# ON ISOLATED SINGULARITIES AND GENERIC REGULARITY OF MIN-MAX CMC
HYPERSURFACES

COSTANTE BELLETTINI & KOBE MARSHALL–STEVENS

###### Abstract

In compact Riemannian manifolds of dimension 3 or higher with positive Ricci
curvature, we prove that every constant mean curvature hypersurface produced
by the Allen–Cahn min-max procedure in [BW20a] (with constant prescribing
function) is a local minimiser of the natural area-type functional around each
isolated singularity. In particular, every tangent cone at each isolated
singularity of the resulting hypersurface is area-minimising. As a
consequence, for any real $\lambda$ we show, through a surgery procedure, that
for a generic 8-dimensional compact Riemannian manifold with positive Ricci
curvature there exists a closed embedded smooth hypersurface of constant mean
curvature $\lambda$; the minimal case ($\lambda=0$) of this result was
obtained in [CLS22].

###### Contents

  1. 1 Introduction
    1. 1.1 Notation
    2. 1.2 Allen–Cahn min-max
    3. 1.3 Strategy
    4. 1.4 Structure and remarks
  2. 2 Surgery procedure for isolated singularities with regular tangent cone
  3. 3 Signed distance and the one-dimensional profile
    1. 3.1 Singular behaviour of the distance function
    2. 3.2 Level sets of the signed distance function
    3. 3.3 Approximation with the one-dimensional profile
  4. 4 Relating local properties of the energy to the geometry
    1. 4.1 Local smoothing of Caccioppoli sets
    2. 4.2 Local energy minimisation
    3. 4.3 Recovery functions for local geometric properties
  5. 5 Construction of paths
    1. 5.1 Paths with energy drop fixing a ball
      1. 5.1.1 Choosing radii for local properties
      2. 5.1.2 Defining the shifted functions
      3. 5.1.3 Energy calculations and continuity of paths
    2. 5.2 Sliding the one-dimensional profile
    3. 5.3 Paths to local energy minimisers
  6. 6 Proof of Theorems 1, 2 & 3
    1. 6.1 Proof of Theorem 3
    2. 6.2 Proof of Theorems 1 & 2

##  1 Introduction

The Allen–Cahn min-max procedure in [BW20a], with constant prescribing
function, shows that in a compact $8$-dimensional Riemannian manifold there
exists a quasi-embedded hypersurface of constant mean curvature, with a
singular set consisting of finitely many points (see Subsection 1.2 below for
a precise description). One may thus conjecture the existence of a smoothly
embedded constant mean curvature hypersurface in all $8$-dimensional
Riemannian manifolds under some assumption on the metric; for example a
genericity assumption. As a first step, we resolve this for manifolds with
positive Ricci curvature:

######  Theorem 1.

Let $N$ be a smooth compact $8$-dimensional manifold and
$\lambda\in\mathbb{R}$. There is an open and dense subset, $\mathcal{G}$, of
the smooth metrics with positive Ricci curvature such that for each
$g\in\mathcal{G}$, there exists a closed embedded smooth hypersurface of
constant mean curvature $\lambda$ in $(N,g)$.

We actually prove more general results valid in higher dimensions, showing the
generic existence of a closed embedded hypersurface of constant mean
curvature, with singular set of codimension $7$, containing no isolated
singularities with regular tangent cones. Indeed, Theorem 1 is a consequence
of the following:

######  Theorem 2.

Let $N$ be a smooth compact manifold of dimension $n+1\geq 3$ and
$\lambda\in\mathbb{R}$. There is a dense subset, $\mathcal{G}$, of the smooth
metrics with positive Ricci curvature such that for each $g\in\mathcal{G}$,
there exists a closed embedded hypersurface of constant mean curvature
$\lambda$, smooth away from a closed singular set of Hausdorff dimension at
most $n-7$, containing no isolated singularities with regular tangent cones in
$(N,g)$.

######  Remark 1.

Let $\mathrm{Met}_{\mathrm{Ric}_{g}>0}^{k,\alpha}(N)$ for each $k\geq 1$ and
$\alpha\in(0,1)$ denote the open subset of Riemannian metrics of regularity
$C^{k,\alpha}$ on $N$ with positive Ricci curvature. In the proof of Theorem 2
we in fact establish that there exists a dense set,
$\mathcal{G}_{k}\subset\mathrm{Met}^{k,\alpha}_{\mathrm{Ric}_{g}>0}(N)$, such
that for each $g\in\mathcal{G}_{k}$, the same existence conclusion of Theorem
2 holds.

The focus of the present work concerns the generic regularity of constant mean
curvature hypersurfaces in ambient dimensions $8$ or higher and is related to
previous work concerning minimal hypersurfaces. We now briefly summarise the
works on generic regularity for minimal hypersurfaces in ambient dimension
$8$:

  * •

The existence of the Simons cone, introduced in [Sim68], showed that stable
minimal hypersurfaces may admit isolated singularities in dimension $8$.

  * •

The generic regularity of area-minimisers in each non-zero homology class was
established in [Sma93], using the fundamental result of [HS85].

  * •

In [CLS22], using the Almgren–Pitts min-max procedure and [HS85] (in
particular the extension in [Liu19]), the existence of smooth minimal
hypersurfaces was established in manifolds equipped with a generic metric of
positive Ricci curvature.

  * •

In [LW21] it was shown that every $8$-dimensional closed manifold equipped
with a generic metric (with no curvature assumption) admits a smooth minimal
hypersurface. See also [LW22], where it is shown that for a generic metric,
every embedded locally stable minimal hypersurface is smooth in dimension $8$.

Both [Sma93] and [CLS22] exploit local foliations by area-minimising
hypersurfaces, provided by [HS85], allowing for a surgery procedure to be
established in order to perturb the metric locally. Recently an analogous
foliation was established in [Les23] for constant mean curvature hypersurfaces
that locally minimise a prescribed mean curvature functional to at least one
side. Such a foliation, to one side of the hypersurface, provides a natural
way to perturb away an isolated singularity via a surgery procedure.

In order to produce such a foliation, one needs to establish that the tangent
cones to isolated singularities of a candidate hypersurface are area-
minimising; we establish this for all hypersurfaces of constant mean curvature
arising from the Allen–Cahn min-max procedure in [BW20a] (with constant
prescribing function) in manifolds with positive Ricci curvature.

The Allen–Cahn min-max procedure in [BW20a] produces in the first instance a
quasi-embedded hypersurface of constant mean curvature with a possibly non-
empty singular set of codimension at least $7$. The results of [BW22] then
establish that in manifolds with positive Ricci curvature the above
hypersurface of constant mean curvature in fact attains the min-max value (in
a manner made precise below) and is embedded, with the above dimension bound
on the singular set. We thus work with this hypersurface as a candidate to
perturb away isolated singularities via a surgery procedure.

For a compact Riemannian manifold $(N,g)$ and $\lambda\in\mathbb{R}$ we define
the $\mathcal{F}_{\lambda}$ functional on a Caccioppoli set, $F\subset N$, by

$\mathcal{F}_{\lambda}(F)=\mathrm{Per}_{g}(F)-\lambda\mathrm{Vol}_{g}(F).$

Recall that, as shown in [BW20b, Proposition B.1], smooth constant mean
curvature hypersurfaces are locally $\mathcal{F}_{\lambda}$-minimising. Our
main technical result shows that this $\mathcal{F}_{\lambda}$-minimisation
also holds in sufficiently small balls around isolated singularities for
constant mean curvature hypersurfaces produced by the Allen–Cahn min-max in
manifolds with positive Ricci curvature:

######  Theorem 3.

Let $(N,g)$ be a smooth compact Riemannian manifold of dimension $n+1\geq 3$,
with positive Ricci curvature, and $\lambda\in\mathbb{R}$. The one-parameter
Allen–Cahn min-max in [BW20a], with constant prescribing function $\lambda$,
produces a closed embedded hypersurface of constant mean curvature $\lambda$,
smooth away from a closed singular set of Hausdorff dimension at most $n-7$,
locally $\mathcal{F}_{\lambda}$-minimising in balls around each isolated
singularity. Precisely, this hypersurface arises as the boundary of a
Caccioppoli set, $E\subset N$, and for each isolated singularity,
$p\in\partial E\setminus\partial^{*}E$, there exists an $r>0$ such that

$\mathcal{F}_{\lambda}(E)=\inf_{G\in\mathcal{C}(N)}\\{\mathcal{F}_{\lambda}(G)\>|\>G\setminus
B_{r}(p)=E\setminus B_{r}(p)\\},$

where $\mathcal{C}(N)$ is the set of Caccioppoli sets in $N$. Consequently,
the hypersurface has area-minimising tangent cones at each isolated
singularity.

######  Remark 2.

In the case that $\lambda=0$, Theorem 3 shows that minimal hypersurfaces
produced by the Allen–Cahn min-max procedure in [Gua18] in manifolds with
positive Ricci curvature are in fact locally area-minimising (to both sides),
as opposed to just one-sided homotopy minimising as obtained via the
Almgren–Pitts min-max procedure in the results of [CLS22]. These minimising
properties immediately pass to tangent cones. Stable regular minimal cones
that do not minimise area are known to exist, for example the Simons cone

$C^{1,5}=\left\\{(x,y)\in\mathbb{R}^{2}\times\mathbb{R}^{6}\>|\>5|x|^{2}=|y|^{2}\right\\},$

is stable and one-sided area-minimising, but is not area-minimising to the
other side (see [Law91]). Such a cone is not explicitly ruled out in [CLS22]
from arising as a tangent cone to a min-max minimal hypersurface at an
isolated singularity. Theorem 3 precludes such tangent cones. We also note
that in the Allen–Cahn framework, obtaining an absolute area-minimisation
property, as opposed to a homotopic minimisation property, appears to be
natural; indeed the relevant space where the min-max is carried out is
$W^{1,2}(N)$, which is contractible.

The present work fits into the broader context of generic regularity for
solutions to geometric variational problems. In addition to the works on
generic regularity for minimal hypersurfaces in dimension $8$ as discussed
above, we provide a non-exhaustive summary of work on generic regularity for
geometric variational problems:

  * •

In [CMS23a] and [CMS23b] the generic regularity of area-minimising minimal
hypersurfaces is established, in various settings, up to ambient dimension
$10$.

  * •

In [Whi85] and [Whi19] it is shown that for a generic ambient metric, every
$2$-dimensional surface (integral current or flat chain mod $2$) without
boundary that minimises area in its homology class has support equal to a
smoothly embedded minimal surface.

  * •

In [Moo06] and [Moo17] it is shown that for a generic ambient metric,
parameterised $2$-dimensional minimal surfaces are free of branch points.

  * •

In [CCMS20] and [CCMS22] an analogy was established between mean curvature
flow with generic initial data and the generic regularity of area-minimising
hypersurfaces.

  * •

In [FROS20] the generic regularity of free boundaries for the obstacle problem
is established up to ambient dimension $4$.

###  1.1 Notation

We now collect some notation and definitions that will be used throughout the
paper:

  * •

Unless otherwise stated, throughout the paper we let $(N^{n+1},g)$ be a
compact (with empty boundary), Riemannian manifold of dimension $n+1\geq 3$,
with positive Ricci curvature, $\mathrm{Ric}_{g}>0$. We will implicitly assume
$N$ is connected.

  * •

We let $M\subset N$ be a non-empty, smooth, two-sided, separating, embedded
hypersurface of constant mean curvature $\lambda\in\mathbb{R}$, with closed
singular set, $\mathrm{Sing}(M)=\overline{M}\setminus M$, of Hausdorff
dimension at most $n-7$. As $M$ is a separating hypersurface, we may write
$N\setminus\overline{M}$ as two disjoint open sets, $E$ and $N\setminus E$,
with common boundary $\overline{M}$.

  * •

We say that $p\in\mathrm{Sing}(M)$ is an isolated singularity of $M$ if there
exists some $R_{p}>0$ such that
$\mathrm{Sing}(M)\cap\overline{B_{R_{p}}}(p)=\\{p\\}$, i.e. such that
$M\cap\overline{B_{R_{p}}}(p)$ is smooth. Moreover, we say that a multiplicity
one tangent cone, $C_{p}$, to $M$ at an isolated singularity,
$p\in\mathrm{Sing}(M)$, is a regular tangent cone if
$\mathrm{Sing}(C_{p})=\\{0\\}$, where $0$ here denotes the origin in
$\mathbb{R}^{n+1}$. We note that the tangent cone to $M$ at an isolated
singularity with regular tangent cone is necessarily unique by the work of
[Sim83].

  * •

A measurable set $F\subset N$ is a Caccioppoli set if

$\mathrm{Per}_{g}(F)=\sup\left\\{\int_{E}div_{g}\varphi\>\bigg{|}\>\varphi\in\Gamma(TN),||\varphi||_{\infty}\leq
1\right\\}<\infty,$

where $div_{g}$ is the divergence with respect to the metric $g$, $\Gamma(TN)$
is the set of vector fields on $N$ and $||\cdot||_{\infty}$ denotes the
supremum norm. We denote by $\mathcal{C}(N)$ the set of Caccioppoli sets in
$N$.

  * •

By De Giorgi’s Structure Theorem we have that the distributional derivative
$D_{g}\chi_{F}$ (a Radon measure) of a Caccioppoli set $F$ is given by
$D_{g}\chi_{F}=-\nu_{F}\mathcal{H}^{n}\llcorner\partial^{*}F$, where
$\partial^{*}F$ is the reduced boundary of $F$ (an $n$-rectifiable set),
$\mathcal{H}^{n}$ is the $n$-dimensional Hausdorff measure, $\nu_{F}$ is the
normal direction to $\partial^{*}F$ pointing inside $F$ defined
$\mathcal{H}^{n}$-a.e. and $\chi_{F}$ is the indicator function of the set
$F$. Our main reference for Caccioppoli sets will be [Mag12].

  * •

We define the following prescribed mean curvature functional on measurable
subsets of $N$: for a measurable set $F\subset N$ we let

$\mathcal{F}_{\lambda}(F)=\mathrm{Per}_{g}(F)-\lambda\mathrm{Vol}_{g}(F)+\frac{\lambda}{2}\mathrm{Vol}_{g}(N),$

where $\mathrm{Vol}_{g}$ denotes the $\mathcal{H}^{n+1}$ measure with respect
to the metric $g$. This definition differs from that of Section 1 by the
addition of the constant $\frac{\lambda}{2}\mathrm{Vol}_{g}(N)$. Note however
this does not affect the set of critical points of the functional
$\mathcal{F}_{\lambda}$ and is made purely for convenience of notation in
forthcoming computations.

  * •

With the above two definitions in mind, we take throughout $\nu$ to be the
global unit normal to $M$ pointing into $E$ and write $M=\partial^{*}E$; by
viewing $M=\partial^{*}E$ as the reduced boundary of the Caccioppoli set $E$
we have that $E$ is a critical point for $\mathcal{F}_{\lambda}$.

  * •

We will use the notions of integral currents and varifolds throughout the
paper; a reference for the notation and definitions used here may be found in
[Sim84].

  * •

Let $\mathrm{dist}_{N}$ denote the Riemannian distance on $N$ and define the
function $d_{\overline{M}}$ on $N$ by setting
$d_{\overline{M}}(\cdot)=\mathrm{dist}_{N}(\cdot,\overline{M})$; we then have
that $d_{\overline{M}}$ is Lipschitz on $N$ (with Lipschitz constant equal to
$1$) and, as $N$ is complete, for each $x\in N$ there exists a geodesic
realising the value $d_{\overline{M}}(x)$. Let $d(N)=\sup_{x,y\in
N}d_{N}(x,y)$ be the diameter of $N$, which is finite as $N$ is closed.

  * •

We fix $R_{l}>0$ such that for every $R\in(0,R_{l})$ and each point $p\in N$
we have that the ball $B_{R}(p)\subset N$ of radius $R$ centred at a point
$p\in N$ is $2$-bi-Lipschitz diffeomorphic, via a geodeisic normal coordinate
chart, to the Euclidean ball,
$B^{\mathbb{R}^{n+1}}_{R}(0)\subset\mathbb{R}^{n+1}$ of radius $R$ centred at
the origin in $\mathbb{R}^{n+1}$.

  * •

For $\varepsilon\in(0,1)$ we denote the Allen–Cahn energy of a function $u\in
W^{1,2}(N)$ by

$\mathcal{E}_{\varepsilon}(u)=\frac{1}{2\sigma}\int_{N}e_{\varepsilon}(u)=\frac{1}{2\sigma}\int_{N}\frac{\varepsilon}{2}|\nabla
u|^{2}+\frac{W(u)}{\varepsilon},$

where $W$ is a $C^{2}$ double well potential with non-degenerate minima at
$\pm 1$, $c_{W}\leq W^{\prime\prime}(t)\leq C_{W}$ for constants
$c_{W},C_{W}>0$ for all $t\in\mathbb{R}\setminus[-2,2]$ and
$\sigma=\int_{-1}^{1}\sqrt{W(t)/2}\,dt$. We then consider the following
functional, which we shall refer to simply as the energy, defined on functions
$u\in W^{1,2}(N)$ by

$\mathcal{F}_{\varepsilon,\lambda}(u)=\mathcal{E}_{\varepsilon}(u)-\frac{\lambda}{2}\int_{N}u.$

###  1.2 Allen–Cahn min-max

Let $(N^{n+1},g)$ a closed, connected Riemannian manifold of dimension
$n+1\geq 3$. The Allen–Cahn min-max procedure in [BW20a] produces a
hypersurface with mean curvature prescribed by an arbitrary non-negative
Lipschitz function, and provides sharp dimension bounds on the singular set.

