{F}$, i.e.,
$\mathrm{d}\mathfrak{F}_{(y_{0},x_{0})}:\mathfrak{Y}\times\mathfrak{X}\rightarrow\mathfrak{Z}$
is surjective for any $(y_{0},x_{0})\in\mathfrak{F}^{-1}(z_{0})$;

  3. ($F_{3}$)

Let
$\iota:\mathfrak{F}^{-1}\left(z_{0}\right)\rightarrow\mathfrak{Y}\times\mathfrak{X}$
be the canonical embedding and
$\pi_{\mathfrak{Y}}:\mathfrak{Y}\times\mathfrak{X}\rightarrow\mathfrak{Y}$ be
the projection of the first coordinate. Then
$\pi_{\mathfrak{Y}}\circ\iota:\mathfrak{F}^{-1}(z_{0})\rightarrow\mathfrak{Y}$
is $\sigma-$ proper, i.e.,
$\mathfrak{F}^{-1}(z_{0})=\bigcup_{k=1}^{\infty}C_{k}$, where $C_{k}$ is a
closed subset of $\mathfrak{F}^{-1}(z_{0})$ and
$\pi_{\mathfrak{Y}}\circ\iota|_{C_{k}}$ is proper for all $k\in\mathbb{N}$.

Then, the set $\\{y\in\mathfrak{V}:z_{0}\ \mbox{is a regular value of}\
\mathfrak{F}(y,\cdot)\\}$ is an open dense subset of $\mathfrak{V}$.

We must verify the hypotheses in Theorem B for our vectorial energy. Namely,
($E_{1}$) follows directly from the definition, and ($E_{2}$) is obtained
using the subcritical growth condition (${{\rm W}_{1}}$). The most delicate
part is to show ($E_{3}$). For this, we need to use the geometric nature of
the problem for small values of the relaxation parameter and the volume. Also,
condition (${{\rm W}_{2}}$) is used to apply Theorem C for the proof of the
genericity part of our main theorem.

We now explain in detail how to construct the maps in ($E_{3}$). Before, we
need the following definition

######  Definition 10.

Associated to each potential $\boldsymbol{W}\in\mathcal{W}^{+}_{N}$, let us
consider the functions $\phi_{i}:\mathbb{R}^{m}\rightarrow\mathbb{R}$ and
$\Phi_{i}:{M}\rightarrow\mathbb{R}$ given by
$\phi_{i}=\mathrm{d}_{\boldsymbol{W}}(\mathbf{p}_{i},\cdot)$ and
$\Phi_{i}=\phi_{i}\circ\mathbf{u}$ for each $i=1,\dots,N$. We define the
transformation

$\mathbf{\Phi}:L_{g}^{1}(M,\mathbb{R}^{m})\rightarrow
L_{g}^{1}(M,\mathbb{R}^{N})\quad\mbox{given
by}\quad\mathbf{\Phi}(\mathbf{u})=(\Phi_{1},\dots,\Phi_{N}).$ (6)

Let us denote $\mathbf{u}^{\prime}=\mathbf{\Phi}(\mathbf{u})$ and
$\mathbf{v}^{\prime}=\boldsymbol{\mathcal{V}}_{g}(\mathbf{u}^{\prime})$. For
the sake of simplicity, we fix the convention
$\mathbf{v}=\mathbf{v}^{\prime}$.

First, a result of S. Baldo [9] proves the $\Gamma$-convergence of (1) to the
weighted multi-perimeter (4) in a bounded subset of the Euclidean space with
flat metric $(\mathbb{R}^{n},\delta)$. We need to extend this result to the
case of non-flat background metrics. Moreover, we use (${{\rm W}_{3}}$) to
further extend this for the case of a sequence of functions with a bounded
energy. Then, we will use the approximating family given by the
$\Gamma$-convergence as the map $\Psi_{L}$, the so-called photography map. We
also need to use the parallelizability to guarantee that this map is a one-to-
one continuous bijection.

######  Proposition 11.

Let $(M^{n},g)$ be a closed Riemannian manifold and
$\boldsymbol{W}\in\mathcal{W}^{+}_{N,3}$. For any family
$\\{\mathbf{u}_{\varepsilon}\\}_{\varepsilon>0}$ satisfying
$\boldsymbol{\mathcal{E}}_{\varepsilon}(\mathbf{u}_{\varepsilon})\leqslant E$
for some $E>0$, there exists $\mathbf{u}_{0}\in{L}_{g}^{1}(M,\mathbb{R}^{m})$
such that, up to a subsequence, it follows

$\lim_{\varepsilon\rightarrow
0}\|\mathbf{u}_{\varepsilon}-\mathbf{u}_{0}\|_{L_{g}^{1}(M,\mathbb{R}^{m})}=0\quad\mbox{and}\quad\lim_{\varepsilon\rightarrow
0}\boldsymbol{\mathcal{E}}_{\varepsilon}(\mathbf{u}_{\varepsilon})=\boldsymbol{\mathcal{P}}_{g}(\mathbf{\Omega}_{0}),$

where
$\mathbf{\Omega}_{0}=\cup_{i=1}^{N-1}(\mathbf{\Phi}\circ\mathbf{u}_{0})^{-1}(\mathbf{p}_{i})$.
Moreover, there exists $\mathbf{\Omega}\in\mathcal{C}_{g}(M,\mathbb{R}^{N})$
minimizing the multi-perimeter with constraint
$\sum_{i=1}^{N}\mathrm{v}_{g}(\Omega_{i})\mathbf{p}_{i}=\mathrm{v}_{i}$, and
such that $\|\nabla_{g}\mathbf{u}_{0}\|(M)\leqslant E$ and

$\mathbf{u}_{0}=\sum_{i=1}^{N}\mathbf{p}_{i}\chi_{\Omega_{i}}\in
BV_{g}(M,\mathbb{R}^{m}),$ (7)

where $BV_{g}(M,\mathbb{R}^{m})=\left\\{\mathbf{u}\in
L_{g}^{1}(M,\mathbb{R}^{m}):|\mathbf{u}|\in BV_{g}(M)\right\\}$. Conversely,
for any $\mathbf{\Omega}\in\mathcal{C}_{g}(M,\mathbb{R}^{N})$ and
$\mathbf{u}_{0,\mathbf{\Omega}}\in{L}_{g}^{1}(M,\mathbb{R}^{m})$ of the form
(7), there exists a sequence
$\\{\mathbf{u}_{\varepsilon,\mathbf{\Omega}}\\}_{\varepsilon>0}$ satisfying
$\lim_{\varepsilon\rightarrow
0}\|\mathbf{u}_{\varepsilon,\mathbf{\Omega}}-\mathbf{u}_{0,\mathbf{\Omega}}\|_{{L}_{g}^{1}(M,\mathbb{R}^{m})}=0$.
In particular, it follows that $\Gamma$-$\lim_{\varepsilon\rightarrow
0}\boldsymbol{\mathcal{E}}_{\varepsilon}=\boldsymbol{\mathcal{P}}_{g}$ in
$L_{g}^{1}(M,\mathbb{R}^{m})$.

Second, we build the right-inverse homotopy $\Psi_{R}$ to the photography map
by composing the barycenter map together with the projection provided by the
Nash embedding theorem. To control the range of the barycenter map, we need to
use some deep results of the isoperimetric theory for clusters.

More precisely, we prove that from an isoperimetric weighted cluster enclosing
a small volume we can build another cluster with small diameter which almost
the same volume and perimeter of the original cluster. Such result is called
selecting a large subdomain. Although we will apply this result to the
perimeter in (4), which is related to the potential $\boldsymbol{W}$, this
result holds for a broader class of weighted perimeters, which we call
immiscible (see Remark 8) and are of independent interest. Indeed, we use
(${{\rm W}_{0}}$) to guarantee that $\boldsymbol{\mathcal{P}}_{g}$ is
immiscible. For results concerning properties of isoperimetric clusters, see
[42, 40, 64, 4, 54, 60].

######  Proposition 12.

Let $(M^{n},g)$ be a closed Riemannian manifold and
$\\{\mathbf{\Omega}_{k}\\}_{k\in\mathbb{N}}\subset\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{3})$
be a sequence of isoperimetric immiscible weighted $3$-clusters. There exists
another sequence
$\\{\mathbf{\Omega}_{k}^{\prime}\\}_{k\in\mathbb{N}}\subset\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{3})$
such that

  * (i)

$\lim_{k\rightarrow\infty}{\mathrm{v}_{g}({\mathbf{\Omega}}_{k}\triangle{\mathbf{\Omega}}_{k}^{\prime})}{\mathrm{v}_{g}({\mathbf{\Omega}}_{k})}^{-1}=0$;

  * (ii)

$\lim_{k\rightarrow\infty}{\mathrm{v}_{g}(\mathbf{\Omega}_{k}^{\prime})}{\mathrm{v}_{g}(\mathbf{\Omega}_{k})}^{-1}=1$;

  * (iii)

$\lim_{k\rightarrow\infty}{\boldsymbol{\mathcal{P}}^{\alpha}_{g}(\mathbf{\Omega}_{k}^{\prime})}{\boldsymbol{\mathcal{P}}^{\alpha}_{g}(\mathbf{\Omega}_{k})}^{-1}=1$;

  * (iv)

$\lim_{k\rightarrow\infty}\operatorname{diam}_{g}(\widetilde{{\Omega}}_{k}^{\prime})=0$.

The proofs of Proposition 11 and Proposition 12 are independent of each other.
However, in the proof of Proposition 11, it is convenient to assume that the
limiting weighted cluster has small diameter, for $N=3$, we use directly [38]
which implies that if we are under the small volumes regime then the small
diameter property holds. There is an alternative strategy to prove our main
results without relying on the full strength of [38]. The price to pay is to
extend Proposition 11 to the Riemannian setting without the small diameter
property and to make a more complicated definition of the photography map with
a consequent more involved proof of its well posedness and continuity.

Beyond its geometrical relevance, the study of the ACH equation dates back to
the theory of phase separation for binary fluids in an alloy [32, 2, 19], or
more generally in the van der Walls theory of phase transition [63]. Using
this analogy, System ($ACH_{\varepsilon,\mathbf{v},m,N,g}$) is used to
describe the free energy of a multiphasic mixture of interacting fluids, where
the coefficients $\omega_{ij}$ measure the surface tension between two pure
states of the system. To be more physically realistic, the energy should have
a non-local ferromagnetic Kac potential instead of the norm of the gradient
[29, 30]; these models have applications in several areas like the Ising
process, and dislocations in elastic materials exhibiting microstructure [21].

Let us explain the connection of our result to differential geometry. An
ancient problem in this field is to determine the number of critical points of
the perimeter functional $\mathcal{P}_{g}$ on a general Riemannian manifold.
Indeed, S.-T. Yau [66] conjectured that any closed Riemannian manifold
contains infinitely many critical points of the perimeter. This conjecture was
recently solved affirmatively for generic metrics [43, 41] and for the
remaining cases when $3\leqslant n\leqslant 7$ [61]. Recall that critical
points of the perimeter are called minimal hypersurfaces and they must have
zero mean curvature. When there is a volume constraint, one has the equivalent
problem of minimizing $\mathcal{P}_{g}(\Omega)$ under the condition
${\mathrm{v}}_{g}(\Omega)=\mathrm{v}$. In this case, the reduced boundary of
$\Omega$ is called a CMC (Constant Mean Curvature) hypersurface, or a CMC
boundary. The convergence of the ACH energy to the perimeter functional was
recently used to construct minimal hypersurfaces in any closed Riemannian
manifold [31, 27] as an alternative approach to min-max methods [3, 58]. For
CMC boundaries, it is only known the min-max construction in [67, 25, 24, 68].
It is not known whether Yau’s conjecture also holds true. In this context, let
us consider a slightly more general problem. We aim to study the asymptotic
behavior of $\boldsymbol{\eta}(0,\mathbf{v}):=\lim_{\varepsilon\rightarrow
0}\boldsymbol{\eta}(\varepsilon,\mathbf{v})$, which coincides with the number
of critical points of the multi-perimeter with volume constraint $\mathbf{v}$,
denoted by $\mathfrak{N}_{0,\mathbf{v}}$. We aim to study the abundance of
elements in this set. It is already known that
$\mathfrak{N}_{0,\mathbf{v}}\neq\emptyset$, then it is natural to ask if in
any closed manifold one has $\boldsymbol{\eta}(0,\mathbf{v})=\infty$.

If we can pass to the limit as $\varepsilon\rightarrow 0$ in Theorem (i), we
would obtain the first result in this direction for the case of weighted
clusters with two interior chambers under the small volume condition. In the
scalar case, by combining the results on [16] and [11], it follows that
${\eta}(0,\mathbf{v})\geqslant 1$, that is, there exists at least one almost
embedded CMC boundary in the small volume regime, which is the pioneering
existence result using phase transition methods.

At last, we compare our proof to the one in the scalar case. The strategy to
prove Theorem A relies also on the combination of several results from the
realm of PDEs, algebraic topology, isoperimetric problem, minimal surface
theory, and phase transition approximations. First, classical results [50, 49,
33, 62] (see also [57, 56] for the geometric case) state that the energy
$\mathcal{E}_{\varepsilon}$ is a singular perturbation (relaxation) of the
perimeter functional $\mathcal{P}_{g}$. This analogy creates an unexpected and
surprising bridge between the theory of nonlinear PDEs from phase transition
and the study of qualitative properties for CMC boundaries (or minimal
hypersurfaces). The heuristics is that the action of the double-well potential
induces the decomposition $M=\Omega_{1}\cup\Omega_{2}\cup\Sigma_{12}$, where
$\Sigma_{12}=\partial^{*}\Omega_{1}=\partial^{*}\Omega_{2}$ is the limit
interface, known to be a critical point of the perimeter, that is, the mean
curvature of the interface $H_{\Sigma_{12}}$ is constant, and
$\Omega_{1}:=\\{x\in M:\lim_{\varepsilon\rightarrow
0}{\mathrm{u}}_{\varepsilon}(x)=1\\}$ and $\Omega_{2}:=\\{x\in
M:\lim_{\varepsilon\rightarrow 0}u_{\varepsilon}(x)=0\\},$ where
$\mathrm{u}_{\varepsilon}$ is a solution to
($ACH_{\varepsilon,\mathrm{v},g}$). In other terms, in the singular limit,
there is a relation between the nodal sets of solutions to
($ACH_{\varepsilon,\mathrm{v},g}$) and CMC boundaries. In this proof, it is
also used that isoperimetric regions of small volume have a small diameter,
and the asymptotic expansion of the isoperimetric profile function [52, 51,
18, 53, 8].

The rest of the paper is divided as follows. In Section 2, we introduce some
definitions and results from the abstract Lusternik–Schnirelmann and infinite-
dimensional Morse theories, which we use to sketch the proof of Theorem B. In
Section 3, we show that we can select a large subdomain from isoperimetric
weighted clusters in Proposition 12. In Section 4, we prove the convergence
and approximation results in Proposition 11. In Section 5, we apply Theorem B
to prove the multiplicity part of Theorem Theorem 1. In Section 6, based on
Theorem C, we prove the generic nondegeneracy part of Theorem Theorem 1.

Acknowledgments. This project started when the first-named author was a
visiting Ph.D. student and the third-named author was a visiting fellow in the
Department of Mathematics at Princeton University, whose hospitality they
greatly acknowledge. This work was partially supported by Fundação de Amparo à
Pesquisa do Estado de São Paulo (FAPESP), Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq), Fulbright Commission in Brazil, and
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior–Brasil (CAPES).
J.H.A. was supported by FAPESP #2020/07566-3 and #2021/15139-0, Fulbright
#G-1-00001, and CAPES #88882.440505/2019-01. J.C. was supported by CAPES
#88882.377936/2019-01. S.N. was supported by FAPESP #2021/05256-0 and
#2018/22938-4, CNPq #305726/2017-0, and CNPq #12327/2021-8. P.P. was supported
by FAPESP #2016/23746-6 and CNPq #313773/2021-1. R.R. was supported by CAPES
#88882.377954/2019-01.

##  2\. Abstract photography method

In this section, we present some standard results from Lusternik–Schnirelmann
and infinite-dimensional Morse theories which will be used in the abstract
photography method [14, 15]. We use the approach developed in [12, 13, 44],
which is suitable for problems arising from PDEs.

