# Min-max construction of minimal surfaces with a fixed angle at the boundary

Luigi De Masi  L. De Masi: SISSA Via Bonomea 265, 34136 Trieste, Italy
<EMAIL_ADDRESS>and  Guido De Philippis  G. De Philippis: Courant Institute
of Mathematical Sciences, New York University, 251 Mercer St., New York, NY
10012, USA<EMAIL_ADDRESS>
###### Abstract.

We prove the existence of minimal surfaces in a bounded convex subset of
$\mathbb{R}^{3}$ $\mathcal{M}$ intersecting the boundary of $\mathcal{M}$ with
a fixed contact angle. The proof is based on a min-max construction in the
spirit of Almgren-Pitts for the capillarity functional.

###### Key words and phrases:

Minimal surfaces, capillarity functional, min-max techniques

Acknowledgments. G.D.P has been partially supported by the NSF grant DMS
2055686 and by the Simons Foundation.

##  1\. Introduction

###  1.1. Background and main result

Minimal surfaces, i.e. critical points of the area functional, are a
fundamental object in Geometric Analysis and the problem of finding a minimal
surface (possibly with given side constraints) in a given Riemannian manifold
$\mathcal{M}$ has been a fundamental topic of research in the last 50 years.

When the topology of the ambient manifold $\mathcal{M}$ is non-trivial,
existence result can be often obtained via minimization of the area functional
in suitable homology classes. On the other hand, when the topology is trivial,
it is easy to see that non-trivial minimizers do not exist and one has to rely
on different methods.

Since the seminal work of Birkhoff about the existence of closed geodesic in
Riemannian manifolds, the use of min-max techniques has been a successful tool
to show existence of critical points for a variety of functionals.

The implementation of min-max techniques for the construction of minimal
surfaces has been started in the seminal work of Almgren and Pitts and deeply
relies on techniques of Geometric Measure Theory. One of the fundamental
outcomes of the theory has been to show the existence of a $d$ dimensional
minimal surface in every $(d+1)$-dimensional manifold, [22, 3, 5]. (To be
precise the minimal surface is allowed to have a “small” singular set when
$d\geq 7$) . Since their seminal work, the strategy has been suitably modified
to show the existence of surfaces with prescribed curvature [27], given
boundary [4] as well as free boundary, [4]. Furthermore these techniques have
been as well used to show the existence of _infinitely many_ minimal surfaces
in a given Riemannian manifold [19, 14, 26]

The main result of this paper is to show how Almgren-Pitts techniques (and
more precisely their revised version by Colding-De Lellis, [3]) allows the
existence of minimal surfaces with _prescribed contact angle at the boundary_
in convex subsets of $\mathbb{R}^{3}$.

######  Theorem 1.1.

Let $\mathcal{M}\subset\mathbb{R}^{3}$ be a convex bounded open set with
$C^{2,\alpha}$ boundary. Then, for all $\theta\in(0,\pi/2)$ there exists a
minimal surface $\Sigma\subset\mathcal{M}$ with
$\partial\Sigma\subset\partial\mathcal{M}$ and

$\eta_{\Sigma}\cdot N=\sin\theta\qquad\text{on $\partial\Sigma$}$

where $\eta_{\Sigma}$ is the exterior co-normal field to $\Sigma$ at
$\partial\Sigma$ and $N$ is the exterior unit normal to $\partial\mathcal{M}$.

###  1.2. Sketch of the argument and structure of the paper

The main idea of the proof, which has already appeared in the Master Thesis of
the first author [6], where some preliminary steps of this program were
carried out, is to apply Almgren-Pitts/Colding-De Lellis techniques to the
_capillarity energy_ :

(1.1)
$F_{a}(\Omega)=\mathcal{H}^{2}(\partial_{i}{\Omega})+\cos\theta\mathcal{H}^{2}(\partial_{b}\Omega),$

where $\Omega\subset\mathcal{M}$ and
$\partial_{i}{\Omega}=\partial\Omega\cap{\kern
0.0pt\mathcal{M}}^{\mathrm{o}}$,
$\partial_{b}\Omega=\partial\Omega\cap\partial\mathcal{M}$. Note indeed that a
(smooth) stationary point of (1.1) shall satisfy

$H_{\Sigma}=0\text{ in ${\kern
0.0pt\mathcal{M}}^{\mathrm{o}}$,}\qquad\eta_{\Sigma}\cdot
N=\sin\theta\quad\text{on $\partial\Sigma\cap\partial\mathcal{M}$}$

where $\Sigma=\partial_{i}{\Omega}$. With respect to the case of the pure area
functional, or to the free boundary one (when $\theta=\pi/2$) two main new
difficulties arise. First, a priori, the capillarity functional is only
defined for surfaces which are boundary of a set and it is not clear how to
extend it to class of rectifiable varifold, a technical step which is
instrumental to the proof of Theorem 1.1. The key idea here is to “decouple”
the functional and initially look at the interior and boundary part as
independent. Once this has been done, the pull-tight procedure can be carried
on and one can show the existence of a stationary varifold which satisfies a
suitable notion of “angle condition” which a slight modification on the on
proposed by Kagaya and Tonegawa in [15]. This will be done in Section 3 and 4.

The second key step is to show regularity of the obtained varifold; here we
start by following the stategy of [22, 3, 5], and showing that the stationary
varifold found in the previous step is $\varepsilon$-minimizing in annuli and
thus it admits replacements there, see Section 5 and 7. Existence of
replacements then follows by the the regularity theory of [8, 9] together with
the curvature estimate proved for $2$-dimensional stable critical point of the
capillarity functional. These estimates, recalled in Section 6 can be easily
obtained if one assumes an a priori area growth on the surface (a condition
which is met on our construction) and have been recently established in full
generality in [13]. Existence of replacements implies that the obtained
varifold is induced by a smooth surface outside a finite set of points and the
proof follows the one in [4]. In order to remove the final singular points we
instead rely on a new argument which strongly uses that $\theta<\pi/2$. Under
this assumption we can show that stable surface is indeed graphical and
satisfies a suitable free boundary problem, from which we can deduce the
desired regularity. This is done in Section 8.

We conclude the introduction by commenting the restriction to open convex
subsets of $\mathbb{R}^{3}$. First of all, it will be clear from the proof
that we can endow $\mathcal{M}$ with any (smooth) Riemaniann metric for which
the boundary is still convex and obtain the same result. Convexity on the
other hand is used in showing that the “interior” part of $\Sigma$ does not
touch the boundary of $\mathcal{M}$ and it looks plausible that by following
the argument of [17], the assumption can be dispensed. The restriction to
dimension $3$ appears instead only in one (crucial) point of our proof, namely
in showing curvature estimates for stable capillarity surfaces in Section 6.
We conjecture that similar estimates should hold true in higher dimension as
well and if this is the case, the same arguments of the paper will show the
existence of $d$-dimensional surfaces into convex sets of $(d+1)$-dimensional
manifolds. We refer the reader to the forthcoming PhD thesis of the first
author for some of the mentioned generalizations, [7].

While completing this paper, we learned that Chao Li, Xin Zhou and Jonathan Zu
[16] had just obtained a similar result with related techniques.

##  2\. Notations

###  2.1. Basic notations

We work in $\mathbb{R}^{3}$. If $x\in\mathbb{R}^{3}$, we denote by $B_{r}(x)$
the closed ball with center $x$ and radius $r$ Moreover we set
$B_{r}:=B_{r}(0)$. If $0<s<r$ we call $\operatorname*{An}(x,s,r)$ the open
annulus ${\kern 0.0ptB_{r}(x)}^{\mathrm{o}}\setminus B_{s}(x)$ and
$\mathcal{AN}_{\rho}({x})$ the set of open annuli $\operatorname*{An}(x,s,r)$
such that $r<\rho$. For any $\lambda>0$ and $x\in\mathbb{R}^{3}$ we consider
the dilation map

(2.1) $D_{{x,r}}(y)=\frac{1}{r}(y-x).$

For every $A\subset\mathbb{R}^{3}$, we denote by $\mathbf{1}_{A}$ the
indicator function of $A$, by $\mkern
1.0mu\overline{\mkern-1.0muA\mkern-1.0mu}\mkern 1.0mu$ and ${\kern
0.0ptA}^{\mathrm{o}}$ respectively the closure and the interior of $A$ in the
euclidean topology. We denote by $S$ a generic $m$-dimensional linear subspace
(or $m$-plane) of $\mathbb{R}^{3}$ and we write $S^{\perp}$ for the orthogonal
complement of $S$ in $\mathbb{R}^{3}$. We denote by $P_{S}$ the orthogonal
projection on $S$. If $X$ is a $C^{1}$ vector field, we call
$\operatorname{div}_{S}X$ the scalar product $P_{S}\cdot DX$. If
$\tau_{1},\dots,\tau_{m}$ is an orthonormal basis of $S$, by simple
computations one has

$\operatorname{div}_{S}X(x)=\sum_{i=1}^{m}{D_{\tau_{i}}X(x)}\cdot{\tau_{i}}.$

If $\Gamma$ is a $C^{1}$ $m$-dimensional sub-manifold in $\mathbb{R}^{3}$ and
$x\in\Gamma$, we write $T_{x}\Gamma$ for the tangent space to $\Gamma$ at $x$.

We work on a compact domain $\mathcal{M}\subset\mathbb{R}^{3}$ with
$C^{2,\alpha}$ boundary $\partial\mathcal{M}$ for some $\alpha\in(0,1)$. We
will assume also that $\partial\mathcal{M}$ is uniformly convex, that is the
principal curvatures of $\partial\mathcal{M}$ satisfy an uniform positive
lower bound. We write $N(x)$ for the exterior unit normal vector to
$\partial\mathcal{M}$ at $x$. If $\Omega\subset\mathcal{M}$ is open in the
relative topology of $\mathcal{M}$ and has finite perimeter, we call
$\partial_{i}\Omega$ and $\partial_{b}\Omega$ respectively the internal and
the external reduced boundary of $\Omega$, that is

$\partial_{i}\Omega=\partial^{*}\Omega\cap{\kern
0.0pt\mathcal{M}}^{\mathrm{o}},\qquad\partial_{b}\Omega=\partial^{*}\Omega\cap\partial\mathcal{M},$

where $\partial^{*}\Omega$ denotes the reduced boundary of $\Omega$. If
$\partial_{i}\Omega$ and $\partial_{b}\Omega$ are sufficiently regular
$2$-submanifolds of $\mathbb{R}^{3}$, we denote by $\gamma(\Omega)$ their
common boundary (as manifolds).

We work with several classes of vector fields on $\mathcal{M}$, which we
denote with the letter $\mathfrak{X}$ with subscripts based on their behavior
on $\partial\mathcal{M}$:

$\displaystyle\mathfrak{X}(\mathcal{M})=C^{1}(\mkern
1.0mu\overline{\mkern-1.0mu\mathcal{M}\mkern-1.0mu}\mkern
1.0mu,\mathbb{R}^{3}),\qquad\mathfrak{X}_{t}(\mathcal{M})=\\{X\in\mathfrak{X}(\mathcal{M})\mid
X(x)\in T_{x}\partial\mathcal{M},\,\,\forall x\in\partial\mathcal{M}\\},$
$\displaystyle\mathfrak{X}_{\perp}(\mathcal{M})=\\{X\in\mathfrak{X}(\mathcal{M})\mid
X(x)\in(T_{x}\partial\mathcal{M})^{\perp},\,\,\forall
x\in\partial\mathcal{M}\\}$
$\displaystyle\mathfrak{X}_{0}(\mathcal{M})=\\{X\in\mathfrak{X}(\mathcal{M})\mid
X(x)=0,\,\,\forall
x\in\partial\mathcal{M}\\}\quad\mathfrak{X}_{c}(\mathcal{M})=\\{X\in\mathfrak{X}(\mathcal{M})\mid\operatorname{supp}X\subset\joinrel\subset{\kern
0.0pt\mathcal{M}}^{\mathrm{o}}\\}.$

###  2.2. Measures

If $A\subset\mathbb{R}^{3}$, $\mathscr{M}(A,\mathbb{R}^{m})$ is the space of
$\mathbb{R}^{m}$-valued Radon measures on $A$ and $\mathscr{M}^{+}(A)$ is the
space of positive Radon measures on $A$. If
$\mu\in\mathscr{M}(A,\mathbb{R}^{m})$ we denote by $|\mu|$ the total variation
measure of $\mu$. If $B\subset\mathbb{R}^{3}$ is Borel, we write $\mu\llcorner
B$ for the restriction of the measure $\mu$ to $B$. We endow
$\mathscr{M}(A,\mathbb{R}^{m})$ with the weak*-topology: i.e. we say that a
sequence of Radon measures $\\{\sigma_{j}\\}_{j}$ converges to $\mu$
($\sigma_{j}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}\mu$) if

$\lim_{j\to\infty}\int f\mathop{}\\!\mathrm{d}\sigma_{j}=\int
f\mathop{}\\!\mathrm{d}\mu\qquad\forall f\in C_{c}(A,\mathbb{R}^{m}).$

If $\mu\in\mathscr{M}^{+}(\mathbb{R}^{3})$, $x\in\mathbb{R}^{3}$,
$m\in\mathbb{N}$, we define the upper and the lower $m$-densities of $\mu$ at
$x$:

$\Theta^{*}_{m}(\mu,x)=\limsup_{r\to
0}\frac{\mu\big{(}B_{r}(x)\big{)}}{r^{m}}\qquad\Theta_{*m}(\mu,x)=\liminf_{r\to
0}\frac{\mu\big{(}B_{r}(x)\big{)}}{r^{m}}.$

If the above limits coincide, then we define the $m$-density of $\mu$ at $x$
as their common value, which we denote by $\Theta_{m}(\mu,x)$. For any $s>0$,
we denote by $\mathcal{H}^{s}$ the $s$-dimensional Hausdorff measure.

###  2.3. Varifolds

If $1\leq m\leq 3$ we call $G(m,3)$ the Grassmannian of the un-oriented
$m$-dimensional linear subspaces (or $m$-planes) of $\mathbb{R}^{3}$. If
$A\subset\mathbb{R}^{3}$ we denote by $G_{m}(A):=A\times G(m,3)$ the trivial
Grassmannian bundle over $A$.

An _$m$ -varifold_ on $A$ is a positive Radon measure on $G_{m}(A)$. We denote
by $\mathcal{V}_{m}(A)$ the set of all $m$-varifolds on $A$ and we endow
$\mathcal{V}_{m}(A)$ with the topology of the weak*-convergence of Radon
measures, i.e. we say that
$V_{j}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}V$ if

$\lim_{j\to\infty}\int_{G_{m}(A)}\varphi(x,S)\mathop{}\\!\mathrm{d}V_{j}(x,S)=\int_{G_{m}(A)}\varphi(x,S)\mathop{}\\!\mathrm{d}V(x,S)\qquad\forall\varphi\in
C_{c}(G_{m}(A)).$

An $m$-rectifiable measure $\mu=\theta\mathcal{H}^{m}\llcorner M$ in
$\mathbb{R}^{3}$ (where $M$ is $m$-rectifiable and $\theta\geq 0$ belongs to
$L^{1}_{\operatorname{loc}}(M,\mathcal{H}^{m})$) induces the $m$-varifold

$V=\theta\mathcal{H}^{m}\llcorner M\otimes\delta_{T_{x}M},$

where $T_{x}M$ is the approximate tangent space of $M$ at $x$. A varifold that
is induced by a rectifiable measure is called a _rectifiable varifold_. If $V$
is an $m$-varifold, the _mass_ $\lVert V\rVert$ (or _total variation_) of $V$
is the positive Radon measure defined as

$\lVert V\rVert(A)=V(G_{m}(A))\qquad\forall A\subset\Omega\mbox{ Borel}.$

If $V$ is the $m$-varifold induced by the rectifiable measure
$\mu=\theta\mathcal{H}^{m}\llcorner M$, then

$\lVert V\rVert(B)=\int_{B}\theta(x)\mathop{}\\!\mathrm{d}\mathcal{H}^{m}(x).$

By slight abuse of notation, we often denote $\operatorname{supp}\lVert
V\rVert$ by $\operatorname{supp}V$. If $V\in\mathcal{V}_{m}(\mathcal{M})$ and
if $\psi:\mathcal{M}\to\mathcal{M}$ is a diffeomorphism, the _push forward_
$\psi_{\sharp}V$ of $V$ through $\psi$ is the varifold in
$\mathcal{V}_{m}(\mathcal{M})$ such that, $\forall\varphi\in
C_{c}\big{(}G_{m}(\mathcal{M})\big{)}$,

(2.2)
$\int_{G_{m}\left(\mathcal{M}\right)}\varphi(y,T)\mathop{}\\!\mathrm{d}\psi_{\sharp}V(y,T)=\int_{G_{m}(\mathcal{M})}J_{S}\psi(x)\varphi\big{(}\psi(x),\mathop{}\\!\mathrm{d}\psi_{x}(S)\big{)}\mathop{}\\!\mathrm{d}V(x,S),$

where $J_{S}\psi(x)$ is the Jacobian of $\psi$ relative to the $m$-plane $S$,
i.e.

$J_{S}\psi(x)=\sqrt{\det\big{(}(\mathop{}\\!\mathrm{d}\psi_{x})_{|S}^{*}\circ\big{(}(\mathop{}\\!\mathrm{d}\psi_{x})_{|S}\big{)}}.$

We notice that this _is not_ the push forward of measures previous defined
(which is denoted by the different symbol $f_{\\#}\mu$). In fact, the push
forward of varifolds is defined in this way in order to ensure the validity of
the area formula: indeed if $V$ is induced by a rectifiable set $M$, then
$\psi_{\sharp}V$ is induced by $\psi(M)$. If
$V\in\mathcal{V}_{m}(\mathbb{R}^{n})$, we say that
$C\in\mathcal{V}_{m}(\mathbb{R}^{n})$ is a _blow-up_ of $V$ at $x$ or a
_tangent varifold_ to $V$ at $x$ if there exists a sequence $r_{j}\downarrow
0$ such that

$(D_{{x,r_{j}}})_{\sharp}V\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}C.$

We write $\operatorname{Tan}(V,x)$ for the set of tangent varifold to $V$ at
the point $x$.

If $V\in\mathcal{V}_{m}(\mathcal{M})$ and if $X\in\mathfrak{X}(\mathcal{M})$
the _first variation_ $\delta V(X)$ of $V$ with respect to $X$ is

$\delta
V[X]=\left.\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\Big{(}\left\lVert(\psi_{t})_{\sharp}V\right\rVert(\mathbb{R}^{3})\Big{)}\right|_{t=0}$

where $\psi_{t}$ is the flow map of $X$ at the time $t$. The following _first
variation formula_ holds:

$\delta
V[X]=\int_{G_{k}(\mathcal{M})}\operatorname{div}_{S}X(x)\,\mathop{}\\!\mathrm{d}V(x,S).$

We now define the class of varifolds with _bounded first variation_ :

######  Definition 2.1.

We say that a varifold $V\in\mathcal{V}_{m}(\mathcal{M})$ has _bounded first
variation_ in $\mathcal{M}$ if

(2.3) $\sup\\{|\delta V[X]|\mid X\in\mathfrak{X}(\mathcal{M})\,,\max|X|\leq
1\\}<+\infty.$

If (2.3) holds with a proper subset of $\mathfrak{X}(\mathcal{M})$(e.g.
$\mathfrak{X}_{c}(\mathcal{M})$, $\mathfrak{X}_{0}(\mathcal{M})$…) in place of
$\mathfrak{X}(\mathcal{M})$, we say that $V$ has bounded first variation with
respect to this subset.

Therefore $V$ has bounded first variation if $\delta
V\in\mathscr{M}(\mathcal{M},\mathbb{R}^{3})$. If $V$ has bounded first
variation, then by Lebesgue decomposition there exist
$|\delta^{s}{V}|\in\mathscr{M}^{+}(\mathcal{M})$, a
$|\delta^{s}{V}|$-measurable function $\eta:\mathcal{M}\to\mathbb{R}^{3}$ and
a $\lVert V\rVert$-measurable function $H:\mathcal{M}\to\mathbb{R}^{3}$ such
that

$\delta V[X]=-\int_{\mathcal{M}}{H}\cdot{X}\,\mathop{}\\!\mathrm{d}\lVert
V\rVert+\int_{\mathcal{M}}{X}\cdot{\eta}\mathop{}\\!\mathrm{d}|\delta^{s}{V}|\qquad\forall
X\in\mathfrak{X}(\mathcal{M})$

where $|\delta^{s}{V}|$ is the singular part of $|\delta V|$ with respect to
$\lVert V\rVert$:

$|\delta^{s}{V}|=|\delta V|\llcorner Z\qquad
Z=\Big{\\{}x\in\mathcal{M}\mid\limsup_{r\to 0}\frac{|\delta
V|(B_{r}(x))}{\lVert V\rVert(B_{r}(x))}=+\infty\Big{\\}}.$

Since the previous formula is similar to the corresponding one for smooth
surfaces, we call $H$ the _generalized mean curvature_ of $V$,
$|\delta^{s}{V}|$ the _boundary measure_ of $V$, the set $Z$ is the _boundary_
of $V$ and $\eta$ is the _unit co-normal_ of $V$.

######  Definition 2.2.

We say that $V\in\mathcal{V}_{m}(\mathcal{M})$ has _generalized mean
curvature_ $H$ with respect to $\mathfrak{X}_{c}(\mathcal{M})$ (respectively
$\mathfrak{X}_{0}(\mathcal{M})$, $\mathfrak{X}_{t}(\mathcal{M})$) if for each
$X\in\mathfrak{X}_{c}(\mathcal{M})$ (respectively
$\mathfrak{X}_{0}(\mathcal{M})$, $\mathfrak{X}_{t}(\mathcal{M})$) the
following formula holds:

(2.4)
$\int_{G_{k}(\mathcal{M})}\operatorname{div}_{S}X(x)\,\mathop{}\\!\mathrm{d}V(x,S)=-\int_{\mathcal{M}}{H}\cdot{X}\,\mathop{}\\!\mathrm{d}\lVert
V\rVert.$

Thus $V$ has _generalized mean curvature_ with respect to
$\mathfrak{X}_{c}(\mathcal{M})$ (respectively $\mathfrak{X}_{0}(\mathcal{M})$,
$\mathfrak{X}_{t}(\mathcal{M})$) if has bounded variation with respect to
$\mathfrak{X}_{c}(\mathcal{M})$ (respectively $\mathfrak{X}_{0}(\mathcal{M})$,
$\mathfrak{X}_{t}(\mathcal{M})$) and $\delta V$ has no singular part with
respect to $\lVert V\rVert$ when we test with vector fields in
$\mathfrak{X}_{c}(\mathcal{M})$ (respectively $\mathfrak{X}_{0}(\mathcal{M})$,
$\mathfrak{X}_{t}(\mathcal{M})$).