We recall this procedure in the case relevant to this work, in which the
metric is assumed to have positive Ricci curvature, and the prescribing
function is a non-negative constant $\lambda$; for producing a candidate
hypersurface of constant mean curvature $\lambda<0$ one can consider
$-\lambda$ in the results below. The constant mean curvature hypersurfaces
produced will, after proving Theorem 3, be candidates for our surgery
procedure established in Section 2.

For $\varepsilon\in(0,1)$ there exist two constant functions,
$a_{\varepsilon}$ and $b_{\varepsilon}$, on $N$, stable critical points of
$\mathcal{F}_{\varepsilon,\lambda}$ with $-1<a_{\varepsilon}<-1+c\varepsilon$,
$1<b_{\varepsilon}<1+c\varepsilon$ and $a_{\varepsilon}\rightarrow-1$,
$b_{\varepsilon}\rightarrow 1$ on $N$ as $\varepsilon\rightarrow 0$, where
$c>0$ depends on $W$ and $\lambda$. These functions are constructed in [BW20a,
Section 5] by means of the negative gradient flow, through constant functions,
of $\mathcal{F}_{\varepsilon,\lambda}$ starting at $\pm 1$. In particular,
there are continuous paths of functions in $W^{1,2}(N)$ connecting
$a_{\varepsilon}$ to $-1$ and $b_{\varepsilon}$ to $1$, provided by the
negative gradient flow of $\mathcal{F}_{\varepsilon,\lambda}$, with energy
along these paths bounded from above by
$\mathcal{F}_{\varepsilon,\lambda}(-1)$ and
$\mathcal{F}_{\varepsilon,\lambda}(1)$ respectively.

For every $\varepsilon\in(0,1)$ a min-max critical point, $u_{\varepsilon}\in
W^{1,2}(N)$, of $\mathcal{F}_{\varepsilon,\lambda}$ may be constructed, with
$\sup_{N}|u_{\varepsilon}|$ uniformly bounded and
$\mathcal{E}_{\varepsilon}(u_{\varepsilon})$ uniformly bounded from above and
below by positive constants. This is done by applying a mountain pass lemma,
for paths between the two stable critical points $a_{\varepsilon}$ and
$b_{\varepsilon}$, based on the fact that $\mathcal{F}_{\varepsilon,\lambda}$
satisfies a Palais–Smale condition. The Morse index of the $u_{\varepsilon}$
will then automatically be equal $1$ by virtue of the fact that if
$\mathrm{Ric}_{g}>0$ then, as noted in [BW20a, Remark 6.7], $a_{\varepsilon}$
and $b_{\varepsilon}$ are the only stable critical points of
$\mathcal{F}_{\varepsilon,\lambda}$.

By general principles, the bounds above imply that there exist a sequence
$\varepsilon_{j}\rightarrow 0$, a non-zero Radon measure, $\mu$, on $N$ and a
function, $u_{\infty}\in BV(N)$ with $u_{\infty}=\pm 1$ for a.e. $x\in N$,
such that for the min-max critical points,
$\\{u_{\varepsilon_{j}}\\}_{j=1}^{\infty}$, we have as
$\varepsilon_{j}\rightarrow 0$ that
$e_{\varepsilon_{j}}(u_{\varepsilon_{j}})\rightarrow\mu$ weakly as measures
and $u_{\varepsilon_{j}}\rightarrow u_{\infty}$ strongly in $L^{1}(N)$.
Defining $E=\\{u_{\infty}=1\\}$, we note that $E$ is a Caccioppoli set with
its reduced boundary $\partial^{*}E\subset\mathrm{Spt}\mu$; moreover, as
$\mathrm{Ric}_{g}>0$ we have that $E\neq\emptyset$ by [BW20a, Remark 6.7].

In [BW20a], relying on the combined works of [HT00] and [RT07], it is then
established that $\mu$ is the weight measure of an integral $n$-varifold $V$
with the following properties:

  * •

$V=V_{0}+V_{\lambda}$.

  * •

$V_{0}$ is a (possibly zero) stationary integral $n$-varifold on $N$ with
$\mathrm{Sing}(V_{0})$ empty if $2\leq n\leq 6$, $\mathrm{Sing}(V_{0})$
discrete if $n=7$ and $\mathrm{Sing}(V_{0})$ of Hausdorff dimension $\leq n-7$
when $n\geq 8$.

  * •

$V_{\lambda}=|\partial^{*}E|\neq 0$ (by [BW20a, Remark 6.7] as
$\mathrm{Ric}_{g}>0$), the multiplicity one $n$-varifold associated with the
reduced boundary $\partial^{*}E$. Then
$\mathrm{Spt}(V_{\lambda})\setminus\mathrm{Sing}(V_{\lambda})=\overline{\partial^{*}E}\setminus\mathrm{Sing}(V_{\lambda})$
is a quasi-embedded hypersurface of constant mean curvature $\lambda$ with
respect to the unit normal pointing into $E$; moreover,
$\mathrm{Sing}(V_{\lambda})$ is empty if $2\leq n\leq 6$,
$\mathrm{Sing}(V_{\lambda})$ is discrete if $n=7$ and
$\mathrm{Sing}(V_{\lambda})$ is of Hausdorff dimension $\leq n-7$ when $n\geq
8$.

In the above, quasi-embedded means that near every non-embedded point of
$\overline{\partial^{*}E}\setminus\mathrm{Sing}(V_{\lambda})$,
$\overline{\partial^{*}E}$ is the union of two embedded $C^{2,\alpha}$ disks
intersecting tangentially, with each disk lying on one side of the other.
Furthermore, for a varifold $W$ (for example $V_{0}$ and $V_{\lambda}$ above)
by letting $\text{gen-reg}(W)$ be the set of quasi-embedded points of
$\mathrm{\mathrm{Spt}}||W||$ we have defined
$\mathrm{Sing}(W)=\mathrm{\mathrm{Spt}}||W||\setminus\text{gen-reg}(W)$ (thus
for an embedded hypersurface this agrees with the definition in Subsection 1.1
above). For a more detailed description of the definitions and results above
we refer to [BW20a, Sections 3 and 4].

As in [BW22, Theorem 2], the path we exhibit, for all $\varepsilon>0$
sufficiently small, in Subsection 5.2 with the upper energy bounds provided by
Lemma 5 (depicted by the dashed lines in Figure 2) between $1$ and $-1$, along
with short paths of constant functions connecting $1$ to $b_{\varepsilon}$ and
$-1$ to $a_{\varepsilon}$, proves that $V_{0}=0$ (i.e. that the min-max
procedure produces no minimal piece in manifolds with positive Ricci
curvature). Using this we then note that as we have
$e_{\varepsilon_{j}}\rightarrow\mu$ as $\varepsilon_{j}\rightarrow 0$ and
$E=\\{u_{\infty}=1\\}$ we have

$\mathcal{F}_{\varepsilon_{j},\lambda}(u_{\varepsilon_{j}})\rightarrow\mathcal{F}_{\lambda}(E)\text{
as }\varepsilon_{j}\rightarrow 0,$ (1)

i.e. that the constant mean curvature hypersurface attains the min-max value.
In the proof of Theorem 3, under the contradiction assumption that our
candidate hypersurface produced by the above procedure does not satisfy a
local minimisation property, we will exploit (1) by constructing continuous
paths of functions in $W^{1,2}(N)$ for all $\varepsilon>0$ sufficiently small,
admissible in the min-max construction above, with energy along the paths
bounded above by a value strictly below $\mathcal{F}_{\lambda}(E)$
(independently of $\varepsilon$); thus violating the min-max characterisation
of $E$.

In fact, by [BW22, Theorem 4], for $\lambda\neq 0$ we have that
$\partial^{*}E$ is embedded (rather than quasi-embedded). Moreover,
$\partial^{*}E$ is connected, has index $1$ and is separating in the sense
that $N\setminus\partial E$ may be written as the union of two disjoint open
sets whose common boundary is $\partial E$. The same results hold in the case
that $\lambda=0$ by combining [Gua18, Theorem A] with [Bel20, Theorem 1.8].

To summarise, we know that for $(N^{n+1},g)$ a closed, connected Riemannian
manifold of dimension $n+1\geq 3$, with positive Ricci curvature, the
properties of $M$ as stated in Subsection 1.1 hold for any constant mean
curvature hypersurface produced from the Allen–Cahn min-max construction of
[BW20a] with prescribing function taken to be a constant.

###  1.3 Strategy

The main strategy of the present work can be split into three distinct steps:

  1. 1.

Surgery procedure: We show how to perturb constant mean curvature
hypersurfaces that are locally $\mathcal{F}_{\lambda}$-minimising around
isolated singularities with regular tangent cones, resulting in a smooth
hypersurface with constant mean curvature.

  2. 2.

Functions to geometry: We relate the local geometric behaviour of
hypersurfaces produced by the Allen–Cahn min-max procedure to the
$\varepsilon\rightarrow 0$ energy properties of specific $W^{1,2}(N)$
functions.

  3. 3.

Paths of functions: By exhibiting an admissible min-max path, we establish the
energy properties for the functions from Step $2$ hold, which enables us to
conclude that hypersurfaces generated through the Allen-Cahn min-max procedure
in positive Ricci curvature are locally $\mathcal{F}_{\lambda}$-minimising
around isolated singularities.

We now sketch these steps in more detail:

1\. Surgery procedure: As mentioned in Section 1, a local foliation around a
hypersurface provides a natural way to perturb away an isolated singularity
via a surgery procedure. Using the work of [HS85], in both [Sma93] and
[CLS22], isolated singularities with regular tangent cones to locally area-
minimising hypersurfaces are perturbed away by a “cut-and-paste” gluing along
with a conformal change of the initial metric.

Figure 1: In both graphics the innermost two thinly dotted curves depict an
annulus around an isolated singularity of a constant mean curvature
hypersurface and the outermost thinly dotted curves depict the boundary of the
ball in which the foliations will be defined. In the left-hand graphic the
lower solid curve depicts an isolated singularity (assumed to have regular
tangent cone) of a constant mean curvature hypersurface and the upper solid
curve depicts, under the assumption that the lower singular hypersurface is
locally $\mathcal{F}_{\lambda}$-minimising, a smooth constant mean curvature
hypersurface in the one-sided foliation provided by [Les23]. The solid curve
in the right-hand graphic depicts the smooth hypersurface constructed by
gluing both of the hypersurfaces in the left-hand graphic. This gluing is done
in such a way that the hypersurface outside of the larger ball in the annulus
agrees with the singular one and inside of the smaller ball in the annulus
agrees with the hypersurface provided by the foliation; with the thick dashed
lines depicting the pieces of the hypersurfaces in the left-hand graphic not
included in the construction in the right. The resulting construction in the
right-hand graphic is then, after a suitable metric perturbation, the desired
smooth constant mean curvature hypersurface.

We take a similar approach here. Near an isolated singularity with regular
tangent cone of a locally $\mathcal{F}_{\lambda}$-minimising hypersurface of
constant mean curvature $\lambda$, in [Les23] it is shown that there is a
foliation around this hypersurface (to either side) by smooth hypersurfaces of
constant mean curvature $\lambda$. In Section 2, using this foliation, we
establish a surgery procedure to perturb away isolated singularities with
regular tangent cones of locally $\mathcal{F}_{\lambda}$-minimising constant
mean curvature hypersurfaces.

This is achieved by first constructing, via a “cut-and-paste” gluing, a smooth
hypersurface close in Hausdorff distance to the original one inside a chosen
ball (which may be taken arbitrarily small), with constant mean curvature
$\lambda$ outside of an annulus and inside which the original hypersurface is
assumed to be smooth; this construction is depicted in Figure 1.

It then remains to perturb the metric inside of this annulus so that the newly
constructed hypersurface has constant mean curvature $\lambda$ everywhere.
This is achieved by an appropriate choice of function for conformal change of
the original metric; with the resulting metric arbitrarily close, in the
$C^{k,\alpha}$-norm, to the original. The result is then a smooth hypersurface
of constant mean curvature $\lambda$ with respect to the new metric, agreeing
with the original hypersurface outside of the chosen ball.

2\. Functions to geometry: We aim to show that, when the ambient metric is
assumed to have positive Ricci curvature, the hypersurfaces of constant mean
curvature $\lambda$ produced by the Allen–Cahn min-max procedure in [BW20a]
are in fact locally $\mathcal{F}_{\lambda}$-minimising. In order to do this we
first relate the local $\mathcal{F}_{\lambda}$-minimisation we desire to a
property about the energy of specific $W^{1,2}(N)$ functions defined from such
a hypersurface as the parameter $\varepsilon\rightarrow 0$.

Rather than work directly with the min-max critical points of
$\mathcal{F}_{\varepsilon,\lambda}$ produced in [BW20a], we instead introduce,
in Subsection 3.3, a function $v_{\varepsilon}\in W^{1,2}(N)$, which we call
the one-dimensional profile. This function is constructed by placing a
truncated version of the one-dimensional solution to the Allen–Cahn equation,
$\overline{\mathbb{H}}^{\varepsilon}$, in the normal direction to an
underlying hypersurface, $M$, of constant mean curvature $\lambda$ as in
Subsection 1.1:

$v_{\varepsilon}=\overline{\mathbb{H}}^{\varepsilon}\circ
d^{\pm}_{\overline{M}},$

where here $d^{\pm}_{\overline{M}}$ is the Lipschitz signed distance function
to $\overline{M}$, taking positive values in $E$ and negative values in
$N\setminus E$. The function $v_{\varepsilon}$ is then shown to act as an
approximation of the hypersurface $M$ in the sense that, analogously to (1),
we have

$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\rightarrow\mathcal{F}_{\lambda}(E)\text{
as }\varepsilon\rightarrow 0.$ (2)

In Subsection 4.2, for an isolated singularity, $p\in\mathrm{Sing}(M)$,
$\varepsilon>0$ sufficiently small and radius $\rho>0$, we minimise
$\mathcal{F}_{\varepsilon,\lambda}$ over a class of functions
$\mathcal{A}_{\varepsilon,\rho}(p)$. The set
$\mathcal{A}_{\varepsilon,\rho}(p)$ is, roughly speaking, all $W^{1,2}(N)$
functions agreeing with $v_{\varepsilon}$ outside of the ball of radius $\rho$
centred at $p$. The minimiser of this problem is thus a function,
$g_{\varepsilon}\in W^{1,2}(N)$, that agrees with $v_{\varepsilon}$ outside of
the ball $B_{\rho}(p)$ and is such that

$\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})=\inf_{u\in\mathcal{A}_{\varepsilon,\rho}(p)}\mathcal{F}_{\varepsilon,\lambda}(u).$
(3)

Note that the notation used for $g_{\varepsilon}$ suppresses the dependence on
$p\in\mathrm{Sing}(M)$ and $\rho>0$ used in the construction; in each instance
that the functions $g_{\varepsilon}$ are utilised the choice of isolated
singularity and radius in question will be made explicit. We then produce, in
Subsection 4.3, a sequence of “recovery functions”, admissible in the
minimisation problem that produced $g_{\varepsilon}$ above, for any local
$\mathcal{F}_{\lambda}$-minimiser. Precisely, in the vein of [KS89], for each
local $\mathcal{F}_{\lambda}$-minimiser, $F\in\mathcal{C}(N)$, agreeing with
$E$ outside of $B_{\frac{\rho}{2}}(p)$, we show that there exists a sequence
of functions, $f_{\varepsilon}\in\mathcal{A}_{\varepsilon,\rho}(p)$, for all
$\varepsilon>0$ sufficiently small such that

$\mathcal{F}_{\varepsilon,\lambda}(f_{\varepsilon})\rightarrow\mathcal{F}_{\lambda}(F)\text{
as }\varepsilon\rightarrow 0.$ (4)

As $f_{\varepsilon}\in\mathcal{A}_{\varepsilon,\rho}(p)$ we conclude that by
(3) we have
$\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(f_{\varepsilon})$.
In particular, by (2), (3) and (4), it holds that

$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})+\tau_{\varepsilon}\text{
for some sequence }\tau_{\varepsilon}\rightarrow 0\text{ as
}\varepsilon\rightarrow 0,$ (5)

if and only if

$\mathcal{F}_{\lambda}(E)\leq\mathcal{F}_{\lambda}(F),$

so that $E$ is $\mathcal{F}_{\lambda}$-minimising in $B_{\frac{\rho}{2}}(p)$.
In this manner we have related, via (5), the $\varepsilon\rightarrow 0$ energy
behaviour of specific $W^{1,2}(N)$ functions, namely $v_{\varepsilon}$ and
$g_{\varepsilon}$, defined from a hypersurface as produced by the Allen–Cahn
min-max procedure, to the geometric behaviour of the underlying hypersurface;
precisely, (5) holds if and only if $E$ is locally
$\mathcal{F}_{\lambda}$-minimising. In order to prove Theorem 3 we will turn
our attention, in Step 3 below, to establishing that (5) holds for all
constant mean curvature hypersurfaces produced by the Allen–Cahn min-max
procedure in manifolds with positive Ricci curvature.

In order to produce the “recovery functions” above we first establish a local
smoothing procedure for Caccioppoli sets that are smooth in an annular region.
This is done by another “cut-and-paste” argument using Sard’s Theorem on the
level sets of mollified indicator functions for the Caccioppoli set, full
details of which can be found in Subsection 4.1.