######  Definition 2.1.

Let $X$ a topological space and $Y\subseteq X$ be a closed subset. We define
the Lusternik-Schnirelmann category of $Y$ in $X$ as the extended natural
number $\operatorname{cat}_{X}(Y)$ obtained as the minimum number
$k\in\mathbb{N}\cup\\{\infty\\}$ such that there exist
$\mathcal{U}_{1},\dots,\mathcal{U}_{k}\subseteq X$, open contractible subsets
satisfying $Y\subseteq\bigcup_{i=1}^{k}\mathcal{U}_{i}$. We also denote
$\operatorname{cat}(X):=\operatorname{cat}_{X}(X)$.

######  Definition 2.2.

Let $X,Z$ be topological spaces. We say that $X$ and $Z$ are homotopically
superjacent if there exist continuous maps $\Psi_{L}:X\rightarrow Z$ and
$\Psi_{R}:Z\rightarrow X$ such that $\Psi_{L}\circ\Psi_{R}$ is homotopic to
the identity of $X$.

Now, we present a standard result in Lusternik–Schnirelman’s theory.

######  Lemma 2.3.

If $X$ and $Z$ are homotopically superjacent topological spaces, then
$\operatorname{cat}(X)\leqslant\operatorname{cat}(Z)$.

###### Proof.

Assume that $\operatorname{cat}(Z)=m$, that is, there exist $m$ closed
contractible sets $\\{\widetilde{\mathcal{U}}_{i}\\}_{i=0}^{m}$, such that
$Z\subseteq\cup_{i=1}^{m}\widetilde{\mathcal{U}}_{i}$,
$w_{i}\in\widetilde{\mathcal{U}}_{i}$, and $F_{i}\in
C([0,1]\times\widetilde{\mathcal{U}}_{i},Z)$ for $i=1,\dots,m$ satisfying

$\begin{cases}F_{i}(0,u)=u,\ {\rm if}\ u\in\tilde{\mathcal{U}}_{i}\\\
F_{i}(1,u)=w_{i},\ {\rm if}\ u\in\tilde{\mathcal{U}}_{i}.\end{cases}$

Now, setting $\mathcal{U}_{i}=\Psi_{R}(\widetilde{\mathcal{U}}_{i})$, notice
that $\\{\mathcal{U}_{i}\\}_{i=1}^{m}$ is such that
$X\subseteq\cup_{i=1}^{m}\mathcal{U}_{i}$, where the null-homotopic retraction
$\widetilde{F}_{i}\in C([0,1]\times\mathcal{U}_{i},X)$ is given by
$\widetilde{F}_{i}:=\Psi_{R}\circ F_{i}$. ∎

Let us fix the notation ${\rm Crit}(\mathcal{E})$ for the set of critical
points of $\mathcal{E}$,
$\mathcal{E}^{c}:=\\{u\in\mathfrak{M}:\mathcal{E}(u)\leqslant c\\}$ for
sublevel sets and
$\mathcal{E}^{a}_{b}:=\\{u\in\mathfrak{M}:a\leqslant\mathcal{E}(u)\leqslant
b\\}$ for level regions.

######  Definition 2.4.

Let $\mathfrak{M}$ be a $C^{2}$-Hilbert manifold,
$\mathcal{E}:\mathfrak{M}\rightarrow\mathbb{R}$ be a $C^{1}$-functional, and
$\\{u_{k}\\}_{k\in\mathbb{N}}$ be a sequence in $\mathfrak{M}$. We call
$\\{u_{k}\\}_{k\in\mathbb{N}}$ a Palais–Smale sequence at level
$c\in\mathbb{R}$, if $\mathcal{E}(u_{k})\rightarrow c$, and
$\quad\|\mathrm{d}\mathcal{E}(u_{k})\|_{T_{u_{k}}^{*}\mathfrak{M}}\rightarrow
0$ as $k\rightarrow\infty$.

######  Definition 2.5.

Let $\mathfrak{M}$ be a $C^{2}$-Hilbert manifold and
$\mathcal{E}:\mathfrak{M}\rightarrow\mathbb{R}$ be a $C^{1}$-functional. We
say that $\mathcal{E}$ satisfies the Palais–Smale condition, if every
Palais–Smale sequence has a convergent subsequence in the strong topology of
$\mathfrak{M}$.

Let $X$ a topological space, $Y\subseteq X$ be a closed subset, and
$k\in\mathbb{N}$. We denote $H_{\rm AS}^{k}(X,Y)$ by the $k$-th relative
Alexander–Spanier cohomology group of the pair $X,Y$. Here the $k$-th Betti
number is given by the number of generators of $H_{\rm AS}^{k}(X,Y)$ seen as a
$\mathbb{Z}_{2}$-modulo, that is, $\beta_{k}(X,Y):={\rm
rank}_{\mathbb{Z}_{2}}H_{\rm AS}^{k}(X,Y)$ (see [55, Chapter 2]). In the
nondegenerate case, Theorem B can be made more precise by means of Morse
theory, which we describe as follows

######  Definition 2.6.

Let $X$ be a topological space and denote by $H_{\rm AS}^{k}(X)$ its $k$-th
Alexander–Spanier cohomology group with coefficients in $\mathbb{Z}_{2}$. The
Poincaré polynomial $\mathscr{P}_{t}(X)$ of $X$ is defined as the following
power series in the variable $t$,

$\mathscr{P}_{t}(X):=\sum_{k=0}^{\infty}\beta_{k}(X)t^{k}.$ (2.1)

We notice that if $X$ is a compact manifold, then $H^{k}(X)$ is a finite-
dimensional vector space and the formal series (2.1) is actually a polynomial.
The next result is a Poincaré–Morse-type identity, relating the Morse
polynomials of two homotopically superjacent topological spaces.

######  Lemma 2.7.

If $X$ and $Z$ are homotopically superjacent, then

$\mathscr{P}_{t}(Z)=\mathscr{P}_{t}(X)+\mathscr{Z}(t),$ (2.2)

where $\mathscr{Z}(t)$ is a polynomial with nonnegative coefficients.

###### Proof.

Using the homotopy equivalence, we can construct an exact sequence

$0{\longrightarrow}H_{k}(X)\stackrel{{\scriptstyle({\Psi_{L})}_{k}}}{{\longrightarrow}}H_{k}(Z)\stackrel{{\scriptstyle{(\Psi_{R})}_{k}}}{{\longrightarrow}}H_{k}(X){\longrightarrow}0,$

where $({\Psi_{L}})_{k},({\Psi_{R}})_{k}$ is the induced homomorphisms on the
$k$-th homology groups. Additionally, since
$({\Psi_{L}})_{k}\circ({\Psi_{R}})_{k}={\rm id}_{k}$ we get that $H_{k}(X)$ is
homotopic equivalent to a subspace of $H_{k}(Z)$, which gives us that ${\rm
rank}_{\mathbb{Z}_{2}}H_{k}(X)\leqslant{\rm rank}_{\mathbb{Z}_{2}}H_{k}(Z)$
for all $k\in\mathbb{N}$. In terms of Poincaré polynomials, this translates to
(2.2). ∎

######  Lemma 2.8.

Let $\mathfrak{M}$ be a Hilbert manifold and let
$\mathfrak{N}\subset\mathfrak{M}$ be a closed oriented submanifold of
codimension $d$. If $\mathfrak{N}_{0}$ is a closed subset of $\mathfrak{N}$,
then

$\mathscr{P}_{t}(\mathfrak{M},\mathfrak{M}\setminus\mathfrak{N}_{0})=t^{d}\mathscr{P}(\mathfrak{N},\mathfrak{N}\setminus\mathfrak{N}_{0}).$

###### Proof.

This follows from the Thom isomorphism theorem [35, Corollary 4D.9]. Moreover,
it holds even when $\dim\mathfrak{M}=\infty$. ∎

In the following definition, we give the notion of Morse index of a critical
point of $\mathcal{E}$, which is necessary to establish a relation between the
Poincaré polynomial $\mathscr{P}_{t}(X)$ and the number of solutions to the
Euler–Lagrange equation associated to the energy $\mathcal{E}$.

######  Definition 2.9.

Let $u$ be a critical point of $\mathcal{E}$ at level $c\in\mathbb{R}$, that
is, $\mathcal{E}(u)=c$ and $\mathrm{d}\mathcal{E}[u]=0$. We call $u$ an
isolated critical point if there exists a neighborhood $\mathfrak{U}$ of $u$
in $\mathfrak{M}$ such that the only critical point of $\mathcal{E}$ contained
in $\mathfrak{U}$ is $u$. Equivalently, the self-adjoint operator induced by
the quadratic form $\mathrm{d}^{2}\mathcal{E}[u]$ is an isomorphism.

######  Definition 2.10.

Let $\mathfrak{M}$ be a $C^{2}$-Hilbert manifold,
$\mathcal{E}:\mathfrak{M}\rightarrow\mathbb{R}$ be a $C^{1}$-functional and
$u\in\mathfrak{M}$ be an isolated critical point of $\mathcal{E}$ at level
$c\in\mathbb{R}$. We define the formal power series

$i_{t}(u):=\sum_{k=0}^{\infty}\beta_{k}(\mathcal{E}^{c},\mathcal{E}^{c}\setminus\\{u\\})t^{k}.$

This is called the polynomial Morse index of $u$ and the number $i_{1}(u)$ is
called its multiplicity.

######  Definition 2.11.

Let $\mathcal{E}:\mathfrak{M}\rightarrow\mathbb{R}$ be a $C^{2}$-functional.
We call $u$ nondegenerate if the bilinear form $\mathrm{d}^{2}\mathcal{E}[u]$
is nondegenerate. In this case, we have that $i_{t}(u)=t^{\mu(u)}$, where
$\mu(u)$ is the numerical Morse index of $u$, that is, the dimension of the
maximal subspace on which the bilinear form $\mathrm{d}^{2}\mathcal{E}[u]$ is
negative-definite.

######  Definition 2.12.

Let $\mathfrak{M}$ be a $C^{2}$-Hilbert manifold,
$\mathcal{E}:\mathfrak{M}\rightarrow\mathbb{R}$ be a $C^{1}$-functional and
$u\in\mathfrak{M}$ be an isolated critical point of $\mathcal{E}$ at level
$c$. We say that $u$ is topologically nondegenerate, if $i_{t}(u)=t^{\mu(u)}$,
for some $\mu(u)\in\mathbb{N}$.

For the sake of completeness, we now present the strategy of proof of the
first cornerstone abstract result in this manuscript, namely the photography
theorem.

###### Proof of Theorem B.

Using ($E_{1}$) and ($E_{2}$), one can invoke the Lusternik–Schnirelman lemma
[65, Theorem 5.20] to obtain that
$\\#\mathcal{E}^{c}\geqslant\operatorname{cat}(\mathcal{E}^{c})$. Moreover,
combining ($E_{3}$) with the last inequality, we have
$\operatorname{cat}(\mathcal{E}^{c})\geqslant\operatorname{cat}(X)$, which
concludes the proof of the first part of the theorem.

For the remaining part, we set $\mathfrak{C}_{0}:=\overline{\Psi_{L}(X)}$.
Thus, $\mathfrak{C}_{0}$ is non-contractible in $\mathcal{E}^{c}$ since
$\operatorname{cat}(X)>1$.

Claim 1: There exists $c^{*}\in\mathbb{R}$ such that $c^{*}>c$ and
$\mathfrak{C}_{0}$ is contractible in $\mathcal{E}^{c^{*}}$.

Indeed, let $u_{0}\in{\rm Crit}(\mathcal{E})\setminus\mathfrak{C}_{0}$ and
define $\mathfrak{C}_{1}:=\left\\{tu_{0}+(1-t)u:t\in[0,1]\ \mbox{and}\
u\in\mathfrak{C}_{0}\right\\}$. Notice that $\mathfrak{C}_{1}$ is compact,
contractible, and $0\notin\mathfrak{C}_{1}$, which implies that the

$\mathfrak{C}_{2}:=\left\\{t(w)w:w\in\mathfrak{C}_{1}\ \mbox{and}\
t(w)={\Psi_{R}}\left({w}{\|w\|^{-1}}\right)\|w\|^{-1}\right\\}$

is well-defined and satisfies
$\mathfrak{C}_{1}\subseteq\mathfrak{C}_{2}\subseteq{\rm Crit}(\mathcal{E})$;
thus, the result follows setting $c^{*}:=\max_{\mathfrak{C}_{2}}\mathcal{E}$.

The second part of the proof is divided into a sequence of steps, which
connects the algebraic structure of the Poincaré polynomial
$\mathscr{P}_{t}(X)$ with the set of critical points ${\rm
Crit}(\mathcal{E})$.

Step 1: For $\tau>0$ and $c\in(\tau,\infty)$ a noncritical level of
$\mathcal{E}$, we have
$\mathscr{P}_{t}(\mathcal{E}^{c},\mathcal{E}^{\tau})=t\mathscr{P}_{t}({\rm
Crit(\mathcal{E}^{c})})$.

This clearly follows from Lemma 2.8.

Step 2: For $a,b\in\mathbb{R}$ with no critical levels inside
$\mathcal{E}^{a}_{b}$, we have
$\mathscr{P}_{t}(\mathcal{E}^{b},\mathcal{E}^{a})=t\mathscr{P}_{t}(\mathcal{E}^{a}_{b},\mathcal{E}^{a}_{b}\setminus{\rm
Crit}(\mathcal{E})).$

The strategy here is to use standard deformation theory generated by the
gradient flow of $\mathcal{E}$.

Step 3: For $\tau>0$ and $c\in(\tau,\infty)$ a noncritical level of
$\mathcal{E}$, we have
$\mathscr{P}_{t}(\mathcal{E}^{c},\mathcal{E}^{\tau})=t\mathscr{P}_{t}(X)+t\mathscr{Z}(t)$.
In particular, $\mathscr{P}_{t}(\mathfrak{M},\mathcal{E}^{\tau})=t$.

This is a consequence of Steps 1 and 2. Furthermore, since $\mathfrak{M}$ is
contractible, one has ${\rm rank}_{\mathbb{Z}_{2}}H^{k}(\mathfrak{M})=1$ if
$k=0$ and ${\rm rank}_{\mathbb{Z}_{2}}H^{k}(\mathfrak{M})=0$, otherwise.

Step 4: There exists $\tau_{0}>0$ such that for $\tau\in(0,\tau_{0})$ and
$c\in(\tau,\infty)$ a noncritical level of $\mathcal{E}$, we have
$\mathscr{P}_{t}(\mathcal{E}^{c},\mathcal{E}^{\tau})=t\mathscr{P}_{t}(X)+t\mathscr{Z}(t)$.

Indeed, consider the long exact sequence

$\displaystyle
0{\longrightarrow}H_{k}(\mathfrak{M},\mathcal{E}^{\tau})\stackrel{{\scriptstyle
j_{k}}}{{\longrightarrow}}H_{k}(\mathfrak{M},\mathcal{E}^{c})\stackrel{{\scriptstyle\partial_{k}}}{{\longrightarrow}}H_{k-1}(\mathcal{E}^{c},\mathcal{E}^{\tau})\stackrel{{\scriptstyle
i_{k-1}}}{{\longrightarrow}}H_{k-1}(\mathfrak{M},\mathcal{E}^{\tau}){\longrightarrow}0,$

where $i_{k},j_{k},\partial_{k}$ are respectively the monomorphism,
epimorphism, and boundary operator. Using some standard techniques of
homological algebra, we get that ${\rm
rank}_{\mathbb{Z}_{2}}H_{k}(\mathfrak{M},\mathcal{E}^{\tau})=0$ for $k=0,1$
and ${\rm rank}_{\mathbb{Z}_{2}}H_{k}(\mathfrak{M},\mathcal{E}^{\tau})={\rm
rank}_{\mathbb{Z}_{2}}H_{k}(\mathcal{E}^{c},\mathcal{E}^{\tau})$, which by
Step 1 finishes the proof.

Step 5: If ${\rm Crit(\mathcal{E})}$ is discrete, then

$\sum_{u\in\mathfrak{C}_{1}}i_{t}(u)=t\mathscr{P}_{t}(X)+t\left[\mathscr{Z}(t)-1\right]+(1+t)\mathscr{Q}_{1}(t)$
(2.3)

and

$\sum_{u\in\mathfrak{C}_{2}}i_{t}(u)=t^{2}\left[\mathscr{P}_{t}(X)+\mathscr{Z}(t)-1\right]+(1+t)\mathscr{Q}_{2}(t),$
(2.4)

where $\mathscr{Z}(t),\mathscr{Q}_{1}(t)$ and $\mathscr{Q}_{2}(t)$ are
polynomials with nonnegative integer coefficients, and

$\mathfrak{C}_{1}:=\\{u\in{\rm
Crit(\mathcal{E})}:u\in\mathcal{E}^{-1}((\tau,c_{0}])\\}\quad\mbox{and}\quad\mathfrak{C}_{2}:=\\{u\in{\rm
Crit(\mathcal{E})}:u\in\mathcal{E}^{-1}((c_{0},\infty))\\},$

for some $c_{0}\in(\tau,\infty)$.