######  Definition 2.3 (Varifold with free boundary).

If $V\in\mathcal{V}_{k}(\mathcal{M})$ has generalized mean curvature with
respect to $\mathfrak{X}_{t}(\mathcal{M})$, we say that $V$ _has free
boundary_ at $\partial\mathcal{M}$.

###  2.4. Capillarity free energy

######  Definition 2.4 (Capillarity free energy).

If $\Omega\subset\mathcal{M}$ is relatively open with finite perimeter and if
$a\in[0,1)$, the _$a$ -capillarity energy_ $F_{a}$ applied to $\Omega$ is
defined as

$F_{a}(\Omega)=\mathcal{H}^{2}(\partial_{i}{\Omega})+a\mathcal{H}^{2}(\partial_{b}\Omega),$

We now define $\mathscr{O}(\mathcal{M})$ as the class of all subsets of
$\mathcal{M}$ which are open in the relative topology of $\mathcal{M}$ and
whose internal and external boundary are $2$-dimensional submanifolds of class
$C^{2}$ with $C^{1}$ boundary, that is

$\mathscr{O}(\mathcal{M})=\\{\Omega\subset\mathcal{M}\mid\Omega\text{ is open
in }\mathcal{M},\,\partial_{i}(\Omega)\text{ and }\partial_{b}(\Omega)\text{
are of class }C^{2}\text{ with }C^{1}\text{ boundary }\gamma(\Omega)\\},$

where we recall that $\partial_{i}\Omega$ and $\partial_{b}\Omega$ are
respectively the internal and the external boundary of $\Omega$ previous
defined and $\gamma(\Omega)$ denotes the common boundary of
$\partial_{i}\Omega$ and $\partial_{b}\Omega$ as manifolds.

If $\Omega\in\mathscr{O}(\mathcal{M})$, standard computations yield the first
variation of $F_{a}$ with respect to $X\in\mathfrak{X}_{t}(\mathcal{M})$:

$\delta_{F_{a}}\Omega[X]=-\int_{\partial_{i}\Omega}{X}\cdot{H}\mathop{}\\!\mathrm{d}\mathcal{H}^{2}+\int_{\gamma(\Omega)}{\eta+a\zeta}\cdot{X}\mathop{}\\!\mathrm{d}\mathcal{H}^{1},$

where $H$ is the mean curvature vector of $\partial_{i}\Omega$, $\eta$ is the
exterior unit conormal vector to $\partial_{i}\Omega$ and $\zeta$ is the
exterior unit conormal vector to $\partial_{b}\Omega$.

In the following we need to extend our definition of capillarity functional to
the set of pairs of varifolds
$\mathcal{V}_{2}(\mathcal{M})\times\mathcal{V}_{2}(\partial\mathcal{M})$.

######  Definition 2.5 (Capillarity functional for pairs of varifolds).

If
$(V,W)\in\mathcal{V}_{2}(\mathcal{M})\times\mathcal{V}_{2}(\partial\mathcal{M})$
and if $a\in[0,1)$, the _$a$ -capillarity functional_ $F_{a}$ applied to
$(V,W)$ is defined as

$F_{a}(V,W)=\lVert V\rVert(\mathcal{M})+a\lVert W\rVert(\partial\mathcal{M}).$

##  3\. Contact angle condition for varifolds

We now discuss a notion of contact angle condition for varifolds. Our
condition follows the one given by Kagaya and Tonegawa in [15], but it is
weaker since it allows the “boundary part” to be just a $2$-varifold on
$\partial\mathcal{M}$ (that is a positive Radon measure on
$\partial\mathcal{M}$), whereas the one in [15] is more restrictive, since it
fixes the boundary part to be a subset of $\partial\mathcal{M}$. We state our
condition for pairs $(V,W)$ where $V\in\mathcal{V}_{2}(\mathcal{M})$ and
$W\in\mathcal{V}_{2}(\partial\mathcal{M})$.

######  Definition 3.1 (Contact angle condition).

Given $\theta\in[0,\frac{\pi}{2})$ we say that the pair
$(V,W)\in\mathcal{V}_{2}(\mathcal{M})\times\mathcal{V}_{2}(\partial\mathcal{M})$
satisfies the contact angle condition $\theta$ if there exists a $\lVert
V\rVert$-measurable vector field $H\in L^{1}(\mathcal{M},\lVert V\rVert)$ such
that, for every $X\in\mathfrak{X}_{t}(\mathcal{M})$, it holds

(3.1)
$\begin{split}\delta_{F_{a}}(V,W)[X]&=\int_{G_{2}(\mathcal{M})}\operatorname{div}_{S}X(x)\mathop{}\\!\mathrm{d}V(x,S)+a\int_{G_{2}(\partial\mathcal{M})}\operatorname{div}_{\partial\mathcal{M}}X(x)\mathop{}\\!\mathrm{d}W(x,T_{x}\partial\mathcal{M})\\\
&=-\int_{\mathcal{M}}{X(x)}\cdot{H(x)}\mathop{}\\!\mathrm{d}\lVert
V\rVert(x),\end{split}$

where $a=\cos\theta$. Since we test only with vector fields in
$\mathfrak{X}_{t}(\mathcal{M})$, we assume that $H(x)\in
T_{x}\partial\mathcal{M}$ for $\lVert V\rVert$-a.e. $x\in\partial\mathcal{M}$.

######  Definition 3.2 (Stationary pairs).

If $\theta\in[0,\frac{\pi}{2})$ and $a=\cos\theta$, we say that the pair
$(V,W)\in\mathcal{V}_{2}(\mathcal{M})\times\mathcal{V}_{2}(\partial\mathcal{M})$
is _stationary_ for $F_{a}$ if it satisfies the contact angle condition (3.1)
with $H=0$. If $U\subset\mathcal{M}$ is relatively open in $\mathcal{M}$, we
say that $(V,W)$ is stationary in $U$ if it satisfies (3.1) with $H=0$ for
every vector field $X\in\mathfrak{X}_{t}$ that is compactly supported in $U$.

We also give the definition of stability for pairs with respect to $F_{a}$.

######  Definition 3.3 (Stable pairs).

If $\theta\in[0,\frac{\pi}{2})$ and $a=\cos\theta$, we say that the pair
$(V,W)\in\mathcal{V}_{2}(\mathcal{M})\times\mathcal{V}_{2}(\partial\mathcal{M})$
is _stable_ for $F_{a}$ if it satisfies

(3.2) $\delta^{2}_{F_{a}}(V,W)[X]\geq 0\qquad\forall
X\in\mathfrak{X}_{t}(\mathcal{M}),$

where $\delta^{2}_{F_{a}}$ is the second variation of $F_{a}$. If
$U\subset\mathcal{M}$ is relatively open in $\mathcal{M}$, we say that $(V,W)$
is stable in $U$ if it satisfies (3.2) for every vector field
$X\in\mathfrak{X}_{t}$ that is compactly supported in $U$.

######  Remark 3.1.

To understand the intuition behind the previous definitions, we recall that
that a $C^{1}$-surface $\Sigma\subset\mathcal{M}$ with $C^{1}$-boundary
$\partial\Sigma\subset\partial\mathcal{M}$ has fixed contact angle $\theta$ at
$\partial\mathcal{M}$ if its unit exterior conormal $\eta_{\Sigma}$ satisfies
${\eta(x)}\cdot{N(x)}\equiv\sin\theta$ for every $x\in\partial\Sigma$ (where
$N$ is the exterior unit normal vector to $\partial\mathcal{M}$).

If $\Omega\in\mathscr{O}(\mathcal{M})$, then it easily seen that the surface
$\partial_{i}\Omega\subset\mathcal{M}$ has fixed contact angle $\theta$ if and
only if either $\eta+a\zeta\perp\partial\mathcal{M}$ or
$\eta-a\zeta\perp\partial\mathcal{M}$, where $a=\cos\theta$, and $\eta$ and
$\zeta$ are the unit conormal respectively of $\partial_{i}\Omega$ and
$\partial_{b}\Omega$; thus $\partial_{i}\Omega$ has fixed contact angle
$\theta$ if and only if

(3.3)
$\int_{\partial_{i}\Omega}\operatorname{div}_{\partial_{i}\Omega}X\mathop{}\\!\mathrm{d}\mathcal{H}^{2}\pm
a\int_{\partial_{b}\Omega}\operatorname{div}_{\partial\mathcal{M}}X\mathop{}\\!\mathrm{d}\mathcal{H}^{2}=-\int_{\partial_{i}\Omega}{X}\cdot{H}\mathop{}\\!\mathrm{d}\mathcal{H}^{2},$

where $\pm$ must be constantly $+$ or $-$ and $H$ is the mean curvature of
$\partial_{i}\Omega$. Moreover, if $H\equiv 0$, then $\partial_{i}\Omega$ is a
minimal surface which meets $\partial\mathcal{M}$ with angle $\theta$.

###  3.1. Bounded first variation and Monotonicity formula

All the results of this section are proved in Appendix A. We first state that,
if
$(V,W)\in\mathcal{V}_{2}(\mathcal{M})\times\mathcal{V}_{2}(\partial\mathcal{M})$
has contact angle $\theta$, then $V+aW$ has bounded first variation in
$\mathcal{M}$.

######  Proposition 3.1.

Let
$(V,W)\in\mathcal{V}_{2}(\mathcal{M})\times\mathcal{V}_{2}(\partial\mathcal{M})$
have contact angle $\theta$. Then $V+aW$ has bounded first variation. More
precisely, there exist a positive Radon measure $\sigma_{V}$ on
$\partial\mathcal{M}$ and a continuous vector field $\tilde{H}$ such that

(3.4)
$\begin{split}&\int_{G_{2}(\mathcal{M})}\operatorname{div}_{S}X(x)\mathop{}\\!\mathrm{d}\big{(}V(x,S)+aW(x,S)\big{)}=-\int_{\mathcal{M}}{X}\cdot{H}\mathop{}\\!\mathrm{d}\lVert
V\rVert\\\
&\qquad-\int_{\mathcal{M}}{X}\cdot{\tilde{H}}\mathop{}\\!\mathrm{d}(\lVert
V\rVert+a\lVert
W\rVert)+\int_{\partial\mathcal{M}}{X}\cdot{N}\mathop{}\\!\mathrm{d}\sigma_{V}\qquad\forall
X\in\mathfrak{X}(\mathcal{M}),\end{split}$

where $\tilde{H}(x)=-N(x)\operatorname{div}_{\partial\mathcal{M}}N(x)$ for
every $x\in\partial\mathcal{M}$. Moreover, the following global and local
estimates hold:

(3.5) $\sigma_{V}(\partial\mathcal{M})\leq c\lVert
V\rVert(\mathcal{M})+\int_{\mathcal{M}}|H|\mathop{}\\!\mathrm{d}\lVert
V\rVert;$ (3.6) $\sigma_{V}\big{(}B_{r/2}(x_{0})\big{)}\leq\frac{c}{r}\lVert
V\rVert\big{(}B_{r}(x_{0})\big{)}+\int_{B_{r}(x_{0})}|H|\mathop{}\\!\mathrm{d}\lVert
V\rVert\qquad\forall x_{0}\in\partial\mathcal{M},\,\forall r\leq
R(\mathcal{M}),$

where the constant $c=c(\mathcal{M})$ depends only on the second fundamental
form of $\partial\mathcal{M}$ and $R(\mathcal{M})$ is the minimum radius of
curvature of $\partial\mathcal{M}$.

As in [15], a monotonicity formula at points on $\partial\mathcal{M}$ holds
for a pair $(V,W)$ that satisfies the contact angle condition. The following
monotonicity is slightly different from the one in [15], since it does not
involve reflections over $\partial\mathcal{M}$.

######  Proposition 3.2.

Suppose
$(V,W)\in\mathcal{V}_{2}(\mathcal{M})\times\mathcal{V}_{2}(\partial\mathcal{M})$
satisfies the contact angle condition with $H\in L^{p}(\mathcal{M},\lVert
V\rVert)$ for some $p\in(2,+\infty)$. Then there exists
$\Lambda=\Lambda\big{(}p,\mathcal{M},(\lVert V\rVert+a\lVert
W\rVert)(\mathcal{M}),\lVert H\rVert_{L^{p}}\big{)}>0$ such that, for all
$x_{0}\in\partial\mathcal{M}$ the function

(3.7) $\rho\longmapsto e^{\Lambda\rho}\left[\bigg{(}\frac{\lVert
V\rVert\big{(}B_{\rho}(x_{0})\big{)}+a\lVert
W\rVert\big{(}B_{\rho}(x_{0})\big{)}}{\rho^{2}}\bigg{)}^{\frac{1}{p}}+\Lambda\rho^{1-\frac{2}{p}}\right]$

is monotone increasing.

The proof is a straightforward modification of the one of [20, Corollary 4.8]
and is reported in Appendix A. If $H\in L^{\infty}(\mathcal{M},\lVert
V\rVert)$, we have the following result.

######  Proposition 3.3.

Suppose
$(V,W)\in\mathcal{V}_{2}(\mathcal{M})\times\mathcal{V}_{2}(\partial\mathcal{M})$
satisfies the contact angle condition with $H\in L^{\infty}(\mathcal{M},\lVert
V\rVert)$. Then there exists $\Lambda=\Lambda(\mathcal{M},\lVert
H\rVert_{L^{\infty}})>0$ such that, for all $x_{0}\in\partial\mathcal{M}$ the
function

(3.8) $\rho\longmapsto e^{\Lambda\rho}\frac{\lVert
V\rVert\big{(}B_{\rho}(x)\big{)}+a\lVert
W\rVert\big{(}B_{\rho}(x_{0})\big{)}}{\rho^{2}}$

is monotone increasing.

As above, the proof is very similar to the one of [20, Corollary 4.9] and is
sketched in Appendix A.

##  4\. Existence of non-trivial stationary pairs

###  4.1. Sweepouts

In this section we define the basic elements of the min-max procedure.

######  Definition 4.1.

A family $\\{\Sigma_{t}\\}_{t\in[0,1]}$ is a _parametrized smooth family of
surfaces_ if it satisfies the following properties:

  1. (1)

For every $t\in[0,1]$ $\Sigma_{t}$ is a closed subset of $\mathcal{M}$ of
finite $\mathcal{H}^{2}$-measure;

  2. (2)

For every $t\in[0,1]$ there exists a finite set $S_{t}$ such that
$\Sigma_{t}\setminus S_{t}$ is a smooth surface with boundary
$\partial\Sigma_{t}\subset\partial\mathcal{M}$ which is smooth outside of
$S_{t}$

  3. (3)

if $t\to s$, then $\sup_{x\in\Sigma_{t}}\operatorname*{dist}(x,\Sigma_{s})\to
0$;

  4. (4)

if $U\subset\joinrel\subset\mathcal{M}\setminus S_{s}$, then $\Sigma_{t}$
converges in the smooth sense to $\Sigma_{s}$ in $U$ as $t\to s$;

  5. (5)

the function $t\mapsto\mathcal{H}^{2}(\Sigma_{t})$ is continuous.

A first elementary consequence of the definition of parametrized family is
that the area of the surfaces converge also locally. This is useful in the
following, then we record it in the following lemma.

######  Lemma 4.1.

Let $\\{\Sigma_{t}\\}_{t\in[0,1]}$ be a smooth parametrized family of
surfaces. Then, for every open set $U\subset\mathcal{M}$, it holds

(4.1) $\lim_{t\to s}\mathcal{H}^{2}(\Sigma_{t}\cap
U)=\mathcal{H}^{2}(\Sigma_{s}\cap U).$

###### Proof.

For every $\varepsilon>0$, there exists
$W\subset\joinrel\subset\mathcal{M}\setminus S_{s}$ such that
$\mathcal{H}^{2}(\Sigma_{s}\cap W)>\mathcal{H}^{2}(\Sigma_{s})-\varepsilon$.
By smooth convergence in $W$, we have that

(4.2) $\lim_{t\to s}\mathcal{H}^{2}(\Sigma_{t}\cap U\cap
W)=\mathcal{H}^{2}(\Sigma_{s}\cap U\cap W),\qquad\lim_{t\to
s}\mathcal{H}^{2}(\Sigma_{t}\cap(W\setminus
U))=\mathcal{H}^{2}(\Sigma_{s}\cap(W\setminus U))$

Since $\mathcal{H}^{2}(\Sigma_{t})\to\mathcal{H}^{2}(\Sigma_{s})$ by the
choice of $W$ we have also $|\mathcal{H}^{2}(\Sigma_{t}\cap U\setminus
W)-\mathcal{H}^{2}(\Sigma_{s}\cap U\setminus W)|<\varepsilon$ for $|t-s|$
sufficiently small. This implies that $|\mathcal{H}^{2}(\Sigma_{t}\cap
U)-\mathcal{H}^{2}(\Sigma_{s}\cap U)|<\varepsilon$ for $|t-s|$ sufficiently
small. ∎

We often refer to a parametrized smooth family of surfaces simply as a family
of surfaces.

######  Definition 4.2 (Sweepout).

A pair $(\Sigma_{t},\Gamma_{t})_{t\in[0,1]}$ of parametrized family of
surfaces with (finite) singular sets $S_{t}$ and $G_{t}$ is called a
_sweepout_ if there exists a family of open sets
$\\{\Omega_{t}\\}_{t\in[0,1]}$ in the relative topology of $\mathcal{M}$ that
satisfy the following properties:

  1. (1)

for every $t\in[0,1]$ $\Sigma_{t}\subset\mathcal{M}$ and
$\Gamma_{t}\subset\partial\mathcal{M}$;

  2. (2)

for every $t\in[0,1]$
$\Sigma_{t}\cap\Gamma_{t}=\partial\Sigma_{t}=\partial\Gamma_{t}$ (here the
boundary is intended in the sense of differential topology);

  3. (3)

for every $t\in[0,1]$, $\partial_{i}\Omega_{t}\setminus\Sigma_{t}\subset
S_{t}$;

  4. (4)

for every $t\in[0,1]$,
$\big{(}\partial_{b}\Omega_{t}\setminus(\Sigma_{t}\cup\Gamma_{t})\big{)}\cup\big{(}(\Sigma_{t}\cap\partial\mathcal{M})\setminus\partial_{b}\Omega_{t}\big{)}\cup\big{(}\Gamma_{t}\setminus\partial_{b}\Omega_{t}\big{)}\subset
S_{t}\cup G_{t}$;

  5. (5)

$\Omega_{0}=\emptyset$ and $\Omega_{1}=\mathcal{M}$;

  6. (6)

if $t\to s$, then
$\operatorname{Vol}(\Omega_{t}\setminus\Omega_{s})+\operatorname{Vol}(\Omega_{s}\setminus\Omega_{t})\to
0$.

As it is stated in [5, Proposition 0.4] sweepouts always exist.

######  Definition 4.3 (Pair compatible with open set).

A pair of surfaces $(\Sigma,\Gamma)$ and an open set $\Omega$ are said to be
_compatible_ if the properties 1, 2, 3, 4 and 6 of Definition 4.2 holds with
$(\Sigma,\Gamma)$, $\Omega$, $S$, $G$ in place of
$(\Sigma_{t},\Gamma_{t})_{t\in[0,1]}$, $\\{\Omega_{t}\\}_{t\in[0,1]}$,
$S_{t}$, $G_{t}$.

######  Definition 4.4.

Two sweepouts $(\Sigma_{t},\Gamma_{t})_{t\in[0,1]}$ and
$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})_{t\in[0,1]}$ are _homotopic_ if there
exists a parametrized family of surfaces
$(\Sigma_{(t,s)},\Gamma_{(t,s)})_{(t,s)\in[0,1]^{2}}$ such that
$\Sigma_{(t,0)}=\Sigma_{t}$, $\Gamma_{(t,0)}=\Gamma_{t}$,
$\Sigma_{(t,1)}=\tilde{\Sigma}_{t}$ and $\Gamma_{(t,1)}=\tilde{\Gamma}_{t}$
for every $t\in[0,1]$.

######  Definition 4.5.

A non-empty set of sweepouts $\mathscr{S}$ is _homotopically closed_ if it
contains the homotopy class of all its elements.

From now on, with $\mathscr{S}$ we always denote a homotopically closed set of
sweepouts.

######  Definition 4.6.

If $\mathscr{S}$ is an homotopically closed set of sweepouts, the _min-max
value of $\mathscr{S}$_ is

(4.3)
${m_{0}}(\mathscr{S})=\inf\\{\max_{t\in[0,1]}F_{a}(\Sigma_{t},\Gamma_{t})\mid(\Sigma_{t},\Gamma_{t})_{t\in[0,1]}\in\mathscr{S}\\}$

######  Definition 4.7.

A sequence
$\\{(\Sigma_{t}^{i},\Gamma_{t}^{i})_{t\in[0,1]}\\}^{i\in\mathbb{N}}$ of
sweepouts in $\mathscr{S}$ is a _minimizing sequence_ if

(4.4)
$\lim_{i\to\infty}\max_{t\in[0,1]}F_{a}(\Sigma_{t}^{i},\Gamma_{t}^{i})={m_{0}}(\mathscr{S}).$

If $(\Sigma_{t}^{i},\Gamma_{t}^{i})_{t\in[0,1]}^{i\in\mathbb{N}}$ is a
minimizing sequence of sweepouts, a sequence
$(\Sigma^{i},\Gamma^{i}):=(\Sigma_{t_{i}}^{i},\Gamma_{t_{i}}^{i})$ (where
$t_{i}\in[0,1]$ for every $i\in\mathbb{N}$) is a _min-max sequence_ if

(4.5)
$\lim_{i\to\infty}F_{a}\big{(}\Sigma^{i},\Gamma^{i}\big{)}={m_{0}}(\mathscr{S}),$

###  4.2. Pull-tight procedure

We now want to prove that there exists a minimizing sequence of sweepouts such
that every min-max sequence converges to a pair which is stationary for the
capillarity functional. From now on, we consider fixed the homotopically
closed set of sweepouts $\mathscr{S}$ and we call
${m_{0}}={m_{0}}(\mathscr{S})$.

######  Theorem 4.2 (Pull-tight procedure).

There exists a minimizing sequence of sweepouts
$\\{(\Sigma_{t}^{k},\Gamma_{t}^{k})\\}^{k\in\mathbb{N}}$ such that every min-
max sequence converges, up to subsequences, to a pair
$(V,W)\in\mathcal{V}_{2}(\mathcal{M})\times\mathcal{V}_{2}(\partial\mathcal{M})$
which is stationary for the capillarity functional.