We emphasise that this local smoothing procedure we exhibit for Caccioppoli
sets is not, in and of itself, sufficient to establish Theorem 1. The reason
for this is that the local smoothing we produce records no information about
the mean curvature near the isolated singularity, and this lack of information
on the mean curvature makes it difficult to perturb the metric. The foliation
of [Les23] however ensures there exists a sequence of smooth hypersurfaces of
constant mean curvature $\lambda$ converging to the singular hypersurface,
allowing for the perturbation to be constructed in the manner as described in
Step 1.

3\. Paths of functions: Similarly to the strategy employed in previous works
on hypersurfaces produced by the Allen–Cahn min-max procedure, for example in
[Bel20], [BW20a], and [BW22], the proof of Theorem 3 is achieved by exhibiting
a suitable continuous path in $W^{1,2}(N)$.

Under the assumption that a hypersurface produced by the min-max procedure
violates a desired property, one basic idea is to exploit its min-max
characterisation as follows. If a path admissible in the min-max procedure may
be produced, with energy along this path bounded above by a constant strictly
less than the min-max value, then one contradicts the assumption that such a
hypersurface arose from the min-max. Thus, the desired property must hold for
all hypersurfaces produced by the min-max procedure.

We first emphasise that the paths we construct in $W^{1,2}(N)$ reflect the
underlying geometry imposed by the assumption of positive Ricci curvature. We
denote the super-level sets and level sets of the signed distance function,
for each $s\in\mathbb{R}$, by

$E(s)=\\{x\in N\>|\>d^{\pm}_{\overline{M}}(x)>s\\}\text{ and
}\Gamma(s)=\\{x\in N\>|\>d^{\pm}_{\overline{M}}(x)=s\\},$

respectively; so that $E(0)=E$ and $\Gamma(0)=\overline{M}$. Formally
computing we have, for almost every $s\in\mathbb{R}$, that

$\frac{d}{ds}\mathcal{H}^{n}(\Gamma(s))=-\int_{\Gamma(s)}H(x,s)\,d\mathcal{H}^{n}(x)\text{
and }\frac{d}{ds}\mathrm{Vol}_{g}(E(s))=\mathcal{H}^{n}(\Gamma(s)),$ (6)

where $H(x,s)$ denotes the mean curvature of the level set $\Gamma(s)$ at a
point $x$. By denoting $m=\min_{N}\mathrm{Ric}_{g}>0$, the assumption of
positive Ricci curvature implies the following relation between the mean
curvature of $M$ and the level sets $\Gamma(s)$:

$\begin{cases}H(x,s)\geq\lambda+ms\text{ for }s>0\\\ H(x,0)=\lambda\\\
H(x,s)\leq\lambda+ms\text{ for }s<0\end{cases},$ (7)

see Subsection 3.2. Therefore, by (6) and (7), for each
$t\in\mathbb{R}\setminus\\{0\\}$ we compute that

$\displaystyle\mathcal{F}_{\lambda}(E(t))-\mathcal{F}_{\lambda}(E)$
$\displaystyle=\int_{0}^{t}\frac{d}{ds}\mathcal{H}^{n}(\Gamma(s))-\lambda\frac{d}{ds}\mathrm{Vol}_{g}(E(s))\,ds$
$\displaystyle=\int_{0}^{t}\int_{\Gamma(s)}\lambda-H(x,s)\,d\mathcal{H}^{n}(x)\,ds<0,$

and so from the assumption of positive Ricci curvature we conclude that for
each $t\in\mathbb{R}\setminus\\{0\\}$ we have

$\mathcal{F}_{\lambda}(E(t))<\mathcal{F}_{\lambda}(E).$ (8)

We then consider the continuous path of sliding functions,
$v^{t}_{\varepsilon}\in W^{1,2}(N)$, produced by sliding the zero level set of
$v_{\varepsilon}$ from $\overline{M}$ to $\Gamma(t)$ for each
$t\in\mathbb{R}$; that is

$v^{t}_{\varepsilon}=\overline{\mathbb{H}}^{\varepsilon}\circ(d^{\pm}_{\overline{M}}-t),$

so that for each $t\in\mathbb{R}$ we have
$\\{v^{t}_{\varepsilon}=0\\}=\Gamma(t)$,
$v^{0}_{\varepsilon}=v_{\varepsilon}$, $v^{2d(N)}_{\varepsilon}=-1$ and
$v^{-2d(N)}=+1$ (whenever $\varepsilon>0$ is sufficiently small). The
geometric relation (8), induced by the positive Ricci curvature assumption,
then translates to the level of functions and allows us to compute, in Lemma
5, that the $v^{t}_{\varepsilon}$ have the following property:

$\mathcal{F}_{\varepsilon,\lambda}(v^{t}_{\varepsilon})\leq\mathcal{F}_{\lambda}(v_{\varepsilon})+E(\varepsilon)\text{
where }E(\varepsilon)\rightarrow 0\text{ as }\varepsilon\rightarrow 0.$

As the nodal sets of the $v^{t}_{\varepsilon}$ are the level sets of the
signed distance function to $\overline{M}$, the functions
$v^{t}_{\varepsilon}$ may be seen as a path of functions analogous to a sweep-
out of $N$ by the level sets of the signed distance function to
$\overline{M}$.

Thus, the path provided by the sliding functions, along with the energy
reducing paths from $-1$ and $+1$ provided by negative gradient flow of the
energy to $a_{\varepsilon}$ and $b_{\varepsilon}$ respectively, provides a
“recovery path” for the value $\mathcal{F}_{\lambda}(E)$; this path connects
$a_{\varepsilon}$ to $b_{\varepsilon}$, passing through $v_{\varepsilon}$,
with the maximum value of the energy along this path approximately
$\mathcal{F}_{\lambda}(E)$ (by virtue of (2)). Approximate upper energy bounds
along this path are depicted by the thick dashed lines in Figure 2 below. In
this manner, as mentioned in Subsection 1.2, such a path establishes that the
Allen–Cahn min-max procedure in positive Ricci curvature produces no minimal
piece.

$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})$$a_{\varepsilon}$$b_{\varepsilon}$$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\frac{\eta}{2}$$v_{\varepsilon}^{t_{0}}$$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\eta$$g_{\varepsilon}$$v_{\varepsilon}^{-t_{0}}$$v_{\varepsilon}^{t_{0},0}$$v_{\varepsilon}^{-t_{0},0}$$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\tau$$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\eta-\tau$$g_{\varepsilon}^{t_{0},2}$$g_{\varepsilon}^{-t_{0},2}$
Figure 2: The solid curve depicts approximate (i.e. up to the addition of a
term that converges to zero as $\varepsilon\rightarrow 0$) upper energy bounds
along the path taken from $a_{\varepsilon}$ to $b_{\varepsilon}$ constructed
for the proof of Theorem 3 in Subsection 6.1. The horizontal axis identifies
some specific functions in the path and the vertical axis depicts the
approximate upper bound on the energy of the path between each identified
function. The thick dashed lines depict an approximate upper energy bound
along the path of functions in Lemma 5.

We now exhibit (for all $\varepsilon>0$ sufficiently small) a continuous path
in $W^{1,2}(N)$ between $a_{\varepsilon}$ and $b_{\varepsilon}$ which, under
the assumption that $E$ is not $\mathcal{F}_{\lambda}$-minimising a small ball
around an isolated singularity, contradicts the min-max characterisation of
$E$ and proves Theorem 3. We emphasise that this path is constructed with
energy bounded above by a value strictly below
$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})$ (independently of
$\varepsilon$); approximate upper energy bounds are depicted by the solid
curve in Figure 2 above. We thus conclude, by the arguments in Step 2, that
the Allen–Cahn min-max procedure in positive Ricci curvature produces a
hypersurface which is locally $\mathcal{F}_{\lambda}$-minimising.

The approximate upper energy bounds along the portions of the path we
construct are computed explicitly in Section 5. However, in this section, we
now sketch the various path constructions and motivate the upper energy bounds
as a diffuse reflection of the underlying geometry.

First, for a given isolated singularity, $p\in\mathrm{Sing}(M)$, we are able
to continuously deform the sets $E(t_{0})$ (for a fixed $t_{0}>0$ sufficiently
small) locally around $p$ so that the resulting deformation agrees with $E$
inside of a fixed ball $B_{r_{0}}(p)$ (for some $r_{0}>0$ determined only by
the area of $M$). This is done by exploiting (8) in such a way that the
$\mathcal{F}_{\lambda}$-energy of the deformations remain a fixed amount below
$\mathcal{F}_{\lambda}(E)$. At the diffuse level this is replicated by placing
$\overline{\mathbb{H}}^{\varepsilon}$ in the normal direction to the
deformations as in the construction of $v_{\varepsilon}$, producing a
$W^{1,2}(N)$ continuous path of functions with controlled energy; depicted in
Figure 3 below.

$v_{\varepsilon}$

Figure 3: In both graphics above the lower thin dashed horizontal line depicts
$\overline{M}$, the zero level set of the function $v_{\varepsilon}$, and the
upper thick dashed horizontal line depicts the zero level set,
$\Gamma(t_{0})$, of $v^{t_{0}}_{\varepsilon}=v^{t_{0},1}_{\varepsilon}$ which
is deformed in the construction of the shifted functions. In the left-hand
graphic the solid lines depict various zero level sets of the
$v^{t_{0},s}_{\varepsilon}$ as we vary $s$ from $0$ to $1$. In the right-hand
graphic the solid line depicts the zero level set of
$v^{t_{0},0}_{\varepsilon}$ and the thick dashed circle depicts the boundary
of the ball, $B_{r_{0}}(p)$, in which
$v^{t_{0},0}_{\varepsilon}=v_{\varepsilon}$.

Specifically, we produce a continuous path of shifted functions,
$s\in[0,1]\rightarrow v^{t_{0},s}_{\varepsilon}\in W^{1,2}(N)$, which satisfy
$v^{t_{0},1}_{\varepsilon}=v^{t_{0}}_{\varepsilon}$ on $N$ and
$v^{t_{0},0}_{\varepsilon}=v_{\varepsilon}$ inside $B_{r_{0}}(p)$.
Furthermore, there exists an $\eta>0$ such that for each $s\in[0,1]$ we have
the following upper energy bound

$\mathcal{F}_{\varepsilon,\lambda}(v^{t_{0},s}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\eta,$

as depicted in Figure 2. In this manner we have exhibited a continuous path in
$W^{1,2}(N)$ from $v^{t_{0}}_{\varepsilon}$ to a function
$v^{t_{0},0}_{\varepsilon}$, equal to $v_{\varepsilon}$ in a fixed ball
$B_{r_{0}}(p)$, with the energy along this path a fixed amount below the min-
max value.

Next we construct a continuous path of functions, from
$v^{t_{0},0}_{\varepsilon}$ to the local energy minimiser
$g_{\varepsilon}\in\mathcal{A}_{\varepsilon,\frac{R}{2}}(p)$ (recall Step 2)
for a fixed $R\in(0,r_{0})$. This is done in such a way that we only alter the
functions inside $B_{R}(p)$; thus, in the following description we will only
consider functions in the ball $B_{R}(p)$. The radius $R>0$ here is chosen
sufficiently small based on the energy drop, $\eta>0$, above achieved outside
of $B_{r_{0}}(p)$, ensuring that the energy along the constructed path will
remain a fixed amount below the min-max value.

Figure 4: The graphics above depict various stages of the transition, in
$W^{1,2}(N)$, between $v_{\varepsilon}$ and
$\min\\{v_{\varepsilon},g_{\varepsilon}\\}$ in $B_{R}(p)$. The first row
depicts schematics of this transition and the second row depicts the geometry
of the zero level sets of the local path functions; in this way, each column
of the figure contains a schematic for the local path function with a
corresponding depiction of the geometry of its zero level set below it. For
the first row, in all three of the images the solid black line depicts the
local path function in question (equal to $v_{\varepsilon}$ and
$\min\\{v_{\varepsilon},g_{\varepsilon}\\}$ in the left-hand and right-hand
graphic respectively), the thick dashed curve depicts portions of
$\min\\{v_{\varepsilon},g_{\varepsilon}\\}$ that are not yet included in the
path, the thin dashed curve depicts the diffuse sweep-out function in
question, and the thick dashed upper and lower horizontal lines depict the
functions $\pm 1$ respectively. Notice that the local path function depicted
in the middle graphic includes portions of $v_{\varepsilon}$ that lie to the
right of the diffuse sweep-out function, portions of
$\min\\{v_{\varepsilon},g_{\varepsilon}\\}$ that lie to the left of the
diffuse sweep-out function and uses the diffuse sweep-out function itself to
interpolate between $v_{\varepsilon}$ and
$\min\\{v_{\varepsilon},g_{\varepsilon}\\}$. The explicit construction of the
local path functions replicating the behaviour of these schematics involves
taking various maxima and minima of the functions in the diffuse sweep-out,
$v_{\varepsilon}$ and $g_{\varepsilon}$; the diffuse sweep-out allows for this
choice of maxima and minima (corresponding geometrically to a choice of zero
level set) to be made continuously. For the second row, in all three graphics
the outer solid circle depicts the boundary of $B_{R}(p)$ and the thick dashed
inner circle depicts the boundary of $B_{\frac{R}{2}}(p)$. In the left-hand
graphic the solid line depicts the zero level set of $v_{\varepsilon}$, which
is $\overline{M}$, in the right-hand graphic the solid curve depicts the zero
level set of $\min\\{g_{\varepsilon},v_{\varepsilon}\\}$, and in the middle
graphic the solid curve depicts the zero level set of a local path function in
the transition between $v_{\varepsilon}$ and
$\min\\{v_{\varepsilon},g_{\varepsilon}\\}$. Each the thin dashed curves in
the three graphics depict a given hypersurface in the “planar” sweep-out of
$B_{R}(p)$, each separating $B_{R}(p)$ into two open sets (one above and one
below it). The zero level set of the function in the local path associated to
this hypersurface is chosen to be the portions of the zero level set of
$\min\\{v_{\varepsilon},g_{\varepsilon}\\}$ that are beneath the hypersurface,
the portions of $\overline{M}$ that are above the hypersurface, and when the
the hypersurface lies between the nodal sets of $v_{\varepsilon}$ and
$\min\\{v_{\varepsilon},g_{\varepsilon}\\}$, the hypersurface itself is chosen
as the zero level set. The same ideas described above are used for the
construction of the portion of the path from
$\min\\{v_{\varepsilon},g_{\varepsilon}\\}$ to $g_{\varepsilon}$ in
$B_{R}(p)$.

In order to construct this local path from $v_{\varepsilon}$ to
$g_{\varepsilon}$ inside of $B_{R}(p)$ we utilise a sweep-out of the ball by
images of Euclidean planes via a geodesic normal coordinate chart; depicted by
the thin dashed curves in the second row of Figure 4 above. The hypersurfaces
in this sweep-out are used to continuously transition from our hypersurface
and the local $\mathcal{F}_{\lambda}$-minimiser at the diffuse level;
precisely, the planes facilitate the construction of a path between the
diffuse representatives $v_{\varepsilon}$ and $g_{\varepsilon}$.

######  Remark 3.

In the setting of the Almgren–Pitts min-max homotopy sweep-outs of $N$ by
cycles are considered, as opposed to continuous paths in $W^{1,2}(N)$ in the
Allen–Cahn min-max. However, there does not necessarily exist a homotopy of
cycles between our hypersurface and any local
$\mathcal{F}_{\lambda}$-minimiser. We overcome this for the Allen–Cahn min-max
by directly exploiting the topology of $W^{1,2}(N)$, showing that the local
path may be seen as a diffuse analogue of [CLS22, Lemma 1.12], and
illustrating why we guarantee local $\mathcal{F}_{\lambda}$-minimisation (see
Remark 2).

In the same manner as in the construction of $v_{\varepsilon}$, by placing
$\overline{\mathbb{H}}^{\varepsilon}$ in the normal direction to the
hypersurfaces in this “planar” sweep-out we construct a sweep-out at the
diffuse level that is continuous in $W^{1,2}(N)$; the diffuse sweep-out thus
acts as an approximation for the underlying “planar” sweep-out. The diffuse
sweep-out of $B_{R}(p)$ is utilised twice, first for the construction of a
path from $v_{\varepsilon}$ to $\min\\{g_{\varepsilon},v_{\varepsilon}\\}$,
and second for the construction of a path from
$\min\\{g_{\varepsilon},v_{\varepsilon}\\}$ to $g_{\varepsilon}$. By taking a
combination of maxima and minima of functions in the diffuse sweep-out,
$v_{\varepsilon}$ and $g_{\varepsilon}$ (which ensure the resulting functions
are in $W^{1,2}(N)$), we are able to produce a local path from
$v_{\varepsilon}$ to $g_{\varepsilon}$; see Figure 4 above for a description
of this construction.