In fact, by ($E_{2}$) we know that $\mathcal{E}$ satisfies the (PS)-condition,
thus using Morse theory, it follows

$\sum_{u\in\mathfrak{C}_{1}}i_{t}(u)=\mathscr{P}_{t}(\mathcal{E}^{c},\mathcal{E}^{\tau})+(1+t)\mathscr{Q}_{1}(t),$

which by Step 3 implies (2.3). The same holds for (2.4).

Finally, since $\mathcal{E}$ does not have any nonzero solution below the
level $\tau>0$, by Step 5 we get ${\rm
Crit(\mathcal{E})}=\mathfrak{C}_{1}\mathring{\cup}\mathfrak{C}_{2}$. ∎

######  Remark 2.13.

If we count the critical points with their multiplicity, then by Theorem B it
follows that there are at least $2\mathscr{P}_{1}(X)-1$. In fact, whenever the
critical points are isolated the result follows from Morse’s relation (5);
otherwise there are infinitely many of them.

##  3\. Riemannian isoperimetric theory for weighted clusters

In this section, we prove Proposition 12. Here we denote by
$\mathcal{B}^{g}_{r}(x)$ the geodesic ball of radius $r>0$ with center $x\in
M$, and we omit the metric when necessary. Recall the notation established in
the introduction (see Definitions 7 and 9). Also, we denote $f_{1}\sim f_{2}$
as $s\rightarrow 0$, if $\lim_{s\rightarrow 0}f_{1}(s)/f_{2}(s)=1$, or
$f_{1}=f_{2}+\mathrm{o}(1)$ as $s\rightarrow 0$, where $\mathrm{o}(1)$ is the
standard little-o notation from Landau’s formalism. It is implicit that we
assume the natural association between vectors $\mathbf{v}\in\mathbb{R}^{N}$
and weighted clusters
$\mathbf{\Omega}\in\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{N})$ such that
$\mathbf{v}=\mathbf{v}_{g}(\mathbf{\Omega})$ and
$\mathrm{v}=|\mathbf{v}|=\mathrm{v}_{g}(\widetilde{{\Omega}})=\sum_{i=1}^{N-1}\mathrm{v}_{i}$,
where $\mathrm{v}_{i}=\mathrm{v}_{g}(\Omega_{i})$ for $i=1,\dots,N-1$.

######  Definition 3.1.

We say that a Riemannian manifold $(M^{n},g)$ is of bounded geometry if, there
exist $\mathrm{v}_{0}>0$ and $\kappa\in\mathbb{R}$ such that ${\rm
Ric}_{g}\geqslant(n-1)\kappa$ $($in the sense of bilinear forms$)$ and
$\inf_{p\in
M}\mathrm{v}_{g}(\mathcal{B}^{g}_{1}(p))\geqslant\mathrm{v}_{0}>0$.

######  Remark 3.2.

If $(M^{n},g)$ is a closed Riemannian manifold, we have that its injectivity
radius is well-defined and it holds ${\rm inj}_{g}>0$. Also, there exist
$v_{0}>0$ and $\kappa,b\in\mathbb{R}$ such that ${\rm
Ric}_{g}\geqslant(n-1)\kappa$ $($in the sense of bilinear forms$)$, $|{\rm
Sec}_{g}|\leqslant b$ and $\mathrm{v}_{g}(\mathcal{B}^{g}_{1}(x))>v_{0}$ for
all $x\in M$, where ${\rm Ric}_{g}$ and ${\rm Sec}_{g}$ are the Ricci and
scalar curvatures, respectively. In particular, closed manifolds are always of
bounded geometry.

We need to introduce some concepts and results from geometric measure theory,
which will be used in the proof of the main proposition and can be found at
[1, 23].

Let us start with some notations and concepts relative to varifolds.

######  Definition 3.3.

Let $(M^{n},g)$ be a Riemannian manifold. For any $d\in\mathbb{N}$ with
$1\leqslant d\leqslant n-1$, we say that $V$ is a $d$-dimensional varifold in
$M$, if $V$ is a nonnegative extended real valued Radon measure on the
Grassmannian manifold $G_{d}(M)$. For every $d=1,\dots,n-1$, we denote by
$\mathbf{V}_{d}(M)$ be the space of all $d$-dimensional varifolds over $M$
endowed with the weak topology induced by $C_{0}(G_{d}(M))$ the space of
continuous compactly supported functions on $G_{d}(M)$ endowed with the
compact open topology.

######  Definition 3.4.

Let $(M^{n},g)$ be a Riemannian manifold and $V\in\mathbf{V}_{d}(M)$. We say
that the nonnegative Radon measure on $M$, denoted by $\|V\|$, is the weight
of $V$, if $\|V\|=\pi_{\\#}(V)$, where $\pi$ indicates the natural fiber
bundle projection $\pi:G_{d}(M)\rightarrow M$, that is,
$\|V\|(B):=V(\pi^{-1}(B))$ for all $B\subset G_{d}(M)$ Borel set.

We also present the concept of density for a measure.

######  Definition 3.5.

Let $(M^{n},g)$ be a Riemannian manifold and $\nu$ be a Borel regular measure
on $M$. We define the $d$-lower density and $d$-upper density of $\nu$ at
$x\in M$, respectively by

$\Theta_{*}^{d}(\nu,x):=\liminf_{r\rightarrow
0^{+}}\frac{\nu(\mathcal{B}_{r}(x))}{\omega_{d}r^{d}}\quad{\rm
and}\quad\Theta^{*}_{d}(\nu,x):=\limsup_{r\rightarrow
0^{+}}\frac{\nu(\mathcal{B}_{r}(x))}{\omega_{d}r^{d}},$

where $\omega_{d}$ is the $d$-dimensional Hausdorff measure of the unit
Euclidean ball. Also, when the $d$-lower and $d$-upper densities coincide, we
set

$\Theta^{d}(\nu,x):=\Theta^{*}_{d}(\nu,x)=\Theta_{*}^{d}(\nu,x)=\lim_{r\rightarrow
0^{+}}\frac{\nu(\mathcal{B}_{r}(x))}{\omega_{d}r^{d}}.$

We call $\Theta^{d}(\nu,x)$ the $d$-density of $\nu$ at $x\in M$.

We present the following definition for the first variation of a varifold.

######  Definition 3.6.

Let $(M^{n},g)$ be a Riemannian manifold. For any $V\in\mathbf{V}_{d}(M)$ its
first variation in the direction of the smooth vector field
$X\in\mathfrak{X}_{c}(M)$ is the linear function
$\delta_{g}V:\mathfrak{X}_{c}(M)\rightarrow\mathbb{R}$ defined as

$\displaystyle\delta_{g}V(X)$ $\displaystyle:=$ $\displaystyle\int_{\xi\in
G_{d}(M)}{\rm div}_{S}X\mathrm{d}V(\xi),$

where $S\subset T_{x}M$ is a $d$-dimensional subspace of $T_{x}M$ such that
$\xi=(x,S)\in G_{d}(M)$, $\pi_{S}:T_{x}M\rightarrow S$ is the orthogonal
projection with respect to the metric $g$, $(e_{1},...,e_{n})$ is an
orthonormal basis of $(T_{\pi(\xi)}M,g_{\pi(\xi)})$, and ${\rm
div}_{S}X=\sum_{i=1}^{d}\langle\nabla_{\tilde{e}_{i}}X,\tilde{e}_{i}\rangle$
with $\\{\tilde{e}_{1},\dots,\tilde{e}_{d}\\}$ an orthonormal basis over $S$.

######  Definition 3.7.

Let $(M^{n},g)$ be a Riemannian manifold. A Borel subset $B\subset M$ is said
to be $d$-countably rectifiable, if ${\rm dim}_{\mathcal{H}}(B)=d$, and there
exist a countable collection $\\{f_{k}\\}_{k\in\mathbb{N}}\subset{\rm
Lip}(\mathbb{R}^{d},M)$ of Lipschitz continuous such that
$\mathcal{H}^{d}(B\setminus\cup_{k=0}^{\infty}f_{k}(\mathbb{R}^{d}))=0$.

Next, we give the definition of integral varifolds.

######  Definition 3.8.

Let $(M^{n},g)$ be a Riemannian manifold, $1\leqslant d\leqslant n-1$,
$B\subseteq M$ a $d$-countably rectifiable $\mathcal{H}_{g}^{d}$-measurable
subset, and $\theta:B\rightarrow\mathbb{N}$ a positive locally integrable
Borel map. We define a varifold $V(B,\theta,g)\in\mathbf{V}_{d}(M)$ as follows

$V(B,\theta,g)(S):=\int_{\\{x\in B:(x,T_{x}B)\in
S\\}}\theta(x)\mathrm{d}\mathcal{H}_{g}^{d}(x)\quad{\rm for\ all}\quad S\in
G_{d}(M).$

We say that $V\in\mathbf{V}_{d}(M)$ is a $d$-integral varifold, if there
exists a $d$-countably rectifiable $\mathcal{H}_{g}^{d}$-measurable subset
$B\subseteq M$ and a Borel map $\theta:B\rightarrow\mathbb{N}\setminus\\{0\\}$
such that $V=V(B,\theta,g)$. The set of all $d$-integral varifolds over $M$
will be denoted by $\mathbf{IV}_{d}(M)$.

Next, we have the definition of the pushforward of a varifold.

######  Definition 3.9.

Let $(M_{1}^{n_{1}},g_{1})$ and $(M_{2}^{n_{2}},g_{2})$ be Riemannian
manifolds and $F:M_{1}\rightarrow M_{2}$ be a smooth map. If
$V\in\mathbf{V}_{d}(M_{1})$, then $F$ induces a natural Borel regular measure
on $G_{d}(M_{2})$ given by

$F_{\\#}(V)(B):=\int_{\\{(x,S):(F(x),\mathrm{d}F_{x}(S))\in
B\\}}\sum_{i=1}^{d}\langle\mathrm{d}F_{x}\circ\pi_{S}(e_{i}),e_{i}\rangle_{g_{2}}\mathrm{d}V(x,S),$

for any Borel subset $B\subset G_{d}(M_{2})$, where $\\{e_{1},...,e_{d}\\}$ is
an orthonormal basis of $S\subset T_{x}M_{1}$. The measure $F_{\\#}(V)$ is a
varifold when $F$ is a proper map. In this case, $F_{\\#}(V)$ is called the
pushforward varifold of $V$ by $F$.

In this fashion, if $V(B,\theta,g_{1})$ is an integral varifold in $M$ and
$F:M\rightarrow N$ is a diffeomorphism, then we have that $(F(B),\theta\circ
F^{-1},g_{2})$ is an integral varifold in $N$ that coincides with
$F_{\\#}(V)$. Given a vector field $X\in\mathfrak{X}_{c}(M)$, the one-
parameter family of diffeomorphisms generated by $X$ is defined by
$\Phi_{t}(x)=\Phi(t,x)$ where $\Phi:\mathbb{R}\times M\rightarrow M$ is the
unique solution of the ODE system below

$\displaystyle\begin{cases}\frac{\partial\Phi}{\partial t}=X(\Phi),\\\
\Phi(0,x)=x.\end{cases}$

With respect to the last definition, we have the following alternative
formulation of the first variation of a varifold.

######  Proposition 3.10.

Let $(M^{n},g)$ be a Riemannian manifold. For any $V\in\mathbf{IV}_{d}(M)$ and
$X\in\mathfrak{X}_{c}(M)$, then the first variation of $V$ along $X$ is given
by the following formula

$\displaystyle\delta_{g}V(X)=\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}\|(\Phi_{t})_{\\#}V\|(M)=\int_{M}{\rm
div}_{T_{x}B}X\mathrm{d}\|V\|,$

where $B=\operatorname{supp}\|V\|$ and $\Phi_{t}$ is the one-parameter family
generated by $X$.

It is possible to define a generalized version of the mean curvature for
integral varifolds.

######  Definition 3.11.

Let $(M^{n},g)$ be a Riemannian manifold. We say that $V\in\mathbf{IV}_{d}(M)$
has bounded generalized mean curvature vector, if there exists a constant
$C\geqslant 0$ such that

$|\delta_{g}V(X)|\leqslant C\int_{M}|X|\mathrm{d}\|V\|\quad{\rm for\ all}\quad
X\in\mathfrak{X}_{c}(M).$ (3.1)

######  Remark 3.12.

Observe that with (3.1), by the Riesz representation theorem and Radon–Nikodym
decomposition theorem, it is straightforward to prove the existence of a
measurable vector field $H_{g}\in L^{\infty}_{g}(M,TM)$ such that

$\delta_{g}V(X)=\int_{M}\langle X,H_{g}\rangle\mathrm{d}\|V\|\quad{\rm for\
all}\quad X\in\mathfrak{X}_{c}(M),$

In this case, $H_{g}$ is the generalized mean curvature and
$\|H_{g}\|_{\infty}$ denotes the $L_{g}^{\infty}(M,TM)$-norm of the
generalized mean curvature.

In the following, we present the notion of concentrated measure and relative
perimeter.

######  Definition 3.13.

Let $(M^{n},g)$ be a Riemannian manifold. For any $A\subset M$, let us define
the Borel measure $(\mathcal{H}_{g}^{n-1}\llcorner
A)(B):=\mathcal{H}_{g}^{n-1}(A\cap B)$ for every Borel set $B\subset M$. We
also define the relative perimeter of an $N$-cluster $\mathbf{\Omega}$ as
$\boldsymbol{\mathcal{P}}^{\alpha}_{g}(\mathbf{\Omega},A)=\sum_{i,j=1}^{N}\alpha_{ij}\mathcal{H}^{n-1}_{g}(\Sigma_{ij}\cap
A)$. As usual we will adopt the following notational convention
$\boldsymbol{\mathcal{P}}^{\alpha}_{g}(\mathbf{\Omega}):=\boldsymbol{\mathcal{P}}^{\alpha}_{g}(\mathbf{\Omega},M)$.

We can prove a version of the monotonicity formula for the reduced boundary of
the interior chambers of a cluster. For this, we will use the notations in
Definition 7.

######  Lemma 3.14.

Let $(M^{n},g)$ be a Riemannian manifold and
$\mathbf{\Omega}\in\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{N})$ be a weighted
cluster. Assume that there exists $b\in\mathbb{R}$ and $0<r_{0}=r_{0}(b)<{\rm
inj}_{g}(M)$ satisfying $r_{0}\cot_{b}(r_{0})>0$, where $b>0$ is given in
Remark 3.2. Then, for every $x\in\Sigma_{ij},$ for some
$i,j\in\\{1,\cdots,N\\}$, there exists $0<C_{1}=C_{1}(b,r_{0},x)\leqslant 1$
such that the function
$\boldsymbol{\Upsilon}_{g}:[0,r_{0}]\rightarrow\mathbb{R}$ defined as

$\boldsymbol{\Upsilon}_{g}(r)=\boldsymbol{\mathcal{P}}^{\alpha}_{g}(\mathbf{\Omega},\mathcal{B}^{g}_{r}(x))r^{-(n-1)}e^{\frac{\|H_{g}\|_{\infty}}{C_{1}}r}$

is monotone nondecreasing. Moreover,
$\Theta^{n-1}(\mathcal{H}_{g}^{n-1}\llcorner\partial^{*}\widetilde{{\Omega}},x)$
exists and the following inequality holds

$\boldsymbol{\mathcal{P}}^{\alpha}_{g}(\mathbf{\Omega},\mathcal{B}^{g}_{r}(x))\geqslant\omega_{n-1}\sum_{i,j=1}^{N}\alpha_{ij}\Theta^{n-1}(\mathcal{H}_{g}^{n-1}\llcorner\Sigma_{ij},x)r^{n-1}e^{-\frac{\|H_{g}\|_{\infty}}{C_{1}}r}.$

In particular, when ${\rm inj}_{g}(M)>0$, the constant $C_{1}\in(0,1]$ could
be chosen to be independent of $x$ and depending just on $b$ and ${\rm
inj}_{g}$.