###### Proof.

We follow the construction in [3]. Let us fix a minimizing sequence of
sweepouts $\\{(\Sigma_{t}^{k},\Gamma_{t}^{k})\\}^{k\in\mathbb{N}}$. The proof
is based on the construction of a deformation of the minimizing sequence; this
deformation reduce the mass of the surfaces of a quantity that depends on how
much the pair is “far” from being stationary. This deformed minimizing
sequence has the desired property. We call

(4.6)
$A^{2{m_{0}}}=\\{(V,W)\in\mathcal{V}_{2}(\mathcal{M})\times\mathcal{V}_{2}(\partial\mathcal{M})\mid\lVert
V\rVert(\mathcal{M})\leq 2{m_{0}},\lVert
W\rVert(\partial\mathcal{M})\leq\mathcal{H}^{2}(\partial\mathcal{M})\\}$

and

(4.7) $A_{\infty}=\\{(V,W)\in A^{2{m_{0}}}\mid(V,W)\text{ is stationary for
}F_{a}\\}.$

Since the weak*-topology is metrizable on bounded sets, we choose a distance
which induces it on $\\{V\in\mathcal{V}_{2}(\mathcal{M})\mid\lVert
V\rVert(\mathcal{M})\leq 2{m_{0}}\\}$ and
$\\{W\in\mathcal{V}_{2}(\partial\mathcal{M})\mid\lVert
W\rVert(\partial\mathcal{M})\leq\mathcal{H}^{2}(\partial\mathcal{M})\\}$ and
we call $\mathcal{D}$ the distance on the product space $A^{2{m_{0}}}$ which
induces the product topology. Up to scaling $\mathcal{D}$, we can assume that
$\operatorname*{diam}A^{2{m_{0}}}=1$. Moreover $A^{2{m_{0}}}$ is compact since
it is a product of compact spaces. Since the property of being stationary is
preserved under weak*-convergence of varifolds, it follows that $A_{\infty}$
is also a compact subset of $A^{2{m_{0}}}$. We call $\mathcal{B}_{r}(V,W)$ the
ball of radius $r$ and center $(V,W)$ with respect the distance $\mathcal{D}$.

For every $i\in\mathbb{N}$, we call

(4.8) $A_{i}=\Big{\\{}(V,W)\in
A^{2{m_{0}}}\mid\frac{1}{2^{i+1}}\leq\mathcal{D}\big{(}(V,W),A_{\infty}\big{)}\leq\frac{1}{2^{i}}\Big{\\}}.$

Since every $A_{i}$ is closed, it is compact.

  1. Step 1:

We claim that for every $i\in\mathbb{N}$ there exists a constant $c_{i}>0$
such that, for every $(V,W)\in A_{i}$, there exists
$X_{(V,W)}\in\mathfrak{X}_{t}(\mathcal{M})$ such that $|X_{(V,W)}(x)|\leq 1$
for every $x\in\mathcal{M}$ and

(4.9)
$\begin{split}\delta_{F_{a}}(V,W)[X_{(V,W)}]=&\int_{G_{2}(\mathcal{M})}\operatorname{div}_{S}X_{(V,W)}(x)\mathop{}\\!\mathrm{d}V(x,S)+a\int_{G_{2}(\partial\mathcal{M})}\operatorname{div}_{\partial\mathcal{M}}X_{(V,W)}(x)\mathop{}\\!\mathrm{d}W(x,T_{x}\partial\mathcal{M})\\\
\leq&-c_{i}.\end{split}$

Indeed, suppose by contradiction that the claim is false for some
$i\in\mathbb{N}$; thus there exists a sequence of pairs $(V_{j},W_{j})_{j}$ in
$A_{i}$ such that for every $j$ and for every
$X\in\mathfrak{X}_{t}(\mathcal{M})$ such that $|X|\leq 1$ we have

(4.10)
$\Big{|}\int_{G_{2}(\mathcal{M})}\operatorname{div}_{S}X(x)\mathop{}\\!\mathrm{d}V_{j}(x,S)+a\int_{G_{2}(\partial\mathcal{M})}\operatorname{div}_{\partial\mathcal{M}}X(x)\mathop{}\\!\mathrm{d}W_{j}(x,T_{x}\partial\mathcal{M})\Big{|}\leq\frac{1}{j}.$

By compactness of $A_{i}$, $(V_{j},W_{j})$ converges, up to subsequences, to
$(V,W)\in A_{i}$. By the previous equation, it follows that $(V,W)$ is
stationary for $F_{a}$, which contradicts $(V,W)\in A_{i}$. This proves the
claim.

  2. Step 2:

The map $(V,W)\in A^{2{m_{0}}}\setminus A_{\infty}\mapsto
X_{(V,W)}\in\mathfrak{X}_{t}(\mathcal{M})$ is not necessarily continuous. We
want now to use it to construct a continuous one with respect the
$C^{1}$-topology in $\mathfrak{X}_{t}(\mathcal{M})$ that satisfies an
inequality on the first variation of $F_{a}$ similar to (4.9) in a
neighborhood of $(V,W)$.

We first observe that, by continuity of the first variation under varifold
convergence, for every $i\in\mathbb{N}$ and every $(V,W)\in A_{i}$ there
exists $r_{(V,W)}\in(0,\frac{1}{2^{i+2}})$ such that

(4.11)
$\int_{G_{2}(\mathcal{M})}\operatorname{div}_{S}X_{(V,W)}(x)\mathop{}\\!\mathrm{d}\tilde{V}(x,S)+a\int_{G_{2}(\partial\mathcal{M})}\operatorname{div}_{\partial\mathcal{M}}X_{(V,W)}(x)\mathop{}\\!\mathrm{d}\tilde{W}(x,T_{x}\partial\mathcal{M})\leq-\frac{c_{i}}{2}$

whenever $\mathcal{D}\big{(}(V,W),(\tilde{V},\tilde{W})\big{)}\leq r_{(V,W)}$.
Thus the family

(4.12) $\mathcal{F}_{i}=\\{\mathcal{B}_{r_{(V,W)}}(V,W)\mid(V,W)\in A_{i}\\}$

(we recall that $\mathcal{B}_{r}(V,W)$ is the ball of center $(V,W)$ and
radius $r$ with respect the distance $\mathcal{D}$) is an open cover of
$A_{i}$ and by compactness there exists a finite number $J_{i}$ of pairs
$\\{(V_{i}^{j},W_{i}^{j})\\}^{j=1,\dots,J_{i}}$ such that

(4.13)
$A_{i}\subset\bigcup_{j=1}^{J_{i}}\mathcal{B}_{\frac{r_{i,j}}{2}}(V_{i}^{j},W_{i}^{j})$

(where we call $r_{i,j}:=r_{(V_{i}^{j},W_{i}^{j})}$). For simplicity of
notation, we call
$\mathcal{B}_{i,j}=\mathcal{B}_{r_{i,j}}(V_{i}^{j},W_{i}^{j})$ and
$\tilde{\mathcal{B}}_{i,j}=\mathcal{B}_{\frac{r_{i,j}}{2}}(V_{i}^{j},W_{i}^{j})$.
We underline that by the choice of $r_{(V,W)}$ it follows that

(4.14) $A_{h}\cap\mathcal{B}_{i,j}=\emptyset\qquad\text{whenever }|i-h|\geq
2.$

By the assumptions, the family

(4.15) $\mathcal{F}=\\{\tilde{\mathcal{B}}_{i,j}\mid
i\in\mathbb{N},j=1,\dots,J_{i}\\}$

is a locally finite covering of $A^{2{m_{0}}}\setminus A_{\infty}$. We can
construct a partition of the unity $\psi_{i,j}$ with respect to this covering
and define

(4.16)
$Y_{(V,W)}=\mathcal{D}\big{(}(V,W),A_{\infty}\big{)}\sum_{\begin{subarray}{c}i\in\mathbb{N}\\\
j=1,\dots,J_{i}\end{subarray}}\psi_{i,j}(V,W)X_{(V_{i}^{j},W_{i}^{j})}\qquad\forall(V,W)\in
A^{2{m_{0}}}.$

The map $(V,W)\mapsto Y_{(V,W)}$ is continuous with respect the $C^{1}$
topology of $\mathfrak{X}_{t}(\mathcal{M})$ and $Y_{(V,W)}=0$ if $(V,W)\in
A_{\infty}$. If we define for every $(V,W)\in A^{2{m_{0}}}$

(4.17)
$\rho(V,W)=\min\Big{\\{}\frac{r_{i,j}}{2}\mid(V,W)\in\tilde{\mathcal{B}}_{i,j}\Big{\\}},$

by definition we have $\rho(V,W)\leq\frac{1}{2^{i+3}}$ whenever $(V,W)\in
A_{i}$. Hence, if $(V,W)\in A_{h}$, for every
$(\tilde{V},\tilde{W})\in\mathcal{B}_{\rho(V,W)}(V,W)$ we have that

(4.18)
$\begin{split}\delta_{F_{a}}(\tilde{V},\tilde{W})(Y_{(V,W)})&=\mathcal{D}\big{(}(\tilde{V},\tilde{W}),A_{\infty}\big{)}\sum_{\begin{subarray}{c}|i-h|\leq
1\\\
j=1,\dots,J_{i}\end{subarray}}\psi_{i,j}(\tilde{V},\tilde{W})\delta_{F_{a}}(\tilde{V},\tilde{W})(X_{(V_{i}^{j},W_{i}^{j})})\\\
&\leq-\frac{1}{2^{h+2}}\min\\{c_{h-1},c_{h},c_{h+1}\\},\end{split}$

by (4.11), since $\mathcal{B}_{\rho(V,W)}(V,W)\subset\mathcal{B}_{i,j}$ for
every $i,j$ such that $(V,W)\in\tilde{\mathcal{B}}_{i,j}$.

By compactness again, there exist two continuous functions
$d\colon[0,1]\to(0,1]$ and $f\colon(0,1]\to(0,+\infty)$ such that

(4.19) $\displaystyle d(s)\leq 2^{-(i+3)}\qquad\forall
s\in[2^{-(i+1)},2^{-i}],$ (4.20)
$\displaystyle\delta_{F_{a}}(\tilde{V},\tilde{W})(Y_{(V,W)})\leq-f\big{(}\mathcal{D}((V,W),A_{\infty})\big{)}\quad\text{whenever}\quad\mathcal{D}\big{(}(V,W),(\tilde{V},\tilde{W})\big{)}<d\big{(}\mathcal{D}((V,W),A_{\infty})\big{)}.$

  3. Step 3:

We now want to use the vector field-valued map constructed to deform the
minimizing sequence of sweepouts
$(\Sigma_{t}^{k},\Gamma_{t}^{k})^{k\in\mathbb{N}}$ in a new minimizing
sequence $(\tilde{\Sigma}_{t}^{k},\tilde{\Gamma}_{t}^{k})^{k\in\mathbb{N}}$ in
$\mathscr{S}$.

If $(V,W)\in A^{2{m_{0}}}$, we define
$(V_{t},W_{t})=\big{(}(\psi_{(V,W),t})_{\sharp}V,(\psi_{(V,W),t})_{\sharp}W\big{)}$,
where $\psi_{(V,W),t}$ is the flow map of $Y_{(V,W)}$ at the time $t$. Thus
(4.20) yields the existence of a maximum $t_{0}(V,W)>0$ such that, for any
$t\in[0,t_{0}(V,W)]$ we have

(4.21)
$\delta_{F_{a}}(V_{t},W_{t})(Y_{(V,W)})\leq-f\big{(}\mathcal{D}((V,W),A_{\infty})\big{)}\quad\text{and}\quad\mathcal{D}((V_{t},W_{t}),(V,W))\leq
d\big{(}\mathcal{D}((V,W),A_{\infty})\big{)}.$

Again by compactness of the sets $A_{i}$ there exists a continuous function
$T\colon(0,1]\to(0,1]$ such that, for every $(V,W)\in A^{2{m_{0}}}$ and for
every $t\in[0,T\big{(}\mathcal{D}((V,W),A_{\infty})\big{)}]$, it holds

(4.22)
$\displaystyle\delta_{F_{a}}(V_{t},W_{t})(Y_{(V,W)})\leq-f\big{(}\mathcal{D}((V,W),A_{\infty})\big{)},$
(4.23) $\displaystyle\mathcal{D}((V_{t},W_{t}),(V,W))\leq
d\big{(}\mathcal{D}((V,W),A_{\infty})\big{)}\stackrel{{\scriptstyle\eqref{eq:definition_distance_deformation_pulltight}}}{{<}}\frac{\mathcal{D}((V,W),A_{\infty})}{4}.$

Hence, setting $\beta=\mathcal{D}((V,W),A_{\infty})$

(4.24)
$F_{a}(V_{T(\beta)},W_{T(\beta)})-F_{a}(V,W)\leq\int_{0}^{T(\beta)}\delta_{F_{a}}(V_{t},W_{t})(Y_{(V,W)})\mathop{}\\!\mathrm{d}t\leq-T(\beta)f\big{(}\mathcal{D}((V,W),A_{\infty})\big{)}.$

We can re-normalize the flows $\psi_{(V,W),t}$ by defining
$\Psi_{(V,W),t}(x)=\psi_{(V,W),T(\beta)t}$, where again
$\beta=\mathcal{D}((V,W),A_{\infty})$. Thus, re-defining
$(V_{t},W_{t})=\big{(}(\Psi_{(V,W),t})_{\sharp}V,\Psi_{(V,W),t})_{\sharp}W\big{)}$
we have that there exists $g\colon(0,1]\to(0,+\infty)$ such that $\lim_{s\to
0}g(s)=0$ and

(4.25)
$F_{a}(V_{1},W_{1})-F_{a}(V,W)\leq-g\big{(}\mathcal{D}((V,W),A_{\infty})\big{)},\qquad\mathcal{D}\big{(}(V_{1},W_{1}),(V,W)\big{)}<\frac{\mathcal{D}((V,W),A_{\infty})}{4}.$

Now we choose a minimizing sequence of sweepouts
$\\{(\Sigma_{t}^{k},\Gamma_{t}^{k})\\}^{k\in\mathbb{N}}$ in $\mathscr{S}$ such
that

(4.26)
$\max_{t\in[0,1]}F_{a}(\Sigma_{t}^{k},\Gamma_{t}^{k})\leq{m_{0}}(\mathscr{S})+\frac{1}{k}\qquad\forall
k\in\mathbb{N}.$

Let us fix $k\in\mathbb{N}$. We would like to deform every
$(\Sigma_{t}^{k},\Gamma_{t}^{k})$ by the flow
$\Psi_{(\Sigma_{t}^{k},\Gamma_{t}^{k}),1}$, but since the map

(4.27) $Y_{k}\colon t\in[0,1]\mapsto
Y_{(\Sigma_{t}^{k},\Gamma_{t}^{k})}\in\mathfrak{X}_{t}(\mathcal{M})$

is only continuous, the family of pairs
$t\mapsto(\Psi_{(\Sigma_{t}^{k},\Gamma_{t}^{k}),1})_{\sharp}(\Sigma_{t}^{k},\Gamma_{t}^{k})$
may not belong to $\mathscr{S}$. Therefore we have to smooth the map $Y_{k}$.
To this aim let us consider a smooth map
$Z_{k}\colon[0,1]\to\mathfrak{X}_{t}(\mathcal{M})$. If $\sup_{t\in[0,1]}\lVert
Y_{k}-Z_{k}\rVert_{C^{1}}$ is sufficiently small and if $Z_{k}(t)=0$ when
$Y_{k}(t)=0$, then for every $t\in[0,1]$ we have

(4.28)
$F_{a}(\tilde{\Sigma}_{t}^{k},\tilde{\Gamma}_{t}^{k})-F_{a}(\Sigma_{t}^{k},\Gamma_{t}^{k})\leq-\frac{g\big{(}\mathcal{D}((\Sigma_{t}^{k},\Gamma_{t}^{k}),A_{\infty})\big{)}}{2},\qquad\mathcal{D}\big{(}(\tilde{\Sigma}_{t}^{k},\tilde{\Gamma}_{t}^{k}),(\Sigma_{t}^{k},\Gamma_{t}^{k})\big{)}\leq\frac{\mathcal{D}\big{(}(\Sigma_{t}^{k},\Gamma_{t}^{k}),A_{\infty}\big{)}}{4}.$

where
$(\tilde{\Sigma}_{t}^{k},\tilde{\Gamma}_{t}^{k})=\Phi_{1}(t)_{\sharp}(\Sigma_{t}^{k},\Gamma_{t}^{k})$
and $\Phi_{1}(t)$ is the flow map of $Z_{k}(t)$ evaluated at the time $1$.
Hence $(\tilde{\Sigma}_{t}^{k},\tilde{\Gamma}_{t}^{k})_{t\in
I}\in\mathscr{S}$. By repeating the same procedure for every $k\in\mathbb{N}$
we obtain a new sequence of sweepouts
$(\tilde{\Sigma}_{t}^{k},\tilde{\Gamma}_{t}^{k})^{k\in\mathbb{N}}$. By (4.28)
we have that this sequence is again a minimizing sequence.

  4. Step 4:

We claim that
$(\tilde{\Sigma}_{t}^{k},\tilde{\Gamma}_{t}^{k})^{k\in\mathbb{N}}$ has the
desired property, that is, every min-max sequence converges to a stationary
pair for $F_{a}$. To prove this, suppose by contradiction that the claim is
false and let
$(\tilde{\Sigma}_{t_{k}}^{k},\tilde{\Gamma}_{t_{k}}^{k})^{k\in\mathbb{N}}$ a
min-max sequence such that

(4.29)
$\limsup_{k\to\infty}\mathcal{D}((\tilde{\Sigma}_{t_{k}}^{k},\tilde{\Gamma}_{t_{k}}^{k}),A_{\infty})=\ell>0.$

Up to passing to a subsequence we can assume that the above upper limit is in
fact a limit. By the second relation in (4.28), we have that, for $k$
sufficiently large,

(4.30)
$\mathcal{D}\big{(}(\Sigma_{t_{k}}^{k},\Gamma_{t_{k}}^{k}),A_{\infty}\big{)}\geq\frac{\ell}{2}.$

By the first inequality in (4.28) we get

(4.31) $F_{a}(\Sigma_{t_{k}}^{k},\Gamma_{t_{k}}^{k})\geq
F_{a}(\tilde{\Sigma}_{t_{k}}^{k},\tilde{\Gamma}_{t_{k}}^{k})+\frac{1}{2}g\Big{(}\frac{\ell}{2}\Big{)}\xrightarrow{k\to\infty}{m_{0}}+\frac{1}{2}g\Big{(}\frac{\ell}{2}\Big{)}>{m_{0}}.$

Thus

(4.32)
$\liminf_{k\to\infty}\max_{t\in[0,1]}F_{a}(\Sigma_{t}^{k},\Gamma_{t}^{k})\geq\liminf_{k\to\infty}F_{a}(\Sigma_{t_{k}}^{k},\Gamma_{t_{k}}^{k})\geq{m_{0}}+\frac{1}{2}g\Big{(}\frac{\ell}{2}\Big{)}$

which contradicts the fact that the sequence $(\Sigma_{t}^{k},\Gamma_{t}^{k})$
was minimizing in $\mathscr{S}$.

∎

In order to prove that the limit pair of every min-max sequence is non-trivial
we need the following proposition.

######  Proposition 4.3.

If $\mathscr{S}$ is an homotopically closed set of sweepouts, then
$a\mathcal{H}^{2}(\partial\mathcal{M})<{m_{0}}(\mathscr{S})$.

###### Proof.

We want to show that there exists $\eta>0$ such that, for every sweepout
$(\Sigma_{t},\Gamma_{t})_{t\in[0,1]}\in\mathscr{S}$, it holds
$\max_{t\in[0,1]}F_{a}(\Sigma_{t},\Gamma_{t})\geq
a\mathcal{H}^{2}(\partial\mathcal{M})+\eta$.

To do so, we are going to prove that there exists $\varepsilon>0$ and a
differentiable function $f\colon[0,\varepsilon)\to\mathbb{R}^{+}$ such that,
for every sweepout $(\Sigma_{t},\Gamma_{t})$, $f(0)=0$, $f^{\prime}(s)>0$ for
$s\in(0,\varepsilon)$ and

(4.33) $F_{a}(\Sigma_{t},\Gamma_{t})\geq
a\mathcal{H}^{2}(\partial\mathcal{M})+f(|\mathcal{M}\setminus\Omega_{t}|)\qquad\forall
t\in[0,1]\text{ s.t. }|\mathcal{M}\setminus\Omega_{t}|\leq\varepsilon.$

This clearly implies the result. To prove the existence of such $f$, we write
$F_{a}$ in a different way, and next we use the relative isoperimetric
inequality to obtain a lower bound on $F_{a}(\Sigma_{t},\Gamma_{t})$.

We fix a vector field $X\in\mathfrak{X}(\mathcal{M})$ such that $|X(x)|\leq 1$
for all $x\in\mathcal{M}$ and $X(x)=N(x)$ (the exterior unit normal to
$\mathcal{M}$) for all $x\in\partial\mathcal{M}$; the $C^{1}$-norm of $X$
depends only on $\mathcal{M}$. Using $X$ we can write
$F_{a}(\Sigma_{t},\Gamma_{t})$ as

(4.34)
$\begin{split}F_{a}(\Sigma_{t},\Gamma_{t})=&\mathcal{H}^{2}(\Sigma_{t})+a\mathcal{H}^{2}(\Gamma_{t})\\\
=&\mathcal{H}^{2}(\Sigma_{t})+a\int_{\Sigma_{t}\cup\Gamma_{t}}{X}\cdot{\nu_{\Sigma}}\mathop{}\\!\mathrm{d}\mathcal{H}^{2}-a\int_{\Sigma_{t}}{X}\cdot{\nu_{\Sigma}}\mathop{}\\!\mathrm{d}\mathcal{H}^{2}\\\
=&\int_{\Sigma_{t}}(1-a{X}\cdot{\nu_{\Sigma}})\mathop{}\\!\mathrm{d}\mathcal{H}^{2}+a\int_{\Omega_{t}}\operatorname{div}X(x)\mathop{}\\!\mathrm{d}x.\end{split}$

Since up to sets of $\mathcal{H}^{2}$-measure $0$ it holds
$\Sigma_{t}\supset\partial\Omega_{t}\cap{\kern
0.0pt\mathcal{M}}^{\mathrm{o}}$, by the relative isoperimetric inequality
there exists a constant $c(\mathcal{M})>0$ such that

(4.35)
$\mathcal{H}^{2}(\Sigma_{t})\geq\mathcal{H}^{2}(\partial\Omega_{t}\cap{\kern
0.0pt\mathcal{M}}^{\mathrm{o}})\geq
c(\mathcal{M})\min\\{|\Omega_{t}|,|\mathcal{M}\setminus\Omega_{t}|\\}^{\frac{2}{3}}.$

Thus, by (4.34) and $1-aX\cdot\nu_{\Sigma}\geq 1-a$, for every $t\in[0,1]$ we
have

(4.36)
$\begin{split}F_{a}(\Sigma_{t},\Gamma_{t})\geq&c(\mathcal{M})(1-a)\min\\{|\Omega_{t}|,|\mathcal{M}\setminus\Omega_{t}|\\}^{\frac{2}{3}}+a\int_{\mathcal{M}}\operatorname{div}X(x)\mathop{}\\!\mathrm{d}x-a\int_{\mathcal{M}\setminus\Omega_{t}}\operatorname{div}X(x)\mathop{}\\!\mathrm{d}x\\\
=&c(\mathcal{M})(1-a)\min\\{|\Omega_{t}|,|\mathcal{M}\setminus\Omega_{t}|\\}^{\frac{2}{3}}+a\mathcal{H}^{2}(\partial\mathcal{M})-a\int_{\mathcal{M}\setminus\Omega_{t}}\operatorname{div}X(x)\mathop{}\\!\mathrm{d}x.\end{split}$

If $\varepsilon>0$ is sufficently small, for all $t\in[0,1]$ such that
$|\mathcal{M}\setminus\Omega_{t}|\leq\varepsilon$, the previous inequality
becomes

(4.37)
$\begin{split}F_{a}(\Sigma_{t},\Gamma_{t})\geq&a\mathcal{H}^{2}(\partial\mathcal{M})+c_{1}|\mathcal{M}\setminus\Omega_{t}|^{\frac{2}{3}}-c_{2}|\mathcal{M}\setminus\Omega_{t}|\\\
=&a\mathcal{H}^{2}(\partial\mathcal{M})+f(|\mathcal{M}\setminus\Omega_{t}|),\end{split}$

where $c_{1},c_{2}$ are positive constants depending on $\mathcal{M},a$.
Clearly $f\in C^{1}\big{(}(0,\varepsilon)\big{)}$, $f(0)=0$ and, up to
choosing $\varepsilon>0$ even smaller, we have and $f^{\prime}(s)>0$ for all
$s\in(0,\varepsilon)$. ∎

##  5\. Existence of almost minimizing min-max sequences

We now introduce the notion of _almost minimizing_ pair $(\Sigma,\Gamma)$.