We note that
$\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(\min\\{v_{\varepsilon},g_{\varepsilon}\\})\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})$
(by local energy minimisation of $g_{\varepsilon}$) and that $R\in(0,r_{0})$
is chosen based on $\eta>0$ to ensure that the total energy contribution of
the diffuse sweep-out functions in the ball is at most $\frac{\eta}{2}$. As a
consequence of these two facts, the energy in $B_{R}(p)$ of any local path
function can be estimated to be at most the energy of $v_{\varepsilon}$ in
$B_{R}(p)$ plus $\frac{\eta}{2}$; from there one obtains the energy estimate
in the whole of $N$. Specifically, we produce a continuous path of local
functions, $s\in[-2,2]\rightarrow g^{t_{0},s}_{\varepsilon}\in W^{1,2}(N)$,
which satisfy $g^{t_{0},-2}_{\varepsilon}=v^{t_{0},0}_{\varepsilon}$ on $N$,
$g^{t_{0},2}=g_{\varepsilon}$ in $B_{R}(p)$ and are all equal to
$v^{t_{0},0}_{\varepsilon}$ outside of $B_{R}(p)$. Furthermore, for each
$s\in[-2,2]$ we have the following upper energy bound

$\mathcal{F}_{\varepsilon,\lambda}(g^{t_{0},s}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\frac{\eta}{2},$

as depicted in Figure 2. In this manner we have exhibited a continuous path in
$W^{1,2}(N)$ from $v^{t_{0},0}_{\varepsilon}$ to a function
$g^{t_{0},2}_{\varepsilon}$, changing $v^{t_{0},0}_{\varepsilon}$ only inside
of $B_{R}(p)$ (from $v_{\varepsilon}$ to the local energy minimiser
$g_{\varepsilon}$).

In order to establish that $E$ is locally $\mathcal{F}_{\lambda}$-minimising,
we now argue by contradiction and assume that there exists a $\tau>0$ such
that

$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\geq\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})+\tau\text{
for all }\varepsilon>0\text{ sufficiently small.}$ (9)

Note that by the results discussed in Step 2, contradicting (9) will establish
that $M$ is $\mathcal{F}_{\lambda}$-minimising in $B_{\frac{R}{4}}(p)$ for
$R>0$ as chosen above (as (5) must hold). Using the contradiction assumption,
in addition to keeping the energy of the local path functions a fixed amount,
$\frac{\eta}{2}$, below the min-max value, as
$g^{t_{0},2}_{\varepsilon}=g_{\varepsilon}$ in $B_{R}(p)$, we thus also have
that

$\mathcal{F}_{\varepsilon,\lambda}(g^{t_{0},2}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\eta-\tau,$

as depicted in Figure 2. We now directly exploit this extra energy drop
afforded by the contradiction assumption to construct the next portion of the
path.

To this end, we deform the rest of the set $E(t_{0})$ entirely onto $E$
outside of $B_{r_{0}}(p)$. This is done in such a way that the deformations
fix the inside of $B_{r_{0}}(p)$, thus preserving a drop in
$\mathcal{F}_{\lambda}$-energy under the assumption that $E$ is not locally
$\mathcal{F}_{\lambda}$-minimising. At the diffuse level this is replicated by
placing $\overline{\mathbb{H}}^{\varepsilon}$ in the normal direction to these
deformations, keeping the functions equal to the local energy minimiser
$g_{\varepsilon}$ (which yields an energy drop by (9)) inside of
$B_{r_{0}}(p)$ and producing a $W^{1,2}(N)$ continuous path of functions with
controlled energy; this is depicted in Figure 5.

$g_{\varepsilon}$

$g_{\varepsilon}$

Figure 5: In both graphics above the lower horizontal lines depict
$\overline{M}$, the upper horizontal line depicts the zero level set,
$\Gamma(t_{0})$, of $v^{t_{0}}_{\varepsilon}=v^{t_{0},1}_{\varepsilon}$ which
is deformed in the construction of the shifted functions and the thick dashed
circle depicts the boundary of the ball, $B_{r_{0}}(p)$, in which the shifted
functions $g^{t,2}_{\varepsilon}=g_{\varepsilon}$. In the left-hand graphic
the solid lines depict various zero level sets of the $g^{t,2}_{\varepsilon}$
as we vary $t$ from $t_{0}$ to $0$. In the right-hand graphic the solid line
depicts the zero level set of $g^{0,2}_{\varepsilon}=g_{\varepsilon}$.

Specifically, we produce (under the assumption that (9) holds) a continuous
path of shifted functions, $t\in[0,t_{0}]\rightarrow g^{t,2}_{\varepsilon}\in
W^{1,2}(N)$, which are equal to the local function $g^{t_{0},2}_{\varepsilon}$
when $t=t_{0}$ (justifying notation), equal to $g_{\varepsilon}$ when $t=0$
and are all equal to $g_{\varepsilon}$ in $B_{r_{0}}(p)$. Furthermore, for
each $t\in[0,t_{0}]$ have the following upper energy bound

$\mathcal{F}_{\varepsilon,\lambda}(g^{t,2}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v^{t,0}_{\varepsilon})-\tau,$

as depicted Figure 2. In this manner we have exhibited a continuous path in
$W^{1,2}(N)$ from $g^{t_{0},2}_{\varepsilon}$ to the local energy minimiser
$g_{\varepsilon}$, only changing $g^{t_{0},2}_{\varepsilon}$ outside of
$B_{r_{0}}(p)$.

To summarise all of the above, as the endpoints of each of the paths described
agree with the start of the next, for all $\varepsilon>0$ sufficiently small
we have exhibited a continuous paths in $W^{1,2}(N)$ connecting
$a_{\varepsilon}$ to the local energy minimiser $g_{\varepsilon}$. We also
demonstrated that the energy along this path is bounded above by a value
strictly below $\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})$
(independently of $\varepsilon$), depicted in Figure 2.

To complete the desired path from $a_{\varepsilon}$ to $b_{\varepsilon}$ it
remains to construct the portion from $g_{\varepsilon}$ to $b_{\varepsilon}$.
By considering $-t_{0}$ instead of $t_{0}$ in each of the paths sketched
above, we ensure that through symmetric (with respect to the underlying
hypersurfaces) deformations of the sets $E(-t_{0})$, the relevant symmetric
portions of the path may constructed with identical upper energy bounds; this
portion of the path is depicted in Figure 2, where the symmetry of the path
with respect to $g_{\varepsilon}$ is made apparent.

To conclude the proof of Theorem 3 we concatenate the path from
$a_{\varepsilon}$ to $g_{\varepsilon}$ and the path from $g_{\varepsilon}$ to
$b_{\varepsilon}$, completing the desired continuous path in $W^{1,2}(N)$
between the two stable critical points $a_{\varepsilon}$ and
$b_{\varepsilon}$. Under the assumption that (9) holds, the upper energy
bounds along this path (depicted by the solid curve in Figure 2) and (2)
ensure that for $\varepsilon>0$ sufficiently small we have

$\text{$\mathcal{F}_{\varepsilon,\lambda}$ energy along the
path}\leq\mathcal{F}_{\lambda}(E)-\min\left\\{\frac{\eta}{4},\frac{\tau}{2}\right\\}.$

As (1) holds, as mentioned in Subsection 1.2, and as the above path is
admissible in the Allen–Cahn min-max construction of $E$, we must contradict
the assumption that (9) holds. We thus conclude that any such $E$ as produced
by the Allen–Cahn min-max procedure in Ricci positive curvature must be such
that $E$ is locally $\mathcal{F}_{\lambda}$-minimising (as (5) holds), proving
Theorem 3. We then establish Theorems 1 and 2 by applying the results
described in Step 1.

###  1.4 Structure and remarks

We now proceed as follows:

  * •

Section 2 recalls the result of [Les23] and uses it to perturb away isolated
singularities of constant mean curvature hypersurfaces with regular tangent
cones.

  * •

Section 3 analyses the signed distance function and introduces the one-
dimensional profile, before showing that it approximates the underlying
hypersurface in a suitable sense.

  * •

Section 4 establishes a procedure to locally smooth Caccioppoli sets which are
smooth in annular regions. This procedure is then used to relate local energy
minimisation to the local geometric behaviour of the hypersurface.

  * •

Section 5 provides constructions of the various continuous paths in
$W^{1,2}(N)$ along with calculations of upper bounds of the energy along these
paths.

  * •

Section 6 ties together the results of the previous sections in order to prove
the main results.

The following remarks should be kept in mind throughout:

######  Remark 4.

As mentioned above, in [BW20b, Proposition B.1] it is shown that smooth
constant mean curvature hypersurfaces are locally
$\mathcal{F}_{\lambda}$-minimising. In proving Theorem 3 we aim to show that
this also holds around isolated singularities for constant mean curvature
hypersurfaces produced by the Allen–Cahn min-max in manifolds with positive
Ricci curvature. We may thus restrict to the dimensions in which these objects
may be singular and work under the assumptions that $N$ is of dimension
$n+1\geq 8$.

######  Remark 5.

The assumption of positive Ricci curvature is used only to ensure that the
upper energy bounds on the paths constructed in Section 5 remain a fixed
amount below the min-max value, $\mathcal{F}_{\lambda}(E)$. In particular, we
note that the results of Sections 2, 3 and 4 as well as the paths constructed
(but not their upper energy bounds) in Section 5 make no use of the assumption
of positive Ricci curvature.

######  Remark 6.

We make the choice of a positive upper bound on $\varepsilon>0$ finitely many
times throughout the construction of the paths in the proof of Theorem 3,
ultimately constructing the paths for all $\varepsilon>0$ smaller than a fixed
positive constant. The specific choice of upper bound utilised in each
instance may change, but we implicitly assume that a correct upper bound for
which the desired property holds is used in each case. This remark will apply
each time we choose $\varepsilon>0$ sufficiently small.

### Acknowledgements

CB was supported in part by the Engineering and Physical Sciences Research
Council [EP/S005641/1].

KMS was supported by the Engineering and Physical Sciences Research Council
[EP/N509577/1], [EP/T517793/1]. He would like to thank Fritz Hiesmayr,
Konstantinos Leskas and Myles Workman for many enlightening discussions. He is
also grateful to Otis Chodosh for interesting email exchanges related to some
aspects of this work, and to Marco Badran for comments on a preliminary
version of the manuscript.

##  2 Surgery procedure for isolated singularities with regular tangent cone

We show how the recent result of [Les23] can be combined with a local
perturbation of the metric to regularise a hypersurface of constant mean
curvature around an isolated singularity with area-minimising regular tangent
cone. We first collect some notation and definitions, phrased in notation in
keeping with this paper, before stating the main theorem from [Les23] and
using it to establish the surgery procedure.

We reset notation for this section, letting $(N^{n+1},g)$ be a Riemannian
manifold with no curvature assumption. Throughout this section
$T=(\partial[A])\llcorner B_{1}(p)$ will denote the current associated to a
Caccioppoli set, $A\in\mathcal{C}(N)$, restricted to a ball $B_{1}(p)$ about a
point $p\in N$, for which the following properties hold:

  * •

$\mathrm{Spt}(T)$ is connected.

  * •

$\mathrm{Sing}(T)=\\{p\\}$ (so that $p$ is an isolated singularity of $T$).

  * •

$T$ has a regular tangent cone at $p$.

For a given $\lambda\in\mathbb{R}$ we fix a choice of $0<r_{1}<r_{2}<1$
sufficiently small so that the following properties hold:

  * •

$\Gamma_{0}=\partial(T\llcorner B_{r_{1}}(p))$ is a closed embedded, connected
$(n-1)$-dimensional sub-manifold of $\partial B_{r_{1}}(p)$.

  * •

$\mathrm{Spt}(T)\cap\partial B_{r_{1}}(p)$ is a transverse intersection.

  * •

$\overline{A\cap B_{r_{2}}(p)}$ has connected complement in $B_{r_{2}}(p)$.

Let $\phi_{j}:\Gamma_{0}\rightarrow\partial B_{r_{1}}(p)$ denote $C^{2}$ maps
with

$|\phi_{j}-i_{\Gamma_{0}}|_{C^{2}}\leq\frac{1}{j},$

and for $\Gamma_{j}=(\phi_{j})_{*}\Gamma_{0}$ we assume that $\Gamma_{j}\cap
A^{o}\neq\emptyset$. Here we denote by $i_{\Gamma_{0}}$ the identity map on
$\Gamma_{0}$, $(\phi_{j})_{*}\Gamma_{0}$ the push-forward of $\Gamma_{0}$ by
the map $\phi_{j}$ and $A^{o}$ the interior of $A$. We now recall the
definitions used in [Les23]:

######  Definition 1.

For $\lambda\in\mathbb{R}$ we say that $T=(\partial[A])\llcorner B_{1}(p)$ is
one-sided minimising for $\mathcal{F}_{\lambda}$ in $B_{1}(p)$ if both of the
following properties hold:

  * •

$A$ is a critical point of $\mathcal{F}_{\lambda}$ in $B_{1}(p)$.

  * •

We have that

$\mathcal{F}_{\lambda}(A)\leq\mathcal{F}_{\lambda}(X).$

for any integral $n$-current $X$ with $\mathrm{Spt}(X)\subset\overline{A\cap
B_{1}(p)}$ and $\partial X=\partial T$.

Using the above notation and definition, [Les23] then proves the following
foliation result, generalising the results of [HS85] to the case of constant
mean curvature hypersurfaces.

######  Theorem 4.

[Les23] Let $\lambda\in\mathbb{R}$, with $T$ as defined above such that in
addition the following two properties hold:

  * •

$T$ is a critical point for $\mathcal{F}_{\lambda}$ in $B_{1}(p)$.

  * •

$T$ is one-sided minimising for $\mathcal{F}_{\lambda}$ in $B_{1}(p)$.

Then, for every $j\geq 1$, there exist integral $n$-currents, $S_{j}$, that
minimise $\mathcal{F}_{\lambda}$ in $\overline{A\cap B_{r_{2}}(p)}$, subject
to the boundary condition $\partial S_{j}=\Gamma_{j}$, that satisfy the
following properties:

  * •

$\mathrm{Spt}(S_{j})\subset\overline{B_{r_{1}}(p)}$.

  * •

There exist sets of finite perimeter $B_{j}$, with
$\overline{B_{j}}\subset\overline{A\cap B_{r_{1}}(p)}$, such that
$S_{j}=\partial[B_{j}]\llcorner B_{r_{1}}(p)$.

  * •

$\Gamma_{j}=\mathrm{Spt}(S_{j})\cap\partial B_{r_{1}}(p)$.

  * •

Each $S_{j}$ is a critical point of $\mathcal{F}_{\lambda}$ in $B_{r_{1}}(p)$.

  * •

For the measures associated to the supports of the $S_{j}$ and $T$ we have
$\mu_{S_{j}}\rightarrow\mu_{T\llcorner B_{r_{1}}(p)}$. Thus, in particular on
compact sets we have that $\mathrm{Spt}(S_{j})\rightarrow\mathrm{Spt}(T)$ in
the Hausdorff distance.

  * •

The $\mathrm{Spt}(S_{j})$ are smooth hypersurfaces. ∎

Using Theorem 4 we now establish the desired surgery procedure, the proof of
which is similar to [CLS22, Proposition 4.1].

######  Proposition 1.

Let $(N^{n+1},g)$ be a Riemannian manifold and $T=(\partial[A])\llcorner
B_{1}(p)$ be a current associated to $A\in\mathcal{C}(N)$ with the properties
as stated above and satisfying the hypotheses of Theorem 4. Given
$r\in(0,r_{1})$ and any $\varepsilon>0$ there exists a current $\tilde{T}$ and
metric $\tilde{g}$ with the following properties:

  * •

$\mathrm{Sing}(\tilde{T})=\emptyset$.

  * •

$\tilde{T}$ is a critical point of $\mathcal{F}_{\lambda}$ in $B_{1}(p)$ with
respect to the metric $\tilde{g}$.

  * •

$\mathrm{Spt}(\tilde{T})\setminus B_{r}(p)=\mathrm{Spt}(T)\setminus B_{r}(p)$
and $\partial(\tilde{T}\llcorner B_{r}(p))=\partial(T\llcorner B_{r}(p))$.

  * •

$d_{\mathcal{H}}(\mathrm{Spt}(T),\mathrm{Spt}(\tilde{T}))<\varepsilon$, where
here $d_{\mathcal{H}}$ denotes the Hausdorff distance.

  * •

$||\tilde{g}-g||_{C^{k,\alpha}}<\varepsilon$ for any $k\geq 1$ and
$\alpha\in(0,1)$, with $g=\tilde{g}$ on $N\setminus B_{r}(p)$.

###### Proof.

The case $\lambda=0$ is precisely the content of [CLS22, Proposition 4.1]. We
thus consider the case $\lambda\in\mathbb{R}\setminus\\{0\\}$.

As $\mathrm{Sing}(T)=\\{p\\}$, for each $r\in(0,r_{1})$ we have that
$(\mathrm{Spt}(T)\cap B_{r}(p))\setminus\\{p\\}$ is smooth. We apply Theorem 4
to see that there exists some sequence, $S_{j}$, of smooth constant mean
curvature hypersurfaces such that the $S_{j}$ converge as currents to
$T\llcorner B_{r}(p)$. Allard’s Theorem, (see [Sim84, Chapter 5] and [HS85,
Lemma 1.14]) implies that we may write (for $j$ sufficiently large) the
intersection of $\mathrm{Spt}(S_{j})$ with the annulus
$A(p,\frac{r}{4},\frac{3r}{4})=B_{\frac{3r}{4}}(p)\setminus\overline{B_{\frac{r}{4}}(p)}$
as a smooth graph over $\mathrm{Spt}(T)\cap A(p,\frac{r}{4},\frac{3r}{4})$.