###### Proof.

It follows the same ideas in [53]. ∎

We can formulate the last lemma in terms of an inequality involving integral
varifolds. Let us fix the standard notation
$\cot_{b}(t)={\cos(\sqrt{b}t)}{\sin(\sqrt{b}t)^{-1}}$.

######  Lemma A ([53]).

Let $(M^{n},g)$ be a Riemannian manifold and $V\in\mathbf{IV}_{d}(M)$ be an
integral varifold with bounded generalized mean curvature vector $H_{g}$. Fix
$y\in M$ and $0<r_{0}<{\rm inj}_{g}$ satisfying $r_{0}\cot_{b}(r_{0})>0$.
There exists a constant $0<C_{2}=C_{2}(b)\leqslant 1$ such that for
$0<r_{1}<r_{2}<r_{0}$, it follows

$\displaystyle\frac{\|V\|(\mathcal{B}_{r_{2}}(y))}{r_{2}^{d}}-\frac{\|V\|(\mathcal{B}_{r_{1}}(y))}{r_{1}^{d}}$
$\displaystyle\geqslant\frac{1}{C_{2}}\int_{\mathcal{B}_{r_{2}}(y)}\frac{\langle
H_{g},u\nabla_{g}u\rangle}{d}\left(\frac{1}{m(r)^{d}}-\frac{1}{r_{2}^{d}}\right)\mathrm{d}\|V\|$
$\displaystyle+\frac{1}{C_{2}}\int_{\mathcal{B}_{r_{2}}(y)\setminus\mathcal{B}_{r_{1}}(y)}\frac{|\nabla_{g}^{\bot}r|^{2}}{r^{d}}\mathrm{d}\|V\|.$

Here $u(x)=\mathrm{d}_{g}(x,y)$,
$\nabla_{g}^{\bot}r=\pi_{T_{x}B^{\bot}}(\nabla_{g}u(x))$, and
$m(r):=\max\\{u(x),r_{1}\\}$, where $\pi_{T_{x}B^{\bot}}$ is the standard
orthogonal projection on $T_{x}B^{\bot}$.

######  Remark 3.15.

Notice that the optimal $r_{0}$ in the preceding theorem is given by the first
positive zero of the function $t\mapsto t\cot_{b}(t)$, if $b>0$, and
$r_{0}=\infty$, if $b\leqslant 0$.

We present the definition of a varifold tangent, also called a tangent cone.

######  Definition 3.16.

Let $(M^{n},g)$ be a Riemannian manifold. Let $V\in\mathbf{IV}_{d}(M)$ and
$\Sigma={\rm supp}(\|V\|)$. Fix a point $x\in\Sigma$, and for every radius
$r>0$ consider the translated and rescaled pushforward varifold
$\Sigma_{x,r}:=\frac{\Sigma-x}{r}=\\{y:x+ry\in\Sigma\\}$. Using the uniform
boundedness on the mass and the compactness theorem for integral varifolds
[1], we can assume that $\Sigma_{x,r}$ converges as $r\rightarrow 0$ to an
integral varifold, called the varifold tangent, which for convenience will be
denoted by ${\rm VarTan}_{x}(V)=\Sigma_{x}\in\mathbf{IV}_{d}(M)$. This is also
called a tangent cone.

######  Remark 3.17.

At a regular point $x\in\Sigma$, that is, a point in a neighborhood of which
$\Sigma$ is smooth, the tangent cone ${\rm VarTan}_{x}(V)$ is of course unique
and it is given by the tangent space $T_{x}\Sigma$ in the classical sense of
differential geometry $($counted with the appropriate multiplicity, depending
upon the chosen variational framework$)$.

Next, we present the right notion of convergence of $N$-clusters which is
usually called flat convergence in the literature. The flat convergence is
essentially the $\mathrm{L}^{1}_{loc}$ convergence of the characteristics
functions of a sequence of Caccioppoli sets.

######  Definition 3.18.

Let $(M^{n},g)$ be a Riemannian manifold,
$\mathbf{\Omega}_{1},\mathbf{\Omega}_{2}\in\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{N})$,
and $A\subset M$ be an open subset. We define the relative flat distance
between $\mathbf{\Omega}_{1}$ and $\mathbf{\Omega}_{2}$ in $A$ given by
$\mathrm{d}_{\mathcal{F},g,A}\left(\mathbf{\Omega}_{1},\mathbf{\Omega}_{2};A\right):=\mathrm{\sum}_{i=1}^{N-1}\mathrm{v}_{g}(A\cap({\Omega}_{1i}\triangle{\Omega}_{2i}))$,
where $\triangle$ stands for the symmetric difference between two sets. When
$A=M$, we denote $\mathrm{d}_{\mathcal{F},g,A}=\mathrm{d}_{\mathcal{F},g}$. We
say that a sequence
$\\{\mathbf{\Omega}_{k}\\}_{k\in\mathbb{N}}\subset\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{N})$
locally converges to $\mathbf{\Omega}$, and denote
$\mathbf{\Omega}_{k}\stackrel{{\scriptstyle{\rm
loc}}}{{\rightarrow}}\mathbf{\Omega}$, if for every compact set $K\subset M$,
we have
$\mathrm{d}_{\mathcal{F},g,K}(\mathbf{\Omega}_{k},\mathbf{\Omega})\rightarrow
0$ as $k\rightarrow\infty$. If
$\mathrm{d}_{\mathcal{F},g}(\mathbf{\Omega}_{k},\mathbf{\Omega})\rightarrow 0$
as $k\rightarrow\infty$, we say that $\mathbf{\Omega}_{k}$ converges to
$\mathbf{\Omega}$ and we denote
$\mathbf{\Omega}_{k}\rightarrow\mathbf{\Omega}$.

The main difference between the scalar and vectorial cases is captured below.

######  Remark 3.19.

Recall that when $N=2$, the formula for the isoperimetric profile is given by
$\mathcal{I}_{(\mathbb{R}^{n},\delta)}(\mathrm{v})=c_{n}\mathrm{v}^{(n-1)/n}$,
where $c_{n}=n^{1/n}\pi^{1/2}\Gamma(n/2+1)^{1/n}$, with
$\Gamma(z)=\int_{0}^{\infty}e^{-t}t^{z-1}\mathrm{d}t$ the standard Gamma
function, is the best isoperimetric constant in the Euclidean space. In
contrast with this case, when $N>2$, an explicit formula for the multi-
isoperimetric profile function is not known; this sets a substantial
difficulty in our approach.

Nevertheless, it is not hard to check the validity of the following result,
which will be enough for our purposes here. Henceforth, we will use the
following notations $\mathrm{max}(\alpha)=\max_{1\leqslant i,j\leqslant
N}\alpha_{ij}$ and $\mathrm{min}(\alpha)=\min_{1\leqslant i,j\leqslant
N}\alpha_{ij}$.

######  Lemma 3.20.

Let $(M^{n},g)$ be a Riemannian manifold of bounded geometry. There exist
$\bar{\mathrm{v}}=\bar{\mathrm{v}}(n,\kappa,\mathrm{v}_{0})>0,C_{0}=C_{0}(n,N,\kappa,\mathrm{v}_{0},\alpha)>0$
and $\widetilde{C}_{0}=\widetilde{C}_{0}(n,N,\kappa,\mathrm{v}_{0},\alpha)>0$
such that

$C_{0}\sum_{i=1}^{N}|\mathrm{v}_{i}|^{\frac{n-1}{n}}\leqslant\boldsymbol{\mathcal{I}}^{\alpha}_{(M,g)}(\mathbf{v})\leqslant\widetilde{C}_{0}\sum_{i=1}^{N}|\mathrm{v}_{i}|^{\frac{n-1}{n}}\quad{\rm
for\ all}\quad\mathbf{v}\in(0,\bar{\mathrm{v}})^{N}.$

###### Proof.

Using ($\alpha_{0}$), the result is obtained as a consequence of the following
direct computation

$\displaystyle
c_{n}|\mathbf{v}|^{\frac{n-1}{n}}\sim{\mathcal{I}}_{(M,g)}\left(|\mathbf{v}|\right)$
$\displaystyle{\leqslant}\boldsymbol{\mathcal{I}}^{\alpha}_{(M,g)}(\mathbf{v})$
$\displaystyle\leqslant\mathrm{max}(\alpha)\inf\left\\{\sum_{i,j=1}^{N}\mathcal{H}_{g}^{n-1}(\Sigma_{ij}):{\mathbf{\Omega}}\in\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{N})\
\mbox{and}\ \mathbf{v}_{g}(\mathbf{\Omega})=\mathbf{v}\right\\}$
$\displaystyle\leqslant\mathrm{max}(\alpha)\inf\left\\{\sum_{i=1}^{N}\mathcal{P}_{g}(\Omega_{i}):{\mathbf{\Omega}}\in\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{N})\
\mbox{and}\ \mathbf{v}_{g}(\mathbf{\Omega})=\mathbf{v}\right\\}$
$\displaystyle=\mathrm{max}(\alpha)\sum_{i=1}^{N}\mathcal{I}_{(M,g)}(\mathrm{v}_{i})\sim\mathrm{max}(\alpha)c_{n}\sum_{i=1}^{N}|\mathrm{v}_{i}|^{\frac{n-1}{n}}.$

The first inequality is derived by the same computations in [4, Chapter VI
(7)]. ∎

Next, we prove some preliminary results. Firstly, let us present a comparison
principle.

######  Lemma 3.21.

Let $(M^{n},g)$ be a Riemannian manifold of bounded geometry. There exist
constants $C_{3}=C_{3}(n,\kappa),C_{4}=C_{4}(n,\kappa)>0$, and
$r_{1}=\overline{r}_{1}(n,\kappa)>0$ such that, for every
$0<r<\overline{r}_{1}$, it follows

$\mathrm{v}_{0}C_{3}r^{n}{\leqslant}\mathrm{v}_{g}(\mathcal{B}_{r}(x)){\leqslant}C_{4}r^{n},$

where $v_{0},\kappa>0$ are defined in Definition 3.1. More explicitly, we have

$\overline{r}_{1}(n,\kappa):=\min\left\\{1,e^{\frac{\sqrt{(n-1)|\kappa|}}{n}}\right\\}\quad{\rm
and}\quad C_{3}(n,\kappa)={e^{-\sqrt{(n-1)|\kappa|}}}.$

###### Proof.

The proof is a consequence of the Bishop–Gromov volume comparison. ∎

We also have the following result about the boundedness of isoperimetric
weighted clusters. In particular, we will have that
$\mathrm{diam}(\widetilde{\Omega})<\infty,$ which from now on we will use
without mentioning.

######  Lemma 3.22.

Let $(M^{n},g)$ be a Riemannian manifold of bounded geometry and
$\mathbf{\Omega}\in\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{N})$ be an
isoperimetric weighted cluster. Then $\widetilde{\Omega}$ is bounded.

###### Proof.

The proof is a line-by-line adaptation of the one in [60, Theorem 3]. ∎

We now provide an estimate of the generalized mean curvature of the reduced
boundary of the chambers of an isoperimetric immiscible weighted cluster,
which is a direct consequence of the Heintze–Karcher inequality. We notice
that the fact that the fluids are immiscible, i.e., the coefficients matrix
$\alpha$ satisfies the strict triangle inequality ($\alpha_{0}$), plays a
major role in this regularity result.

######  Lemma 3.23.

Let $(M^{n},g)$ be a Riemannian manifold of bounded geometry and
${\mathbf{\Omega}}\in\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{N})$ be an
isoperimetric weighted cluster.

  * (i)

For $\mathcal{H}^{n-1}_{g}$-almost every point
$x\in\cup_{i,j=1}^{N}\Sigma_{ij}$, there exists a neighborhood $U$ of $x$ in
$M$ and indexes $i_{x},j_{x}\in\\{1,\cdots,N\\}$ such that
$\Sigma_{i_{x}j_{x}}\cap U$ is smooth.

  * (ii)

The generalized mean curvature for regular points of the reduced boundary is
piecewise constant. Furthermore, there exists $C_{5}=C_{5}(n,\kappa)>0$ such
that

$\|H_{g}\|_{\infty}\leqslant
C_{5}{\boldsymbol{\mathcal{P}}^{\alpha}_{g}(\mathbf{\Omega})}{\mathrm{v}_{g}(\widetilde{\mathbf{{\Omega}}})^{-1}}.$
(3.2)

###### Proof.

We start recalling the regularity results [40, Collorary 4.8], which can be
stated in the following

  1. (1)

For each $i\in\\{1,\cdots,N\\}$, the set $\Omega_{i}^{(1)}$ is open in $M$,

  2. (2)

For $\mathcal{H}_{g}^{n-1}$-a.e. point $p$, there exists
$r_{p}=r_{p}(p,\mathbf{\Omega})>0$ such that
$\mathcal{B}^{g}_{r_{p}}(p)\cap\Omega_{i}\neq\emptyset$ holds for only two
different index $i_{p},j_{p}\in\\{1,\cdots,N\\}$,

  3. (3)

For all $i,j\in\\{1,\cdots,N\\}$, the set $\Sigma_{ij}$ is a smooth surface
with constant mean curvature up to a $\mathcal{H}_{g}^{n-1}$-null set.

So (i) follows directly from (3). To prove the desired estimate on the
generalized mean curvature in (ii), we apply (3) and the classical
Heintze–Karcher inequality [MR2229062, Theorem IX.3.2] to obtain the estimate
(3.2) and that the mean curvature is piecewise constant. ∎

It is well known that in Euclidean spaces the isoperimetric sets ($N=2$) are
round balls. Although this property can be extended to some particular types
of manifolds, this is not true in a general context. In fact, even simpler
properties such as the connectedness of isoperimetric regions do not hold in
any manifold. Indeed, in [34, Theorem 1.1], the author showed that there are
Cartan–Hadamard manifolds with $n=2,3$ that contain isoperimetric regions that
are not connected. Nonetheless, this property is true in the Euclidean space
for immiscible clusters. This fact is fundamental to proving the desired
diameter estimate.

######  Lemma 3.24.

If
$\mathbf{\Omega}\in\mathcal{C}_{\delta}^{\alpha}(\mathbb{R}^{n},\mathbb{R}^{N})$
is an isoperimetric immiscible weighted cluster, then $\tilde{\Omega}\subset
M$ is connected.

###### Proof.

The proof is a straightforward adaptation of [47, Lemma 9.3] which relies in
the regularity result [42, Corollary 30.3], we instead rely in Lemma 3.23. ∎

We now state a generalized abstract compactness result for a sequence of
weighted clusters. These results can be found in several forms in the
literature, we have chosen to state it in the more general fashion that we
know, that is, in the context of $\mathrm{RCD}$ spaces. One interested reader
can consult [7] and the references therein to see the definition and
properties of these spaces. For what concern our work, we will always apply
the theorem for manifolds of bounded geometry, which are well-known examples
of $\mathrm{RCD}$ spaces. In fact, the heuristics behind the $\mathrm{RCD}$
spaces is to define a metric space of bounded geometry, which means that given
the right notion of curvature and measurability in the metric space we require
that it satisfies the properties of Definition 3.1.

Before the statement of the theorem, we need to define a strong notion of
convergence of weighted clusters.

######  Definition 3.25.

Let $\\{(M^{n}_{k},g_{k})\\}_{k\in\mathbb{N}}$ be a sequence of closed
manifolds converging in the pmGH sense to a closed manifold
$(M^{n}_{\infty},g_{\infty})$. Assume that there exists a locally compact
separable metric space $(Z,\mathrm{d}_{Z})$ on which $M_{\infty}$ and $M_{k}$,
for all $k\in\mathbb{N}$, are isometrically embedded. We say that a sequence
of clusters
$\\{\mathbf{\Omega}_{k}\\}_{k\in\mathbb{N}}\subset\mathcal{C}_{g_{k}}^{\alpha}(M_{k},\mathbb{R}^{N})$
converges in the $L^{1}$-strong sense to a cluster
$\mathbf{\Omega}\in\mathcal{C}_{g_{\infty}}^{\alpha}(M_{\infty},\mathbb{R}^{N})$
when

  * (i)

$\mathbf{{v}}_{g}(\mathbf{\Omega}_{k})\rightarrow\mathbf{v}_{g}(\mathbf{\Omega})$
as $k\rightarrow\infty$;

  * (ii)

$\chi_{\Omega_{ki}}\mathcal{H}_{g}^{n}\rightharpoonup\chi_{\Omega_{i}}\mathcal{H}_{d_{Z}}^{n}$
as $k\rightarrow\infty$ for all $i\in\\{1,\cdots N\\}$ with respect to the
duality with continuous bounded functions with bounded support on $Z$.