######  Definition 5.1.

Let $\varepsilon>0$, $U\subset\mathcal{M}$ be open and let $(\Sigma,\Gamma)$
be a pair of surfaces that is compatible with an open set
$\Omega\subset\mathcal{M}$ (in the sense of Definition 4.3). We say that
$(\Sigma,\Gamma)$ is _$\varepsilon$ -almost minimizing_ in $U$ if there exists
no parametrized pair of surfaces $(\Phi_{t},\Psi_{t})_{t\in[0,1]}$ and no
family of open sets $\\{\Omega_{t}\\}_{t\in[0,1]}$ such that:

  1. (1)

$(\Phi_{t},\Psi_{t})_{t\in[0,1]}$ and $\\{\Omega_{t}\\}_{t\in[0,1]}$ satisfy
properties 1, 2, 3, 4 and 6 of Definition 4.2;

  2. (2)

$\Omega_{0}=\Omega$ and $\Omega_{t}\setminus U=\Omega\setminus U$ for every
$t\in[0,1]$;

  3. (3)

$\Phi_{0}=\Sigma$ and $\Phi_{t}\setminus U=\Sigma\setminus U$ for every
$t\in[0,1]$;

  4. (4)

$\Psi_{0}=\Gamma$ and $\Psi_{t}\setminus U=\Gamma\setminus U$ for every
$t\in[0,1]$;

  5. (5)

$F_{a}(\Phi_{t},\Psi_{t})\leq F_{a}(\Sigma,\Gamma)+\frac{\varepsilon}{8}$ for
every $t\in[0,1]$;

  6. (6)

$F_{a}(\Phi_{1},\Psi_{1})\leq F_{a}(\Sigma,\Gamma)-\varepsilon$.

A sequence of pairs $\\{(\Sigma^{k},\Gamma^{k})\\}^{k}$ is _almost minimizing_
in $U$ if there exists a sequence $\varepsilon_{k}\downarrow 0$ such that each
$(\Sigma^{k},\Gamma^{k})$ is $\varepsilon_{k}$-almost minimizing in $U$.

Roughly speaking, a pair is almost minimizing in $U$ if it cannot be
continuously deformed in $U$ into one other pair with $F_{a}$ smaller than
$\varepsilon$ without passing through a pair with $F_{a}$ larger than
$\varepsilon/8$. The main result of this section is the following.

######  Proposition 5.1.

Let $\mathscr{S}$ a homotopically closed set of sweepouts; then there exist a
min-max sequence
$(\Sigma^{k},\Gamma^{k})^{k\in\mathbb{N}}=(\Sigma_{t_{k}}^{k},\Gamma_{t_{k}}^{k})^{k\in\mathbb{N}}$
and a function $r\colon\mathcal{M}\to(0,+\infty)$ that satisfy:

  1. (1)

$(\Sigma^{k},\Gamma^{k})$ converges, up to subsequences, to a pair
$(V,W)\in\mathcal{V}_{2}(\mathcal{M})\times\mathcal{V}_{2}(\partial\mathcal{M})$
which is stationary for $F_{a}$;

  2. (2)

There exists $\varepsilon_{k}\downarrow 0$ such that for every
$x\in\mathcal{M}$ and for every annulus
$\operatorname*{An}\in\mathcal{AN}_{r(x)}({x})$, there exists $N\in\mathbb{N}$
such that the sequence $(\Sigma^{k},\Gamma^{k})^{k>N}$ is
$\varepsilon_{k}$-almost minimizing in $\operatorname*{An}$.

The strategy of the proof is as follows: we have an homotopically closed set
of sweepouts and a minimizing sequence
$\\{(\Sigma_{t}^{k},\Gamma_{t}^{k})\\}^{k\in\mathbb{N}}$ such that any min-max
sequence converges to a stationary varifold, which is given by Theorem 4.2. We
prove that, if there were no almost minimizing min-max sequence, then the
minimizing sequence can be deformed in another one
$(\tilde{\Sigma}_{t}^{k},\tilde{\Gamma}_{t}^{k})^{k\in\mathbb{N}}$ such that
$\lim_{k}\max_{t\in[0,1]}F_{a}(\tilde{\Sigma}_{t}^{k},\tilde{\Gamma}_{t}^{k})<{m_{0}}$
and this would be a contradiction.

###  5.1. Deforming a sweepout near a non-almost minimizing slice

If a slice $(\Sigma_{t_{0}},\Gamma_{t_{0}})$ of a sweepout is not
$\varepsilon$-almost minimizing in some open set $U$, it can be deformed in
another pair $(\tilde{\Sigma},\tilde{\Gamma})$, with
$F_{a}(\tilde{\Sigma},\tilde{\Gamma})\leq
F_{a}(\Sigma_{t_{0}},\Gamma_{t_{0}})-\varepsilon$ by a parametrized family of
pairs $(\Phi_{t},\Psi_{t})_{t\in[0,1]}$ such that
$(\Phi_{0},\Psi_{0})=(\Sigma_{t_{0}},\Gamma_{t_{0}})$ and
$(\Phi_{1},\Psi_{1})=(\tilde{\Sigma},\tilde{\Gamma})$.

We cannot simply replace $(\Sigma_{t_{0}},\Gamma_{t_{0}})$ with
$(\tilde{\Sigma},\tilde{\Gamma})$, first of all because in this way we no
longer have a smooth parametrized family of pairs and because we need to
deform also the slices for times near $t_{0}$ to build the contradiction
argument stated above. Hence we need the following lemma, which is similar to
[5, Lemma 3.1] and [4, Lemma 5.1].

######  Lemma 5.2.

Let $(\Sigma_{t},\Gamma_{t})_{t\in[0,1]}$ be a sweepout and suppose
$t_{0}\in(0,1)$ is such that $(\Sigma_{t_{0}},\Gamma_{t_{0}})$ is not
$\varepsilon$-almost minimizing in $U\subset\mathcal{M}$ for some
$\varepsilon>0$ and some open set $U$. Then for every open set
$V\subset\mathcal{M}$ with $U\subset\joinrel\subset V$, there exists
$\bar{\eta}>0$ such that, whenever we choose $\eta\leq\bar{\eta}$ and $0\leq
t_{0}-\eta<a<a^{\prime\prime}<b^{\prime\prime}<b<t_{0}+\eta\leq 1$, there
exists a sweepout $(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})_{t\in[0,1]}$ that
is homotopic to $(\Sigma_{t},\Gamma_{t})_{t\in[0,1]}$ and satisfies the
following properties:

  1. (1)

$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})=(\Sigma_{t},\Gamma_{t})$ for every
$t\in[0,a]\cup[b,1]$;

  2. (2)

$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})\setminus
V=(\Sigma_{t},\Gamma_{t})\setminus V$ for every $t\in[0,1]$;

  3. (3)

$F_{a}(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})<F_{a}(\Sigma_{t},\Gamma_{t})+\frac{\varepsilon}{4}$
for every $t\in[0,1]$;

  4. (4)

$F_{a}(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})<F_{a}(\Sigma_{t},\Gamma_{t})-\frac{\varepsilon}{2}$
for every $t\in[a^{\prime\prime},b^{\prime\prime}]$.

###### Proof.

The number $\bar{\eta}$ will be chosen at the end of the proof.

  1. Step 1:

The first step in the proof is to build the new sweepout
$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})$; this is achieved first of all by
“deforming the time inside $U$”, to have that the slice is frozen on
$(\Sigma_{t_{0}},\Gamma_{t_{0}})$ in an open neighborhood of $t_{0}$, whereas
in $\mathcal{M}\setminus V$ nothing changes. While
$(\Sigma_{t_{0}},\Gamma_{t_{0}})$ remains frozen in $U$, we deform it therein
by the family of $(\Phi_{t},\Psi_{t})$ given by Definition 5.1 which
contradicts the $\varepsilon$-almost minimality of
$(\Sigma_{t_{0}},\Gamma_{t_{0}})$. Of course, this requires some
technicalities.

Let $\\{\Omega_{t}\\}_{t\in[0,1]}$ be the family of open sets relative to
$(\Sigma_{t},\Gamma_{t})$. Let us fix $V$ and choose two open sets $A,B$ such
that

(5.1) $U\subset\joinrel\subset A\subset\joinrel\subset B\subset\joinrel\subset
V$

and such that both $\Sigma_{t_{0}}$ and $\Gamma_{t_{0}}$ are smooth surfaces
(with eventually non-empty smooth boundary) in $C:=B\setminus A$ (this is
possibile since the sets of singularities $S_{t_{0}},G_{t_{0}}$ of
$\Sigma_{t_{0}}$ and $\Gamma_{t_{0}}$ are finite). Let us fix two non-negative
smooth functions $\chi_{A},\chi_{B}$ such that

     * •

$\chi_{A}\in C_{c}^{\infty}(B)$, $\chi_{B}\in
C_{c}^{\infty}(\mathcal{M}\setminus\mkern
1.0mu\overline{\mkern-1.0muA\mkern-1.0mu}\mkern 1.0mu)$;

     * •

$\chi_{A}(x)+\chi_{B}(x)\equiv 1$ for all $x\in\mathcal{M}$.

Since $\partial\mathcal{M}$ is $C^{2,\alpha}$, there exists $r>0$ such that
the distance function $\operatorname*{dist}(x,\partial\mathcal{M})$ is
$C^{2,\alpha}$ if $\operatorname*{dist}(x,\partial\mathcal{M})\leq 2r$. Let
$d(\cdot)$ a smooth approximation of
$\operatorname*{dist}(\cdot,\partial\mathcal{M})$ that coincides with
$\operatorname*{dist}(\cdot,\partial\mathcal{M})$ whenever
$\operatorname*{dist}(x,\partial\mathcal{M})\leq r$. By the regularity of
$\partial\mathcal{M}$ and $d$, there exists a smooth function $F\colon
x\in\mathcal{M}\mapsto\varphi_{x}\in C^{2,\alpha}(B_{r},B_{r})$ such that each
$\varphi_{x}$ is a diffeomorphism and

(5.2) $\varphi_{x}\big{(}B_{r}\cap
D_{{x,1}}(\mathcal{M})\big{)}=B_{r}\cap\\{x_{1}\geq-d(x)\\}\qquad\forall
x\in\mathcal{M},$

where $D_{{x,\lambda}}$ is the dilation map defined in (2.1). Essentially,
$\varphi_{x}$ is a local chart that depends smoothly on $x$ and maps
$\partial\mathcal{M}$ in $\\{x_{1}=-d(x)\\}$.

By a suitable choice of $A,B,\bar{\eta}$ and by smooth convergence of
$\Sigma_{t}\to\Sigma_{t_{0}}$ in $C$, for every
$t\in(t_{0}-\bar{\eta},t_{0}+\bar{\eta})$ $\Sigma_{t}\cap C$ can be written as
$\psi_{t}(\Sigma_{t_{0}})$ for a smooth 1-parameter family of diffeomorphisms
such that $\psi_{t_{0}}=\operatorname{Id}\llcorner_{\Sigma_{t_{0}}}$.
Moreover, if $\bar{\eta}$ is sufficiently small, then $\psi_{t}(x)\in
B_{r}(x)$ for all $t\in(t_{0}-\bar{\eta},t_{0}+\bar{\eta})$ and for all
$x\in\mathcal{M}$.

All the previous arguments show that, for all
$s,t\in(t_{0}-\bar{\eta},t_{0}+\bar{\eta})$ and all $\lambda\in[0,1]$, it is
well defined

(5.3)
$f_{t,s,\lambda}(x)=\varphi_{x}^{-1}\Big{(}\chi_{B}(x)\varphi_{x}(\psi_{t}(x))+\chi_{A}(x)\big{[}(1-\lambda)\varphi_{x}(\psi_{t}(x))+\lambda\varphi_{x}(\psi_{s}(x))\big{]}\Big{)}\quad\forall
x\in\mathcal{M}.$

In other words, $f_{t,s,\lambda}(\Sigma_{t_{0}}\cap C)$ coincides with
$\Sigma_{t}$ in a neighborhood of $B$ and changes smoothly in a convex
combination of $\Sigma_{t}$ and $\Sigma_{s}$ in a neighborhood $A$. This
definition ensures that $\partial[f_{t,s,\lambda}(\Sigma_{t_{0}}\cap
C)]\subset\partial\mathcal{M}$ for all
$\Sigma,t\in(t_{0}-\bar{\eta},t_{0}+\bar{\eta})$ and all $\lambda\in[0,1]$.
Moreover, since $f_{t,s,\lambda}(\Sigma_{t_{0}})$ is essentially a convex
combination of $\Sigma_{t}$ and $\Sigma_{s}$ and by smooth convergence of the
$\Sigma_{t}$ in $C$, $\bar{\eta}$ can be chosen so that every
$f_{t,s,\lambda}(\Sigma_{t_{0}}\cap C)$ is diffeomorphic to
$\Sigma_{t_{0}}\cap C$ and

(5.4) $\mathcal{H}^{2}(f_{t,s,\lambda}(\Sigma_{t_{0}}\cap
C))<\mathcal{H}^{2}(\Sigma_{t}\cap C)+\frac{\varepsilon}{32}\qquad\forall
t,s,\in(t_{0}-\bar{\eta},t_{0}+\bar{\eta}),\,\forall\lambda\in[0,1].$

In a similar way we can define $g_{t,s,\lambda}$ which satisfies the same
properties of $f_{t,s,\lambda}$ with $\Gamma_{t}$ in place of $\Sigma_{t}$, in
particular

(5.5) $\mathcal{H}^{2}(g_{t,s,\lambda}(\Gamma_{t_{0}}\cap
C))<\mathcal{H}^{2}(\Gamma_{t}\cap C)+\frac{\varepsilon}{32}\qquad\forall
t,s,\in(t_{0}-\bar{\eta},t_{0}+\bar{\eta}),\,\forall\lambda\in[0,1].$

Moreover we have that $g_{t,s,\lambda}(\Gamma_{t_{0}}\cap
C)\subset\partial\mathcal{M}$ for all
$s,t\in(t_{0}-\bar{\eta},t_{0}+\bar{\eta})$ and all $\lambda\in[0,1]$. We
underline that by the construction of $f_{t,s,\lambda}(\Sigma_{t_{0}}\cap C)$
and $g_{t,s,\lambda}(\Gamma_{t_{0}}\cap C)$, it follows that their boundaries
coincide and are contained in $\partial\mathcal{M}$. Again in a similar way,
we can define the family of open sets $\Xi_{t,s,\lambda}$ relative to
$f_{t,s,\lambda}(\Sigma_{t_{0}})$ and $g_{t,s,\lambda}(\Sigma_{t_{0}})$.

We now fix
$a,a^{\prime},a^{\prime\prime},b^{\prime\prime},b,b^{\prime}\in(0,1)$ such
that
$t_{0}-\bar{\eta}<a<a^{\prime}<a^{\prime\prime}<b^{\prime\prime}<b^{\prime}<b<t_{0}+\bar{\eta}$
and two smooth functions $\beta,\gamma\colon[0,1]\to[0,1]$ such that

     * •

$\beta(t)=0$ for every $t\in[0,a]\cup[b,1]$ and $\beta(t)\equiv 1$ for every
$t\in[a^{\prime},b^{\prime}]$;

     * •

$\gamma(t)=t$ for every $t\in[0,a]\cup[b,1]$, $\gamma(t)=t_{0}$ for every
$t\in[a^{\prime},b^{\prime}]$ and $\gamma$ is monotone non-decreasing.

We now use the hypotheses that $(\Sigma_{t_{0}},\Gamma_{t_{0}})$ is not
$\varepsilon$-almost minimizing in $U$, that is there exists a parametrized
pair of surfaces $(\Phi_{t},\Psi_{t})_{t\in[0,1]}$ which satisfies the
properties of Definition 5.1. Let us consider another smooth function
$\omega\colon[0,1]\to[0,1]$ such that $\omega(t)=0$ for every
$t\in[0,a^{\prime}]\cup[b^{\prime},1]$ and $\omega(t)=1$ for every
$t\in[a^{\prime\prime},b^{\prime\prime}]$. We now define the new sweepout
$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})_{t\in[0,1]}$ as follows:

     * •

$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})=(\Sigma_{t},\Gamma_{t})$ for every
$t\in[0,a]\cup[b,1]$;

     * •

$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})\setminus
B=(\Sigma_{t},\Gamma_{t})\setminus B$ for every $t\in[0,1]$;

     * •

$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})\cap
C=(f_{t,\gamma(t),\beta(t)}(\Sigma_{t_{0}}\cap
C),g_{t,\gamma(t),\beta(t)}(\Gamma_{t_{0}}\cap C))$ for every $t\in(a,b)$;

     * •

$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})\cap
A=(\Sigma_{\gamma(t)},\Gamma_{\gamma(t)})$ for every
$t\in[a,a^{\prime}]\cup[b^{\prime},b]$;

     * •

$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})\cap
A=(\Phi_{\omega(t)},\Psi_{\omega(t)})$ for every
$t\in(a^{\prime},b^{\prime})$.

In a similar way are defined the open sets $\tilde{\Omega}_{t}$ relative to
$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})$. By definition it is clear that
$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})_{t\in[0,1]}$ is homotopic to
$(\Sigma_{t},\Gamma_{t})_{t\in[0,1]}$ and satisfies the first two conditions
of the lemma.

  2. Step 2:

The next step is to estimate $F_{a}(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})$ to
complete the proof of the lemma. Since
$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})=(\Sigma_{t},\Gamma_{t})$ for every
$t\in[0,a]\cup[b,1]$, we have to estimate
$F_{a}(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})$ only for $t\in(a,b)$.

     * •

Let us consider $t\in(a,a^{\prime}]\cup[b^{\prime},b)$; We have that
$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})\setminus
B=(\Sigma_{t},\Gamma_{t})\setminus B$. In $C$ we have

(5.6) $\displaystyle\mathcal{H}^{2}(\tilde{\Sigma}_{t}\cap
C)=\mathcal{H}^{2}(f_{t,\gamma(t),\beta(t)}(\Sigma_{t_{0}}\cap
C))<\mathcal{H}^{2}(\Sigma_{t}\cap C)+\frac{\varepsilon}{32},$ (5.7)
$\displaystyle\mathcal{H}^{2}(\tilde{\Gamma}_{t}\cap
C)=\mathcal{H}^{2}(g_{t,\gamma(t),\beta(t)}(\Gamma_{t_{0}}\cap
C))<\mathcal{H}^{2}(\Gamma_{t}\cap C)+\frac{\varepsilon}{32},$

where the inequalities follow by (5.4) and (5.5) and by $\gamma(t)\in(a,b)$
for $t\in(a,b)$. In $A$ we have

(5.8) $\displaystyle|\mathcal{H}^{2}(\tilde{\Sigma}_{t}\cap
A)-\mathcal{H}^{2}(\Sigma_{t}\cap A)|=|\mathcal{H}^{2}(\Sigma_{\gamma(t)}\cap
A)-\mathcal{H}^{2}(\Sigma_{t}\cap A)|<\frac{\varepsilon}{32}$ (5.9)
$\displaystyle|\mathcal{H}^{2}(\tilde{\Gamma}_{t}\cap
A)-\mathcal{H}^{2}(\Gamma_{t}\cap A)|=|\mathcal{H}^{2}(\Gamma_{\gamma(t)}\cap
A)-\mathcal{H}^{2}(\Gamma_{t}\cap A)|<\frac{\varepsilon}{32}$

by Lemma 4.1 if $\bar{\eta}$ is chosen sufficiently small. Gathering these
estimates we get

(5.10)
$F_{a}(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})<F_{a}(\Sigma_{t},\Gamma_{t})+\frac{\varepsilon}{8}\qquad\forall
t\in(a,a^{\prime}]\cup[b^{\prime},b).$

     * •

Now we consider
$t\in(a^{\prime},a^{\prime\prime})\cup(b^{\prime\prime},b^{\prime})$. The
estimates in $\mathcal{M}\setminus B$ and in $C$ are the same of the case
$t\in(a,a^{\prime}]\cup[b^{\prime},b)$, the only difference being in $A$. Thus
we have to estimate only in $A$. Therein we have

(5.11) $\begin{split}\mathcal{H}^{2}(\tilde{\Sigma}_{t}\cap
A)+a\mathcal{H}^{2}(\tilde{\Gamma}_{t}\cap
A)&=\mathcal{H}^{2}(\Phi_{\omega(t)}\cap
A)+a\mathcal{H}^{2}(\Psi_{\omega(t)}\cap A)\\\
&<\mathcal{H}^{2}(\Sigma_{t_{0}}\cap A)+a\mathcal{H}^{2}(\Gamma_{t_{0}}\cap
A)+\frac{\varepsilon}{8}\\\ &<\mathcal{H}^{2}(\Sigma_{t}\cap
A)+a\mathcal{H}^{2}(\Sigma_{t}\cap
A)+\frac{\varepsilon}{8}+\frac{\varepsilon}{16}\end{split}$

where the first inequality follows by the fact that $(\Phi_{s},\Psi_{s})$
contradicts the $\varepsilon$-almost minimality of
$(\Sigma_{t_{0}},\Gamma_{t_{0}})$ in $U$ (recall that
$(\Phi_{s},\Psi_{s})\setminus U=(\Sigma_{t_{0}},\Gamma_{t_{0}})\setminus U$),
whereas the second inequality is true by Lemma 4.1 if we choose $\bar{\eta}$
sufficiently small.