Explicitly, let $u_{j}\in C^{2}(\mathrm{Spt}(T)\cap
A(p,\frac{r}{4},\frac{3r}{4}))$ denote the graphing function of
$\mathrm{Spt}(S_{j})$ over $\mathrm{Spt}(T)\cap A(p,\frac{r}{4},\frac{3r}{4})$
and let $\varphi$ be a smooth cutoff function taking values in $[0,1]$ such
that $\varphi=1$ on $B_{\frac{3r}{8}}(p)$ and $\varphi=0$ outside
$B_{\frac{5r}{8}}(p)$. We then denote by $T+\varphi u_{j}$ the image of the
normal graph of the function $\varphi u_{j}$ over $\mathrm{Spt}(T)\cap
A(p,\frac{r}{4},\frac{3r}{4})$.

We now define $\widetilde{T}=(T\setminus B_{r}(p))\cup(S_{j}\cap
B_{\frac{r}{4}})\cup((T+\varphi u_{j})\cap A(p,\frac{r}{4},\frac{3r}{4}))$ for
$j$ large enough to ensure that $\mathrm{Spt}(S_{j})$ is smooth and graphical
as above. Note that the $\mathrm{Spt}(\widetilde{T})$ is smooth as
$\mathrm{Spt}(S_{j})$, $\mathrm{Spt}(T)\setminus B_{r}(p)$ and
$(\mathrm{Spt}(T)+\varphi u_{j})\cap A(p,\frac{r}{4},\frac{3r}{4})$ are
smooth; hence $\mathrm{Sing}(\widetilde{T})=\emptyset$. By construction we
ensure that $\mathrm{Spt}(\tilde{T})\setminus
B_{r}(p)=\mathrm{Spt}(T)\setminus B_{r}(p)$ and $\partial(\tilde{T}\llcorner
B_{r}(p))=\partial(T\llcorner B_{r}(p))$. Note that for $j$ sufficiently large
we have that
$d_{\mathcal{H}}(\mathrm{Spt}(T),\mathrm{Spt}(\tilde{T}))<\varepsilon$ by the
properties in the conclusion of Theorem 4.

Let $H_{g}(x)$ denote the mean curvature of the hypersurface $\widetilde{T}$
with respect to the metric $g$ at a point $x\in\widetilde{T}$. By
construction, $H_{g}$ may not be equal to $\lambda$ only on
$A(p,\frac{r}{4},\frac{3r}{4})$ (as both $T$ and $S_{j}$ are critical points
of $\mathcal{F}_{\lambda}$). It remains to construct a new metric $\tilde{g}$,
close to $g$, and show that $\widetilde{T}$ has constant mean curvature
$\lambda$ with respect to this new metric. In the construction of
$\widetilde{T}$ above, as $\lambda\in\mathbb{R}\setminus\\{0\\}$ and the
graphing functions, $u_{j}$, converge to $0$, we may choose $j$ large enough
to ensure that $H_{g}$ and $\lambda$ have the same sign, so that
$\frac{H_{g}(x)}{\lambda}>0$ for each $x\in\mathrm{Spt}(\widetilde{T})$.

For some smooth function $f$ on $N$ to be determined we set
$\tilde{g}=e^{2f}g$; by standard results for conformal change of metric (e.g.
see [Sak96, Chapter 2]) we then have that the mean curvature, $H_{\tilde{g}}$,
of $\widetilde{T}$ with respect to the metric $\tilde{g}$ satisfies
$H_{\tilde{g}}(x)=e^{-f}(H_{g}(x)+\frac{\partial f}{\partial\nu})$ at each
point $x\in\widetilde{T}$, where $\nu$ is the unit normal on $\widetilde{T}$
(agreeing with $T$ outside of $A(p,\frac{r}{4},\frac{3r}{4})$).

We define a smooth cutoff function, $z$, such that $z(x)\equiv 1$ if
$\mathrm{dist}_{N}(y,\mathrm{Spt}(\widetilde{T})\cap
A(p,\frac{r}{4},\frac{3r}{4}))<\frac{r}{20}$ and $z\equiv 0$ whenever
$\mathrm{dist}_{N}(y,\widetilde{T}\cap
A(p,\frac{r}{4},\frac{3r}{4}))>\frac{r}{10}$. We denote by $\Pi(y)$ the
closest point projection to $\widetilde{T}$ so that $H_{g}(\Pi(y))$ is a well
defined function, supported in a tubular neighbourhood of
$\mathrm{Spt}(\widetilde{T})\cap A(p,\frac{r}{4},\frac{3r}{4})$. We now solve
for $H_{\tilde{g}}=\lambda$ for each $x\in\mathrm{Spt}(\tilde{T})$ by setting
$f(y)=\log\left(\frac{H_{g}(\Pi(y))}{\lambda}\right)z(y)$; note then by
construction $\mathrm{Spt}(f)\subset A(p,\frac{r}{4},\frac{3r}{4})$ and
$\frac{\partial f}{\partial\nu}=0$ on $\mathrm{Spt}(\widetilde{T})\cap
A(p,\frac{r}{4},\frac{3r}{4})$ (as $f$ is constant along normal geodesics to
this region).

Thus we have that $H_{\tilde{g}}(\widetilde{T})=\lambda$ for the choice of $f$
as above, so in particular $\widetilde{T}$ is a critical point of
$\mathcal{F}_{\lambda}$, with the change in metric occurring only in
$A(p,\frac{r}{4},\frac{3r}{4})$. Furthermore, by the smoothness of $z$ and the
fact that the graphing functions, $u_{j}$, are converging to $0$ ensures
$||H_{g}(\Pi(y))-\lambda||_{C^{2,\alpha}}\rightarrow 0$ as
$j\rightarrow\infty$, we see that for each $k\geq 1$ and $\alpha\in(0,1)$ we
have

$||e^{2f}-1||_{C^{k,\alpha}}=\bigg{|}\bigg{|}\left(\frac{H_{g}(\Pi(y))}{\lambda}\right)^{2z(y)}-1\bigg{|}\bigg{|}_{C^{k,\alpha}}\rightarrow
0\text{ as }j\rightarrow\infty,$

and so

$||\tilde{g}-g||_{C^{k,\alpha}}\leq||e^{2f}-1||_{C^{k,\alpha}}||g||_{C^{k,\alpha}}\rightarrow
0\text{ as }j\rightarrow\infty.$

Hence by choosing $j$ sufficiently large we ensure the metric change is
arbitrarily small. ∎

######  Remark 7.

In the proof of Theorem 3 we will establish that the one-parameter Allen–Cahn
min-max procedure of [BW20a] with constant prescribing function $\lambda$, in
a compact Riemannian manifold of dimension $3$ or higher with Ricci positive
curvature, produces a closed embedded hypersurface, $M$, of constant mean
curvature $\lambda$ which is locally $\mathcal{F}_{\lambda}$-minimising around
isolated singular points. For $M$ as above, around isolated singularities with
regular tangent cones the results of [Sim83] imply that $M$ is locally a graph
over its unique tangent cone. Thus, $M$ will satisfy all the required
properties of $T$ specified throughout this section in a sufficiently small
ball around an isolated singular point with regular tangent cone. With this in
hand, as any one-sided minimiser as in Definition 1 is required to be a
Caccioppoli set (see [Liu19], [Les23] for more details), we may thus apply the
results of this section to $M$.

##  3 Signed distance and the one-dimensional profile

###  3.1 Singular behaviour of the distance function

Recall that $M=\partial^{*}E\subset N$ is assumed to be a closed embedded
hypersurface of constant mean curvature $\lambda$, smooth away from a closed
singular set, denoted $\mathrm{Sing}(M)=\overline{M}\setminus M$, of Hausdorff
dimension at most $n-7$. We now adapt some of the analysis in [Bel20, Section
3] to the setting of hypersurfaces of constant mean curvature.

Denoting by $S_{d_{\overline{M}}}$ the set of points in
$N\setminus\overline{M}$ where the distance function $d_{\overline{M}}$ fails
to be differentiable (precisely the set of points $x\in
N\setminus\overline{M}$ such that there exist two or more geodesics realising
$d_{\overline{M}}(x)$), we then have that $d_{\overline{M}}$ is $C^{1}$ on
$N\setminus(\overline{M}\cup\overline{S_{d_{\overline{M}}}})$ and that
$S_{d_{\overline{M}}}$ is countably $n$-rectifiable (by [Alb94]). We now show
that $\overline{S_{d_{\overline{M}}}}$ is a countably $n$-rectifiable set;
this fact will allow us to work with the smooth portions of the signed level
sets of $d_{\overline{M}}$. To establish this we will need the following
lemma; the proof of which is identical to that of [Bel20, Lemma 3.1] replacing
the use of the sheeting theorem in [Wic14] by the one in [BW20c].

######  Lemma 1.

(Geodesic Touching) Let $x\in N\setminus\overline{M}$, then any minimising
geodesic connecting $x$ to $\overline{M}$ (i.e. a geodesic whose length
realises $d_{\overline{M}}(x)$) with endpoint $y\in\overline{M}$ is such that
$y\in M$ (i.e. is a regular point of $M$). ∎

Let $F(y,t)=\exp_{y}(t\nu(y))$ where $y\in M$, $t\in\mathbb{R}$ and $\nu$ the
choice of unit normal to $M$ pointing into $E$. We then have that $F(y,t)$ is
a geodesic emanating from $M$ orthogonally; here we interpret $F(y,t)$ for
$t<0$ by $\exp_{y}(t(\nu^{-}(y)))$ where $\nu^{-}$ is the unit normal to $M$
pointing into $N\setminus E$. We define
$\sigma^{+}(y),\sigma^{-}(y)\in\mathbb{R}$ with $\sigma^{+}(y)>0$ and
$\sigma^{-}(y)<0$ chosen so that $F(y,t)$ is the minimising geodesic between
its endpoint $y$, on $M$, and $F(y,t)$ for all $\sigma^{-}(y)\leq
t\leq\sigma^{+}(y)$ but fails to be the minimising geodesic, between $y$ and
$F(y,t)$, for $t>\sigma^{+}(y)$ or $t<\sigma^{-}(y)$. With this definition we
define the cut locus of $M$ to be

$\mathrm{Cut}(M)=\\{F(y,\sigma^{\pm}(y))\>:\>y\in
M,\sigma^{\pm}(y)<\infty\\}.$ (10)

Standard theory (e.g. [Sak96]) for geodesics characterises the cut locus in
the following manner: if $x=F(y,\sigma^{\pm}(y))\in\mathrm{Cut}(M)$ then
either their exist (at least) two distinct geodesics realising
$d_{\overline{M}}(x)$ or $F:M\times(0,\infty)\rightarrow N$ is such that
$dF(y,\sigma^{\pm}(y))$ is not invertible. We then see that
$\overline{S_{d_{\overline{M}}}}\cap(N\setminus\overline{M})=\mathrm{Cut}(M)$
(c.f. proposition 4.6. in [MM02]) and so in order to establish the countable
$n$-rectifiability of $\overline{S_{d_{\overline{M}}}}$ it is sufficient to
show that $\mathrm{Cut}(M)\setminus S_{d_{\overline{M}}}$ is countably
$n$-rectifiable in $N\setminus\overline{M}$; this fact is the analogue of
proposition 4.9. in [MM02] in our setting (where the surface $M$ may be
singular). Observe that
$\overline{S_{d_{\overline{M}}}}\cap\overline{M}\subset\mathrm{Sing}(M)$ and
as noted above $\mathrm{Sing}(M)$ has zero $\mathcal{H}^{n}$ measure, so this
set does not affect the rectifiability.

The proof of [MM02, Proposition 4.9.] may be adapted (exactly as in [Bel20,
Section 3]) to our setting by virtue of the fact that their arguments are
local to points away from $\overline{M}$, and hence we may apply the arguments
used in their proof in our situation without change. Thus we conclude that
$\overline{S_{d_{\overline{M}}}}$ is countably $n$-rectifiable, consequently
its $\mathcal{H}^{n+1}$ measure is zero.

The level sets of $d_{\overline{M}}$ are smooth on
$N\setminus(\overline{M}\cup\overline{S_{d_{\overline{M}}}})$ by virtue of the
Implicit Function Theorem, invertibility of $F$ and differentiability of
$d_{\overline{M}}$ on this set. The arguments in the proof of [MM02,
Proposition 4.6.] show further that $F$ restricts to a diffeomorphism,

$F:\\{(y,t)\>:\>y\in M,t\in(\sigma^{-}(y),\sigma^{+}(y))\\}\rightarrow
N\setminus(\overline{M}\cup\overline{S_{d_{\overline{M}}}}).$

We then extend $F$ to $M\times\\{0\\}$ by setting $F(y,0)=y$ so that the image
of the extension of $F$ is now
$N\setminus(\mathrm{Sing}(M)\cup\overline{S_{d_{\overline{M}}}})$. Finally, by
defining

$V_{M}=\\{(y,t)\>|\>y\in M,t\in\left(\sigma^{-}(y),\sigma^{+}(y)\right)\\}.$
(11)

we have that $V_{M}$ is diffeomorphic to
$N\setminus(\mathrm{Sing}(M)\cup\overline{S_{d_{\overline{M}}}})$.

######  Remark 8.

For each compact $K\subset M$ the continuity of the functions
$\sigma^{\pm}(x)$ (defined in the paragraph before (10)) on $M$ implies that
there exists some $c_{K}>0$ such that $0<c_{K}<\min_{x\in
K}\\{\sigma^{+}(x),|\sigma^{-}(x)|\\}$. For such a set $K$, as $M$ itself is a
two-sided hypersurface, there is a two-sided tubular neighbourhood of $K$ when
viewed as a subset of $V_{M}$ (the coordinates defined in (11)), given by
$K\times(-c_{K},c_{K})$ with its closure a subset of $V_{M}$. Furthermore, the
image under the map $F$ of this two-sided tubular neighbourhood,
$F(K\times(-c_{K},c_{K}))\subset N$, is such that
$F(K\times(-c_{K},c_{K}))\cap(\mathrm{Sing}(M)\cup\overline{S_{d_{\overline{M}}}})=\emptyset$,
by the definition of $c_{K}$.

######  Remark 9.

We define a projection to $M$ on
$N\setminus(\mathrm{Sing}(M)\cup\overline{S_{d_{\overline{M}}}})$. For each
$y\in N\setminus(\mathrm{Sing}(M)\cup\overline{S_{d_{\overline{M}}}})$ there
exists a unique geodesic in $N$ with endpoint $x\in M$ realising
$d_{\overline{M}}(y)$ (i.e. such that $d_{\overline{M}}(y)=d_{N}(x,y)$). We
denote by $\Pi$ the smooth projection from a point in
$N\setminus(\mathrm{Sing}(M)\cup\overline{S_{d_{\overline{M}}}}))$ to its
unique $M$. Note that we may express this as $\Pi(x)=F\circ\Pi_{V_{M}}\circ
F^{-1}(x)$, where $\Pi_{V_{M}}$ is the smooth projection map on $V_{M}$
sending a point $(x,s)\in V_{M}$ to $(x,0)\in V_{M}$.

###  3.2 Level sets of the signed distance function

We now define the signed distance, corresponding to our choice of unit normal,
$\nu$, to the hypersurface $M$ pointing into $E$ as

$d^{\pm}_{\overline{M}}=\begin{cases}+d_{\overline{M}}(x),\text{ if }x\in E\\\
0,\text{ if }x\in\overline{M}\\\ -d_{\overline{M}}(x),\text{ if }x\in
N\setminus E\end{cases},$

so that the positive sign corresponds to our point lying in $E$. Denoting by
$S_{d^{\pm}_{\overline{M}}}$ the set of points in $N\setminus\overline{M}$
where $d^{\pm}_{\overline{M}}$ fails to be differentiable, we then have that
$\overline{S_{d^{\pm}_{\overline{M}}}}=\overline{S_{d_{\overline{M}}}}$ and so
$\overline{S_{d^{\pm}_{\overline{M}}}}$ is countably $n$-rectifiable with zero
$\mathcal{H}^{n+1}$ measure.

We now consider the level sets

$\Gamma(s)=\\{x\in N\>|\>d^{\pm}_{\overline{M}}=s\\}.$

These level sets, $\Gamma(s)$, of $d^{\pm}_{\overline{M}}$ are smooth in the
open set $N\setminus(\overline{S_{d^{\pm}_{\overline{M}}}}\cup\overline{M})$
by the Implicit Function Theorem, as $d^{\pm}_{\overline{M}}$ is smooth and
the exponential map is invertible on this open set. Note that we have
$\Gamma(s)=\emptyset$ for $|s|>d(N)$, where $d(N)$ is the diameter of $N$.

We use the following notation to refer to the smooth parts of the level sets
of $d^{\pm}_{\overline{M}}$, setting $\widetilde{\Gamma}(0)=M$, and for $s\neq
0$,

$\widetilde{\Gamma}(s)=\Gamma(s)\setminus\overline{S_{d^{\pm}_{\overline{M}}}}.$

As $\overline{S_{d^{\pm}_{\overline{M}}}}$ is countably $n$-rectifiable, and
hence has vanishing $\mathcal{H}^{n+1}$ measure, we may apply the co-area
formula (slicing with $d^{\pm}_{\overline{M}}$) to conclude that

$\text{for almost every }s\in\mathbb{R}\text{ we have
}\mathcal{H}^{n}\left(\Gamma(s)\cap\overline{S_{d^{\pm}_{\overline{M}}}}\right)=0.$
(12)

Recall the diffeomorphism $F:V_{M}\rightarrow N\setminus(\mathrm{Sing}(M)\cup
Cut(M))$ defined by $F(y,t)=\exp_{y}(t\nu(y))$ and the coordinates $V_{M}$ on
$N\setminus(\mathrm{Sing}(M)\cup Cut(M))$ as defined in (11). We equip $V_{M}$
with the pull-back metric via the map $F$, giving the usual induced metric for
$M$ on $M\times\\{0\\}\subset V_{M}$. Note that we have
$F^{-1}(\widetilde{\Gamma}(s))=M\times\\{s\\}\subset V_{M}$. We now work on
$V_{M}$ to establish facts about the level sets of $d^{\pm}_{\overline{M}}$.