######  Proposition 3.26.

Let $\\{(M^{n}_{k},g_{k})\\}_{k\in\mathbb{N}}$ be a sequence of Riemannian
manifold of bounded geometry. Assume that
$\\{\mathbf{\Omega}_{k}\\}_{k\in\mathbb{N}}\subset\mathcal{C}_{g_{k}}^{\alpha}(M_{k},\mathbb{R}^{N})$
is a sequence of bounded weighted clusters such that
$\sup_{k\in\mathbb{N}}(\boldsymbol{\mathcal{P}}^{\alpha}_{g_{k}}(\mathbf{\Omega}_{k})+\mathrm{v}_{g_{k}}(\widetilde{\Omega}_{k}))<\infty$.
Then, up to subsequence, there exists a nondecreasing, possibly unbounded,
sequence $\\{Q_{k}\\}_{k\in\mathbb{N}}\subset\mathbb{N}$ and
$\\{x^{j}_{k}\\}_{k\in\mathbb{N}}\subset M_{k}$ with $1\leqslant j\leqslant
Q_{k}$ for any $k\in\mathbb{N}$, and pairwise disjoint subclusters
$\mathbf{\Omega}_{k}^{j}\subset\mathbf{\Omega}_{k}$ satisfying:

  1. (i)

$\lim_{k\rightarrow\infty}\mathrm{d}_{g_{k}}(x^{j}_{k},x^{\ell}_{k})=\infty$
for any $j\neq\ell<\bar{Q}+1$, where
$\bar{Q}:=\lim_{k\rightarrow\infty}Q_{k}\in\mathbb{N}\cup\\{\infty\\}$;

  2. (ii)

For every $1\leqslant j<\bar{Q}+1$,
$(M_{k},\mathrm{d}_{g_{k}},\mathcal{L}^{n}_{g_{k}},x^{j}_{k})$ converges to
$(M_{\infty}^{j},\mathrm{d}^{j}_{{\infty}},\mathcal{L}^{n}_{g^{j}_{\infty}},x^{j}_{\infty})$
as $k\rightarrow\infty$ in the pmGH sense, where the limit is a pointed
$\operatorname{RCD}(\kappa,n)$ space.

  3. (iii)

There exist subclusters
$\mathbf{\Omega}^{j}_{\infty}\in\mathcal{C}^{\alpha}_{g^{j}_{\infty}}(M^{j}_{\infty},\mathbb{R}^{N})$
such that $\mathbf{\Omega}_{k}^{j}$ converges to
$\mathbf{\Omega}_{\infty}^{j}$ as $k\rightarrow\infty$ in the $L^{1}$-strong
sense, and

$\displaystyle\lim_{k\rightarrow\infty}\mathcal{L}_{g^{j}_{\infty}}^{n}(\Omega_{ik})$
$\displaystyle=\sum_{j=1}^{\tilde{Q}}\mathcal{L}_{g^{j}_{\infty}}^{n}(\Omega_{\infty
i}^{j})\quad{\rm
and}\quad\sum_{j=1}^{\tilde{Q}}\boldsymbol{\mathcal{P}}^{\alpha}_{g^{j}_{\infty}}\left(\mathbf{\Omega}_{\infty}^{j}\right)$
$\displaystyle\leqslant\liminf_{k\rightarrow\infty}\boldsymbol{\mathcal{P}}^{\alpha}_{g_{k}}\left(\mathbf{\Omega}_{k}\right).$

Moreover, if
$\mathbf{\Omega}_{k}\in\mathcal{C}_{g_{k}}^{\alpha}(M_{k},\mathbb{R}^{N})$ is
an isoperimetric cluster in for any $k$, then
$\mathbf{\Omega}_{\infty}^{j}\in\mathcal{C}_{g^{j}_{\infty}}^{\alpha}(M_{\infty}^{j},\mathbb{R}^{N})$
is an isoperimetric clusters for any $j<\tilde{Q}+1$ and
$\boldsymbol{\mathcal{P}}^{\alpha}_{g^{j}_{\infty}}(\mathbf{\Omega}_{\infty}^{j})=\lim_{k\rightarrow\infty}\boldsymbol{\mathcal{P}}^{\alpha}_{g_{k}}(\mathbf{\Omega}_{k}^{j})$
for any $j<\tilde{Q}+1$.

###### Proof.

The proof is carried out along the same lines as the proof of [7, Theorem
1.2], replacing mutatis mutandis, finite perimeter sets by weighted clusters,
and the standard perimeter by the multi-perimeter for clusters, which
satisfies the lower semicontinuity property with respect to the usual
$L^{1}_{loc}$ convergence of clusters chamber by chambers (see Remark 8). ∎

In the light of the last result, our strategy consists in first studying the
limit problem in $(\mathbb{R}^{n},\delta)$. This is the $L^{1}$-strong limit
of the sequence of blow-up metrics or the infinitesimal problem at small
scales. The result below states that any isoperimetric sequence of clusters
can be replaced, up to volume and perimeter, by a new sequence whose diameter
of its elements becomes small enough. Due to the fact that we can choose one
subcluster (which is the original cluster intersected with a smartly chosen
ball) that contributes to almost all volume and perimeter of the original
cluster, this result is usually called _selecting a large subdomain_. The
proof of the following lemma relies crucially on three properties of
isoperimetric clusters, namely, existence, uniqueness up to isometries, and
connectedness.

######  Lemma 3.27.

Let
$\\{\mathbf{\Omega}_{k}\\}_{k\in\mathbb{N}}\subset\mathcal{C}^{\alpha}_{\delta}(\mathbb{R}^{n},\mathbb{R}^{N})$
be a sequence of isoperimetric weighted clusters. There exists another
sequence
$\\{\mathbf{\Omega}_{k}^{\prime}\\}_{k\in\mathbb{N}}\subset\mathcal{C}^{\alpha}_{\delta}(\mathbb{R}^{n},\mathbb{R}^{N})$
such that

  * (i)

$\lim_{k\rightarrow\infty}{\mathrm{v}_{\delta}({\mathbf{\Omega}}_{k}\triangle{\mathbf{\Omega}}_{k}^{\prime})}{\mathrm{v}_{\delta}({\mathbf{\Omega}}_{k})}^{-1}=0$;

  * (ii)

$\lim_{k\rightarrow\infty}{\mathrm{v}_{\delta}(\mathbf{\Omega}_{k}^{\prime})}{\mathrm{v}_{\delta}(\mathbf{\Omega}_{k})}^{-1}=1$;

  * (iii)

$\lim_{k\rightarrow\infty}{\boldsymbol{\mathcal{P}}^{\alpha}_{\delta}(\mathbf{\Omega}_{k}^{\prime})}{\boldsymbol{\mathcal{P}}^{\alpha}_{\delta}(\mathbf{\Omega}_{k})}^{-1}=1$;

  * (iv)

$\lim_{k\rightarrow\infty}\operatorname{diam}_{\delta}(\widetilde{{\Omega}}_{k}^{\prime})=0$.

###### Proof.

Let us fix $\mathrm{v}_{k}:=\mathrm{v}_{\delta}(\widetilde{\Omega}_{k})$ and
define the following sequence of manifolds of bounded geometry

$(M_{k},\mathrm{d}_{\mathrm{g}_{k}},\mathcal{L}_{\mathrm{g}_{k}}^{n}):=(\mathbb{R}^{n},\mathrm{v}_{k}^{-{1}/{n}}\mathrm{d},\mathcal{L}^{n})\quad{\rm
for\ all}\quad k\in\mathbb{N}.$

Hence, applying Proposition 3.26 to the last sequence, we find

$(M_{k},\mathrm{d}_{\mathrm{g}_{k}},\mathcal{L}_{\mathrm{g}_{k}}^{n})\rightarrow(M^{j}_{\infty},\mathrm{d}^{j}_{\infty},\mathcal{L}^{n}_{g^{j}_{\infty}})=(\mathbb{R}^{n},\mathrm{d},\mathcal{L}^{n})\quad{\rm
for\ all}\quad j\in[1,\tilde{Q}+1)\cap\mathbb{N}.$

Also, there exists
$\mathbf{\Omega}^{j}_{\infty}\in\mathcal{C}^{\alpha}_{g^{j}_{\infty}}(M^{j}_{\infty},\mathbb{R}^{N})$
such that $\mathbf{\Omega}_{k}^{j}$ converges to
$\mathbf{\Omega}_{\infty}^{j}$ as $k\rightarrow\infty$ in the $L^{1}$-strong
sense, which allows us to define

$\mathbf{\Omega}_{\infty}\in\mathcal{C}^{\alpha}_{\delta}(\mathbb{R}^{n},\mathbb{R}^{N})\quad\mbox{given
by}\quad\mathbf{\Omega}_{\infty}:=\left(\cup_{j=1}^{\tilde{Q}}\Omega_{\infty
1}^{j},\dots,\cup_{j=1}^{\tilde{Q}}\Omega_{\infty N}^{j}\right).$

Next, setting $g_{k}:=\mathrm{v}_{k}^{-2/n}\delta$ and rescaling the multi-
perimeter accordingly, we have the following

$\boldsymbol{\mathcal{P}}^{\alpha}_{\delta}\left(\mathbf{\Omega}_{\infty}\right)\leqslant\liminf_{k\rightarrow\infty}\boldsymbol{\mathcal{P}}^{\alpha}_{g_{k}}(\mathbf{\Omega}_{k})=\liminf_{k\rightarrow\infty}\frac{\boldsymbol{\mathcal{P}}^{\alpha}_{\delta}\left(\mathbf{\Omega}_{k}\right)}{\mathrm{v}_{k}^{\frac{n-1}{n}}}=\liminf_{k\rightarrow\infty}\frac{\boldsymbol{\mathcal{I}}^{\alpha}_{(\mathbb{R}^{n},\delta)}\left(\mathrm{v}_{g}\left(\mathbf{\Omega}_{k}\right)\right)}{\mathrm{v}_{k}^{\frac{n-1}{n}}},$

where we have used that
$\mathbf{\Omega}_{k}\in\mathcal{C}^{\alpha}_{\delta}(\mathbb{R}^{n},\mathbb{R}^{N})$
is isoperimetric for all $k\in\mathbb{N}$.

We now recall that the Euclidean multi-isoperimetric profile is homogeneous
and continuous. The continuity is stated in [60, Theorem 5] and homogeneity is
a direct consequence of the uniqueness, up to isometries. Using these
properties, we get

$\displaystyle\boldsymbol{\mathcal{P}}^{\alpha}_{\delta}\left(\mathbf{\Omega}_{\infty}\right)\leqslant\liminf_{k\rightarrow\infty}\boldsymbol{\mathcal{I}}^{\alpha}_{(\mathbb{R}^{n},\delta)}\left(\frac{\mathrm{v}_{g}\left(\mathbf{\Omega}_{k}\right)}{\mathrm{v}_{k}}\right)=\boldsymbol{\mathcal{I}}^{\alpha}_{(\mathbb{R}^{n},\delta)}(\lambda_{1},\dots,\lambda_{N-1}),$
(3.3)

where
$\lambda_{i}:=\liminf_{k\rightarrow\infty}{\mathrm{v}_{\delta}\left(\Omega_{ki}\right)}{\mathrm{v}_{k}^{-1}}\geqslant
0.$ Furthermore, it is straightforward to check the following properties

$\sum_{i=1}^{N-1}\lambda_{i}=1,\quad\lambda_{i}=\mathrm{v}_{\delta}(\Omega_{\infty
i})=\sum_{j=1}^{\tilde{Q}}\mathrm{v}_{\delta}(\Omega_{\infty i}^{j}),\quad{\rm
and}\quad\lim_{k\rightarrow\infty}\mathrm{v}_{\delta}(\Omega_{ki}^{j})=\mathrm{v}_{\delta}(\Omega_{\infty
i}^{j}).$

Now, assume by contradiction that $\tilde{Q}>1$ then
$\mathbf{\Omega}_{\infty}\in\mathcal{C}^{\alpha}_{\delta}(\mathbb{R}^{n},\mathbb{R}^{N})$
is an isoperimetric immiscible weighted cluster such that
$\widetilde{\Omega}_{\infty}$ is disconnected, which contradicts Lemma 3.24,
if $\lambda_{i}>0$ for all $i\in\\{1,\cdots,N-1\\}$. When there exists
$\Lambda\subset\\{1,\cdots,N-1\\}$ such that $\lambda_{i}=0$ for $i\in\Lambda$
and $\\#\Lambda\geq 1$, we obtain the same contradiction with 3.24 for a
immiscible weighted cluster having $N-\\#\Lambda$ chambers. Hence, we obtain
that $\tilde{Q}=1$ and there exists a small volume $\mathrm{v}_{*}>0$ such
that for every isoperimetric cluster
$\mathbf{\Omega}\in\mathcal{C}^{\alpha}_{\delta}(\mathbb{R}^{n},\mathbb{R}^{N})$
satisfying $\mathrm{v}_{\delta}(\mathbf{\Omega})\in(0,\mathrm{v}_{*})^{N-1}$,
there exists $x_{\mathbf{\Omega}}\in\mathbb{R}^{n}$, $\mu>0$ and
$r(\mathrm{v})=\mu\mathrm{v}^{1/n}$ such that

$\lim_{\mathrm{v}\rightarrow
0^{+}}\frac{\sum_{i=1}^{N-1}\mathrm{v}_{g}(\Omega_{i}\cap\mathcal{B}^{\delta}_{r(\mathrm{v})}(x_{\mathbf{\Omega}}))}{\mathrm{v}_{g}(\widetilde{\Omega})}=1,$

which proves (ii). To conclude the proof of (i), (iii), and (iv) we can do the
same argument of [53, Theorem 4.2]. ∎

In what follows, we prove the main result in the section. The ideas of the
proof are similar in spirit to the ones in Lemma 3.27. We emphasize that this
is the only part where we need the assumption $N=3$.

###### Proof of Proposition 12.

Let us fix $\mathrm{v}_{k}:=\mathrm{v}_{g}(\widetilde{\Omega}_{k})$ and define
the following sequence of manifolds of bounded geometry

$(M_{k},\mathrm{d}_{\mathrm{g}_{k}},\mathcal{L}_{\mathrm{g}_{k}}^{n}):=(\mathbb{R}^{n},\mathrm{v}_{k}^{-{1}/{n}}\mathrm{d},\mathcal{L}^{n})\quad{\rm
for\ all}\quad k\in\mathbb{N}.$

Hence, applying Proposition 3.26 to the last sequence, we find

$(M_{k},\mathrm{d}_{\mathrm{g}_{k}},\mathcal{L}_{\mathrm{g}_{k}}^{n})\rightarrow(M^{j}_{\infty},\mathrm{d}^{j}_{\infty},\mathcal{L}^{n}_{g^{j}_{\infty}})=(\mathbb{R}^{n},\mathrm{d},\mathcal{L}^{n})\quad{\rm
for\ all}\quad j\in[1,\tilde{Q}+1)\cap\mathbb{N}.$

Also, there exists
$\mathbf{\Omega}^{j}_{\infty}\in\mathcal{C}^{\alpha}_{g^{j}_{\infty}}(M^{j}_{\infty},\mathbb{R}^{3})$
such that $\mathbf{\Omega}_{k}^{j}$ converges to
$\mathbf{\Omega}_{\infty}^{j}$ as $k\rightarrow\infty$ in the $L^{1}$-strong
sense, which allows us to define

$\mathbf{\Omega}_{\infty}\in\mathcal{C}^{\alpha}_{\delta}(\mathbb{R}^{n},\mathbb{R}^{3})\quad\mbox{given
by}\quad\mathbf{\Omega}_{\infty}:=\left(\cup_{j=1}^{\tilde{Q}}\Omega_{\infty
1}^{j},\cup_{j=1}^{\tilde{Q}}\Omega_{\infty
2}^{j},\cup_{j=1}^{\tilde{Q}}\Omega_{\infty 3}^{j}\right).$

Next, setting $g_{k}:=\mathrm{v}_{k}^{-2/n}g$ and rescaling the multi-
perimeter accordingly, we have the following

$\boldsymbol{\mathcal{P}}^{\alpha}_{\delta}\left(\mathbf{\Omega}_{\infty}\right)\leqslant\liminf_{k\rightarrow\infty}\boldsymbol{\mathcal{P}}^{\alpha}_{g_{k}}(\mathbf{\Omega}_{k})=\liminf_{k\rightarrow\infty}\frac{\boldsymbol{\mathcal{P}}^{\alpha}_{g}\left(\mathbf{\Omega}_{k}\right)}{\mathrm{v}_{k}^{\frac{n-1}{n}}}=\liminf_{k\rightarrow\infty}\frac{\boldsymbol{\mathcal{I}}^{\alpha}_{(M,g)}\left(\mathrm{v}_{g}\left(\mathbf{\Omega}_{k}\right)\right)}{\mathrm{v}_{k}^{\frac{n-1}{n}}},$
(3.4)

where we have used that
$\mathbf{\Omega}_{k}\in\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{3})$ is
isoperimetric for all $k\in\mathbb{N}$.