By this and the estimate in $C$ we have

(5.12)
$F_{a}(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})<F_{a}(\Sigma_{t},\Gamma_{t})+\frac{\varepsilon}{4}\qquad\forall
t\in(a^{\prime},a^{\prime\prime})\cup(b^{\prime\prime},b^{\prime}).$

     * •

Now consider the case $t\in[a^{\prime\prime},b^{\prime\prime}]$. Again, the
estimates in $\mathcal{M}\setminus B$ and in $C$ as the same as the other
cases. We racall that, by definition, $\tilde{\Sigma}_{t}\cap A=\Phi_{1}\cap
A$ and $\tilde{\Gamma}_{t}\cap A=\Psi_{1}\cap A$. Thus

(5.13) $\begin{split}\mathcal{H}^{2}(\tilde{\Sigma}_{t}\cap
A)+a\mathcal{H}^{2}(\tilde{\Gamma}_{t}\cap A)&=\mathcal{H}^{2}(\Phi_{1}\cap
A)+a\mathcal{H}^{2}(\Psi_{1}\cap A)\\\ &<\mathcal{H}^{2}(\Sigma_{t_{0}}\cap
A)+a\mathcal{H}^{2}(\Gamma_{t_{0}}\cap A)-\frac{\varepsilon}{2}\\\
&<\mathcal{H}^{2}(\Sigma_{t}\cap A)+a\mathcal{H}^{2}(\Gamma_{t}\cap
A)-\frac{3\varepsilon}{8}.\end{split}$

where the first inequality follows by the definition of $(\Phi_{t},\Psi_{t})$
and the second is true if we choose $\bar{\eta}$ sufficiently small. Gathering
this estimate with the one in $C$, we get

(5.14)
$F_{a}(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})<F_{a}(\Sigma_{t},\Gamma_{t})-\frac{\varepsilon}{4}\qquad\forall
t\in[a^{\prime\prime},b^{\prime\prime}].$

If we choose $\bar{\eta}>0$ so small so that all the previous estimate are
valid, $(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})$ satisfies all the conclusion
of the lemma. It is clear that all the previous arguments remain true whenever
$0<\eta\leq\bar{\eta}$.

∎

###  5.2. Almgren-Pitts combinatorial argument

The aim of this section is to show that there exists a min-max sequence of
pairs which is almost minimizing in at least one of any _admissible pair_ of
open sets, that is a pair of open sets which are at a sufficiently large
distance compared to their diameter. This is used in 5.3 to complete the proof
of Proposition 5.1.

######  Definition 5.2 (Admissible pair of sets).

If $U_{1},U_{2}\subset\mathcal{M}$ are relatively open and non-empty, we say
that $(U_{1},U_{2})$ is an _admissible pair of open sets_ if

(5.15) $\operatorname*{dist}(U_{1},U_{2})\geq
4\min\\{\operatorname*{diam}U_{1},\operatorname*{diam}U_{2}\\}.$

We call $\mathcal{C}$ the family of admissible pairs of open sets.

######  Definition 5.3.

For any $\varepsilon>0$ and $(U_{1},U_{2})\in\mathcal{C}$, we say that
$(\Sigma,\Gamma)$ is $\varepsilon$-almost minimizing in $(U_{1},U_{2})$ if it
is $\varepsilon$-a.m. in at least one of $U_{1},U_{2}$.

We can now state the fundamental result of this subsection.

######  Lemma 5.3 (Almgren-Pitts combinatorial lemma).

Let $\mathscr{S}$ be a homotopically closed family of sweepouts; there exists
a minimizing sequence of sweepouts
$(\Sigma_{t}^{k},\Gamma_{t}^{k})_{t\in[0,1]}^{k\in\mathbb{N}}$ and a min-max
sequence
$(\Sigma^{k},\Gamma^{k})^{k\in\mathbb{N}}:=(\Sigma_{t_{k}}^{k},\Gamma_{t_{k}}^{k})^{k\in\mathbb{N}}$
that satisfies, for any $k\in\mathbb{N}$, the following properties:

  1. (1)

$F_{a}(\Sigma^{k},\Gamma^{k})\geq{m_{0}}-\frac{1}{k}$;

  2. (2)

$F_{a}(\Sigma^{k},\Gamma^{k})$ is $\frac{1}{k}$-a.m. in every
$(U_{1},U_{2})\in\mathcal{C}$;

  3. (3)

$(\Sigma^{k},\Gamma^{k})$ converges to a pair in
$\mathcal{V}_{2}(\mathcal{M})\times\mathcal{V}_{2}(\partial\mathcal{M})$ which
is stationary for $F_{a}$.

Before the proof of Lemma 5.3 we state the following simple and useful
property.

######  Lemma 5.4.

If $(U_{1},U_{2}),(V_{1},V_{2})$ are pairs of non-empty open sets such that

(5.16) $\displaystyle\operatorname*{dist}(U_{1},U_{2})\geq
2\min\\{\operatorname*{diam}U_{1},\operatorname*{diam}U_{2}\\}$ (5.17)
$\displaystyle\operatorname*{dist}(V_{1},V_{2})\geq
2\min\\{\operatorname*{diam}V_{1},\operatorname*{diam}V_{2}\\},$

then there exist two indices $i,j\in\\{1,2\\}$ such that $U_{i}\cap
V_{j}=\emptyset$

###### Proof.

Without loss of generality we can suppose that
$\operatorname*{diam}U_{1}\leq\operatorname*{diam}U_{2}$ and
$\operatorname*{diam}V_{1}\leq\operatorname*{diam}V_{2}$. Moreover we can
suppose $\operatorname*{diam}U_{1}\leq\operatorname*{diam}V_{1}$.

If $U_{1}\cap V_{1}=\emptyset$ there is nothing to prove. Otherwise, there
exists $\xi\in U_{1}\cap V_{1}$. Then, for every $x\in U_{1}$ and $y\in V_{2}$

(5.18) $\begin{split}|x-y|&\geq|y-\xi|-|\xi-x|\\\ &\geq
2\operatorname*{diam}V_{1}-\operatorname*{diam}U_{1}\\\
&\geq\operatorname*{diam}V_{1}\\\ &\geq\operatorname*{diam}U_{1}.\end{split}$

Since $U_{1}$ is relatively open and $\partial\mathcal{M}$ is of class
$C^{2,\alpha}$, if $x,y$ both belonged to $U_{1}$, then one would have
$|x-y|<\operatorname*{diam}U_{1}$. This shows that $U_{1}\cap
V_{2}=\emptyset$. ∎

We can now pass to the proof of Lemma 5.3.

###### Proof of Lemma 5.3.

Let $\\{(\Sigma_{t}^{k},\Gamma_{t}^{k})\\}^{k}$ be the minimizing sequence of
sweepouts given by Theorem 4.2; Up to passing to a subsequence, we can assume
that for every $k\in\mathbb{N}$

(5.19)
$\max_{t\in[0,1]}F_{a}(\Sigma_{t}^{k},\Gamma_{t}^{k})\leq{m_{0}}+\frac{1}{8k}.$

This suitable subsequence (not relabeled) is clearly still minimizing and we
are going to prove that is the minimizing sequence stated in the theorem.
Since by Theorem 4.2 every min-max sequence converges to a stationary pair up
to subsequences, we have only to find a min-max sequence with the almost
minimizing property. For any $k\in\mathbb{N}$ let us define the set of times

(5.20) $K_{k}=\Big{\\{}t\in[0,1]\mid
F_{a}(\Sigma_{t}^{k},\Gamma_{t}^{k})\geq{m_{0}}-\frac{1}{k}\Big{\\}}.$

$K_{k}$ is clearly compact. To prove the existence of the min-max sequence
with the desired properties, we argue by contradiction: if there exists
$k\in\mathbb{N}$ such that for every $t\in K_{k}$
$(\Sigma_{t}^{k},\Gamma_{t}^{k})$ is neither $\frac{1}{k}$-almost in $U_{1,t}$
nor in $U_{2,t}$ for some $(U_{1,{t}},U_{2,{t}})\in\mathcal{C}$, we construct
a sweepout $(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})_{t\in[0,1]}$ that is
homotopic to $(\tilde{\Sigma}_{t}^{k},\tilde{\Gamma}_{t}^{k})_{t\in[0,1]}$ and
satisfies
$\max_{t\in[0,1]}F_{a}(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})<{m_{0}}$, which
is absurd. Thus we fix such a $k\in\mathbb{N}$. Since $k$ is fixed, we drop
the subscripts and superscripts $k$.

Let us assume, by contradiction, that for any $t\in K$ there exists
$(U_{1,{t}},U_{2,{t}})\in\mathcal{C}$ such that $(\Sigma_{t},\Gamma_{t})$ is
neither $\frac{1}{k}$-almost in $U_{1,t}$ nor in $U_{2,t}$.

  1. Step 1:

As first step, we want to show that there exists a covering
$\\{J_{j}\\}_{j=1}^{M}$ of $K$ made of closed intervals with the following
properties:

     * •

For every $j=1,\dots,M$, there exist $(a_{j},b_{j})\subset J_{j}$, an open set
$W_{j}\subset\mathcal{M}$ and a sweepout
$(\tilde{\Sigma}_{t}^{j},\tilde{\Gamma}_{t}^{j})_{t\in[0,1]}$ that is
homotopic to $(\Sigma_{t},\Gamma_{t})_{t\in[0,1]}$ and satisfies:

       1. (1)

$(\tilde{\Sigma}_{t}^{j},\tilde{\Gamma}_{t}^{j})=(\Sigma_{t},\Gamma_{t})$ for
every $t\in[0,1]\setminus J_{j}$;

       2. (2)

$(\tilde{\Sigma}_{t}^{j},\tilde{\Gamma}_{t}^{j})\setminus
W_{j}=(\Sigma_{t},\Gamma_{t})\setminus W_{j}$ for every $t\in[0,1]$;

       3. (3)

$F_{a}(\tilde{\Sigma}_{t}^{j},\tilde{\Gamma}_{t}^{j})<F_{a}(\Sigma_{t},\Gamma_{t})+\frac{1}{4k}$
for every $t\in[0,1]$;

       4. (4)

$F_{a}(\tilde{\Sigma}_{t}^{j},\tilde{\Gamma}_{t}^{j})<F_{a}(\Sigma_{t},\Gamma_{t})-\frac{1}{2k}$
for every $t\in[a_{j},b_{j}]$.

     * •

(5.21) $K\subset\bigcup_{j=1}^{M}(a_{j},b_{j})$

     * •

For every $j=1,\dots,M$, $J_{j}$ intersects at most $J_{{j-1}}$ and
$J_{{j+1}}$;

     * •

If $i\neq j$ and $J_{i}\cap J_{j}\neq\emptyset$, then $W_{i}\cap
W_{j}=\emptyset$.

By the assumptions, for every $t\in K$, $i=1,2$ and for any open set
$V_{i,t}\supset\supset U_{i,t}$, let $\bar{\eta_{t}}$ the number given by
Lemma 5.2 and for every $0<\eta_{t}\leq\bar{\eta_{t}}$ let us define
$I_{t,\eta_{t}}=[t-\eta_{t},t+\eta_{t}]$. We choose the sets $V_{i,t}$ such
that

(5.22) $\operatorname*{dist}(V_{1,t},V_{2,t})\geq
2\min\\{\operatorname*{diam}V_{1,t},\operatorname*{diam}V_{2,t}\\}.$

By compactness of $K$, there exists a finite set of points $t_{1}<\dots<t_{m}$
such that

(5.23) $K\subset\bigcup_{j=1}^{m}{\kern
0.0pt(I_{t_{j},{\bar{\eta}}_{t_{j}}})}^{\mathrm{o}}.$

Since for every $t_{j}$ we can choose $\eta_{j}\leq\bar{\eta}_{t_{j}}$, we can
obtain that each $I_{t_{j},\eta_{j}}$ intersects at most
$I_{t_{j-1},\eta_{j-1}}$ and $I_{t_{j+1},\eta_{j+1}}$ and still

(5.24) $K\subset\bigcup_{j=1}^{m}{\kern
0.0pt(I_{t_{j},\eta_{j}})}^{\mathrm{o}}.$

For every $j=1,\dots,m$ and $i=1,2$, we denote $V_{j}^{i}:=V_{i,t_{j}}$ and
$I_{j}:=I_{t_{j},\eta_{j}}$.

Starting from this covering, we want to construct the covering
$\\{J_{j}\\}_{j=1}^{M}$ of $K$. We begin by defining $J_{1}$ and $W_{1}$:

     * •

If $I_{1}\cap I_{2}=\emptyset$, then we set $J_{1}=I_{1}$, $W_{1}=V_{1}^{1}$;

     * •

If $I_{1}\cap I_{2}\neq\emptyset$, up to relabeling the sets $V_{j}^{i}$, by
Lemma 5.4 we can suppose that $V_{1}^{1}\cap V_{2}^{1}=\emptyset$. Then we set
$J_{1}=I_{1}$ and $W_{1}=V_{1}^{1}$.

Now, in any of the two above cases, by Lemma 5.4, we can assume that
$V_{1}^{1}\cap V_{2}^{1}=\emptyset$; to choose $J_{2},W_{2}$, we have to
consider several cases:

    1. (1)

If $I_{2}\cap I_{3}=\emptyset$, then we set $J_{2}=I_{2}$ and
$W_{2}=V_{2}^{1}$;

    2. (2)

If $I_{2}\cap I_{3}\neq\emptyset$, by Lemma 5.4 there exists two indices
$i,\ell$ such that $V_{2}^{i}\cap V_{3}^{\ell}=\emptyset$. Without loss of
generality we can suppose $\ell=1$.

      1. (a)

If $i=1$, we set $J_{2}=I_{2}$, $W_{2}=V_{2}^{1}$;

      2. (b)

If $i=2$, we can find two closed intervals $L,H$ such that

(5.25) ${\kern 0.0ptL}^{\mathrm{o}}\cup{\kern 0.0ptH}^{\mathrm{o}}={\kern
0.0ptI_{2}}^{\mathrm{o}}\qquad L\cap I_{3}=\emptyset\qquad H\cap
I_{1}=\emptyset$

(the first condition implies that $L,H$ cover $I_{2}$ and that their interiors
overlap in the middle of $I_{2}$). We then set $J_{2}=L$, $W_{2}=V_{2}^{1}$,
$J_{3}=H$, $W_{3}=V_{2}^{2}$.

To choose the next interval and open set, we proceed as follows:

     * •

In the case 1, we choose $J_{3},W_{3}$ as we have done for $J_{1},W_{1}$
(since $I_{3}\cap I_{2}=\emptyset$);

     * •

In the case 2a we can choose $J_{3},W_{3}$ as we have done for $J_{2},W_{2}$;

     * •

In the case 2b we can choose $J_{4},W_{4}$ as we have done for $J_{2},W_{2}$.

Thus we can continue this procedure until we complete the choice of all the
intervals $I_{j}$; it is clear that, by construction, each each $J_{j}$
intersects at most $J_{j-1}$ and $J_{j+1}$ and that, if $J_{j}\cap
J_{\ell}\neq\emptyset$, then $W_{j}\cap W_{\ell}=\emptyset$. Moreover, by
(5.24) and the above construction, we have $K\subset\bigcup_{j=1}^{M}{\kern
0.0ptJ_{j}}^{\mathrm{o}}$. Hence for every $j=1,\dots,M$ we can choose,
$a_{j},b_{j}\in{\kern 0.0ptJ_{j}}^{\mathrm{o}}$ such that

(5.26) $K\subset\bigcup_{j=1}^{M}(a_{j},b_{j}).$

This complete the definition of the covering $\\{J_{j}\\}_{j=1}^{M}$ and the
open sets $W_{j}$; we have now to define the sweepouts
$(\tilde{\Sigma}_{t}^{j},\tilde{\Gamma}_{t}^{j})_{t\in[0,1]}$ for
$j=1,\dots,M$.

For every $j=1,\dots,M$ there exists $\ell\in\\{1,\dots,m\\}$ and
$i\in\\{0,1\\}$ such that $(a_{j},b_{j})\subset{\kern
0.0ptJ_{j}}^{\mathrm{o}}\subset I_{\ell}=I_{t_{\ell},\eta_{\ell}}$,
$W_{j}=V_{\ell}^{i}$ and $(\Sigma_{t_{\ell}},\Gamma_{t_{\ell}})$ is not
$\frac{1}{k}$-almost minimizing in $W_{j}$. Thus, by Lemma 5.2 there exists a
sweepout $(\tilde{\Sigma}_{t}^{j},\tilde{\Gamma}_{t}^{j})_{t\in[0,1]}$ that is
homotopic to $(\Sigma_{t},\Gamma_{t})_{t\in[0,1]}$ and such that

    1. (1)

$(\tilde{\Sigma}_{t}^{j},\tilde{\Gamma}_{t}^{j})=(\Sigma_{t},\Gamma_{t})$ for
every $t\in[0,1]\setminus J_{j}$;

    2. (2)

$(\tilde{\Sigma}_{t}^{j},\tilde{\Gamma}_{t}^{j})\setminus
W_{j}=(\Sigma_{t},\Gamma_{t})\setminus W_{j}$ for every $t\in[0,1]$;

    3. (3)

$F_{a}(\tilde{\Sigma}_{t}^{j},\tilde{\Gamma}_{t}^{j})<F_{a}(\Sigma_{t},\Gamma_{t})+\frac{1}{4k}$
for every $t\in[0,1]$;

    4. (4)

$F_{a}(\tilde{\Sigma}_{t}^{j},\tilde{\Gamma}_{t}^{j})<F_{a}(\Sigma_{t},\Gamma_{t})-\frac{1}{2k}$
for every $t\in[a_{j},b_{j}]$,

which are the required conditions to complete the proof of the first step.

  2. Step 2:

We now want to “glue together” the deformations
$(\tilde{\Sigma}_{t}^{j},\tilde{\Gamma}_{t}^{j})$ in an unique sweepout
$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})_{t\in[0,1]}$ that is homotopic to
$(\Sigma_{t},\Gamma_{t})_{t\in[0,1]}$ and satisfies
$\max_{t\in[0,1]}F_{a}(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})<{m_{0}}$, which
is a contradiction.

We define the competitor sweeoput
$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})_{t\in[0,1]}$ as follows:

     * •

If $t\notin\bigcup_{j=1}^{M}J_{j}$, then
$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})=(\Sigma_{t},\Gamma_{t})$;

     * •

If $t\in J_{j}\setminus(J_{j+1}\cup J_{j-1})$ then
$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})=(\tilde{\Sigma}_{t}^{j},\tilde{\Gamma}_{t}^{j})$;

     * •

If $t\in J_{j}\cap J_{j+1}$, then

(5.27) $\displaystyle\tilde{\Sigma}_{t}\cap W_{j}=\tilde{\Sigma}_{t}^{j}\cap
W_{j},\quad\tilde{\Sigma}_{t}\cap W_{j+1}=\tilde{\Sigma}_{t}^{j+1}\cap
W_{j+1},\quad\tilde{\Sigma}_{t}\setminus(W_{j}\cup
W_{j+1})=\Sigma_{t}\setminus(W_{j}\cup W_{j+1});$ (5.28)
$\displaystyle\tilde{\Gamma}_{t}\cap W_{j}=\tilde{\Gamma}_{t}^{j}\cap
W_{j},\quad\tilde{\Gamma}_{t}\cap W_{j+1}=\tilde{\Gamma}_{t}^{j+1}\cap
W_{j+1},\quad\tilde{\Gamma}_{t}\setminus(W_{j}\cup
W_{j+1})=\Gamma_{t}\setminus(W_{j}\cup W_{j+1});$

These are all the possible cases, since $t$ can belong to at most two
consecutive intervals. Clearly
$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})_{t\in[0,1]}$ is well-defined, because
by construction of the intervals $J_{j}$ and open sets $W_{j}$, if $J_{j}\cap
J_{\ell}\neq\emptyset$, then $W_{j}\cap W_{\ell}=\emptyset$; Since every
$(\tilde{\Sigma}_{t}^{j},\tilde{\Gamma}_{t}^{j})_{t\in[0,1]}$ is homotopic to
$(\Sigma_{t},\Gamma_{t})_{t\in[0,1]}$, also the sweepout
$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})_{t\in[0,1]}$ is homotopic to
$(\Sigma_{t},\Gamma_{t})_{t\in[0,1]}$. The open sets $\tilde{\Omega}_{t}$
relative to the pairs $(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})$ are defined in
the same way, starting from the open sets $\tilde{\Omega}_{t}^{j}$ relative to
$(\tilde{\Sigma}_{t}^{j},\tilde{\Gamma}_{t}^{j})$.

We have now to estimate $F_{a}(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})$ to get
the contradiction:

    1. (1)

If $t\notin K$, then $t$ can be in at most two of the intervals $J_{j}$; thus
$(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})$ coincides with
$(\tilde{\Sigma}_{t}^{j},\tilde{\Gamma}_{t}^{j})$ and
$(\tilde{\Sigma}_{t}^{j+1},\tilde{\Gamma}_{t}^{j+1})$ respectively in the two
open sets $W_{j},W_{j+1}$ and coincides with $(\Sigma_{t},\Gamma_{t})$ in
$\mathcal{M}\setminus(W_{j}\cup W_{j+1})$. Since for every $j$ it holds
$F_{a}(\tilde{\Sigma}_{s}^{j},\tilde{\Gamma}_{s}^{j})<F_{a}(\Sigma_{s},\Gamma_{s})+\frac{1}{4k}$
for every $s\in[0,1]$, we have

(5.29)
$F_{a}(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})<F_{a}(\Sigma_{t},\Gamma_{t})+\frac{2}{4k}<m_{0}-\frac{1}{k}+\frac{1}{2k}=m_{0}-\frac{1}{2k},$

where the second inequality is given by $t\notin K$, thus
$F_{a}(\Sigma_{t},\Gamma_{t})<{m_{0}}-\frac{1}{k}$ .