We choose local coordinates, $(x_{1},\dots,x_{n},s)$, on $V_{M}$ such that
$\frac{\partial}{\partial x_{1}},\dots,\frac{\partial}{\partial x_{n}}$ are a
local orthonormal frame around a point $x_{0}\in M$ such that
$\frac{\partial}{\partial s}$ is the unit speed of geodesics with constant
base-point in $M$. The pullback metric (via $F$) then induces a volume form on
$V_{M}$, and hence, at a point $(x_{0},s_{0})\in V_{M}$, an area element,
$\theta(x,s)$, on the set $M\times\\{s_{0}\\}\subset V_{M}$. We set
$\theta(x,s)=0$ for $(x,s)\in(M\times\mathbb{R})\setminus V_{M}$. We then have
that by the structure of $V_{M}$ and $F$ that

$\int_{M}\theta(x,s)dx_{1}\cdots
dx_{n}=\mathcal{H}^{n}(\widetilde{\Gamma}(s)).$ (13)

Note that as the pull-back metric gives the induced metric for $M$ on
$M\times\\{0\\}$ we have that

$\int_{M}\theta(x,0)dx_{1}\cdots dx_{n}=\mathcal{H}^{n}(M).$

As the volume form is smooth on $V_{M}$ we have that the induced area element,
$\theta(x,s)$, is continuous in both variables, in particular we have for any
$x\in M$ that

$\theta(x,s)\rightarrow\theta(x,0)\text{ as }s\rightarrow 0.$ (14)

We have from [Gra04, Theorem 3.11.] that

$\frac{\partial}{\partial s}\log(\theta(x,s))=-H(x,s),$ (15)

so that

$\partial_{s}\theta(x,s)=-H(x,s)\theta(x,s).$ (16)

Here $H(x,s)$ denotes the mean curvature of the the pullback of $\Gamma(s)$ to
$V_{M}$ at the point $(x,s)\in V_{M}$. Note that $H(x,0)=\lambda$ as $M$ is a
hypersurface of constant mean curvature $\lambda$.

We also recall the Ricatti equation, [Gra04, Corollary 3.6.], in the following
form

$\frac{\partial}{\partial s}H(x,s)\geq\min_{N}\mathrm{Ric}_{g}.$ (17)

Let us hereafter denote $m=\min_{N}\mathrm{Ric}_{g}>0$. We then have that

$\begin{cases}H(x,s)\geq\lambda+ms\text{ for }s>0\\\ H(x,0)=\lambda\\\
H(x,s)\leq\lambda+ms\text{ for }s<0\end{cases}.$ (18)

Combining (15), (18) and applying the Fundamental Theorem of Calculus we see
that for each $t\in\mathbb{R}$ we have

$\log(\theta(x,t))\leq-\int_{0}^{t}(ms+\lambda)\>ds,$

from which we conclude that

$\theta(x,t)\leq e^{-t\left(\frac{mt}{2}+\lambda\right)}.$ (19)

Noting that the quadratic $-t\left(\frac{mt}{2}+\lambda\right)$ is maximised
for $t=-\frac{\lambda}{m}$, from (19) we see that

$\theta(x,s)\leq e^{\frac{\lambda^{2}}{2m}}.$ (20)

We then apply the Dominated Convergence Theorem to see that by (14) and (20)
we have

$\int_{M}\theta(x,s)dx_{1}\cdots
dx_{n}\rightarrow\int_{M}\theta(x,0)dx_{1}\cdots dx_{n},$

so in particular by (13)

$\mathcal{H}^{n}(\widetilde{\Gamma}(s))\rightarrow\mathcal{H}^{n}(M)\text{ as
}s\rightarrow 0.$ (21)

Let us record for later use that (21) above implies that

${\mathrm{ess\,inf}}_{s\in[-2\varepsilon\Lambda_{\varepsilon},2\varepsilon\Lambda_{\varepsilon}]}\mathcal{H}^{n}(\widetilde{\Gamma}(s))\rightarrow\mathcal{H}^{n}(M)\text{
as }\varepsilon\rightarrow 0,$ (22)

and

${\mathrm{ess\,sup}}_{s\in[-2\varepsilon\Lambda_{\varepsilon},2\varepsilon\Lambda_{\varepsilon}]}\mathcal{H}^{n}(\widetilde{\Gamma}(s))\rightarrow\mathcal{H}^{n}(M)\text{
as }\varepsilon\rightarrow 0.$ (23)

###  3.3 Approximation with the one-dimensional profile

Let $\overline{\mathbb{H}}^{\varepsilon}$ denote the smooth increasing
truncation of the one-dimensional solution to the Allen–Cahn equation as in
[Bel20]. Note that in particular if $s>2\varepsilon\Lambda_{\varepsilon}$ then
$\overline{\mathbb{H}}^{\varepsilon}(s)=1$ and if
$s<-2\varepsilon\Lambda_{\varepsilon}$ then
$\overline{\mathbb{H}}^{\varepsilon}(s)\equiv-1$; thus as
$|\overline{\mathbb{H}}^{\varepsilon}(s)|=1$ for all $|s|\geq
2\varepsilon\Lambda_{\varepsilon}$ where
$\Lambda_{\varepsilon}=3|\log(\varepsilon)|$, we have that
$e_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon}(s))=0$ for $|s|\geq
2\varepsilon\Lambda_{\varepsilon}$. As shown in [Bel20, Section 2.2], we have

$\mathcal{E}_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon})\rightarrow
1\text{ as }\varepsilon\rightarrow 0,$ (24)

where the specific convergence is such that for fixed $\beta>0$ we have

$1-\beta\varepsilon^{2}\leq\mathcal{E}_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon})\leq
1+\beta\varepsilon^{2}.$ (25)

We define the one-dimensional profile to be

$v_{\varepsilon}(x)=\overline{\mathbb{H}}^{\varepsilon}(d^{\pm}_{\overline{M}}(x)).$

As $\overline{\mathbb{H}}^{\varepsilon}$ is smooth and the distance function
is Lipschitz, it follows that $v_{\varepsilon}\in W^{1,2}(N)$. We now prove
that (2) holds, showing that $v_{\varepsilon}$ acts as an Allen–Cahn
”approximation” of $M$ in the sense that it recovers the
$\varepsilon\rightarrow 0$ limit of the energies of the critical points
obtained by the min-max in [BW20a]. We will exploit this fact directly in the
construction of the paths in Section 5, eventually allowing us to establish
Theorem 3.

By the co-area formula to $d^{\pm}_{\overline{M}}$, noting that $|\nabla
d^{\pm}_{\overline{M}}|=1$, we compute, similarly to [BW22, Section 3.6], that

$\displaystyle\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})$
$\displaystyle=\mathcal{E}_{\varepsilon}(v_{\varepsilon})-\frac{\lambda}{2}\int_{N}v_{\varepsilon}$
$\displaystyle=\frac{1}{2\sigma}\int_{N}\frac{\varepsilon}{2}|\nabla
v_{\varepsilon}|^{2}+\frac{W(v_{\varepsilon})}{\varepsilon}-\frac{\lambda}{2}\int_{N}v_{\varepsilon}$
$\displaystyle=\frac{1}{2\sigma}\int_{\mathbb{R}}\int_{\Gamma(t)}\frac{\varepsilon}{2}\left(\left(\overline{\mathbb{H}}^{\varepsilon}\right)^{\prime}(t)\right)^{2}+\frac{W(\overline{\mathbb{H}}^{\varepsilon}(t))}{\varepsilon}d\mathcal{H}^{n}dt-\frac{\lambda}{2}\int_{\mathbb{R}}\int_{\Gamma(t)}\overline{\mathbb{H}}^{\varepsilon}(t)d\mathcal{H}^{n}dt.$

Using (12) and the definition of $e_{\varepsilon}$ as in Subsection 1.1, we
then have that

$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})=\frac{1}{2\sigma}\int_{\mathbb{R}}e_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon}(t))\mathcal{H}^{n}(\widetilde{\Gamma}(t))dt-\frac{\lambda}{2}\int_{\mathbb{R}}\overline{\mathbb{H}}^{\varepsilon}(t)\mathcal{H}^{n}(\widetilde{\Gamma}(t))dt.$

From the properties of $\overline{\mathbb{H}}^{\varepsilon}$ as stated above
we may obtain the following bounds

$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\geq{\mathrm{ess\,inf}}_{s\in[-2\varepsilon\Lambda_{\varepsilon},2\varepsilon\Lambda_{\varepsilon}]}\mathcal{H}^{n}(\widetilde{\Gamma}(s))\mathcal{E}_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon})-\frac{\lambda}{2}\int_{-2\varepsilon\Lambda_{\varepsilon}}^{\infty}\mathcal{H}^{n}(\widetilde{\Gamma}(t))dt+\frac{\lambda}{2}\int_{-\infty}^{-2\varepsilon\Lambda_{\varepsilon}}\mathcal{H}^{n}(\widetilde{\Gamma}(t))dt,$

and also

$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\leq{\mathrm{ess\,sup}}_{s\in[-2\varepsilon\Lambda_{\varepsilon},2\varepsilon\Lambda_{\varepsilon}]}\mathcal{H}^{n}(\widetilde{\Gamma}(s))\mathcal{E}_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon})-\frac{\lambda}{2}\int_{2\varepsilon\Lambda_{\varepsilon}}^{\infty}\mathcal{H}^{n}(\widetilde{\Gamma}(t))dt+\frac{\lambda}{2}\int_{-\infty}^{2\varepsilon\Lambda_{\varepsilon}}\mathcal{H}^{n}(\widetilde{\Gamma}(t))dt.$

Observe that

$\mathrm{Vol}_{g}(E)=\mathcal{H}^{n+1}(\\{x\in
N\>|\>d^{\pm}_{\overline{M}}>0\\})=\int_{0}^{\infty}\mathcal{H}^{n}(\widetilde{\Gamma}(t))dt=\lim_{\varepsilon\rightarrow
0}\int_{\pm
2\varepsilon\Lambda_{\varepsilon}}^{\infty}\mathcal{H}^{n}(\widetilde{\Gamma}(t))dt,$

and

$\mathrm{Vol}_{g}(N\setminus E)=\mathcal{H}^{n+1}(\\{x\in
N\>|\>d^{\pm}_{\overline{M}}<0\\})=\int_{-\infty}^{0}\mathcal{H}^{n}(\widetilde{\Gamma}(t))dt=\lim_{\varepsilon\rightarrow
0}\int^{\pm
2\varepsilon\Lambda_{\varepsilon}}_{-\infty}\mathcal{H}^{n}(\widetilde{\Gamma}(t))dt.$

Combining these two identities with (22), (23) and (24) in the above two
bounds on $\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})$ we conclude
that

$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\rightarrow\mathcal{H}^{n}(M)-\lambda\mathrm{Vol}_{g}(E)+\frac{\lambda}{2}\mathrm{Vol}_{g}(N)=\mathcal{F}_{\lambda}(E)\text{
as }\varepsilon\rightarrow 0,$

as desired.

##  4 Relating local properties of the energy to the geometry

This section relates local behaviour of the energy of the one-dimensional
profile, $v_{\varepsilon}=\overline{\mathbb{H}}^{\varepsilon}\circ
d^{\pm}_{\overline{M}}$, to the local geometric properties of $E$. Recall that
$M=\partial^{*}E\subset N$ is assumed to be a closed embedded hypersurface of
constant mean curvature $\lambda$, smooth away from a closed singular set,
denoted $\mathrm{Sing}(M)=\overline{M}\setminus M$, of Hausdorff dimension at
most $n-7$. Furthermore, $M$ separates $N\setminus\overline{M}$ into open
sets, $E$ and $N\setminus E$, with common boundary $\overline{M}$ and
$E\in\mathcal{C}(N)$ with $M=\partial^{*}E$. For an isolated singularity
$p\in\overline{M}$ we fix throughout this section some
$0<r_{1}<r_{2}<\min\\{R_{p},R_{l}\\}$ such that
$M\cap\overline{B_{r_{2}}(p)}\setminus B_{r_{1}}(p)$ is smooth.

###  4.1 Local smoothing of Caccioppoli sets

We first establish a procedure to locally perturb Caccioppoli sets that are
assumed to be smooth in an annular region to ensure that they are smooth in
the entire ball.

######  Proposition 2.

Suppose that $F\in\mathcal{C}(N)$ is such that $F\setminus
B_{r_{1}}(p)=E\setminus B_{r_{1}}(p)$. Fix some
$\tilde{r}_{2}\in(r_{1},r_{2})$, then, for each $\delta>0$, there exists
$F_{\delta}\in\mathcal{C}(N)$ with the following properties:

  * •

$\partial F_{\delta}$ is smooth in $B_{r_{2}}(p)$.

  * •

$F_{\delta}\setminus B_{\tilde{r}_{2}}(p)=E\setminus B_{\tilde{r}_{2}}(p)$.

  * •

$|\mathrm{Per}_{g}(F_{\delta})-\mathrm{Per}_{g}(F)|\leq\delta$.

  * •

$Vol_{g}(F_{\delta}\Delta F)\leq\delta$.

Thus, in particular we have that $F_{\delta}$ agrees with $F$ outside of
$B_{\tilde{r}_{2}}(p)$ and is such that

$|\mathcal{F}_{\lambda}(F_{\delta})-\mathcal{F}_{\lambda}(F)|\leq(1+\lambda)\delta.$

######  Remark 10.

Though Proposition 2 is phrased in our setting for a constant mean curvature
hypersurface, the proof makes no use of the variational assumption on $M$.
Thus, the same result holds for any Caccioppoli set which satisfies the same
properties as $M$, without any condition on the mean curvature.

###### Proof.

Recall that, as $r_{2}<R_{l}$, $B_{r_{2}}(p)$ is $2$-bi-Lipschitz to the
Euclidean ball $B_{r_{2}}^{\mathbb{R}^{n+1}}(0)$ via some a geodeisic normal
coordinate chart, $\phi$, with $\phi(p)=0$ and so that
$\frac{1}{2}\leq\sqrt{|g|}\leq 2$ on $B_{r_{2}}(p)$. We consider a radially
symmetric mollifier $\rho\in C^{\infty}_{c}(B^{\mathbb{R}^{n+1}}_{1}(0))$ such
that $\int_{\mathbb{R}^{n+1}}\rho=1$ and for each $\theta>0$ define
$\rho_{\theta}(x)=\frac{1}{\theta^{n+1}}\rho(\frac{x}{\theta})$. We fix
$\tilde{r}_{2}\in(r_{1},r_{2})$ and consider, for
$\theta<r_{2}-\tilde{r}_{2}$, the function
$s_{\theta}=(\chi_{\phi(F)}*\rho_{\theta})\circ\phi\in
C^{\infty}(B_{\tilde{r}_{2}}(p))$. We now show that $s_{\theta}$ approximates
$\chi_{F}$ in the BV norm. First, by standard properties of mollifiers we have

$\displaystyle||\chi_{F}-s_{\theta}||_{L^{1}(B_{\tilde{r}_{2}}(p)))}$
$\displaystyle=\int_{B_{\tilde{r}_{2}}(p)}|\chi_{F}-(\chi_{\phi(F)}*\rho_{\theta})\circ\phi|d\mathcal{H}^{n}_{g}$
$\displaystyle=\int_{\phi(B_{\tilde{r}_{2}}(p))}|\chi_{\phi(F)}-(\chi_{\phi(F)}*\rho_{\theta})|\sqrt{|g|}d\mathcal{L}^{n}$
$\displaystyle\leq
2||\chi_{\phi(F)}-(\chi_{\phi(F)}*\rho_{\theta})||_{L^{1}(\phi(B_{\tilde{r}_{2}}(p)))}\rightarrow
0\text{ as }\theta\rightarrow 0.$

Hence by the lower semi-continuity of the perimeter we have that

$\mathrm{Per}_{g}(F)=|D_{g}\chi_{F}|(B_{\tilde{r}_{2}}(p))\leq\liminf_{\theta\rightarrow
0}|D_{g}s_{\theta}|(B_{\tilde{r}_{2}}(p)).$

We let $X\in\Gamma^{1}_{c}(TB_{\tilde{r}_{2}}(p))$ with $|X|_{g}\leq 1$ and
compute

$\displaystyle\int_{B_{\tilde{r}_{2}}(p)}s_{\theta}div_{g}Xd\mathcal{H}^{n}_{g}$
$\displaystyle=\int_{\phi(B_{\tilde{r}_{2}}(p))}(\chi_{\phi(F)}*\rho_{\theta})\partial_{i}(\sqrt{|g|}\hat{X}^{i})d\mathcal{L}^{n}$
$\displaystyle=\int_{\phi(B_{\tilde{r}_{2}}(p))}\chi_{\phi(F)}\partial_{i}(\rho_{\theta}*\sqrt{|g|}\hat{X}^{i})d\mathcal{L}^{n}$
$\displaystyle=\int_{B_{\tilde{r}_{2}}(p)}\chi_{F}div_{g}Y\leq|D_{g}\chi_{F}|(B_{\tilde{r}_{2}}(p)),$

where here $\hat{X}=\hat{X}^{i}\partial_{i}^{\phi}$ with
$\hat{X}^{i}=X^{i}\circ\phi^{-1}$, and $Y=Y^{i}\partial_{i}^{\phi}$ with
$Y^{i}=\frac{1}{\sqrt{|g|}}(\rho_{\theta}*\sqrt{|g|}\hat{X}^{i})\circ\phi$ (so
$|Y|_{g}\leq 1$). As the choice of $X$ above was arbitrary here we thus
conclude that

$\lim_{\theta\rightarrow
0}|D_{g}s_{\theta}|(B_{\tilde{r}_{2}}(p))=|D_{g}\chi_{F}|(B_{\tilde{r}_{2}}(p)).$

Arguing identically to the proof of [Mag12, Theorem 13.8] we conclude that the
level sets of $s_{\theta}$ (which have smooth boundary for almost every
$t\in(0,1)$ by Sard’s Theorem), $L^{t}_{\theta}=\\{s_{\theta}>t\\}$ for a.e.
$t\in(0,1)$ provide a sequence of open sets with smooth boundary such that
$Vol_{g}(L^{t}_{\theta}\Delta F)\rightarrow 0$ and
$Per(L^{t}_{\theta};A)\rightarrow Per(F;A)$ for each open set $A\subset
B_{\tilde{r}_{2}}(p)$, whenever $Per(F;\partial A)=0,$ as $\theta\rightarrow
0$.