Notice that for small volumes, we can define

$\widehat{\mathbf{\Omega}}_{k}:=\exp_{x_{k}^{*}}(\mathbf{\Omega}^{\prime}_{k})=\left(\exp_{x_{k}^{*}}({\Omega}^{\prime}_{k1}),\exp_{x_{k}^{*}}({\Omega}^{\prime}_{k2}),M\setminus\exp_{x_{k}^{*}}(\widetilde{\Omega}^{\prime}_{k})\right)\in\mathcal{C}^{\alpha}_{g}(M,\mathbb{R}^{3}),$

where $x_{k}^{*}\in M$ is fixed once at all and
$\mathbf{\Omega}^{\prime}_{k}\in\mathcal{C}^{\alpha}_{g}(\mathbb{R}^{n},\mathbb{R}^{3})$
are obtained in Lemma 3.27 when applied to a sequence of Euclidean
isoperimetric clusters
$\mathbf{\Omega}^{*}_{k}\in\mathcal{C}^{\alpha}_{g}(\mathbb{R}^{n},\mathbb{R}^{3})$
satisfying
$\mathbf{\mathrm{v}}_{g}(\mathbf{\Omega}^{*}_{k})\sim\mathbf{\mathrm{v}}_{g}(\mathbf{\Omega}_{k})$
as $k\rightarrow\infty$. Thus, combining (3.4) with by (i)–(iv) of Lemma 3.27
and the asymptotic expansion of the metric $g$ in exponential normal
coordinates yields

$\displaystyle\liminf_{k\rightarrow\infty}\frac{\boldsymbol{\mathcal{I}}^{\alpha}_{(M,g)}(\mathrm{v}_{g}(\mathbf{\Omega}_{k}))}{\mathrm{v}_{k}^{\frac{n-1}{n}}}$
$\displaystyle\leqslant$
$\displaystyle\liminf_{k\rightarrow\infty}\frac{\boldsymbol{\mathcal{P}}^{\alpha}_{g}(\widehat{\mathbf{\Omega}}_{k})}{\mathrm{v}_{k}^{\frac{n-1}{n}}}=\liminf_{k\rightarrow\infty}\frac{\boldsymbol{\mathcal{I}}^{\alpha}_{(\mathbb{R}^{n},\delta)}(\mathrm{v}_{g}(\mathbf{\Omega}_{k}))}{\mathrm{v}_{k}^{\frac{n-1}{n}}}.$
(3.5)

Whence, we straightforwardly get

$\boldsymbol{\mathcal{P}}^{\alpha}_{\delta}\left(\mathbf{\Omega}_{\infty}\right)\leqslant\liminf_{k\rightarrow\infty}\frac{\boldsymbol{\mathcal{I}}^{\alpha}_{(\mathbb{R}^{n},\delta)}\left(\mathrm{v}_{g}\left(\mathbf{\Omega}_{k}\right)\right)}{\mathrm{v}_{k}^{\frac{n-1}{n}}}=\boldsymbol{\mathcal{I}}^{\alpha}_{(\mathbb{R}^{n},\delta)}\left(\lambda_{1},\cdots,\lambda_{N-1}\right),$

where we used the same argument as in (3.3). As before, it is easy to verify
the following properties

$\lambda_{1}+\lambda_{2}=1,\quad\lambda_{i}=\mathrm{v}_{\delta}(\Omega_{\infty
i})=\sum_{j=1}^{\tilde{Q}}\mathrm{v}_{\delta}(\Omega_{\infty i}^{j}),\quad{\rm
and}\quad\lim_{k\rightarrow\infty}\mathrm{v}_{\delta}(\Omega_{ki}^{j})=\mathrm{v}_{\delta}(\Omega_{\infty
i}^{j}).$

Therefore, by the exact same argument as in Lemma 3.27, we get that
$\tilde{Q}=1$. ∎

As a consequence, of Proposition 12, we finish this section with the
asymptotic expansion for the immiscible weighted multi-isoperimetric profile.
Namely, we show that the weighted multi-isoperimetric profile function for a
manifold is asymptotic to the one in the Euclidean case for small volumes.

######  Corollary 3.28.

If $(M^{n},g)$ is a closed Riemannian manifold, then

${\boldsymbol{\mathcal{I}}}^{\alpha}_{(M,g)}(\mathrm{v}_{1},\mathrm{v}_{2},\mathrm{v}_{3})\sim{\boldsymbol{\mathcal{I}}}^{\alpha}_{(\mathbb{R}^{n},\delta)}(\mathrm{v}_{1},\mathrm{v}_{2},\mathrm{v}_{3})\quad{\rm
as}\quad|(\mathrm{v}_{1},\mathrm{v}_{2},\mathrm{v}_{3})|\rightarrow 0.$

##  4\. Convergence results for the vectorial energy

This section is devoted to providing the proof of Proposition 11. In
particular, we show the $\Gamma$-convergence of the vectorial energy to the
perimeter functional and a compactness result for solutions to
($ACH_{\varepsilon,\mathbf{v},m,N,g}$) with bounded energy. For this, our
methods are similar in spirit to the ones in [9]. However, we include some
proofs for the sake of completeness.

######  Remark 4.1.

Henceforth, we are under the small volume condition, that is,
$0<\mathrm{v}_{g}(\widetilde{{\Omega}})\ll 1$, where
$\widetilde{{\Omega}}=\cup_{i=1}^{N}\Omega_{i}$ and whenever it is needed we
assume that $N=3$. Thus, for $N=2$, using [38, Theorem 1.1], we get that there
exist $x\in\widetilde{{\Omega}}$ and $0<r\ll 1$ such that
$\widetilde{{\Omega}}\subset\mathcal{B}^{g}_{r}(x)$ and
$\exp_{x}:\mathcal{B}^{g}_{r}(x)\rightarrow B_{r}(0)\subset\mathbb{R}^{n}$ is
a bi-Lipschitz diffeomorphism, with Lipschitz constant independent of the
point. This allows us to consider the same definitions in the Euclidean space
for the context of manifolds, at least in a ball of a sufficiently small
radius.

Let us fix some notation. We say that a family indexed on the positive real
numbers $\\{u_{\varepsilon}\\}_{\varepsilon>0}$ converges to $u_{0}$, if for
all $\\{\varepsilon_{k}\\}_{k\in\mathbb{N}}$ such that
$\varepsilon_{k}\rightarrow 0$ as $k\rightarrow\infty$, it follows that
$\\{u_{\varepsilon_{k}}\\}_{k\in\mathbb{N}}$ converges to $u_{0}$. We
introduce the notion of $\Gamma$-convergence for operators.

######  Definition 4.2.

Let $\\{\mathfrak{B}_{\varepsilon}\\}_{\varepsilon>0}\subset\mathfrak{B}_{0}$
be Banach spaces and
$\mathcal{E}_{\varepsilon}:\mathfrak{B}_{\varepsilon}\rightarrow\mathbb{{R}}$
be a sequence of operators. We say that
$\\{\mathcal{E}_{\varepsilon}\\}_{\varepsilon>0}$ $\Gamma$-converges to
$\mathcal{E}_{0}:\mathfrak{B}_{0}\rightarrow{\mathbb{R}}$ as
$\varepsilon\rightarrow 0$, if for all $u_{0}\in\mathfrak{B}_{0}$, it follows

  * (i)

For every sequence $\\{u_{\varepsilon}\\}_{\varepsilon>0}$ converging to
$u_{0}$, we have that
$\mathcal{E}_{0}(u_{0})\leqslant\liminf_{\varepsilon\rightarrow
0}\mathcal{E}_{\varepsilon}\left(u_{\varepsilon}\right)$;

  * (ii)

There exists a sequence $\\{u_{\varepsilon}\\}_{\varepsilon>0}$ converging to
$u_{0}$ such that
$\mathcal{E}_{0}(u_{0})\geqslant\limsup_{\varepsilon\rightarrow
0}\mathcal{E}_{\varepsilon}\left(u_{\varepsilon}\right)$.

The operator $\mathcal{E}_{0}$ is called the $\Gamma$-limit of
$\\{\mathcal{E}_{\varepsilon}\\}_{\varepsilon>0}$. Here, we fix the notation
$\Gamma$-$\lim_{\varepsilon\rightarrow
0}\mathcal{E}_{\varepsilon}=\mathcal{E}_{0}$.

Before proving our main proposition, let us start with some preliminary
results. First, we have two lemmas concerning the vectorial transformation
(see (6)).

######  Lemma 4.3.

Let $(M^{n},g)$ be a closed Riemannian manifold and
$\boldsymbol{W}\in\mathcal{W}^{+}_{N,3}$. If
$\mathbf{u}\in{H}_{g}^{1}(M,\mathbb{R}^{m})\cap{L}_{g}^{\infty}(M,\mathbb{R}^{m})$,
then $\Phi{\circ}\mathbf{u}$ is locally Lipschitz continuous and
$\Phi{\circ}\mathbf{u}\in{W}_{g}^{1,1}(M,\mathbb{R}^{m})$. Moreover, the
following inequality holds

$\int_{M}\left|\nabla_{g}\left(\Phi{\circ}\mathbf{u}\right)\right|\mathrm{d}\mathcal{L}^{n}_{g}\leqslant\int_{M}\boldsymbol{W}^{1/2}(\mathbf{u})|\nabla_{g}\mathbf{u}|\mathrm{d}\mathcal{L}^{n}_{g}.$

###### Proof.

It is a straightforward adaptation of [9, Proposition 2.1] to the Riemannian
context. ∎

######  Lemma 4.4.

Let $(M^{n},g)$ be a closed Riemannian manifold and
$\boldsymbol{W}\in\mathcal{W}^{+}_{N,3}$. Assume that
$\mathbf{u}\in{BV}_{g}(M,\mathbb{R}^{m})$ and
$\boldsymbol{W}(\mathbf{u}(x))=0$ almost everywhere. Then, there exists a
weighted cluster $\mathbf{\Omega}\in\mathcal{C}_{g}(M,\mathbb{R}^{N})$ such
that

$\mathbf{u}(x)=\sum_{i=1}^{N}\mathbf{p}_{i}\chi_{\Omega_{i}}(x).$

###### Proof.

Let us define $\Omega_{i}:=(\Phi{\circ}\mathbf{u})^{-1}(\mathbf{p}_{i})$ for
$i=1,\dots,N$ and recall that $\mathbf{p}_{N}=0$. ∎

Next, we present the definition of the supremum of measures.

######  Definition 4.5.

Let $(M^{n},g)$ be a closed manifold and $\nu_{1},\dots,\nu_{N}$ be regular
positive Borel measures on $M$. Let us define its supremum
$\bigvee_{i=1}^{N}\nu_{i}$ as the smallest regular positive measure which is
greater than or equal to $\nu_{1},\dots,\nu_{N-1}$ on all Borel subsets of
$M$. In other words, we have

$\left(\bigvee_{i=1}^{N}\nu_{i}\right)(\mathrm{A})=\sup\left\\{\sum_{i=1}^{N}\nu_{i}\left({A}_{i}\right):\
\begin{aligned} (A_{1},\dots,&A_{N})\ \textrm{is a partition of $A$}\\\
&\mbox{by open sets}\end{aligned}\right\\}.$

We also have the measure-theoretic auxiliary result below.

######  Lemma 4.6.

Let $(M,g)$ be a closed manifold, $\nu$ be a regular positive Borel measure on
$M$, $B_{1},\dots,B_{\ell}$ be disjoint Borel subsets of $M$ of finite
measure, and $c_{i}^{\ell}$, where $i=1,\dots,\ell$ and $\ell=1,\ldots,N$ be
positive coefficients. If
$\nu^{\ell}({A})=\sum_{i=1}^{k}c_{i}^{\ell}\nu(A\cap{B}_{i})$ and
$\nu({A})=\sum_{i=1}^{k}\left(\max_{\ell}c_{i}^{\ell}\right)\nu(A\cap{B}_{i})$,
then $\nu=\bigvee_{i=1}^{N}\nu^{\ell}$.

###### Proof.

The proof is easy and is left to the reader. ∎

In the following result, we compute the supremum of the measures induced by
the functions $\phi_{i}\circ\mathbf{u}$. This explains the appearance of
correcting term $\mathrm{d}_{\boldsymbol{W}_{N}}$ on the $\Gamma$-limit of the
sequence of relaxed energy functionals. For more details on this degenerate
metric, we refer to [62].

######  Lemma 4.7.

Let $(M^{n},g)$ be a closed Riemannian manifold and
$\boldsymbol{W}\in\mathcal{W}_{N,3}^{+}$. Assume that
$\mathbf{\Omega}\in\mathcal{C}_{g}(M,\mathbb{R}^{N})$ and $\nu_{i}$ are the
Borel measures given by
$\nu_{i}:A\mapsto\int_{A}\left|\nabla_{g}\left(\phi_{i}{\circ}\mathbf{u}\right)\right|\mathrm{d}\mathcal{L}_{g}^{n}$
for all $i=1,\dots,N$. Then, it follows that
$\mathcal{H}^{n-1}_{g}(\partial^{*}\Omega_{i})<\infty$ and

$\left(\bigvee_{i=1}^{N}\nu_{i}\right)(M)=\frac{1}{2}\sum_{i,j=1}^{N}\omega_{ij}\mathcal{H}_{g}^{n-1}(\Sigma_{ij}).$

###### Proof.

It is a straightforward adaptation of [9, Proposition 2.2] to the Riemannian
context. ∎

A polygonal domain is every set that is the closure of an open set and whose
topological boundary is contained in the union of a finite number of
hyperplanes of $M$ (in the sense of Remark 4.1). The next lemma will allow us
to consider a cluster such that each component is a polygonal domain.

######  Lemma 4.8.

Let $(M^{n},g)$ be a closed Riemannian manifold and
$\boldsymbol{W}\in\mathcal{W}_{N,3}^{+}$. Assume that
$\mathbf{\Omega}\in\mathcal{C}_{g}(M,\mathbb{R}^{N})$ is such that
$0<\mathrm{v}_{g}(\widetilde{{\Omega}})\ll 1$. Then, there exists a sequence
$\\{\mathbf{\Omega}_{k}\\}_{k\in\mathbb{N}}\subset\mathcal{C}_{g}(M,\mathbb{R}^{N})$
converging to $\mathbf{\Omega}$ such that

  * (i)

Each chamber $\Omega_{ki}$ is a polygonal domain for any $i=1,\cdots,N$ and
$k\in\mathbb{N}$;

  * (ii)

If $\mathbf{u}_{k}(x)=\sum_{i=1}^{N}\mathbf{p}_{i}\chi_{\Omega_{ki}}(x)$ and
$\mathbf{u}(x)=\sum_{i=1}^{N}\mathbf{p}_{i}\chi_{\Omega_{i}}(x)$, then
$\mathbf{u}_{k}\rightarrow\mathbf{u}$ in ${L}_{g}^{1}(M,\mathbb{R}^{m})$ as
$k\rightarrow\infty$;

  * (iii)

$\int_{M}\mathbf{u}_{k}\mathrm{d}\mathcal{L}_{g}^{n}=\int_{M}\mathbf{u}\mathrm{d}\mathcal{L}_{g}^{n}=\mathbf{v}$
for any $k\in\mathbb{N}$; and

  * (iv)

$\lim_{k\rightarrow\infty}\bigvee_{i=1}^{N}\int_{M}|\nabla_{g}(\phi_{i}{\circ}\mathbf{u}_{k})|\mathrm{d}\mathcal{L}_{g}^{n}=\bigvee_{i=1}^{N}\int_{M}|\nabla_{g}(\phi_{i}{\circ}\mathbf{u})|\mathrm{d}\mathcal{L}_{g}^{n}$.

###### Proof.

Using Proposition 12, it is straightforward adaptation of [9, Appendix A] to
Riemannian context. ∎

The last lemma has to do with the tubular neighborhood of smooth boundaries on
a Riemannian manifold.