    2. (2)

If $t\in K$, then there exists $j$ such that $t$ belongs to at least one
$(a_{j},b_{j})$ and at most in another $J_{\ell}$ for $\ell=j-1$ or
$\ell=j+1$. Hence, by the properties of the sweepouts
$(\tilde{\Sigma}_{s}^{h},\tilde{\Gamma}_{s}^{h})_{s\in[0,1]}$ previously
defined one has

(5.30) $\displaystyle\mathcal{H}^{2}(\tilde{\Sigma}_{t}^{j}\cap
W_{j})+a\mathcal{H}^{2}(\tilde{\Gamma}_{t}^{j}\cap
W_{j})<\mathcal{H}^{2}(\Sigma_{t}\cap W_{j})+a\mathcal{H}^{2}(\Gamma_{t}\cap
W_{j})-\frac{1}{2k}$ (5.31)
$\displaystyle\mathcal{H}^{2}(\tilde{\Sigma}_{t}^{\ell}\cap
W_{\ell})+a\mathcal{H}^{2}(\tilde{\Gamma}_{t}^{\ell}\cap
W_{\ell})<\mathcal{H}^{2}(\Sigma_{t}\cap
W_{\ell})+a\mathcal{H}^{2}(\Gamma_{t}\cap W_{\ell})+\frac{1}{4k}.$

Thus

(5.32) $F_{a}(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})\leq
F_{a}(\Sigma_{t},\Gamma_{t})-\frac{1}{4k}\leq{m_{0}}+\frac{1}{8k}-\frac{1}{4k}={m_{0}}-\frac{1}{8k},$

where we used $F_{a}(\Sigma_{t},\Gamma_{t})\leq{m_{0}}+\frac{1}{8k}$.

The estimates (5.29) and (5.32) show that

(5.33) $\max_{t\in[0,1]}F_{a}(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})<{m_{0}},$

which is absurd since $(\tilde{\Sigma}_{t},\tilde{\Gamma}_{t})_{t\in[0,1]}$ is
homotopic to $(\Sigma_{t},\Gamma_{t})_{t\in[0,1]}$.

∎

###  5.3. Proof of Proposition 5.1

We use the min-max sequence $(\Sigma^{k},\Gamma^{k})^{k\in\mathbb{N}}$ given
by Lemma 5.3 to prove Proposition 5.1.

###### Proof of Proposition 5.1.

Let us consider the min-max sequence
$(\Sigma^{k},\Gamma^{k})^{k\in\mathbb{N}}$ given by Lemma 5.3. It converges to
a stationary pair and, for every $k\in\mathbb{N}$, it is $\frac{1}{k}$-almost
minimizing in any pair of open sets $(U_{1},U_{2})\in\mathcal{C}$.

We begin with a simple useful remark: if a surface is $\varepsilon$-a.m. in a
set $U$ and $V\subset U$, then it is $\varepsilon$-a.m. also in $V$. For every
$x\in\mathcal{M}$ and $r>0$, we have $\bigl{(}B_{r}(x),\mathcal{M}\setminus
B_{9r}(x)\bigr{)}\in\mathcal{C}$, provided that $r$ is small enough to have
$\mathcal{M}\setminus B_{9r}(x)\neq\emptyset$.

  * •

If for every $x\in\mathcal{M}$ there exists $\rho(x)>0$ and
$N(x)\in\mathbb{N}$ such that $(\Sigma^{k},\Gamma^{k})$ is
$\frac{1}{k}$-almost minimizing in $B_{\rho(x)}(x)$ for every $k\geq N(x)$,
then the proof is complete if we set $\varepsilon_{k}=\frac{1}{k}$,
$r(x)=\rho(x)$, since $(\Sigma^{k},\Gamma^{k})$ is $\frac{1}{k}$-a.m. in every
$\operatorname*{An}\in\mathcal{AN}_{r(x)}({x})$ for $k\geq N(x)$.

  * •

If there exists $x\in\mathcal{M}$ and a subsequence
$(\Sigma^{k_{j}},\Gamma^{k_{j}})_{j\in\mathbb{N}}$ such that every
$(\Sigma^{k_{j}},\Gamma^{k_{j}})$ is not $\frac{1}{k_{j}}$-a.m. in
$B_{1/j}(x)$, then for every $j\in\mathbb{N}$ the pair
$(\Sigma^{k_{j}},\Gamma^{k_{j}})$ is $\frac{1}{k_{j}}$-a.m. in
$\mathcal{M}\setminus B_{9/j}(x)$. For every $j\in\mathbb{N}$ we define

(5.34)
$\varepsilon_{j}=\frac{1}{k_{j}};\quad(\tilde{\Sigma}^{j},\tilde{\Gamma}^{j})=(\Sigma^{k_{j}},\Gamma^{k_{j}});\quad
r(y)=\begin{cases}\operatorname*{diam}\mathcal{M}&\text{if }y=x,\\\
\operatorname*{dist}(y,x)&\text{if }y\neq x.\end{cases}$

We claim that the sequence
$(\tilde{\Sigma}^{j},\tilde{\Gamma}^{j})_{j\in\mathbb{N}}$ and the function
$r$ satisfy the conclusion. In fact for every
$\operatorname*{An}(x,s,t)\in\mathcal{AN}_{r(x)}({x})$ there exists
$N\in\mathbb{N}$ such that if $j>N$ then
$\operatorname*{An}(x,s,t)\subset\mathcal{M}\setminus B_{1/9j}(x)$ and by the
initial remark $(\tilde{\Sigma}^{j},\tilde{\Gamma}^{j})$ is
$\frac{1}{k_{j}}$-a.m. in $\operatorname*{An}(x,s,t)$. If $y\neq x$ and
$\operatorname*{An}(y,s,t)\in\mathcal{AN}_{r(y)}({y})$, then
$t<\operatorname*{dist}(y,x)$ and there exists $N\in\mathbb{N}$ such that if
$j>N$ then $\operatorname*{dist}(y,x)-\frac{1}{9j}<t$. Thus
$\operatorname*{An}(y,s,t)\subset\mathcal{M}\setminus B_{1/9j}(x)$ and by the
initial remark $(\tilde{\Sigma}^{j},\tilde{\Gamma}^{j})$ is
$\frac{1}{k_{j}}$-a.m. in $\operatorname*{An}(x,s,t)$.

∎

##  6\. Bernstein’s Theorem, curvature estimates and compactness

In this section we report some properties of (smooth) stable pairs for the
capillarity functional. The second variation of the capillarity functional
$F_{a}$ on a pair of surfaces $(\Sigma,\Gamma)$ in $\mathcal{M}$ induces the
so called stability operator $Q$, which is a quadratic form on the set of
smooth functions with compact support.

######  Proposition 6.1 (Stability operator).

Let $(\Sigma,\Gamma)$ be a pair of surfaces in $\mathcal{M}$. Then for any
$X\in\mathfrak{X}_{t}(\mathcal{M})$ we have

(6.1) $\delta^{2}_{F_{a}}(V,W)[X]=Q(u),$

where $u(x)=X(x)\cdot\nu_{\Sigma}(x)$ for every $x\in\Sigma$, and

(6.2) $Q(u)=\int_{\Sigma}|\nabla
u|^{2}-|A_{\Sigma}|^{2}u^{2}\mathop{}\\!\mathrm{d}\mathcal{H}^{2}-\int_{\partial\Sigma}qu^{2}\mathop{}\\!\mathrm{d}\mathcal{H}^{1},$

where

(6.3) $q=\frac{H_{\partial\mathcal{M}}}{\sin\theta}+\cot\theta
H_{\Sigma}-\kappa_{\partial\Sigma}.$

Here $\kappa_{\partial\Sigma}={D_{\tau}\eta_{\Sigma}}\cdot{\tau}$, where
$\tau$ is a unit tangent vector field on $\partial\Sigma$. In particular,
$(\Sigma,\Gamma)$ is stable for $F_{a}$ if and only if the operator $Q$ is
non-negative definite on $C_{c}^{\infty}(\mathcal{M})$.

For a proof we refer to [23, Appendix] and to [13, section 2.1]. If
$(\Sigma,\Gamma)$ is stationary for $F_{a}$ in a half space, then
$H_{\partial\mathcal{M}}=0$ and $H_{\Sigma}=0$. Moreover, if the half space is
defined by $\\{x_{1}\geq 0\\}$, by testing the stability inequality with the
function

$u=\zeta\Bigl{(}\frac{1-\cos\theta{\nu_{\Sigma}}\cdot{e_{1}}}{\sin\theta}\Bigr{)}$

we obtain the following proposition, compare with the proof of [13, Theorem
1.5].

######  Proposition 6.2 (Stability inequality for Capillarity functional).

Assume that $\mathcal{N}$ is a half-space. Then there exists an universal
constant $c(a)>0$ such that, for every pair $(\Sigma,\Gamma)$ which is
stationary and stable for $F_{a}$ and for every $\zeta\in
C_{c}^{\infty}(\mathbb{R}^{3})$ we have

(6.4)
$\int_{\Sigma}\zeta^{2}|A_{\Sigma}|^{2}\mathop{}\\!\mathrm{d}\mathcal{H}^{2}\leq
c\int_{\Sigma}|\nabla\zeta|^{2}\mathop{}\\!\mathrm{d}\mathcal{H}^{2},$

where $|A_{\Sigma}|^{2}$ is the norm squared of the second fundamental form of
$\Sigma$.

###### Proof.

Without loss of generality, we can assume that $\mathcal{N}=\\{x_{1}\geq
0\\}$. Let us define

(6.5)
$\varphi(x)=\frac{1-\cos\theta{\nu_{\Sigma}(x)}\cdot{e_{1}}}{\sin\theta}$

and let $\zeta\in C_{c}^{\infty}(\mathbb{R}^{3})$ be any test function. The
computations in the first part of the proof of [13, Theorem 1.5] yield (see
[13, (4.26)])

(6.6)
$Q(\zeta\varphi)=\int_{\Sigma}\varphi^{2}|\nabla\zeta|^{2}-\zeta^{2}\varphi\frac{|A_{\Sigma}|^{2}}{\sin\theta}.$

Since

(6.7)
$0<\frac{1-\cos\theta}{\sin\theta}\leq\varphi\leq\frac{1+\cos\theta}{\sin\theta},$

the result easily follows. ∎

The stability inequality is used to obtain a Bernstein-type theorem for
stationary and stable surfaces with respect to the capillarity energy in a
three-dimensional half space.

######  Theorem 6.3 (Bernstein Theorem).

Let us assume that $(\Sigma,\Gamma)$ is stationary and stable for $F_{a}$ in
$\mathcal{N}\setminus\\{0\\}$, where $\mathcal{N}:=\\{x_{1}\geq
0\\}\subset\mathbb{R}^{3}$, and that there exists a constant $M>0$ such that

(6.8) $\sup_{R\in(0,\infty)}\frac{\mathcal{H}^{2}(\Sigma\cap
B_{R}(0))}{R^{2}}\leq M;$

Then $\Sigma$ and $\Gamma$ are half planes and $\Sigma$ forms an angle
$\theta$ with $\partial\mathcal{N}$ such that $a=\cos\theta$.

###### Proof.

The proof is classical and we just point out that it is enough to test (6.4)
with the following logarithmic cut-off function, for $N\in\mathbb{N}$:

(6.9) $\zeta_{N}(x)=\begin{cases}0&\text{if }|x|\leq e^{-2N}\\\
2+\frac{\log(|x|)}{N}&\text{if }e^{-2N}\leq|x|\leq e^{-N}\\\ 1&\text{if
}e^{-N}\leq|x|\leq e^{N}\\\ 2-\frac{\log(|x|)}{N}&\text{if }e^{N}\leq|x|\leq
e^{2N}\\\ 0&\text{if }|x|\geq e^{2N}.\end{cases}$

This, together with the area bound (6.8), gives

(6.10) $\int_{\Sigma\cap\big{(}B_{e^{N}}\setminus
B_{e^{-N}}\big{)}}|A_{\Sigma}|^{2}\mathop{}\\!\mathrm{d}\mathcal{H}^{2}\leq\frac{2cMe^{2}}{N}.$

By sending $N\to\infty$, we obtain $A_{\Sigma}\equiv 0$, hence the conclusion.
∎

As it is well known, Bernstein theorem as above implies local curvature
estimates, see [13, Theorem 1.6] for the (classical) argument.

######  Theorem 6.4 (Curvature estimates for stable surfaces).

There exists a constant $\Lambda=\Lambda(a)>0$ such that, for every
$(\Sigma,\Gamma)$ stationary stable pair for $F_{a}$ in $\mathcal{M}$, it
holds

(6.11)
$|A_{\Sigma}|(x)\operatorname*{dist}\nolimits_{\Sigma}(x,\partial\Sigma\setminus\partial\mathcal{M})\leq\Lambda\qquad\forall
x\in\Sigma.$

A straightforward consequence of the above curvature estimates is the
following compactness result.

######  Theorem 6.5 (Compactness for stable pairs).

Let $U\subset\mathcal{M}$ be open and let $(\Sigma^{k},\Gamma^{k})^{k}$ be a
sequence of stable stationary pairs for $F_{a}$ in $\mathcal{M}$ satisfying

  1. (1)

For every $k\in\mathbb{N}$, $\Sigma_{k}$ and $\Gamma_{k}$ are smooth up to the
boundary;

  2. (2)

every $(\Sigma^{k},\Gamma^{k})$ is stationary and stable for $F_{a}$ in $U$;

  3. (3)

$\sup_{k}F_{a}(\Sigma^{k},\Gamma^{k})<+\infty$;

Then, up to subsequences, $(\Sigma^{k},\Gamma^{k})$ converges in the sense of
varifolds to a stationary stable pair $(V,W)$ of varifolds for $F_{a}$ such
that:

  1. (1)

$W=\Gamma$ where $\Gamma\subset\partial\mathcal{M}$ is a surface with smooth
boundary;

  2. (2)

$V$ is, up to multiplicity, a smooth stable minimal surface $\Sigma$ in $U$
with boundary $\partial\Sigma=\partial\Gamma$ and meets $\partial\mathcal{M}$
with contact angle $\theta$.

  3. (3)

For every open set $U^{\prime}\subset\joinrel\subset U$, the convergences
$\Sigma_{k}\to\Sigma$ and $\Gamma_{k}\to\Gamma$ in $U^{\prime}$ are smooth.

##  7\. Replacements

To prove regularity for $(V,W)$ we follow Almgren and Pitts and introduce the
notion of replacement

######  Definition 7.1 (Replacements).

Let $(V,W)$ be stationary for $F_{a}$ in $\mathcal{M}$ and let
$U\subset\mathcal{M}$ be relatively open; a pair $(\tilde{V},\tilde{W})$ is a
_replacement_ of $(V,W)$ in $U$ if

  1. (1)

$(\tilde{V},\tilde{W})$ is stationary for $F_{a}$ in $\mathcal{M}$ and
$\tilde{V}=V$ and $\tilde{W}=W$ in $\mathcal{M}\setminus U$;

  2. (2)

$\lVert\tilde{V}\rVert(\mathcal{M})=\lVert V\rVert(\mathcal{M})$ and
$\lVert\tilde{W}\rVert(\mathcal{M})=\lVert W\rVert(\mathcal{M})$;

  3. (3)

$\tilde{V}$ and $\tilde{W}$ are induced in $U$ by smooth surfaces
$\tilde{\Sigma}$ and $\tilde{\Gamma}$ with boundary such that the pair
$(\tilde{\Sigma},\tilde{\Gamma})$ is stationary and stable for $F_{a}$ in $U$.

The key result of this section is the existence of replacements in every
annulus $\operatorname*{An}$ in $\mathcal{AN}_{r(x)}({x})$.

######  Proposition 7.1 (Existence of replacements).

Let $(\Sigma^{k},\Gamma^{k})^{k},(V,W),r$ be respectively the min-max
sequence, the pair of varifolds and the function $r$ given by Proposition 5.1.
Let $x\in\mathcal{M}$ and let
$\operatorname*{An}:=\operatorname*{An}(x,r,s)\in\mathcal{AN}_{r(x)}({x})$;
then there exists a min-max sequence
$(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})^{k}$ and a function
$\tilde{r}:\mathcal{M}\to(0,+\infty)$ such that

  * •

$(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})$ converges, up to subsequences, to a
pair $(\tilde{V},\tilde{W})$ which is a replacement for $(V,W)$ in
$\operatorname*{An}=\operatorname*{An}(x,r,s)$;

  * •

For every $y\in\mathcal{M}$, $(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})$ is
almost minimizing in every annulus in $\mathcal{AN}_{\tilde{r}(y)}({y})$;

  * •

$\tilde{r}(x)=r(x)$.

Since the proposition is already proved for points $x\in{\kern
0.0pt\mathcal{M}}^{\mathrm{o}}$ in [5, Proposition 2.6], throughout the
section we assume that $x\in\partial\mathcal{M}$.

The idea to prove the existence of replacements is the following: we fix an
annulus
$\operatorname*{An}=\operatorname*{An}(x,r,s)\in\mathcal{AN}_{r(x)}({x})$;
since every $(\Sigma^{k},\Gamma^{k})$ is, for $k$ large enough,
$\varepsilon_{k}$-a.m. in $\operatorname*{An}$ for the sequence
$\varepsilon_{k}\downarrow 0$ stated by Proposition 5.1, we fix such a $k$ and
consider all homotopic deformations
$(\tilde{\Sigma}_{t}^{k},\tilde{\Gamma}_{t}^{k})_{t\in[0,1]}$ of
$(\Sigma^{k},\Gamma^{k})$ supported in $\operatorname*{An}$ which do not
“pass” through pairs for which $F_{a}$ is larger than
$F_{a}(\Sigma^{k},\Gamma^{k})+\frac{\varepsilon_{k}}{8}$, that is

(7.1) $F_{a}(\tilde{\Sigma}_{t}^{k},\tilde{\Gamma}_{t}^{k})\leq
F_{a}(\Sigma^{k},\Gamma^{k})+\frac{\varepsilon_{k}}{8}\qquad\forall t\in[0,1]$

We can consider a minimizing sequence $(\Sigma_{j}^{k},\Gamma_{j}^{k})_{j}$
for this class of deformations that converges, up to subsequences, to a pair
$(\tilde{V}_{k},\tilde{W}_{k})$, which is stationary for $F_{a}$ in
$\operatorname*{An}$ and coincides with $(\Sigma^{k},\Gamma^{k})$ in
$\mathcal{M}\setminus\operatorname*{An}$, since all deformations are supported
in $\operatorname*{An}$. The main step is to prove that
$(\tilde{V}_{k},\tilde{W}_{k})$ is induced by regular surfaces in
$\operatorname*{An}$. This is based on the fact that if we consider a
sufficiently small ball $B\subset\operatorname*{An}$, the sequence
$(\Sigma_{j}^{k},\Gamma_{j}^{k})_{j}$ is actually minimizing $F_{a}$ in $B$
_without_ the restriction (7.1). This allows us to use the regularity theory
for sets which minimize the capillarity functional $F_{a}$, see [8, Theorem
1.2] and [9, Theorem 1.5], to obtain that $(\tilde{V}_{k},\tilde{W}_{k})$ is
regular inside $\operatorname*{An}$.

Up to subsequences, $(\tilde{V}_{k},\tilde{W}_{k})$ converges as $k\to\infty$,
to a pair of varifolds $(\tilde{V},\tilde{W})$. $(\tilde{V},\tilde{W})$
coincides with $(V,W)$ in $\mathcal{M}\setminus\operatorname*{An}$ and, By
Theorem 6.5, coincides in $\operatorname*{An}$ with a regular pair
$(\tilde{\Sigma},\tilde{\Gamma})$ which is stationary and stable for $F_{a}$.
Thus $(\tilde{V},\tilde{W})$ is the desired replacement.

We fix $x\in\partial\mathcal{M}$ and an annulus
$\operatorname*{An}=\operatorname*{An}(x,r,s)\in\mathcal{AN}_{r(x)}({x})$.
These are fixed throughout the remainder of the section, and when we simply
write $\operatorname*{An}$ we refer to $\operatorname*{An}(x,r,s)$.

###  7.1. $\frac{\varepsilon_{k}}{8}$-homotopic Plateau problem

Let $(\Sigma^{k},\Gamma^{k})^{k}$ be the almost minimizing min-max sequence
given by Proposition 5.1 and for every $k\in\mathbb{N}$ let $\Omega_{k}$ be
the open subset of $\mathcal{M}$ which is compatible with
$(\Sigma^{k},\Gamma^{k})$. We fix $k\in\mathbb{N}$ such that
$(\Sigma^{k},\Gamma^{k})$ is ${\varepsilon_{k}}$-a.m. in $\operatorname*{An}$
and we define the class of pairs
$\mathcal{HP}\big{(}(\Sigma^{k},\Gamma^{k}),\operatorname*{An}\big{)}$.

######  Definition 7.2.

A pair $(\Sigma,\Gamma)$ belongs to
$\mathcal{HP}\big{(}(\Sigma^{k},\Gamma^{k}),\operatorname*{An}\big{)}$ if
there exists a parametrized family of pairs of surfaces
$(\Sigma_{t},\Gamma_{t})_{t\in[0,1]}$ and a family of open sets
$(\Omega_{t})_{t\in[0,1]}$ compatible with $(\Sigma_{t},\Gamma_{t})$ that
satisfy the following properties:

  1. (1)

$(\Sigma_{0},\Gamma_{0})=(\Sigma^{k},\Gamma^{k})$ and
$(\Sigma_{1},\Gamma_{1})=(\Sigma,\Gamma)$;

  2. (2)

$(\Sigma_{t},\Gamma_{t})=(\Sigma^{k},\Gamma^{k})$ in
$\mathcal{M}\setminus\operatorname*{An}$ for every $t\in[0,1]$;

  3. (3)

$F_{a}(\Sigma_{t},\Gamma_{t})\leq
F_{a}(\Sigma^{k},\Gamma^{k})+\frac{\varepsilon_{k}}{8}$ for every $t\in[0,1]$.

Roughly speaking, $(\Sigma,\Gamma)$ belongs to
$\mathcal{HP}\big{(}(\Sigma^{k},\Gamma^{k}),\operatorname*{An}\big{)}$ if it
can be obtained deforming $(\Sigma^{k},\Gamma^{k})$ in $\operatorname*{An}$
without passing through a pair for which $F_{a}$ is larger than
$F_{a}(\Sigma^{k},\Gamma^{k})+\frac{\varepsilon_{k}}{8}$. We now consider the
minimizing problem for $F_{a}$ in the class
$\mathcal{HP}\big{(}(\Sigma^{k},\Gamma^{k}),\operatorname*{An}\big{)}$.