As the exponential map is a radial isometry we have that the geodeisic normal
coordinate chart $\phi$ is such that
$B_{r_{2}}^{\mathbb{R}^{n+1}}(0)=\phi(B_{r_{2}}(p))$ and
$S=\phi\left(M\cap(\overline{B}_{r_{2}}(p)\setminus B_{r_{1}}(p))\right)$ is
smooth in $B_{r_{2}}^{\mathbb{R}^{n+1}}(0)\setminus
B_{r_{1}}^{\mathbb{R}^{n+1}}(0)$. In particular we may choose
$r_{1}<r_{a}<r_{b}<\tilde{r}_{2}$ so that

$T=\\{x+b\nu_{S}(x)\>|\>x\in
S\cap(\overline{B_{r_{b}}^{\mathbb{R}^{n+1}}(0)}\setminus
B_{r_{a}}^{\mathbb{R}^{n+1}}(0)),b\in[-\delta,\delta]\\}\subset\subset
B_{\tilde{r}_{2}}^{\mathbb{R}^{n+1}}(0)\setminus
B_{r_{1}}^{\mathbb{R}^{n+1}}(0),$

for some $\delta>0$. Here above we are denoting the unit normal induced on $S$
from $M$ by $\nu_{S}$; note then that

$F\cap\phi^{-1}(T)=\phi^{-1}(\\{x+b\nu_{S}(x)\in T\>|\>b>0\\}).$ (26)

We consider normal derivatives of the function
$\chi_{\widehat{F}}*\rho_{\theta}$ for $\theta<\delta$ in the tubular
neighbourhood $T$ defined above. For $x+b\nu_{S}(x)\in T$ and $\theta>0$ small
enough so that $\nu_{S}(y)\cdot\nu_{S}(x)\geq\frac{1}{2}$ for any $y\in
B_{2\theta}(x)$ ($S$ is smooth so $\nu_{S}$ varies continuously) we compute
that when $b\in(-\theta,\theta)$ we have

$\displaystyle\nabla(\chi_{\widehat{F}}*\rho_{\theta})(x+b\nu_{S}(x))\cdot\nu_{S}(x)$
$\displaystyle=(\nabla\chi_{\widehat{F}}*\rho_{\theta})(x+b\nu_{s}(x))\cdot\nu_{S}(x)$
$\displaystyle=-((\mathcal{H}^{n}\llcorner
S)\nu_{S}*\rho_{\theta})(x+b\nu_{S}(x))\cdot\nu_{S}(x)$
$\displaystyle=-\left(\int_{S\cap
B_{\theta}(x+b\nu_{S}(x))}\rho_{\theta}(x+b\nu_{S}(x)-y)\nu_{S}(y)\cdot\nu_{S}(x)d\mathcal{H}^{n}(y)\right)$
$\displaystyle\leq-\frac{1}{2}\left(\int_{S\cap
B_{\theta}(x+b\nu_{S}(x))}\rho_{\theta}(x+b\nu_{S}(x)-y)d\mathcal{H}^{n}(y)\right)<0.$

Here we’ve used that $D_{g}\chi_{\widehat{F}}=-(\mathcal{H}^{n}\llcorner
S)\nu_{S}$ in $T$ as $S$ is smooth and by the triangle inequality we have both
$B_{\theta}(x+b\nu_{S}(x))\subset B_{2\theta}(x)$ for $b<\theta$ and that the
strict inequality holds as $S\cap B_{\theta-b}(x)\subset S\cap
B_{\theta}(x+b\nu_{S}(x))$.

Thus for $b\in(-\theta,\theta)$ we have that
$\nabla(\chi_{\widehat{F}}*\rho_{\theta})(x+b\nu_{s}(x))\cdot\nu_{S}(x)<0$ and
$\nabla(\chi_{\widehat{F}}*\rho_{\theta})(x+b\nu_{S}(x))=0$ for
$b\notin(-\theta,\theta)$ (as then
$\chi_{\widehat{F}}*\rho_{\theta}\in\\{0,1\\}$). We conclude that for each
$t\in(0,1)$ and $x\in
S\cap(\overline{B_{r_{b}}^{\mathbb{R}^{n+1}}(0)}\setminus
B_{r_{a}}^{\mathbb{R}^{n+1}}(0))$ there exists a unique
$b_{\theta}^{t}(x)\in(-\theta,\theta)$ such that
$(\chi_{\widehat{F}}*\rho_{\theta})(x+b_{\theta}^{t}(x)\nu_{S}(x))=t$; with
$b^{t}_{\theta}\rightarrow 0$ point-wise on
$S\cap(\overline{B_{r_{b}}^{\mathbb{R}^{n+1}}(0)}\setminus
B_{r_{a}}^{\mathbb{R}^{n+1}}(0))$ as $\theta\rightarrow 0$. Note also that by
the structure of $T$ we have

$L^{t}_{\theta}\cap\phi^{-1}(T)=\phi^{-1}(\\{x+b\nu_{S}(x)\in
T\>|\>b<b^{t}_{\theta}(x)\\}).$ (27)

Consider the function
$h_{\theta}:S\cap(\overline{B_{r_{b}}^{\mathbb{R}^{n+1}}(0)}\setminus
B_{r_{a}}^{\mathbb{R}^{n+1}}(0))\times(-\theta,\theta)\rightarrow T$ defined
by

$h^{t}_{\theta}(x,b)=(\chi_{\widehat{F}}*\rho_{\theta})(x+b\nu_{S}(x))-t.$

By the above calculation we have that $\frac{\partial h_{\theta}}{\partial
b}(x+b\nu_{S}(x))=\nabla(\chi_{\widehat{F}}*\rho_{\theta})(x+b\nu_{S}(x))\cdot\nu_{S}(x)<0$
and so by the Implicit Function Theorem (working in charts in which
$S\cap(\overline{B_{r_{b}}^{\mathbb{R}^{n+1}}(0)}\setminus
B_{r_{a}}^{\mathbb{R}^{n+1}}(0))$ is locally a graph) we have that the
function $b^{t}_{\theta}$ is smooth on
$S\cap(\overline{B_{r_{b}}^{\mathbb{R}^{n+1}}(0)}\setminus
B_{r_{a}}^{\mathbb{R}^{n+1}}(0))$ (noting that
$h^{t}_{\theta}(x,b^{t}_{\theta}(x))=0$ for $x$ in this set).

We now show that the directional derivatives of $b^{t}_{\theta}$ converge to
zero as $\theta\rightarrow 0$. At a point $x\in
S\cap(\overline{B_{r_{b}}^{\mathbb{R}^{n+1}}(0)}\setminus
B_{r_{a}}^{\mathbb{R}^{n+1}}(0))$, denoting directional derivatives by
$\frac{\partial}{\partial x_{i}}$, we compute

$\displaystyle\left|\frac{\Big{(}\frac{\partial h^{t}_{\theta}}{\partial
x_{i}}\Big{)}}{\Big{(}\frac{\partial h^{t}_{\theta}}{\partial
b}\Big{)}}\right|$
$\displaystyle=\left|\frac{\nabla(\chi_{\widehat{F}}*\rho_{\theta})(x+b\nu_{S}(x))\cdot\frac{\partial}{\partial
x_{i}}(x+b\nu_{S}(x))}{\nabla(\chi_{\widehat{F}}*\rho_{\theta})(x+b\nu_{S}(x))\cdot\nu_{S}(x))}\right|$
$\displaystyle=\left|\frac{\int_{S\cap
B_{\theta}(x+b\nu_{S}(x))}\rho_{\theta}(x+b\nu_{S}(x)-y)\nu_{S}(y)\cdot\frac{\partial}{\partial
x_{i}}(x+b\nu_{S}(x))d\mathcal{H}^{n}(y)}{\int_{S\cap
B_{\theta}(x+b\nu_{S}(x))}\rho_{\theta}(x+b\nu_{S}(x)-y)\nu_{S}(y)\cdot\nu_{S}(x)d\mathcal{H}^{n}(y)}\right|$
$\displaystyle\leq 2\sup_{y\in S\cap
B_{\theta}(x+b\nu_{S}(x))}\left|\nu_{S}(y)\cdot\frac{\partial}{\partial
x_{i}}(x+b\nu_{S}(x))\right|\rightarrow 0\text{ as }\theta\rightarrow 0.$

Here we see that the last term converges to zero as $\theta\rightarrow 0$ by
using the triangle inequality, noting that
$\nu_{S}(y)\cdot\frac{\partial}{\partial x_{i}}x\rightarrow 0$ by the
continuity of $\nu_{S}$ and $\nu_{S}(y)\cdot
b\frac{\partial\nu_{S}(x)}{\partial x_{i}}\leq C\theta$ for a constant $C$
depending only on the normal of $S$. We then have from the Implicit Function
Theorem that

$\left|\frac{\partial b^{t}_{\theta}}{\partial
x_{i}}\right|(x)=\left|\frac{\Big{(}\frac{\partial h^{t}_{\theta}}{\partial
x_{i}}\Big{)}}{\Big{(}\frac{\partial h^{t}_{\theta}}{\partial
b}\Big{)}}\right|(x,b^{t}_{\theta}(x))\rightarrow 0\text{ as
}\theta\rightarrow 0.$

We conclude that for $t\in(0,1)$ the function $b_{\theta}^{t}$ defined on
$S\cap(\overline{B_{r_{b}}^{\mathbb{R}^{n+1}}(0)}\setminus
B_{r_{a}}^{\mathbb{R}^{n+1}}(0))$ as above is smooth and in fact converges to
zero uniformly in the $C^{1}$ norm as $\theta\rightarrow 0$.

We fix $\tilde{r}_{a}\in(r_{a},r_{b})$ so that for
$\theta<\tilde{r}_{a}-r_{a}$ we ensure (by the triangle inequality) that
$x+b^{t}_{\theta}(x)\nu_{S}(x)\in B_{\tilde{r}_{a}}(p)$ whenever $x\in\partial
B_{r_{a}}(p)$ (i.e. when $|x|=r_{a}$). Fix a further
$\tilde{r}_{b}\in(\tilde{r}_{a},r_{b})$ so that for
$\theta<r_{b}-\tilde{r}_{b}$ we ensure (by the triangle inequality) that
$x+b^{t}_{\theta}(x)\nu_{S}(x)\in N\setminus B_{\tilde{r}_{b}}(p)$ whenever
$x\in\partial B_{r_{b}}(p)$ (i.e. when $|x|=r_{b}$).

We then define $\eta\in C^{\infty}_{c}(\mathbb{R}^{n+1})$ to be a radially
symmetric cut-off function with the following properties

$\begin{cases}0\leq\eta\leq 1\text{ on }\mathbb{R}^{n+1}\\\ |\nabla\eta|\leq
C_{\eta}\text{ on }\mathbb{R}^{n+1}\\\ \eta(x)=1\text{ if
}|x|\leq\tilde{r}_{a}\\\ \eta(x)=0\text{ if }|x|\geq\tilde{r}_{b},\end{cases}$

for some $C_{\eta}=C_{\eta}(\tilde{r}_{a},\tilde{r}_{b})>0$.

Before proceeding to define the local smoothing we let

$H^{t}_{\theta}=\phi^{-1}\left(\\{x+b\nu_{S}(x)\in
T\>|\>b<\eta(x)b^{t}_{\theta}(x)\\}\right),$

and, slightly abusing notation, we denote

$\partial
H^{t}_{\theta}=\phi^{-1}\left(\left\\{x+\eta(x)b^{t}_{\theta}(x)\nu_{S}(x)\>|\>x\in
S\cap(\overline{B_{r_{b}}^{\mathbb{R}^{n+1}}(0)}\setminus
B_{r_{a}}^{\mathbb{R}^{n+1}}(0))\right\\}\right),$

where the notation for boundary is justified by virtue of the fact that, as
$M$ separates $T$, the ”graph” given by
$\phi^{-1}(x+\eta(x)b^{t}_{\theta}(x)\nu_{S}(x))$ separates $\phi^{-1}(T)$.

We then consider the following hypersurface

$\partial G^{t}_{\theta}=(M\setminus B_{\tilde{r}_{b}}(p))\cup(\partial
L^{t}_{\theta}\cap B_{\tilde{r}_{a}}(p))\cup\left(\partial
H^{t}_{\theta}\cap(B_{\tilde{r}_{b}}(p)\setminus B_{\tilde{r}_{a}}(p))\right)$

which is smooth inside $B_{r_{b}}(p)$ as $H^{t}_{\theta}\cap\partial
B_{\tilde{r}_{a}}(p)\in\partial L^{t}_{\theta}$ and
$H^{t}_{\theta}\cap\partial B_{\tilde{r}_{b}}(p)\in M$. We note also that as
$\overline{M}$ satisfies the geodesic touching property from Lemma 1,
$\partial G^{t}_{\theta}$ also satisfies the conclusions of Lemma 1.

Furthermore, by (26) and (27), $\partial G^{t}_{\theta}$ arises as the
boundary of the open set

$G^{t}_{\theta}=(E\setminus((B_{\tilde{r}_{b}}(p)\cup\phi^{-1}(T)))\cup(L^{t}_{\theta}\cap
B_{\tilde{r}_{b}}(p)\setminus\phi^{-1}(T))\cup(H^{t}_{\theta}\cap\phi^{-1}(T)),$

and hence $G^{t}_{\theta}\setminus B_{\tilde{r}_{2}}(p)=E\setminus
B_{\tilde{r}_{2}}(p)$ as $\phi^{-1}(T)\subset\subset
B_{\tilde{r}_{2}}(p)\setminus B_{r_{1}}(p)$.

We may write

$(L^{t}_{\theta}\Delta
E)\cap\phi^{-1}(T)=\phi^{-1}\left(\left\\{x+b\nu_{S}(x)\in
T\>\bigg{|}\>\begin{cases}0<b<b^{t}_{\theta}(x),\text{ if
}b^{t}_{\theta}(x)>0\\\ b^{t}_{\theta}<b<0,\text{ if
}b^{t}_{\theta}(x)<0\end{cases}\right\\}\right).$

As $0\leq\eta\leq 1$, $\eta(x)b^{t}_{\theta}(x)$ has the same sign as
$b^{t}_{\theta}$ and $|\eta(x)b^{t}_{\theta}(x)|\leq|b^{t}_{\theta}(x)|$.
Therefore, by construction of $H^{t}_{\theta}$, we have that
$(G^{t}_{\theta}\Delta E)\cap\phi^{-1}(T)\subset(L^{t}_{\theta}\Delta
E)\cap\phi^{-1}(T)$. As $G^{t}_{\theta}$ agrees with either $E$ or
$L^{t}_{\theta}$ outside of $\phi^{-1}(T)$ we conclude that
$G^{t}_{\theta}\Delta F\subset L^{t}_{\theta}\Delta F$ and hence
$Vol_{g}(G^{t}_{\theta}\Delta F)\leq Vol_{g}(L^{t}_{\theta}\Delta
F)\rightarrow 0$ as $\theta\rightarrow 0$.