######  Lemma 4.9.

Let $(M^{n},g)$ be a closed Riemannian manifold and $A\subset M$ be an open
subset such that $\partial A$ is a polygonal domain with
$\mathcal{H}_{g}^{n-1}(\partial{A})=0$. If we define

$\mathrm{d}_{A}(x)=\begin{cases}{\rm d}_{g}(x,\partial A),&\text{if}\ x\notin
A,\\\ -{\rm d}_{g}(x,\partial A),&\text{if}\ x\in A.\end{cases}$

Then, there exists a constant $\eta>0$ such that on the set
$\mathcal{D}^{\eta}=\\{x\in M:|\mathrm{d}_{A}(x)|<\eta\\}$, we have
$\mathrm{d}_{A}$ is a Lipschitz continuous function and
$|\nabla_{g}\mathrm{d}_{A}(x)|=1$ for almost all $x\in\mathcal{D}^{\tau}$.
Finally, if $\Sigma^{t}=\\{x\in M:\mathrm{d}_{A}(x)=t\\}$, it follows
$\lim_{t\rightarrow
0}\mathcal{H}_{g}^{n-1}(\Sigma^{t})=\mathcal{H}_{g}^{n-1}(\partial A)$.

###### Proof.

See [9, Lemma 3.3]. ∎

######  Definition 4.10.

Let $(M^{n},g)$ be a closed Riemannian manifold. We define
$\mathbf{d}:M\rightarrow\mathbb{R}^{N-1}$ given by
$\mathbf{d}(x)=(\mathrm{d}_{1}(x),\dots\mathrm{d}_{N-1}(x))$, where

$\displaystyle\mathrm{d}_{i}(x)=\begin{cases}\operatorname{d}_{g}\left(x,\partial\Omega_{i}\right),&{\rm
if}\quad x\notin\Omega_{i}\\\
-\operatorname{d}_{g}\left(x,\partial\Omega_{i}\right),&{\rm if}\quad
x\in\Omega_{i}.\end{cases}$ (4.1)

are the signed distance functions for each $i=1,\dots,N-1$.

Since $M$ is compact, by the Hopf–Rinow theorem, we may assume by simplicity
that for any $i,j=\\{1,\dots,N\\}$ with $i\neq j$, there exists a distance-
minimizing geodesic connecting $\mathbf{p}_{i}$ and $\mathbf{p}_{j}$, that is,
one can find a smooth curve $\mathbf{c}_{ij}$ such that
$\mathbf{c}_{ij}(0)=\mathbf{p}_{i}$, $\mathbf{c}_{ij}(1)=\mathbf{p}_{j}$ and
$\mathrm{d}_{\boldsymbol{W}}(\mathbf{p}_{i},\mathbf{p}_{j})=\int_{0}^{1}\boldsymbol{W}^{1/2}(\mathbf{c}_{ij}(t))|\mathbf{c}_{ij}^{\prime}(t)|\mathrm{d}t$.
Notice that it is also not restrictive to assume
$|\mathbf{c}_{ij}^{\prime}(t)|\neq 0$ for all $t\in(0,1)$. Our next result is
concerned with the following system of ordinary differential equations

$(y_{\varepsilon}^{ij})^{\prime
2}=\frac{\tau+\boldsymbol{W}(\mathbf{c}_{ij}(y_{\varepsilon}^{ij}))}{\varepsilon^{2}|\mathbf{c}_{ij}^{\prime}(y_{\varepsilon}^{ij})|^{2}}\quad{\rm
in}\quad\mathbb{R},$ (4.2)

where $i,j\in\\{1,\dots,N\\}$, $i\neq j$, and $\tau>0$ are fixed.

######  Lemma 4.11.

Let $\boldsymbol{W}\in\mathcal{W}_{N,3}^{+}$. For every $\varepsilon>0$, there
exist a Lipschitz continuous function
$\mathbf{q}_{\varepsilon}:\mathbb{R}^{N-1}\rightarrow\mathbb{R}^{m}$ and
$C_{1},C_{2},C_{3}>0$ constants depending only on $\tau$ such that

$\mathbf{q}_{\varepsilon}\left(t_{1},\ldots,t_{N-2}\right)=\begin{cases}\mathbf{p}_{1},&{\rm
if}\ t_{1}<0;\\\ \mathbf{p}_{i},&{\rm if}\
t_{1}>{C}_{1}\varepsilon,\dots,t_{i-1}>{C}_{1}\varepsilon\ {\rm and}\ t_{i}<0\
{\rm for\ any}\ i=2,\ldots,N-2,\\\ \mathbf{p}_{N},&{\rm if}\
t_{1}>{C}_{1}\varepsilon,\ldots,t_{N-2}>{C}_{1}\varepsilon,\end{cases}$ (4.3)

and $0<|\mathbf{q}_{\varepsilon}|<{C}_{2}$ ,
$|D{\mathbf{q}_{\varepsilon}}|<{C}_{3}\varepsilon^{-1}$ almost everywhere in
$\mathbb{R}^{N-1}$. Moreover, if $j>i$ on the set
$\\{t\in\mathbb{R}^{N-1}:0<t_{i}<{C}_{1}\varepsilon,\ t_{j}<0,\ {\rm and}\
t_{k}>{C}_{1}\varepsilon\ {\rm for\ any}\ k\neq i,j\\}$, then
$\mathbf{q}_{\varepsilon}$ depends only on $t_{i}$, and we can write
$\mathbf{q}_{\varepsilon}(t_{i})=\mathbf{c}_{ij}(y_{\varepsilon}^{ij}(t_{i}))$,
where $y_{\varepsilon}^{ij}$ solves (4.2) for any $t_{i}$ such that
$\mathbf{q}_{\varepsilon}(t_{i})\neq\mathbf{p}_{i})$ $($Notice that if $j=N$,
one can ignore the condition $t_{j}<0$, which makes no sense$)$.

###### Proof.

First, we consider the solutions to (4.2) for $i,j=1,\dots,N-1$, $i\neq j$ and
$\tau>0$ fixed. Notice that the function
$\psi_{\varepsilon}:[0,1]\rightarrow[0,\eta_{\varepsilon}]$ given by

$\psi_{\varepsilon}(t)=\int_{0}^{t}\frac{\varepsilon|\mathbf{c}_{ij}^{\prime}(s)|}{\left[\tau+\boldsymbol{W}\left(\mathbf{c}_{ij}(s)\right)\right]^{1/2}}\mathrm{d}s$

is obviously increasing, where $\eta_{\varepsilon}=\psi_{\varepsilon}(1)$
satisfies
$\eta_{\varepsilon}\leqslant\varepsilon\tau^{-1/2}\ell(\mathbf{c}_{ij})$. Here
$\ell$ denotes the length of a curve. Now, the inverse function
$\widetilde{y}_{\varepsilon}:[0,\eta_{\varepsilon}]\rightarrow[0,1]$ of
$\psi_{\varepsilon}$ satisfies (4.2). We extend the function to the whole real
line by putting

$\widetilde{y}_{\varepsilon}(t)=\left\\{\begin{array}[]{ll}\widetilde{y}_{\varepsilon},&\quad{\rm
if}\quad t\leqslant 0\\\ 1,&\quad{\rm if}\quad 0\leqslant
t\leqslant\eta_{\varepsilon}\\\ 0,&\quad{\rm if}\quad
t\geqslant\eta_{\varepsilon}\end{array}\right.$

Now $\widetilde{y}_{\varepsilon}$ is a Lipschitz-continuous function
satisfying (4.2) in all the points where $\widetilde{y}_{\varepsilon}\neq 1$.
Let us consider

$C_{1}=\max_{i,j=1,\ldots,N}\\{\tau^{-1/2}\ell(\mathbf{c}_{ij})\\},$

Hence, we can define
$\widetilde{\mathbf{q}}_{\varepsilon}(t)=\mathbf{c}_{ij}(\widetilde{y}_{\varepsilon}^{ij}(t_{i}))$
on the set $\\{t\in\mathbb{R}^{N-1}:0<t_{i}<{C}_{1}\varepsilon,\ t_{j}<0,\
{\rm and}\ t_{k}>{C}_{1}\varepsilon\ {\rm for\ any}\ k\neq i,j\\}$ with $j>i$.
We choose

${C}_{2}>\sup\left\\{|y|:y\in\bigcup_{i,j=1}^{N-1}\mathbf{c}_{ij}([0,1])\right\\}.$

Defining

$\widetilde{C}_{3}=\sup_{\begin{subarray}{c}i,j=1,\ldots,N\\\ i\neq j,\
t\in[0,1]\end{subarray}}\left\\{\frac{\left[\tau+\boldsymbol{W}\left(\mathbf{c}_{ij}(t)\right)\right]^{1/2}}{\left|\mathbf{c}_{ij}^{\prime}(t)\right|}\right\\},$

we have that
$|D\widetilde{\mathbf{q}}_{\varepsilon}|\leqslant\widetilde{C}_{3}\varepsilon^{-1}$.
Standard extension results for Lipschitz-continuous functions allow one to
define $\widetilde{\mathbf{q}}_{\varepsilon}$ on the whole $\mathbb{R}^{N-2}$,
with $C_{3}>\widetilde{C}_{3}$ suitably chosen. ∎

######  Remark 4.12.

Notice that the family $\\{\mathbf{q}_{\varepsilon}\\}_{\varepsilon>0}$
constructed above approximates the following map

$\mathbf{q}_{0}\left(t_{1},\ldots,t_{N-1}\right)=\begin{cases}\mathbf{p}_{1},&{\rm
if}\ t_{1}<0;\\\ \mathbf{p}_{i},&{\rm if}\ t_{1}>0,\dots,t_{i-1}>0\ {\rm and}\
t_{i}<0\ {\rm for\ any}\ i=2,\ldots,N-1,\\\ \mathbf{p}_{N},&{\rm if}\
t_{1}>0,\ldots,t_{N-1}>0.\end{cases}$

This approximation will be crucial in the construction of the recovery
sequence.

To construct the recovery sequence in the $\Gamma$-convergence argument, we
will work with the truncation of a sequence of minima.

######  Remark 4.13.

Using the assumption (${{\rm W}_{3}}$), it is easy to verify that the
truncation condition holds. That is, there exists $k_{5}>0$ such that

$\boldsymbol{W}(u)\geqslant\sup_{z\in[-k_{5},k_{5}]^{m}}\boldsymbol{W}(z)\quad{\rm
for\ all}\quad u\notin[-k_{5},k_{5}]^{m}.$ (4.4)

This will be important to localize the problem to a compact set in the proof
of the next result.

The preliminary results stated above to allow us to define a candidate to
$\Gamma$-limit of the sequence $\boldsymbol{\mathcal{E}}_{\varepsilon}$ when
$\varepsilon\rightarrow 0$. Namely, for any $\mathbf{v}\in\mathbb{R}^{m}$, let
us denote

$\boldsymbol{\mathfrak{M}}^{\prime}_{\mathbf{v}}=\\{\mathbf{\Phi}\circ\mathbf{u}\in
BV_{g}(M,\mathbb{R}^{N}):\boldsymbol{W}(\mathbf{u})=0\;{\rm a.e.}\;{\rm
and}\;\boldsymbol{\mathcal{V}}_{g}(\mathbf{u})=\mathbf{v}\\}.$

We can define the following limit functional

$\boldsymbol{\mathcal{E}}_{0}(\mathbf{u}):=\begin{cases}2\bigvee_{i=1}^{N}\int_{M}|\nabla_{g}(\phi_{i}\circ\mathbf{u})|\mathrm{d}\mathcal{L}^{n}_{g},&\
\mbox{if}\ \mathbf{u}\in\boldsymbol{\mathfrak{M}}^{\prime}_{\mathbf{v}}\\\
\infty,&\ \mbox{if}\
\mathbf{u}\in{L}_{g}^{1}(M,\mathbb{R}^{m})\setminus\boldsymbol{\mathfrak{M}}^{\prime}_{\mathbf{v}}.\end{cases}$

Notice the relationship between $\boldsymbol{\mathcal{E}}_{0}$ and the
perimeter functional in (4) is given by Lemma 4.7.

More accurately, based on the last preliminary results we have the
$\Gamma$-convergence theorem for the sequence of relaxed operators.

######  Lemma 4.14.

Let $(M^{n},g)$ be a closed Riemannian manifold and
$\boldsymbol{W}\in\mathcal{W}_{N,3}^{+}$. It follows that
$\Gamma$-$\lim_{\varepsilon\rightarrow
0}\boldsymbol{\mathcal{E}}_{\varepsilon}=\boldsymbol{\mathcal{E}}_{0}$ in
$L_{g}^{1}(M,\mathbb{R}^{m})$.

###### Proof.

Let $\\{\varepsilon_{k}\\}_{k\in\mathbb{N}}\subset\mathbb{R}$ be a fixed
sequence such that $\varepsilon_{k}\rightarrow 0$ as $k\rightarrow\infty$.
Without loss of generality, we may assume
$\lim_{k\rightarrow\infty}\boldsymbol{\mathcal{E}}_{\varepsilon_{k}}(\mathbf{u}_{\varepsilon_{k}})<\infty$;
otherwise the proof readily follows. Now, it is convenient to divide the proof
into two claims.

We first prove the lower continuity part in the definition of
$\Gamma$-convergence.

Claim 1: For any $\\{\mathbf{u}_{\varepsilon_{k}}\\}_{k\in\mathbb{N}}$
converging to $\mathbf{u}_{0}$, it holds
$\boldsymbol{\mathcal{E}}_{0}(\mathbf{u}_{0})\leqslant\liminf_{k\rightarrow\infty}\boldsymbol{\mathcal{E}}_{\varepsilon}(\mathbf{u}_{\varepsilon_{k}})$.

Indeed, by choosing a subsequence
$\\{\mathbf{u}_{{\varepsilon}_{k}}\\}_{k\in\mathbb{N}}$ that converges to
$\mathbf{u}_{0}$ pointwise almost everywhere in $M$, the Fatou’s Lemma implies

$\int_{M}\boldsymbol{W}(\mathbf{u}_{0})\mathrm{d}\mathcal{L}_{g}^{n}\leqslant\liminf_{k\rightarrow\infty}\int_{M}\boldsymbol{W}(\mathbf{u}_{\varepsilon_{k}})\mathrm{d}\mathcal{L}_{g}^{n}\leqslant\liminf_{k\rightarrow\infty}\varepsilon_{k}\boldsymbol{\mathcal{E}}_{\varepsilon_{k}}(\mathbf{u}_{\varepsilon_{k}})=0.$

Thus, $\boldsymbol{W}(\mathbf{u}_{0}(x))=0$ almost everywhere in $M$ since
$\boldsymbol{W}\in\mathcal{W}^{+}_{N}$ is a continuous nonnegative map.

Because of Lemma 4.7, the next step is to prove that

$2\bigvee_{i=1}^{N-1}\int_{M}\left|\nabla_{g}(\phi_{i}{\circ}\mathbf{u}_{0})\right|\mathrm{d}\mathcal{L}_{g}^{n}\leqslant\lim_{k\rightarrow\infty}\int_{M}\left(\varepsilon_{k}\left|\nabla_{g}{\mathbf{u}_{\varepsilon_{k}}}\right|^{2}+{\varepsilon_{k}}^{-1}\boldsymbol{W}(\mathbf{u}_{\varepsilon_{k}})\right)\mathrm{d}\mathcal{L}_{g}^{n}.$
(4.5)

Initially, by using the technical assumption (4.4), we can assume
$\\{\mathbf{u}_{{\varepsilon}_{k}}\\}_{k\in\mathbb{N}}$ to be equibounded.
Otherwise, we can replace each $\mathbf{u}_{{\varepsilon}_{k}}$ by the
$\widetilde{\mathbf{u}}_{{\varepsilon}_{k}}$ obtained by truncating each
scalar component over a compact set, namely for each $i=1,\dots,N$, we
consider the component function
$(\widetilde{\mathbf{u}}_{{\varepsilon}_{k}})_{i}(x):={\rm
sgn}(({\mathbf{u}}_{{\varepsilon}_{k}})_{i}(x))\min\\{|({\mathbf{u}}_{{\varepsilon}_{k}})_{i}(x)|,k_{5}\\}$.
Also, notice that
$\widetilde{\mathbf{u}}_{{\varepsilon}_{k}}\rightarrow\mathbf{u}_{0}$ in
${L}_{g}^{1}(M,\mathbb{R}^{m})$ and the integrals in the right-hand side of
(4.5) decrease, that is,
$\boldsymbol{\mathcal{E}}_{\varepsilon_{k}}(\widetilde{\mathbf{u}}_{{\varepsilon}_{k}})\leqslant\boldsymbol{\mathcal{E}}_{\varepsilon_{k}}({\mathbf{u}}_{{\varepsilon}_{k}})$.