######  Proposition 7.2 ($\frac{\varepsilon_{k}}{8}$-Homotopic Plateau
problem).

Let $(\Sigma_{j}^{k},\Gamma_{j}^{k})_{j}$ be a minimizing sequence of pairs in
$\mathcal{HP}\big{(}(\Sigma^{k},\Gamma^{k}),\operatorname*{An}\big{)}$, that
is

(7.2)
$\lim_{j\to\infty}F_{a}(\Sigma_{j}^{k},\Gamma_{j}^{k})=\inf\Big{\\{}F_{a}(\Sigma,\Gamma)\mid(\Sigma,\Gamma)\in\mathcal{HP}\big{(}(\Sigma^{k},\Gamma^{k}),\operatorname*{An}\big{)}\Big{\\}}$

and let $\Omega_{j}^{k}$ be open sets compatible with
$(\Sigma_{j}^{k},\Gamma_{j}^{k})$. Then, up to subsequences, the following
conclusions hold:

  1. (1)

$(\Sigma_{j}^{k},\Gamma_{j}^{k})$ converge, as $j\to\infty$, to a pair
$(V^{k},W^{k})$ which is stationary and stable for $F_{a}$ in
$\operatorname*{An}$;

  2. (2)

$(V^{k},W^{k})$ is induced, in $\operatorname*{An}$, by a pair of surfaces
$(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})$ which are smooth and minimize
$F_{a}$ on small balls contained in $\operatorname*{An}$;

  3. (3)

$\Omega_{j}^{k}$ converge, as $j\to\infty$, to a set $\tilde{\Omega}^{k}$ of
finite perimeter which is compatible with
$(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})$ in $\operatorname*{An}$.

###  7.2. Proof of proposition 7.2

The proof is quite long and we divide it in several steps. Since $k$ is fixed,
throughout the proof we drop superscripts $k$, thus we write $(\Sigma,\Gamma)$
for $(\Sigma^{k},\Gamma^{k})$, $(\Sigma_{j},\Gamma_{j})$ for
$(\Sigma_{j}^{k},\Gamma_{j}^{k})$, $\Omega_{j}$ for $\Omega_{j}^{k}$, $(V,W)$
for $(V^{k},W^{k})$, $(\tilde{\Sigma},\tilde{\Gamma})$ for
$(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})$ and $\tilde{\Omega}$ for
$\tilde{\Omega}^{k}$.

First of all we notice that of course there exists a subsequence, not
relabeled, $(\Sigma_{j},\Gamma_{j})$ which converges to a pair $(V,W)$ in the
sense of varifolds, since masses are bounded; moreover the open sets
$\Omega_{j}$ compatible with $(\Sigma_{j},\Gamma_{j})$ converge to
$\tilde{\Omega}$, which is of finite perimeter since the masses of the pairs
$(\Sigma_{j},\Gamma_{j})$ are uniformly bounded. We then have to prove that
these objects have the properties stated in the proposition.

####  7.2.1. Step 1: $(V^{k},W^{k})$ is stationary and stable for $F_{a}$ in
$\operatorname*{An}$

We first prove that $(V,W)$ is stationary in $\operatorname*{An}$ for $F_{a}$;
indeed, let us assume by contradiction the existence of a vector field
$X\in\mathfrak{X}_{t}(\mathcal{M})$ with compact support in
$\operatorname*{An}$ such that

(7.3) $\delta_{F_{a}}(V,W)[X]=-\beta<0.$

Since $(\Sigma_{j},\Gamma_{j})$ converge to $(V,W)$, by continuity of first
variation there exists $J\in\mathbb{N}$ such that

(7.4)
$\delta_{F_{a}}(\Sigma_{j},\Gamma_{j})[X]\leq-\frac{\beta}{2}\qquad\forall
j\geq J.$

Since the first variation with respect to $X$ is continuous with respect to
the varifold convergence, there exists $T>0$ such that

(7.5)
$\delta_{F_{a}}(\psi_{t})_{\sharp}(\Sigma_{j},\Gamma_{j})[X]\leq-\frac{\beta}{4}\qquad\forall
t\in[0,T],\forall j\geq J.$

Thus, for every $j\geq J$ we have

(7.6) $F_{a}(\psi_{T})_{\sharp}(\Sigma_{j},\Gamma_{j})\leq
F_{a}(\Sigma_{j},\Gamma_{j})+\int_{0}^{T}\delta_{F_{a}}(\psi_{t})_{\sharp}(\Sigma_{j},\Gamma_{j})[X]\mathop{}\\!\mathrm{d}t\leq
F_{a}(\Sigma_{j},\Gamma_{j})-\frac{T\beta}{4}$

which contradicts the minimality of the sequence $(\Sigma_{j},\Gamma_{j})_{j}$
in $\mathcal{HP}\big{(}(\Sigma^{k},\Gamma^{k}),\operatorname*{An}\big{)}$.

By a similar argument, $(V,W)$ is also stable for $F_{a}$ in
$\operatorname*{An}$.

####  7.2.2. Step 2: In small balls pairs with small $F_{a}$ are in
$\mathcal{HP}\big{(}(\Sigma^{k},\Gamma^{k}),\operatorname*{An}\big{)}$

We show that, if we consider a sufficiently small ball $B$ in
$\operatorname*{An}$ and a sufficiently large $j$, if
$(\tilde{\Sigma},\tilde{\Gamma})$ coincides with $(\Sigma_{j},\Gamma_{j})$
outside of $B$ and it is such that $F_{a}(\tilde{\Sigma},\tilde{\Gamma})\leq
F_{a}(\Sigma_{j},\Gamma_{j})$, then $(\tilde{\Sigma},\tilde{\Gamma})$ belongs
to $\mathcal{HP}\big{(}(\Sigma^{k},\Gamma^{k}),\operatorname*{An}\big{)}$.

######  Lemma 7.3.

Let us assume $y\in\operatorname*{An}$; there exists $j_{0}\in\mathbb{N}$ and
$\rho_{0}>0$ such that $B_{\rho_{0}}(y)\subset\operatorname*{An}$ with the
following property: if $j>j_{0}$, $\rho<\rho_{0}$ and if
$(\tilde{\Sigma},\tilde{\Gamma})$ is a smooth pair of surfaces such that

  1. (1)

$(\tilde{\Sigma},\tilde{\Gamma})\setminus
B_{\rho}(y)=(\Sigma_{j},\Gamma_{j})\setminus B_{\rho}(y)$;

  2. (2)

$F_{a}(\tilde{\Sigma},\tilde{\Gamma})<F_{a}(\Sigma_{j},\Gamma_{j})$,

then
$(\tilde{\Sigma},\tilde{\Gamma})\in\mathcal{HP}\big{(}(\Sigma^{k},\Gamma^{k}),\operatorname*{An}\big{)}$.

###### Proof.

Since the case $y\in{\kern 0.0pt\mathcal{M}}^{\mathrm{o}}$ is covered by [5,
Lemma 4.1], we fix $y\in\operatorname*{An}\cap\partial\mathcal{M}$. Up to
translating, we can assume that $y$ coincides with the origin $0$.

We first assume that there exists $R>0$ such that
$B_{R}\cap\partial\mathcal{M}$ is flat, that is, up to rotating the
coordinates, we assume that
$B_{R}\cap\partial\mathcal{M}=\\{(z_{1},z_{2},z_{3})\in B_{R}\mid z_{3}\geq
0\\}$.

  1. (1)

We begin by considering $j_{0}\in\mathbb{N}$ and $\rho_{0}>0$ such that
$4\rho_{0}\leq R$ and $B_{4\rho_{0}}\subset\operatorname*{An}$. The exact
values of $j_{0}$ and $\rho_{0}$ will be chosen at the end of the proof.

We choose $\rho<\rho_{0}$, $j>j_{0}$; since the sets of singularities
$S_{j},G_{j}$ of $(\Sigma_{j},\Gamma_{j})$ are finite, by Sard’s Lemma there
exists $\tau\in(\rho,2\rho)$ such that each surface of the pair
$(\Sigma_{j},\Gamma_{j})$ intersects $\partial B_{\tau}$ transversally and
smoothly.

We now choose $\delta>0$ so that $2\delta<\min\\{\tau-\rho,2\rho-\tau\\}$, and
we consider a monotone non-decreasing smooth function
$\gamma:[0,+\infty)\to[0,+\infty)$ such that

     * •

$\gamma(t)=t$ if $t\in[0,\tau-2\delta]\cup[\tau+2\delta,+\infty)$;

     * •

$\gamma(t)=\tau$ if $t\in[\tau-\delta,\tau+\delta]$.

We recall, by (2.1), that

(7.7)
$D_{{0,\kappa}}:z\in\mathbb{R}^{3}\longmapsto\frac{1}{\kappa}z\in\mathbb{R}^{3},$

and, for every $t\in[0,1]$ and every $\lambda\geq 0$, we define
$(\Sigma_{j,t},\Gamma_{j,t})$ as follows:

(7.8) $(\Sigma_{j,t},\Gamma_{j,t})\cap\partial
B_{\lambda}=D_{{0,[t\gamma(\lambda)+(1-t)\lambda]/\lambda}}\Bigl{(}(\Sigma_{j},\Gamma_{j})\cap\partial
B_{t\gamma(\lambda)+(1-t)\lambda}\Bigr{)}.$

Roughly speaking, we are stretching $(\Sigma_{j},\Gamma_{j})$ in a
neighbourhood of $\partial B_{\tau}$ and deforming it in a cone.

Whatever is $\varepsilon>0$, if $\delta$ is chosen sufficiently small, we have
that

(7.9) $\displaystyle F_{a}(\Sigma_{j,t},\Gamma_{j,t})\leq
F_{a}(\Sigma_{j},\Gamma_{j})+\varepsilon\qquad\forall t\in[0,1].$

Since $(\tilde{\Sigma},\tilde{\Gamma})=(\Sigma_{j},\Gamma_{j})$ outside
$B_{\rho}$, the same modification can be done on
$(\tilde{\Sigma},\tilde{\Gamma})$ with the same estimates on $F_{a}$. This
allows us to assume, without loss of generality, that
$(\Sigma_{j},\Gamma_{j})$ (and thus $(\tilde{\Sigma},\tilde{\Gamma})$, since
it coincides with $(\Sigma_{j},\Gamma_{j})$ outside $B_{\rho}$) is a cone in
$\operatorname*{An}(0,\tau-\delta,\tau+\delta)$.

  2. (2)

We now define $C,K$ as the cones with base $\Sigma_{j}\cap\partial B_{\tau}$
and $\Gamma_{j}\cap\partial B_{\tau}$ and vertex $0$, that is:

(7.10) $\begin{gathered}C\cap\partial
B_{\lambda}=D_{{0,\tau/\lambda}}\Bigl{(}\Sigma_{j}\cap\partial
B_{\tau}\Bigr{)}\qquad\forall\lambda>0,\\\ K\cap\partial
B_{\lambda}=D_{{0,\tau/\lambda}}\Bigl{(}\Gamma_{j}\cap\partial
B_{\tau}\Bigr{)}\qquad\forall\lambda>0.\end{gathered}$

We can now construct the homotopy beetween $(\Sigma_{j},\Gamma_{j})$ and
$(\tilde{\Sigma},\tilde{\Gamma})$, that is the family of pairs
$\\{(\Sigma_{j,t},\Gamma_{j,t})\\}_{t\in[0,1]}$ such that
$(\Sigma_{j,0},\Gamma_{j,0})=(\Sigma_{j},\Gamma_{j})$ and
$(\Sigma_{j,1},\Gamma_{j,1})=(\tilde{\Sigma},\tilde{\Gamma})$; we define it in
the following way:

     * •

$(\Sigma_{j,t},\Gamma_{j,t})\setminus
B_{\tau}=(\Sigma_{j},\Gamma_{j})\setminus B_{\tau}$ for every $t\in[0,1]$;

     * •

$(\Sigma_{j,t},\Gamma_{j,t})\cap\operatorname*{An}(0,|1-2t|\tau,\tau)=(C,K)\cap\operatorname*{An}(0,|1-2t|\tau,\tau)$
for every $t\in[0,1]$;

     * •

$(\Sigma_{j,t},\Gamma_{j,t})\cap
B_{(1-2t)\tau}=D_{{0,1/(1-2t)}}\Bigl{(}(\Sigma_{j},\Gamma_{j})\cap
B_{\tau}\Bigr{)}$ for $t\in[0,\frac{1}{2}]$;

     * •

$(\Sigma_{j,t},\Gamma_{j,t})\cap
B_{(2t-1)\tau}=D_{{0,1/(2t-1)}}\Bigl{(}(\tilde{\Sigma},\tilde{\Gamma})\cap
B_{\tau}\Bigr{)}$ for $t\in[\frac{1}{2},1]$.

The family of open sets compatible with $(\Sigma_{j,t},\Gamma_{j,t})$ is
defined in the same way.

It is clear that this family of pair of surfaces satisfies conditions 1 and 2
of definition 7.2; we have only to check that condition 3 is satisfied.

  3. (3)

By coarea formula, we have

(7.11)
$\begin{gathered}\mathcal{H}^{2}(\Sigma_{j}\cap\operatorname*{An}(0,\rho,2\rho))=\int_{\rho}^{2\rho}\mathcal{H}^{1}\big{(}\Sigma_{j}\cap\partial
B_{\lambda}\big{)}\mathop{}\\!\mathrm{d}\lambda,\\\
\mathcal{H}^{2}(\Gamma_{j}\cap\operatorname*{An}(0,\rho,2\rho))=\int_{\rho}^{2\rho}\mathcal{H}^{1}\big{(}\Gamma_{j}\cap\partial
B_{\lambda}\big{)}\mathop{}\\!\mathrm{d}\lambda.\end{gathered}$

By Chebyshev inequality we have

(7.12)
$\begin{gathered}\mathcal{H}^{1}\Bigl{(}\Bigl{\\{}\lambda\in[\rho,2\rho]\mid\mathcal{H}^{1}(\Sigma_{j}\cap\partial
B_{\lambda})\geq\frac{3\mathcal{H}^{2}(\Sigma_{j}\cap\operatorname*{An}(0,\rho,2\rho))}{\rho}\Bigr{\\}}\Bigr{)}\leq\frac{\rho}{3},\\\
\mathcal{H}^{1}\Bigl{(}\Bigl{\\{}\lambda\in[\rho,2\rho]\mid\mathcal{H}^{1}(\Gamma_{j}\cap\partial
B_{\lambda})\geq\frac{3\mathcal{H}^{2}(\Gamma_{j}\cap\operatorname*{An}(0,\rho,2\rho))}{\rho}\Bigr{\\}}\Bigr{)}\leq\frac{\rho}{3}.\end{gathered}$

Since by Sard’s Lemma the set of $\lambda\in[\rho,2\rho]$ such that
$\Sigma_{j}$ and $\Gamma_{j}$ intersect $\partial B_{\lambda}$ transversally
has full measure, we can choose $\tau\in[\rho,2\rho]$ such that
$\Sigma_{j}\cap\partial B_{\tau}$ and $\Gamma_{j}\cap\partial B_{\tau}$ are
smooth $(1)$-manifolds and

(7.13) $\begin{gathered}\mathcal{H}^{1}(\Sigma_{j}\cap\partial
B_{\tau})\leq\frac{3\mathcal{H}^{2}(\Sigma_{j}\cap\operatorname*{An}(0,\rho,2\rho))}{\rho},\\\
\mathcal{H}^{1}(\Gamma_{j}\cap\partial
B_{\tau})\leq\frac{3\mathcal{H}^{2}(\Gamma_{j}\cap\operatorname*{An}(0,\rho,2\rho))}{\rho}.\end{gathered}$

Thus we have an estimate for the area of
$C\cap\operatorname*{An}(y,|1-2t|\tau,\tau)$ and
$K\cap\operatorname*{An}(y,|1-2t|\tau,\tau)$:

(7.14)
$\begin{split}\mathcal{H}^{2}(C\cap\operatorname*{An}(y,|1-2t|\tau,\tau))&=\int_{|1-2t|\tau}^{\tau}\mathcal{H}^{1}(K\cap\partial
B_{\lambda})\mathop{}\\!\mathrm{d}\lambda\\\
&=\mathcal{H}^{1}(\Sigma_{j}\cap\partial
B_{\tau})\int_{|1-2t|\tau}^{\tau}\Bigl{(}\frac{\lambda}{\tau}\Bigr{)}\mathop{}\\!\mathrm{d}\lambda\\\
&\leq 6\mathcal{H}^{2}(\Sigma_{j}\cap B_{2\rho}).\end{split}$

In a similar way we get

(7.15) $\mathcal{H}^{2}(K\cap\operatorname*{An}(y,|1-2t|\tau,\tau))\leq
6\mathcal{H}^{2}(\Gamma_{j}\cap B_{2\rho}).$

If $t\in[0,\frac{1}{2}]$ we have

(7.16) $\begin{gathered}\mathcal{H}^{2}(\Sigma_{j,t}\cap
B(0,|1-2t|\tau))=|1-2t|^{2}\mathcal{H}^{2}(\Sigma_{j}\cap
B_{\tau})\leq\mathcal{H}^{2}(\Sigma_{j}\cap B_{2\rho})\\\
\mathcal{H}^{2}(\Gamma_{j,t}\cap
B(0,|1-2t|\tau))=|1-2t|^{2}\mathcal{H}^{2}(\Gamma_{j}\cap
B_{\tau})\leq\mathcal{H}^{2}(\Gamma_{j}\cap B_{2\rho})\end{gathered}$

If $t\in[\frac{1}{2},1]$ we similarly have

(7.17) $\begin{gathered}\mathcal{H}^{2}(\Sigma_{j,t}\cap
B(0,|1-2t|\tau))=|1-2t|^{2}\mathcal{H}^{2}(\tilde{\Sigma}\cap
B_{\tau})\leq\mathcal{H}^{2}(\tilde{\Sigma}\cap B_{2\rho})\\\
\mathcal{H}^{2}(\Gamma_{j,t}\cap
B(0,|1-2t|\tau))=|1-2t|^{2}\mathcal{H}^{2}(\tilde{\Gamma}\cap
B_{\tau})\leq\mathcal{H}^{2}(\tilde{\Gamma}\cap B_{2\rho})\end{gathered}$

  4. (4)

Gathering the estimates (7.14) and (7.16), we obtain

(7.18) $\mathcal{H}^{2}(\Sigma_{j,t}\cap
B_{\tau})+a\mathcal{H}^{2}(\Gamma_{j,t}\cap B_{\tau})\leq
7\Big{[}\mathcal{H}^{2}(\Sigma_{j}\cap
B_{2\rho})+a\mathcal{H}^{2}(\Gamma_{j}\cap B_{2\rho})\Big{]}$

We now recall that $(\Sigma_{j},\Gamma_{j})$ converges to $(V,W)$
($=(V^{k},W^{k})$) which is stationary in $\operatorname*{An}$. By Proposition
3.3, we have that, for a suitable constant $c>0$

(7.19) $\lVert V\rVert(B_{\lambda})+a\lVert W\rVert(B_{\lambda})\leq
c\lambda^{2},$

By varifold convergence $(\Sigma_{j},\Gamma_{j})\to(V,W)$, up to choosing the
radius $\rho_{0}$ such that $\lVert V\rVert(\partial B_{2\rho_{0}})+a\lVert
W\rVert(\partial B_{2\rho_{0}})=0$, there exists $j_{0}\in\mathbb{N}$ such
that for all $j>j_{0}$ and $\rho<\rho_{0}$

(7.20) $\begin{split}\mathcal{H}^{2}(\Sigma_{j}\cap
B_{2\rho})+a\mathcal{H}^{2}(\Gamma_{j}\cap
B_{2\rho})&\leq\mathcal{H}^{2}(\Sigma_{j}\cap
B_{2\rho_{0}})+a\mathcal{H}^{2}(\Gamma_{j}\cap B_{2\rho_{0}})\\\ &\leq
2\Big{[}\lVert V\rVert(B_{2\rho_{0}})+a\lVert W\rVert(B_{2\rho_{0}})\Big{]}\\\
&\leq 2^{3}c\rho_{0}^{2}.\end{split}$

If $\rho_{0}$ is chosen such that $7\cdot
2^{3}c\rho_{0}^{2}<\frac{\varepsilon_{k}}{8}$, we obtain

(7.21) $F_{a}(\Sigma_{j,t},\Gamma_{j,t})\leq
F_{a}(\Sigma_{j},\Gamma_{j})+\frac{\varepsilon_{k}}{8}\qquad\forall
t\in[0,1];$

Thus, with these choices of $j_{0}$ and $\rho_{0}$,
$(\tilde{\Sigma},\tilde{\Gamma})\in\mathcal{HP}\big{(}(\Sigma^{k},\Gamma^{k}),\operatorname*{An}\big{)}$.

  5. (5)

If $\partial\mathcal{M}$ is not flat in a neighborhood of $y$, there exists a
ball $B:=B_{R}(y)$, a neighbourhood $W$ of the origin and a diffeomorphism
$\Phi:B\to W$ such that

(7.22) $\Phi(y)=0\qquad\Phi(B\cap\mathcal{M})=\\{(z_{1},\dots,z_{3})\in W\mid
z_{3}\geq 0\\}.$

By regularity of $\Phi$, there exists a constant $b>0$ such that

(7.23)
$\displaystyle\frac{1}{b}\mathcal{H}^{k}(A)\leq\mathcal{H}^{k}(\Phi(A))\leq
b\mathcal{H}^{k}(A)\qquad\forall A\subset B_{R}(y),\,\,k\in\\{1,\dots,3\\}$
(7.24) $\displaystyle\frac{1}{b}|z-w|\leq|\Phi(z)-\Phi(w)|\leq
b|z-w|\qquad\forall z,w\in B_{R}(y).$

We can repeat all the arguments made above and, up to adjusting the constants,
obtain the thesis.

∎

####  7.2.3. Step 3: $\tilde{\Omega}^{k}$ locally minimizes $F_{a}$ in
$\operatorname*{An}$

######  Lemma 7.4.

Let us assume $y\in\mathcal{M}$ and $0<\rho<\rho_{0}$, where $\rho_{0}$ is
given by Lemma 7.3; then, for every open set $\Xi$ of finite perimeter such
that $\Xi\setminus B_{\rho/2}(y)=\tilde{\Omega}\setminus B_{\rho/2}(y)$, we
have

(7.25) $F_{a}(\tilde{\Omega})\leq F_{a}(\Xi).$

###### Proof.

If $y\in{\kern 0.0pt\mathcal{M}}^{\mathrm{o}}$, then the lemma is proved by
showing that $\tilde{\Omega}$ locally minimizes the perimeter in a small ball
with center $y$; the proof is the same as the one of [5, Lemma 4.2], then we
can assume that $y\in\partial\mathcal{M}$.

Let us assume by contradiction that there exists $\eta>0$ and an open set
$\Xi$ of finite perimeter such that $\Xi\setminus
B_{\rho/2}(y)=\tilde{\Omega}\setminus B_{\rho/2}(y)$ and

(7.26) $F_{a}(\Xi)<F_{a}(\tilde{\Omega})-\eta.$

We want to use this assumption to construct a sequence of competitors for
$F_{a}$ which is better than $\Omega_{j}$.