Next, we have that
$\Phi({\mathbf{u}}_{{\varepsilon}_{k}})\rightarrow\Phi(\mathbf{u}_{0})$ in
${L}_{g}^{1}(M,\mathbb{R}^{N-1})$, which by the continuity and lower
semicontinuity, implies

$\left|\nabla_{g}\left(\phi_{i}{\circ}\mathbf{u}_{0}\right)\right|({A})\leqslant\liminf_{k\rightarrow\infty}\left|\nabla_{g}\left(\phi_{i}{\circ}\mathbf{u}_{\varepsilon_{k}}\right)\right|({A})$

for any $i=1,\dots,N-1$ and $A\subset M$ an open set. Therefore, Lemma 4.3
yields that

$\displaystyle\liminf_{k\rightarrow\infty}\bigvee_{i=1}^{N-1}\left|\nabla_{g}(\phi_{i}\circ\mathbf{u}_{\varepsilon_{k}})\right|(M)$
$\displaystyle\leqslant\liminf_{k\rightarrow\infty}\int_{M}\boldsymbol{W}^{1/2}\left(\mathbf{u}_{\varepsilon_{k}}\right)\left|\nabla_{g}\mathbf{u}_{\varepsilon_{k}}(x)\right|\mathrm{d}\mathcal{L}_{g}^{n}$
$\displaystyle\leqslant\lim_{k\rightarrow\infty}\int_{M}\left(\varepsilon_{k}\left|\nabla_{g}{\mathbf{u}_{\varepsilon_{k}}}\right|^{2}+{\varepsilon_{k}}^{-1}\boldsymbol{W}(\mathbf{u}_{\varepsilon_{k}})\right)\mathrm{d}\mathcal{L}_{g}^{n},$

which finishes the proof of the first claim.

Next, we construct the recovery sequence to conclude the proof.

Claim 2: There exists $\\{\mathbf{u}_{\varepsilon_{k}}\\}_{k\in\mathbb{N}}$
converging to $\mathbf{u}_{0}$ such that
$\boldsymbol{\mathcal{E}}_{0}(\mathbf{u}_{0})\geqslant\limsup_{k\rightarrow\infty}\boldsymbol{\mathcal{E}}_{\varepsilon}\left(\mathbf{u}_{\varepsilon_{k}}\right)$.

First, if $\mathbf{u}_{0}\in{L}_{g}^{1}(M,\mathbb{R}^{m})$ is such that
$\boldsymbol{\mathcal{E}}_{0}(\mathbf{u}_{0})=\infty$, the construction of the
recovery sequence $\\{\mathbf{u}_{\varepsilon_{k}}\\}_{k\in\mathbb{N}}$ is
trivial. Hence, without loss of generality, we may assume
$\mathbf{u}_{0}(x)=\sum_{i=1}^{N}\mathbf{p}_{i}\chi_{\Omega_{i}}(x)$ with
$\mathbf{\Omega}\in\mathcal{C}_{g}(M,\mathbb{R}^{N})$ a weighted cluster.

Fix $\tau>0$ as in (4.2). Using Lemma 4.9, for $0<\varepsilon\ll 1$ small
enough, we have $\left|\nabla_{g}\mathrm{d}_{i}(x)\right|=1$ almost everywhere
on the set $\left\\{x\in M:|\mathrm{d}_{i}(x)|<C_{1}\varepsilon\right\\}$ for
all $i=1,\ldots,N-1$. Let us consider the map

$\widetilde{\mathbf{u}}_{\varepsilon}(x)=(\mathbf{q}_{\varepsilon}\circ\mathbf{d})(x),$
(4.6)

where $\mathbf{q}_{\varepsilon}:\mathbb{R}^{N-1}\rightarrow\mathbb{R}^{m}$ is
the ODE solution in (4.3) and $\mathbf{d}:M\rightarrow\mathbb{R}^{N-1}$ is the
vectorial distance function in (4.1). Denoting
$\Sigma_{i}^{t}=\\{x\in\Omega:\mathrm{d}_{i}(x)=t\\}$ for $t>0$ and
$i=1,\dots,N-1$, and using the coarea formula, we find

$\displaystyle\int_{M}\left|\widetilde{\mathbf{u}}_{\varepsilon}-\mathbf{u}_{0}\right|\mathrm{d}\mathcal{L}_{g}^{n}\leqslant\sum_{i=1}^{N-1}\int_{\left\\{x\in
M:0<\mathrm{d}_{i}(x)<C_{2}\varepsilon\right\\}}\left|\widetilde{\mathbf{u}}_{\varepsilon}-\mathbf{u}_{0}\right|\mathrm{d}\mathcal{L}_{g}^{n}\leqslant
2{C}_{2}\sum_{i=1}^{N-1}\int_{0}^{{C}_{1}\varepsilon}\mathcal{H}_{g}^{n-1}(\Sigma_{i}^{t})\mathrm{d}t,$

which, by using Lemma 4.11, yields
$\widetilde{\mathbf{u}}_{\varepsilon}\rightarrow\mathbf{u}_{0}$ in
${L}_{g}^{1}(M,\mathbb{R}^{m})$. If
$\int_{M}\widetilde{\mathbf{u}}_{\varepsilon}\mathrm{d}\mathcal{L}_{g}^{n}=\mathbf{v}$,
then the proof trivially follows. Otherwise, let us define

$\boldsymbol{\nu}_{\varepsilon}=\int_{M}\widetilde{\mathbf{u}}_{\varepsilon}\mathrm{d}\mathcal{L}_{g}^{n}-\int_{M}\mathbf{u}\mathrm{d}\mathcal{L}_{g}^{n}.$
(4.7)

Notice that the absolute value of each scalar component
$\boldsymbol{\nu}_{\varepsilon}$ is less than or equal to $C_{1}\varepsilon$.

Now let us construct a suitable partition. We assume that
$\Omega_{1}\neq\emptyset$, and let $y_{1}\in\Omega_{1}$ be a point in the
interior of the set $\Omega_{1}$. By definition of
$\widetilde{\mathbf{u}}_{\varepsilon}$, if $0<\varepsilon\ll 1$ is small
enough, the open metric ball
$\mathcal{B}^{g}_{\varepsilon}:=\mathcal{B}^{g}_{\varepsilon^{1/n}}(y_{1})$ is
contained in $\\{x\in
M:\widetilde{\mathbf{u}}_{\varepsilon}(x)=\mathbf{p}_{1}\\}$. We define

$\mathbf{u}_{\varepsilon}(x)=\begin{cases}\widetilde{\mathbf{u}}_{\varepsilon}(x)\quad{\rm
in}\quad M\setminus\mathcal{B}^{g}_{\varepsilon}\\\
\mathbf{p}_{1}+\boldsymbol{\xi}_{\varepsilon}(1-\varepsilon^{1/n}|x-y_{1}|)\quad{\rm
in}\quad\mathcal{B}^{g}_{\varepsilon}.\end{cases}$ (4.8)

Here $\widetilde{\mathbf{u}}_{\varepsilon}$ is given by (4.6), $\omega_{n-1}$
is the volume of the $(n-1)$-dimensional unit Euclidean ball and
$\boldsymbol{\xi}_{\varepsilon}=n\omega_{n-1}\varepsilon^{(1-n)/n}\boldsymbol{\nu}_{\varepsilon}$,
where $\boldsymbol{\nu}_{\varepsilon}$ is defined by (4.7). In this fashion,
it follows that
$\int_{M}\mathbf{u}_{\varepsilon}\mathrm{d}\mathcal{L}_{g}^{n}=\mathbf{v}$ for
$0<\varepsilon\ll 1$ small enough, and it converges in
${L}_{g}^{1}(M,\mathbb{R}^{m})$ to $\mathbf{u}_{0}$.

Finally, let us consider a partition of $M$ given by

  * (i)

$\Omega_{1}^{\varepsilon}=\Omega_{1}\setminus\mathcal{B}^{g}_{\varepsilon}$;

  * (ii)

$\Omega_{i}^{\varepsilon}=\\{x\in\Omega_{i}:\mathrm{d}_{j}(x)>{C}_{1}\varepsilon\
{\rm for}\ j=1,\dots,i-1\\}$ for $i=2,\dots,N-1$;

  * (iii)

$\Omega_{ij}^{\varepsilon}=\\{x\in M:0<\mathrm{d}_{i}(x)<{C}_{1}\varepsilon,\
\mathrm{d}_{j}(x)<0,\ \mathrm{d}_{\ell}(x)>{C}_{1}\varepsilon\ {\rm for}\
\ell\in\\{1,\dots,N-1\\}\setminus\\{i,j\\}\\}$ for $i,j\in\\{1,\dots,N-1\\}$
and $i\neq j$;

  * (iv)

$\Omega_{0}^{\varepsilon}=M\setminus\left(\mathcal{B}^{g}_{\varepsilon}\cup(\cup_{i=1}^{N-1}\Omega_{i}^{\varepsilon})\cup\left(\cup_{i,j=1,i\neq
j}^{N-1}\Omega_{ij}^{\varepsilon}\right)\right)$.

Thus, arguing as in the proof of [9, Theorem 2.5], we find that
$\limsup_{\varepsilon\rightarrow
0^{+}}\boldsymbol{\mathcal{E}}_{\varepsilon}(\mathbf{u}_{\varepsilon})\leqslant\boldsymbol{\mathcal{E}}_{0}\left(\mathbf{u}_{0}\right)$.
This completes the proof of Claim 2, and so the lemma holds true. ∎

From the proof of this result, we can extract the following corollary, which
will be important in the proof of the continuity of the photograph map

######  Corollary 4.15.

Let $(M^{n},g)$ be a closed Riemannian manifold and
$\boldsymbol{W}\in\mathcal{W}_{N,3}^{+}$. For any weighted cluster
$\mathbf{\Omega}\in\mathcal{C}_{g}(M,\mathbb{R}^{N})$, let us define the limit
profile function $\mathbf{u}_{0,\mathbf{\Omega}}:M\rightarrow\mathbb{R}^{m}$
by
$\mathbf{u}_{0,\mathbf{\Omega}}=\sum_{i=1}^{N-1}\mathbf{p}_{i}\chi_{\Omega_{i}}$.
Then, there exists a family
$\\{\mathbf{u}_{\varepsilon,\mathbf{\Omega}}\\}_{\varepsilon>0}$ of Lipschitz
continuous function on $M$ such that
$\mathbf{u}_{\varepsilon,\mathbf{\Omega}}$ converges to
$\mathbf{u}_{0,\mathbf{\Omega}}$ in $L_{g}^{1}(M,\mathbb{R}^{m})$ as
$\varepsilon\rightarrow 0^{+}$, $\mathbf{u}_{\varepsilon,\mathbf{\Omega}}\in
BV_{g}(M,\mathcal{Z})$, where
$\mathcal{Z}=\\{\mathbf{p}_{1},\dots,\mathbf{p}_{N}\\}$, for all
$\varepsilon>0$, and

  * (i)

$\int_{M}|\mathbf{u}_{\varepsilon,\mathbf{\Omega}}|\mathrm{d}\mathcal{L}^{n}_{g}=\int_{M}|\mathbf{u}_{0,\mathbf{\Omega}}|\mathrm{d}\mathcal{L}^{n}_{g}=\mathbf{v}\quad{\rm
for\ all}\quad\varepsilon>0$,

  * (ii)

$\limsup_{\varepsilon\rightarrow
0^{+}}\boldsymbol{\mathcal{E}}_{\varepsilon}(\mathbf{u}_{\varepsilon,\mathbf{\Omega}})\leqslant\boldsymbol{\mathcal{P}}_{g}(\mathbf{\Omega})$.

###### Proof.

To verify (i) and (ii), it is enough to consider the recovery family
$\\{\mathbf{u}_{\varepsilon,\mathbf{\Omega}}\\}_{\varepsilon>0}$ as in (4.8).
In addition, since for any $t\in\mathbb{R}$ and $i=1,\dots,N-1$, each
component satisfies
$(\widetilde{\mathbf{q}}_{\varepsilon})_{i}(t)\leqslant(\widetilde{\mathbf{q}}_{0})_{i}(t)$
and
$(\widetilde{\mathbf{q}}_{\varepsilon})_{i}(t_{i}+{\eta}_{\varepsilon})\geqslant(\widetilde{\mathbf{q}}_{0})_{i}(t_{i})$,
there exists
$\boldsymbol{\zeta}_{\varepsilon}:=\boldsymbol{\zeta}_{\varepsilon,{\mathbf{\Omega}}}\in[0,{\eta}_{\varepsilon}]^{N-1}$
such that the following identity holds

$\int_{M}\left|\widetilde{\mathbf{q}}_{\varepsilon}(\mathbf{d}(y)+\boldsymbol{\zeta}_{\varepsilon})\right|\mathrm{d}\mathcal{L}^{n}_{g}(y)=\int_{M}\left|{\mathbf{q}}_{0}(\mathbf{d}(y))\right|\mathrm{d}\mathcal{L}^{n}_{g}(x)=\int_{M}\left|\mathbf{u}_{0,\mathbf{\Omega}}(\mathbf{d}(y))\right|\mathrm{d}\mathcal{L}^{n}_{g}(y).$
(4.9)

Finally, we define

${\mathbf{q}}_{\varepsilon}(t)=\widetilde{\mathbf{q}}_{\varepsilon}(t+\boldsymbol{\zeta}_{\varepsilon}).$
(4.10)

and so

${\mathbf{u}}_{\varepsilon,\mathbf{\Omega}}(x)={\mathbf{q}}_{\varepsilon}(\mathbf{d}(x))=\widetilde{\mathbf{q}}_{\varepsilon}(\mathbf{d}(x)+\boldsymbol{\zeta}_{\varepsilon}),$

which concludes the proof. ∎

######  Definition 4.16.

The approximating family
$\\{\mathbf{u}_{\varepsilon,\mathbf{\Omega}}\\}_{\varepsilon>0}$ obtained in
Corollary 4.15 is called a Modica–Baldo approximation of the limit profile
$\mathbf{u}_{0,\mathbf{\Omega}}$.

At last, we are ready to fulfill our objective in this section. Indeed, we
need to extend the $\Gamma$-convergence to result for the case of a general
sequence of critical points with a uniformly bounded energy. This result is
new in the context of clusters even in the case of a domain of the Euclidean
space.

###### Proof of Proposition 11.

We consider $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ given by
$\varphi(t)=\int_{B_{t}}\boldsymbol{W}^{1/2}(z)\mathrm{d}z$ and set the
sequence

$\mathbf{w}_{\varepsilon_{k}}(x):=\boldsymbol{\varphi}(\mathbf{u}_{\varepsilon_{k}}(x))=(\varphi(\mathrm{u}_{{\varepsilon_{k}},1}(x)),\dots,\varphi(\mathrm{u}_{{\varepsilon_{k}},m}(x)))\quad{\rm
for\ all}\quad k\in\mathbb{N}.$

Next, we divide the proof into several claims.

Claim 1: $\\{\mathbf{w}_{\varepsilon_{k}}\\}_{k\in\mathbb{N}}$ is bounded in
$L_{g}^{1}(M,\mathbb{R}^{m})$.

In fact, using assumption (${{\rm W}_{3}}$) for $|z|=t$, it is not restrictive
to assume that $t_{0}\geqslant 1$, which yields the inequality below

$\displaystyle\varphi(t)$
$\displaystyle=\int_{B_{t_{0}}}\boldsymbol{W}^{1/2}(z)\mathrm{d}z+\int_{B_{t}\setminus
B_{t_{0}}}\boldsymbol{W}^{1/2}(z)\mathrm{d}z\leqslant\int_{B_{t_{0}}}\boldsymbol{W}^{1/2}(z)\mathrm{d}z+\frac{\sqrt{2k_{4}}}{p_{2}+2}t^{\frac{p_{2}}{2}+1}\quad{\rm
for\ all}\quad t\geqslant t_{0}.$

Moreover, since $p_{2}\leqslant 2(p_{1}-1)$, we have that
$\frac{p_{2}}{2}+1\leqslant p_{1}$, which implies that there exists some
constant $C_{4}>0$ satisfying $|\varphi(t)|\leqslant