Since $\mathbf{1}_{\Omega_{j}}\to\mathbf{1}_{\tilde{\Omega}}$ in $L^{1}$, by
coarea formula the sequence of function

(7.27) $\psi_{j}\colon
t\mapsto\mathcal{H}^{2}\bigl{(}(\Omega_{j}\Delta\tilde{\Omega})\cap\partial
B_{t}\bigr{)}$

converge to $0$ in $L^{1}([0,\rho])$ as $j\to\infty$. Hence, up to
subsequences, we have

(7.28) $\mathcal{H}^{2}\bigl{(}(\Omega_{j}\Delta\tilde{\Omega})\cap\partial
B_{t}\bigr{)}\to 0\quad\mbox{ for a.e. }t\in[0,\rho].$

We define, for each $j\in\mathbb{N}$,

(7.29) $\Xi_{j}=\bigr{(}\Omega_{j}\setminus
B_{\tau}\bigr{)}\cup\bigr{(}\Xi\cap B_{\tau}\bigr{)},$

where $\tau$ has to be chosen in $\Big{(}\frac{\rho}{2},\rho\Big{)}$. We have

(7.30)
$\begin{split}F_{a}(\Xi_{j})\leq&\mathcal{H}^{2}\big{(}(\partial\Omega_{j}\setminus
B_{\tau})\cap{\kern
0.0pt\mathcal{M}}^{\mathrm{o}}\big{)}+\operatorname*{Per}(\Xi,{\kern
0.0ptB_{\tau}}^{\mathrm{o}}\cap{\kern
0.0pt\mathcal{M}}^{\mathrm{o}})+\mathcal{H}^{2}\bigl{(}(\Omega_{j}\Delta\tilde{\Omega})\cap\partial
B_{\tau}\bigr{)}\\\ &+a\mathcal{H}^{2}\big{(}(\partial\Omega_{j}\setminus
B_{\tau})\cap\partial\mathcal{M}\big{)}+a\mathcal{H}^{2}\big{(}\partial^{*}\Xi\cap
B_{\tau}\cap\partial\mathcal{M}\big{)}\end{split}$

If we choose $\tau$ such that (7.28) holds true and such that the perimeter
measure of $\Omega_{j}$ and $\tilde{\Omega}$ do not charge $\partial
B_{\tau}$, we have that

(7.31)
$\begin{split}\limsup_{j\to\infty}\,\bigl{(}F_{a}(\Xi_{j})-F_{a}(\Omega_{j})\bigr{)}\leq&\limsup_{j\to\infty}\,\Bigl{[}\operatorname*{Per}(\Xi,{\kern
0.0ptB_{\tau}}^{\mathrm{o}}\cap{\kern
0.0pt\mathcal{M}}^{\mathrm{o}})+\mathcal{H}^{2}\bigl{(}(\Omega_{j}\Delta\tilde{\Omega})\cap\partial
B_{\tau}\bigr{)}\\\ &\quad+a\mathcal{H}^{2}\big{(}\partial^{*}\Xi\cap
B_{\tau}\cap\partial\mathcal{M}\big{)}\\\
&\quad-\operatorname*{Per}(\Omega_{j},{\kern
0.0ptB_{\tau}}^{\mathrm{o}}\cap{\kern
0.0pt\mathcal{M}}^{\mathrm{o}})-a\mathcal{H}^{2}\big{(}\partial^{*}\Omega_{j}\cap
B_{\tau}\cap\partial\mathcal{M}\big{)}\Bigr{]}\\\
\leq&\operatorname*{Per}(\Xi,{\kern 0.0ptB_{\tau}}^{\mathrm{o}}\cap{\kern
0.0pt\mathcal{M}}^{\mathrm{o}})+a\mathcal{H}^{2}\big{(}\partial^{*}\Xi\cap
B_{\tau}\cap\partial\mathcal{M}\big{)}\\\
&\quad-\liminf_{j\to\infty}\Bigl{[}\operatorname*{Per}(\Omega_{j},{\kern
0.0ptB_{\tau}}^{\mathrm{o}}\cap{\kern
0.0pt\mathcal{M}}^{\mathrm{o}})+a\mathcal{H}^{2}\big{(}\partial^{*}\Omega_{j}\cap
B_{\tau}\cap\partial\mathcal{M}\big{)}\Bigr{]}\\\
\leq&\operatorname*{Per}(\Xi,{\kern 0.0ptB_{\tau}}^{\mathrm{o}}\cap{\kern
0.0pt\mathcal{M}}^{\mathrm{o}})+a\mathcal{H}^{2}\big{(}\partial^{*}\Xi\cap
B_{\tau}\cap\partial\mathcal{M}\big{)}\\\
&\quad-\Bigl{[}\operatorname*{Per}(\tilde{\Omega},{\kern
0.0ptB_{\tau}}^{\mathrm{o}}\cap{\kern
0.0pt\mathcal{M}}^{\mathrm{o}})+a\mathcal{H}^{2}\big{(}\partial^{*}\tilde{\Omega}\cap
B_{\tau}\cap\partial\mathcal{M}\big{)}\Bigr{]}\\\ \leq&-\eta,\end{split}$

where the second inequality holds by (7.28), the third by lower semicontinuity
of the capillarity functional under $L^{1}$ convergence of sets, see [18,
Proposition 19.1] and the last inequality is given by the assumption (7.26).

$\Xi_{j}$ is not an admissible candidate, since it is not necessarily smooth
in $B_{\rho/2}$, hence we have to smooth up these set without increasing
$F_{a}$ too much.

By minimality of the sequence $\\{\partial\Omega_{j}\\}_{j}$ for the homotopic
Plateau problem, we choose $j_{0}\in\mathbb{N}$ such that Lemma 7.3 holds and
so that

(7.32)
$F_{a}(\Omega_{j})\leq\inf\\{F_{a}(\Sigma,\Gamma)\mid(\Sigma,\Gamma)\in\mathcal{HP}\big{(}(\Sigma^{k},\Gamma^{k}),\operatorname*{An}\big{)}\\}+\frac{\eta}{8}\qquad\forall
j\geq j_{0}.$

We choose an interval $(a,b)\subset\joinrel\subset(\tau,\rho)$ such that
$\partial_{i}\Omega_{j}$ and $\partial_{b}\Omega_{j}$ are smooth in
$\operatorname*{An}(y,a,b)$. We can choose
$V\subset\joinrel\subset\operatorname*{An}(y,a,b)$ so that there exists $d>0$,
$(a^{\prime},b^{\prime}$ such that

(7.33) $V=\\{(\xi,\zeta)\mid|\xi|\in(a^{\prime},b^{\prime}),\zeta\in[0,d]\\},$

where $\xi,\zeta$ are normal coordinates in a tubular neighborhood of
$\partial\mathcal{M}$; roughly speacking, in the normal coordinates, $V$ is a
“tube” with base on $\partial\mathcal{M}$.

We call $\gamma=\partial\Omega_{j}\cap\partial\mathcal{M}\cap V$. With a small
deformation supported in $\operatorname*{An}(y,a,b)$ we can “push-in” $\gamma$
in $\operatorname{Int}(\mathcal{M})$ in $V$, that is we can assume that, in
the normal coordinates $\xi,\zeta$,

(7.34) $\partial\Omega_{j}\cap V=\gamma\times[0,d].$

Moreover $d$ and $(a^{\prime},b^{\prime})$ can be chosen small enough so that
the possible gain of $F_{a}$ is as small as we want, and then we can assume,
without loss of generality, that $\Omega_{j}$ is of this form.

We now extend every $\mathbf{1}_{\Xi_{j}}$ to a function $f_{j}\in
BV(\mathbb{R}^{3})$ such that $|Df_{j}|(\partial\mathcal{M})=0$. We now call
$\varphi_{\varepsilon}$ the standard mollifier and

(7.35) $g_{j,\varepsilon}:=\mathbf{1}_{\Xi_{j}}*\varphi_{\varepsilon}.$

For every $j>j_{0}$, by standard approximation procedure [18, Theorem 13.8],
for almost every $t\in(0,1)$ we have that the set
$\Delta_{j,\varepsilon}:=\\{g_{j,\varepsilon}>t\\}\cap\mathcal{M}$ is an open
subset of $\mathcal{M}$ which is smooth in $\mathcal{M}$ and satisfies

(7.36) $\Delta_{j,\varepsilon}\xrightarrow[\varepsilon\to
0]{{L^{1}_{\text{loc}}(\mathcal{M})}}\mathbf{1}_{\Xi_{j}},\qquad\operatorname*{Per}\big{(}\Delta_{j,\varepsilon},{\kern
0.0pt\mathcal{M}}^{\mathrm{o}}\big{)}\xrightarrow[\varepsilon\to
0]{{}}\operatorname*{Per}\big{(}\Xi_{j},{\kern
0.0pt\mathcal{M}}^{\mathrm{o}}\big{)}.$

Since $F_{a}$ is continuous under strict convergence of $BV$-functions, we
obtain

(7.37) $\lim_{\varepsilon\to 0}F_{a}(\Delta_{j,\varepsilon})=F_{a}(\Xi_{j}).$

Hence for every $j>j_{0}$ there exists $\bar{\varepsilon}_{j}>0$ such that the
set $\Delta_{j,\varepsilon_{j}}$ satisfies

(7.38) $F_{a}(\Delta_{j,\varepsilon})\leq
F_{a}(\Xi_{j})+\frac{\eta}{4}\qquad\forall\varepsilon\in(0,\bar{\varepsilon}_{j})\forall
j>j_{0}.$

Since $\Delta_{j,\varepsilon}$ does not coincide with $\Omega_{j}$ outside
$B_{a}(y)$, we have to construct a new sequence of sets $\Psi_{j}$ that
“connect” $\Delta_{j,\varepsilon}$ inside $B_{a}(y)$ with $\Omega_{j}$ outside
$B_{b}(y)$ with a small possible increase of $F_{a}$. This is possible since
both $\Omega_{j}$ and $\Delta_{j,\varepsilon}$ are smooth within
$\operatorname*{An}(y,a,b)$.

We fix $j>j_{0}$ a tubular neighbourhood $U$ of $\partial\Omega_{j}$ with
normal coordinates $(w,z)$; since by mollification
$\partial\Delta_{j,\varepsilon}$ converges smoothly to $\partial\Omega_{j}$ in
$\operatorname*{An}(y,a,b)$ as $\varepsilon\to 0$, there exist functions
$\psi_{j,\varepsilon}$ such that

(7.39) $\Omega_{j}\cap U=\\{(w,s)\mid s<0\\},\qquad\Delta_{j,\varepsilon}\cap
U=\\{(w,s)\mid s<\psi_{j,\varepsilon}(w)\\}\quad\mbox{ in
}\operatorname*{An}(y,a,b)$

and $\psi_{j,\varepsilon}\to 0$ smoothly as $\varepsilon\to 0$.

To join smoothly the two sets we choose a function $\chi_{A}\in
C_{c}^{\infty}(B_{b})$ such that $0\leq\chi_{A}\leq 1$ and $\chi_{A}\equiv 1$
in a neighbourhood of $B_{a}$ and we construct a new family of open sets
$\Theta_{j,\varepsilon}$:

  * •

$\Theta_{j,\varepsilon}\cap B_{a}=\Delta_{j,\varepsilon}\cap B_{a}$;

  * •

$\Theta_{j,\varepsilon}\cap\operatorname*{An}(y,a,b)=\\{(w,s)\mid
s<\chi_{A}(w)\psi_{j,\varepsilon(w)}\\}$;

  * •

$\Theta_{j,\varepsilon}\setminus B_{b}=\Omega_{j}\setminus B_{b}$.

By construction
$\partial_{i}\Theta_{j,\varepsilon},\partial_{b}\Theta_{j,\varepsilon}$ are
smooth outside of a finite set.

By smooth convergence $\psi_{j,\varepsilon}\xrightarrow[\varepsilon\to
0]{{}}0$, for every $j>j_{0}$ there exists
$\varepsilon_{j}\in(0,\bar{\varepsilon}_{j})$ such that

(7.40) $F_{a}(\Theta_{j,\varepsilon_{j}})\leq
F_{a}(\Xi_{j})+\frac{\eta}{2}\leq F_{a}(\Omega_{j})-\frac{\eta}{4}$

if $j$ is sufficiently large, by (7.31). Since by Lemma 7.3
$\Theta_{j,\varepsilon_{j}}\in\mathcal{HP}\big{(}(\Sigma^{k},\Gamma^{k}),\operatorname*{An}\big{)}$,
this is a contradiction with the minimality of the sequence $\Omega_{j}$ in
$\mathcal{HP}\big{(}(\Sigma^{k},\Gamma^{k}),\operatorname*{An}\big{)}$ for
$F_{a}$.

∎

####  7.2.4. Step 4: $(V^{k},W^{k})$ is compatible with $\tilde{\Omega}^{k}$

We have now to show that the pair $(V,W)$ is induced by
$\partial_{i}\tilde{\Omega}$ and $\partial_{b}\tilde{\Omega}$. By [5,
Proposition A.1], we have only to prove that

(7.41)
$\lim_{j\to\infty}\mathcal{H}^{2}(\Sigma_{j})=\mathcal{H}^{2}(\partial_{i}\tilde{\Omega}),\qquad\lim_{j\to\infty}\mathcal{H}^{2}(\Gamma_{j})=\mathcal{H}^{2}(\partial_{b}\tilde{\Omega}).$

We first observe that, since $F_{a}$ is continuous under varifolds convergence
of pairs, we have

(7.42) $\lim_{j\to\infty}F_{a}(\Sigma_{j},\Gamma_{j})=F_{a}(V,W)$

Moreover, since $F_{a}$ is lower semicontinuous under $L^{1}$-convergence of
open sets, it holds

(7.43) $\qquad
F_{a}(\tilde{\Omega})\leq\lim_{j\to\infty}F_{a}(\Sigma_{j},\Gamma_{j}).$

The above inequality cannot be strict, because otherwise there would exist a
sufficiently small ball $B$ such that the same inequality holds and we can
argue as the proof of Lemma 7.4, defining new competitors
$\Xi_{j}=\bigr{(}\Omega_{j}\setminus
B_{\tau}\bigr{)}\cup\bigr{(}\tilde{\Omega}\cap B_{\tau}\bigr{)}$ which would
contradict the minimality of sequence $\Omega_{j}$. Hence

(7.44)
$F_{a}(\tilde{\Omega})=\lim_{j\to\infty}F_{a}(\Sigma_{j},\Gamma_{j})=F_{a}(V,W).$

Since $\mathbf{1}_{\Omega_{j}}\to\mathbf{1}_{\tilde{\Omega}}$ in
$L^{1}(\mathbb{R}^{3})$, we have

(7.45)
$\begin{gathered}\operatorname*{Per}(\tilde{\Omega},\mathcal{M})\leq\liminf_{j\to\infty}\operatorname*{Per}(\Omega_{j},\mathcal{M});\\\
\mathcal{H}^{2}(\partial_{i}\tilde{\Omega})=\operatorname*{Per}(\tilde{\Omega},{\kern
0.0pt\mathcal{M}}^{\mathrm{o}})\leq\liminf_{j\to\infty}\operatorname*{Per}(\Omega_{j},{\kern
0.0pt\mathcal{M}}^{\mathrm{o}})=\lVert V\rVert;\\\
\mathcal{H}^{2}(\partial_{b}\tilde{\Omega})=\operatorname*{Per}(\tilde{\Omega},\partial\mathcal{M})\geq\limsup_{j\to\infty}\operatorname*{Per}(\Omega_{j},\partial\mathcal{M})=\lVert
W\rVert.\end{gathered}$

Let us call

(7.46) $\eta:=\lVert
V\rVert-\mathcal{H}^{2}(\partial_{i}\tilde{\Omega}),\qquad\zeta:=\mathcal{H}^{2}(\partial_{b}\tilde{\Omega})-\lVert
W\rVert.$

(7.45) yields $\eta,\zeta\geq 0$. The statement is proved if we show that
$\eta=\zeta=0$. To do so, we first compute

(7.47) $\begin{split}\eta-a\zeta=&\lVert V\rVert+a\lVert
W\rVert-\big{(}\mathcal{H}^{2}(\partial_{i}\tilde{\Omega})+a\mathcal{H}^{2}(\partial_{b}\tilde{\Omega})\big{)}\\\
=&F_{a}(V,W)-F_{a}(\tilde{\Omega})\\\
\stackrel{{\scriptstyle\eqref{eq:fa_tom=fa_pair}}}{{=}}&\,0,\end{split}$

On the other hand, one has

(7.48) $\begin{split}\eta-\zeta=&\lVert V\rVert+\lVert
W\rVert-\big{(}\mathcal{H}^{2}(\partial_{i}\tilde{\Omega})+\mathcal{H}^{2}(\partial_{b}\tilde{\Omega})\big{)}\\\
=&\lim_{j\to\infty}\Big{(}\mathcal{H}^{2}(\Sigma_{j})+\mathcal{H}^{2}(\Gamma_{j})\Big{)}-\operatorname*{Per}(\tilde{\Omega},\mathcal{M})\\\
=&\lim_{j\to\infty}\operatorname*{Per}(\Omega_{j},\mathcal{M})-\operatorname*{Per}(\tilde{\Omega},\mathcal{M})\\\
\geq&0\end{split}$

gathering the previous two equations we get that $\zeta=0$, hence $\eta=0$, as
desired.

####  7.2.5. Step 5: $\partial_{i}\tilde{\Omega}^{k}$ and
$\partial_{b}\tilde{\Omega}^{k}$ are regular surfaces in $\operatorname*{An}$

By the minimality of $\tilde{\Omega}$ for $F_{a}$ in small balls, we use use
the regularity theorems for minimizers of capillarity functional[8, Theorem
1.2 and Corollary 1.4] to obtain that $V$ and $W$, which are induced by
$\partial_{i}\tilde{\Omega}$ and $\partial_{b}\tilde{\Omega}$, coincide in
$\operatorname*{An}$ with surfaces $\tilde{\Sigma},\tilde{\Gamma}$ that are
smooth.

###  7.3. Construction of replacements: proof of theorem 7.1

###### Proof of the theorem 7.1.

  1. (1)

Since every $(V^{k},W^{k})$ is a stationary stable pair in
$\operatorname*{An}$, by Theorem 6.5 we have that the sequence of pairs
$(V^{k},W^{k})$ converges, up to subsequences, to a pair
$(\tilde{V},\tilde{W})$ which is induced by a pair of regular surfaces in
$\operatorname*{An}$ which are stationary and stable for $F_{a}$ in
$\operatorname*{An}$.

We select a diagonal sequence
$(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})^{k}=(\partial_{i}\Omega_{j(k)}^{k},\partial_{b}\Omega_{j(k)}^{k})$
(where for every $k\in\mathbb{N}$ $\\{\Omega_{j}^{k}\\}_{j}$ is a minimizing
sequence for the $\frac{\varepsilon_{k}}{8}$-homotopic Plateau problem
$\mathcal{HP}\big{(}(\Sigma^{k},\Gamma^{k}),\operatorname*{An}\big{)}$) which
satisfies

(7.49) $F_{a}(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})\leq
F_{a}(\Sigma^{k},\Gamma^{k})\qquad\forall k\in\mathbb{N}$

and such that
$(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})\to(\tilde{V},\tilde{W})$. Since every
$(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})\in\mathcal{HP}\big{(}(\Sigma^{k},\Gamma^{k}),\operatorname*{An}\big{)}$,
we have

(7.50) $F_{a}(\Sigma^{k},\Gamma^{k})-\varepsilon_{k}\leq
F_{a}(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})\leq
F_{a}(\Sigma^{k},\Gamma^{k}),$

we have that $(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})^{k}$ is a new min-max
sequence for $F_{a}$.

  2. (2)

We fix $\eta>0$ and
$\operatorname*{An}^{\prime}=\operatorname*{An}(x,r-\eta,s+\eta)$ such that
$\operatorname*{An}^{\prime}\in\mathcal{AN}_{r(x)}({x})$.

     * •

We set $\tilde{r}(x)=r(x)$; $(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})$ is
$\varepsilon_{k}$-almost minimizing in every annulus in
$\mathcal{AN}_{r(x)}({x})$; indeed, if there exists
$\operatorname*{An}^{\prime\prime}\in\mathcal{AN}_{r(x)}({x})$ such that
$(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})$ is not $\varepsilon_{k}$-almost
minimizing in $\operatorname*{An}^{\prime\prime}$, since for every $k$,
$(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})$ is obtained by deforming
$(\Sigma^{k},\Gamma^{k})$ in $\operatorname*{An}$ without passing through a
pair with $F_{a}$ larger than
$F_{a}(\Sigma^{k},\Gamma^{k})+\varepsilon_{k}/8$, this would imply that also
$(\Sigma^{k},\Gamma^{k})$ is not almost minimizing in an annulus
$\operatorname*{An}^{\prime\prime\prime}\in\mathcal{AN}_{r(x)}({x})$ such that
$\operatorname*{An}^{\prime\prime\prime}\supset\operatorname*{An}\cup\operatorname*{An}^{\prime\prime}$
and this is a contradiction.

     * •

if $y\in\operatorname*{An}^{\prime}$, we set
$\tilde{r}(y)=\min\\{r(y),\operatorname*{dist}(y,\partial\operatorname*{An}^{\prime})\\}$;
In this case $(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})$ is almost minimizing in
every annulus belonging to $\mathcal{AN}_{\tilde{r}(y)}({y})$ because such an
annulus is contained in $\operatorname*{An}^{\prime}$ and
$(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})$ is almost minimizing in
$\operatorname*{An}^{\prime}$.

     * •

if $y\notin\operatorname*{An}^{\prime}$, we set
$\tilde{r}(y)=\min\\{r(y),\operatorname*{dist}(y,\operatorname*{An})\\}$. If
$\operatorname*{An}^{\prime\prime}\in\mathcal{AN}_{\tilde{r}(y)}({y})$ then
$\operatorname*{An}^{\prime\prime}\subset\mathcal{M}\setminus\operatorname*{An}$;
since $(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})$ and $(\Sigma^{k},\Gamma^{k})$
coincide outside $\operatorname*{An}$ and since $(\Sigma^{k},\Gamma^{k})$ is
almost minimizing in every annulus in $\mathcal{AN}_{r(y)}({y})$, we obtain
that $(\tilde{\Sigma}^{k},\tilde{\Gamma}^{k})$ is a.m. in
$\operatorname*{An}^{\prime\prime}$.

  3. (3)

We are left to prove that $(\tilde{V},\tilde{W})$ is a replacement for $(V,W)$
in $\operatorname*{An}=\operatorname*{An}(x,r,s)$. Since