Thus we may choose $\theta>0$ to ensure that

$Vol_{g}(G^{t}_{\theta}\Delta F)\leq\delta.$

We note that we may always ensure that the various radii
$r_{a},\tilde{r}_{a},\tilde{r}_{b}$ and $r_{b}$ as chosen above are done so to
ensure that

$\mathrm{Per}_{g}(F;\partial(B_{\tilde{r}_{b}}(p)\setminus\phi^{-1}(T)))=0,$

so that
$\mathrm{Per}_{g}(L^{t}_{\theta};B_{\tilde{r}_{b}}(p)\setminus\phi^{-1}(T))\rightarrow\mathrm{Per}_{g}(F;B_{\tilde{r}_{b}}(p)\setminus\phi^{-1}(T))$
as $\theta\rightarrow 0$; in particular for $\theta>0$ sufficiently small we
ensure that

$|\mathrm{Per}_{g}(L^{t}_{\theta};B_{\tilde{r}_{b}}(p)\setminus\phi^{-1}(T))-\mathrm{Per}_{g}(F;B_{\tilde{r}_{b}}(p)\setminus\phi^{-1}(T))|\leq\frac{\delta}{2}.$

As we have $G^{t}_{\theta}=E$ outside of $B_{\tilde{r}_{b}}\cup\phi^{-1}(T)$
we have that

$\mathrm{Per}_{g}(G^{t}_{\theta};N\setminus(B_{\tilde{r}_{b}}\cup\phi^{-1}(T)))=\mathrm{Per}_{g}(E;N\setminus(B_{\tilde{r}_{b}}\cup\phi^{-1}(T))).$

Note that as $|\nabla\eta|<C_{\eta}$ and $b^{t}_{\theta}\rightarrow 0$
uniformly in the $C^{1}$ norm we may choose $\theta>0$ potentially smaller to
ensure that

$\sup_{y\in M\cap(B_{r_{b}}(p)\setminus
B_{r_{a}}(p))}|J_{\phi^{-1}\circ(Id+(\eta
b^{t}_{\theta}))\circ\phi}(y)-1|\leq\frac{\delta}{2\mathcal{H}^{n}(M\cap(B_{r_{b}}(p)\setminus
B_{r_{a}}(p)))},$

where here $J_{f}$ denotes the Jacobian of a function $f$ (which depends only
on the $C^{1}$ norm of the function $f$), and so by the area formula we have

$\displaystyle|\mathcal{H}^{n}(\partial
H^{t}_{\theta}\cap\phi^{-1}(T))-\mathcal{H}^{n}(M\cap\phi^{-1}(T))|$
$\displaystyle=\left|\int_{M\cap(B_{r_{b}}(p)\setminus
B_{r_{a}}(p))}(J_{\phi^{-1}\circ(Id+(\eta
b^{t}_{\theta}))\circ\phi}(y)-1)d\mathcal{H}^{n}(y)\right|$
$\displaystyle\leq\frac{\delta}{2}.$

We combine the above facts to compute that for $\theta>0$ sufficiently small
we have

$\displaystyle|\mathrm{Per}_{g}(G^{t}_{\theta})-\mathrm{Per}_{g}(F)|$
$\displaystyle=|\mathrm{Per}_{g}(G^{t}_{\theta};\phi^{-1}(T))-\mathrm{Per}_{g}(F;\phi^{-1}(T))$
$\displaystyle\quad\quad+\mathrm{Per}_{g}(G^{t}_{\theta};N\setminus\phi^{-1}(T))-\mathrm{Per}_{g}(F;N\setminus\phi^{-1}(T))|$
$\displaystyle\leq|\mathcal{H}^{n}(\partial
H^{t}_{\theta}\cap\phi^{-1}(T))-\mathcal{H}^{n}(M\cap\phi^{-1}(T))|$
$\displaystyle\quad\quad+|\mathrm{Per}_{g}(L^{t}_{\theta};B_{\tilde{r}_{b}}(p)\setminus\phi^{-1}(T))-\mathrm{Per}_{g}(F;B_{\tilde{r}_{b}}(p)\setminus\phi^{-1}(T))|$
$\displaystyle\leq\delta$

and so

$|\mathcal{F}_{\lambda}(G_{\theta}^{t})-\mathcal{F}_{\lambda}(F)|\leq(1+\lambda)\delta.$

Setting $F_{\delta}=G^{t}_{\theta}$ for one choice of almost any $t\in(0,1)$
and a choice of $\theta>0$ sufficiently small then provides a Caccioppoli
satisfying the desired conclusions. ∎

######  Remark 11.

As $\partial F_{\delta}$ is smooth in $B_{r_{2}}(p)$ and agrees with $M$
outside of $B_{\tilde{r}_{2}}(p)$, we have that $\partial F_{\delta}$
satisfies the conclusions of Lemma 1. More specifically, for any $x\in
N\setminus\partial F_{\delta}$, any minimising geodesic connecting $x$ to
$\partial F_{\delta}$ has endpoint in a regular point of $\partial
F_{\delta}$.

###  4.2 Local energy minimisation

We now prove the existence of local energy minimisers that agree with our one-
dimensional solution,
$v_{\varepsilon}=d^{\pm}_{\overline{M}}\circ\overline{\mathbb{H}}^{\varepsilon}$,
outside of a fixed ball in $N$. Minimisers of such problems will be used in
the next subsection to conclude local statements about constant mean curvature
hypersurfaces under appropriate assumptions on the behaviour of the energy of
$v_{\varepsilon}$ as $\varepsilon\rightarrow 0$.

######  Lemma 2.

Let $B_{\rho}(q)\subset N$ be a ball, of radius $\rho>0$, centred at a point
$q\in\overline{M}$ and define for $\varepsilon\in(0,1)$

$\mathcal{A}_{\varepsilon,\rho}(q)=\\{u\in W^{1,2}(N)\>|\>|u|\leq
1,u=v_{\varepsilon}\text{ on }N\setminus B_{\rho}(q)\\},$

then there exists $g_{\varepsilon}\in\mathcal{A}_{\varepsilon,\rho}(q)$ such
that

$\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})=\inf_{u\in\mathcal{A}_{\varepsilon,\rho}(q)}\mathcal{F}_{\varepsilon,\lambda}(u).$

###### Proof.

Let
$I_{\varepsilon}=\inf_{u\in\mathcal{A}_{\varepsilon,\rho}(q)}\mathcal{F}_{\varepsilon,\lambda}(u)$
and consider $\\{u_{k}\\}_{k=1}^{\infty}\in\mathcal{A}_{\varepsilon,\rho}(q)$
such that $\mathcal{F}_{\varepsilon,\lambda}(u_{k})\rightarrow
I_{\varepsilon}$. Note that for any $u\in\mathcal{A}_{\varepsilon,\rho}(q)$ we
have

$\mathcal{F}_{\varepsilon,\lambda}(u)=\frac{1}{2\sigma}\int_{N}\varepsilon\frac{|\nabla
u|^{2}}{2}+\frac{W(u)}{\varepsilon}-\frac{\lambda}{2}\int_{N}u\geq\frac{1}{2\sigma}\int_{N}\varepsilon\frac{|\nabla
u|^{2}}{2}-\frac{\lambda}{2}\mathrm{Vol}(N),$

using the fact that $|u|\leq 1$ $\mathcal{H}^{n+1}$-a.e. and $W\geq 0$; hence
for $k$ sufficiently large we have that

$\int_{N}|\nabla
u_{k}|^{2}\leq\frac{4\sigma}{\varepsilon}\left(I_{\varepsilon}+1+\frac{\lambda}{2}\mathrm{Vol}(N)\right),$

so $\sup_{k}||\nabla u_{k}||_{L^{2}(N)}<\infty$. Using the triangle and
Poincaré inequality (as $u_{k}-v_{\varepsilon}\in W^{1,2}_{0}(B_{\rho}(q))$
for each $k\in\mathbb{N}$) we have

$\displaystyle||u_{k}||_{L^{2}(N)}$
$\displaystyle=||u_{k}-v_{\varepsilon}+v_{\varepsilon}||_{L^{2}(N)}\leq||u_{k}-v_{\varepsilon}||_{L^{2}(N)}+||v_{\varepsilon}||_{L^{2}(N)}\leq
C_{P}||\nabla u_{k}-\nabla
v_{\varepsilon}||_{L^{2}(N)}+||v_{\varepsilon}||_{L^{2}(N)}$
$\displaystyle\leq C_{P}(\sup_{k}||\nabla u_{k}||_{L^{2}(N)}+||\nabla
v_{\varepsilon}||_{L^{2}(N)})+||v_{\varepsilon}||_{L^{2}(N)},$

(where $C_{P}$ is the Poincaré constant for $N$) so
$\sup_{k}||u_{k}||_{L^{2}(N)}<\infty$. Thus $\\{u_{k}\\}_{k=1}^{\infty}$ is a
bounded sequence in $W^{1,2}(N)$ and hence, by Rellich-Kondrachov compactness,
there exists $u\in W^{1,2}(N)$ and a sub-sequence of the
$\\{u_{k}\\}_{k=1}^{\infty}$ (not relabelled) such that $u_{k}$ converges to
$u$, weakly in $W^{1,2}(N)$, strongly in $L^{2}(N)$ (hence strongly in
$L^{1}(N)$) and $\mathcal{H}^{n+1}\text{-}a.e.\text{ in }N$. We then have that
$|u|\leq 1$ and $u=v_{\varepsilon}$ on $N\setminus B_{\rho}(q)$ at
$\mathcal{H}^{n+1}$-$a.e.$ point, thus
$u\in\mathcal{A}_{\varepsilon,\rho}(q)$. Combining the weak convergence of the
$u_{k}$ in $W^{1,2}(N)$, almost everywhere convergence of the $u_{k}$ with
Fatou’s lemma (using the continuity of $W$ and that $W\geq 0$) and the strong
$L^{1}(N)$ convergence of the $u_{k}$, we have
$\mathcal{F}_{\varepsilon,\lambda}(u)=I_{\varepsilon}$.

The above arguments show that, for sufficiently small $\varepsilon>0$, the
energy minimisation problem in the class $\mathcal{A}_{\varepsilon,\rho}(q)$
is well posed. Hence, for each $\varepsilon\in(0,1)$ we may produce a function
$g_{\varepsilon}\in W^{1,2}(N)$ such that $|g_{\varepsilon}|\leq 1$ on $N$,
$g_{\varepsilon}=v_{\varepsilon}$ on $N\setminus B_{\rho}(q)$ and
$\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})=\inf_{u\in\mathcal{A_{\varepsilon}}}\mathcal{F}_{\varepsilon,\lambda}(u)$;
in particular
$\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})$.∎

###  4.3 Recovery functions for local geometric properties

Using Proposition 2 and our analysis from Subsection 3.3 we now prove the
following recovery type Lemma relating the energy of $v_{\varepsilon}$ to
local geometric properties of constant mean curvature hypersurfaces. Recall
that for an isolated singularity $p\in\overline{M}$ we fixed
$0<r_{1}<r_{2}<\min\\{R_{p},R_{l}\\}$ such that
$M\cap\overline{B_{r_{2}}(p)}\setminus B_{r_{1}}(p)$ is smooth.

######  Lemma 3.

Let $F\in\mathcal{C}(N)$ be such that $F\setminus B_{r_{1}}(p)=E\setminus
B_{r_{1}}(p)$ and

$\mathcal{F}_{\lambda}(F)=\inf_{G\in\mathcal{C}(N)}\\{\mathcal{F}_{\lambda}(F)\>|\>G\setminus
B_{r_{1}}(p)=E\setminus B_{r_{1}}(p)\\}.$

Then, for $\varepsilon>0$ sufficiently small, there exist functions
$f_{\varepsilon}\in\mathcal{A}_{\varepsilon,r_{2}}(p)$, so that
$|f_{\varepsilon}|\leq 1$ and $f_{\varepsilon}=v_{\varepsilon}$ on $N\setminus
B_{r_{2}}(p)$, such that

$\mathcal{F}_{\varepsilon,\lambda}(f_{\varepsilon})\rightarrow\mathcal{F}_{\lambda}(F).$

Furthermore, the functions $v_{\varepsilon}$ are such that
$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})+\tau_{\varepsilon}$,
for some sequence $\tau_{\varepsilon}\rightarrow 0$ as $\varepsilon\rightarrow
0$, where the $g_{\varepsilon}$ are defined as in Lemma 2 for $B_{r_{2}}(p)$,
if and only if $E$ solves the above minimisation problem.

######  Remark 12.

There exist minimisers to the variational problem in Lemma 3 by [Mag12,
Section 12.5].

###### Proof.

Let $F$ solve the minimisation problem and consider, for each
$\varepsilon,\delta>0$, the functions
$f_{\varepsilon,\delta}=\overline{\mathbb{H}}^{\varepsilon}\circ
d^{\pm}_{\partial F_{\delta}}$ where $F_{\delta}$ is a local smoothing of $F$
as in Proposition 2 and we define the signed distance function to $\partial
F_{\delta}$ by

$d^{\pm}_{\partial F_{\delta}}=\begin{cases}+d_{\partial F_{\delta}}(x),\text{
if }x\in F_{\delta}\\\ 0,\text{ if }x\in\partial F_{\delta}\\\ -d_{\partial
F_{\delta}}(x),\text{ if }x\in N\setminus F_{\delta}\end{cases},$

where here $d_{\partial F_{\delta}}$ is the usual distance function to the set
$\partial F_{\delta}$. Note that the functions
$f_{\varepsilon,\delta}\in\mathcal{A}_{\varepsilon,r_{2}}(p)$ for
$\varepsilon>0$ small enough so that
$2\varepsilon\Lambda_{\varepsilon}<r_{2}-\tilde{r}_{2}$, where $\tilde{r}_{2}$
is chosen as in Proposition 2. To see this we note that as $F_{\delta}=E$
outside of $B_{\tilde{r}_{2}}(p)$ we have that $d_{\partial
F_{\delta}}=d_{\overline{M}}$ on the set $(N\setminus
B_{r_{2}}(p))\cap\\{d_{\partial F_{\delta}}\leq
2\varepsilon\Lambda_{\varepsilon}\\}$ and hence
$f_{\varepsilon,\delta}=v_{\varepsilon}$ on $N\setminus B_{r_{2}}(p)$.

By Remark 11 the hypersurfaces $\partial F_{\delta}$ satisfy the conclusions
of Lemma 1. Hence we may repeat the analysis for $\partial F_{\delta}$ (of the
cut locus, level sets etc.) as we did for $M$ in Subsection 3.3 in order to
conclude that
$\mathcal{F}_{\varepsilon,\lambda}(f_{\varepsilon,\delta})\rightarrow\mathcal{F}_{\lambda}(F_{\delta})$
as $\varepsilon\rightarrow 0$. By Proposition 2 we have that
$|\mathcal{F}_{\lambda}(F_{\delta})-\mathcal{F}_{\lambda}(F)|\leq(1+\lambda)\delta$
and thus by setting
$f_{\varepsilon}=f_{\varepsilon,\frac{\varepsilon}{1+\lambda}}$ we ensure that
this sub-sequence is such that
$\mathcal{F}_{\varepsilon,\lambda}(f_{\varepsilon})\rightarrow\mathcal{F}_{\lambda}(F)$
as desired.

Under the assumption that the functions $v_{\varepsilon}$ are such that
$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})+\tau_{\varepsilon}$,
for some sequence $\tau_{\varepsilon}\rightarrow 0$ as $\varepsilon\rightarrow
0$, and the fact that $f_{\varepsilon}\in\mathcal{A}_{\varepsilon,r_{2}}(p)$
we see that

$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})+\tau_{\varepsilon}\leq\mathcal{F}_{\varepsilon,\lambda}(f_{\varepsilon})+\tau_{\varepsilon},$

where here we are using that
$\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})=\inf_{u\in\mathcal{A}_{\varepsilon,r_{2}}(p)}\mathcal{F}_{\varepsilon,\lambda}(u)\leq\mathcal{F}_{\varepsilon,\lambda}(f_{\varepsilon})$
by construction. Hence, by (2) and the fact that
$\mathcal{F}_{\varepsilon,\lambda}(f_{\varepsilon})\rightarrow\mathcal{F}_{\lambda}(F)$
from the above, letting $\varepsilon\rightarrow 0$ we conclude that

$\mathcal{F}_{\lambda}(E)\leq\mathcal{F}_{\lambda}(F),$

so $E$ solves the Caccioppoli set minimisation above. If $E$ does not solve
the minimisation problem above then there exists a $\zeta>0$ such that

$\mathcal{F}_{\lambda}(E)>\mathcal{F}_{\lambda}(F)+\zeta,$

for any solution $F\in\mathcal{C}(N)$ as above. By (2), the fact that
$\mathcal{F}_{\varepsilon,\lambda}(f_{\varepsilon})\rightarrow\mathcal{F}_{\lambda}(F)$
and
$\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(f_{\varepsilon})$
we have

$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\geq\mathcal{F}_{\varepsilon,\lambda}(f_{\varepsilon})+\frac{\zeta}{2}\geq\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})+\frac{\zeta}{2},$

for all $\varepsilon>0$ sufficiently small. Thus there exists no sequence
$\tau_{\varepsilon}\rightarrow 0$ as $\varepsilon\rightarrow 0$ such that
$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})+\tau_{\varepsilon}$,
concluding the proof of the equivalence. ∎

######  Remark 13.

Lemma 3 combined with the results of [Mag12, Section 28.2] imply that if the
functions $v_{\varepsilon}$ are such that
$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})+\tau_{\varepsilon}$,
for some sequence $\tau_{\varepsilon}\rightarrow 0$ as $\varepsilon\rightarrow
0$, then any tangent cone to the hypersurface $M$ at a point of
$B_{r_{1}}(p)\cap\partial F$ is area-minimising (in the sense that the cone is
a perimeter minimiser in $\mathbb{R}^{n+1}$).

##  5 Construction of paths

In order to prove Theorem 3 we will work under the assumption that the
functions $v_{\varepsilon}=\overline{\mathbb{H}}^{\varepsilon}\circ
d^{\pm}_{\overline{M}}$, defined in Subsection 3.3, are such that
$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\geq\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})+\tau$
for some $\tau>0$, where the $g_{\varepsilon}$ are defined as in Lemma 2 in a
ball centred on an isolated singularity of $M$. We then exhibit a continuous
path in $W^{1,2}(N)$, from $a_{\varepsilon}$ to $b_{\varepsilon}$, with energy
along the path bounded a fixed amount below $\mathcal{F}_{\lambda}(E)$,
independent of $\varepsilon$. In this manner we will have exhibited, see
Section 6, an admissible path in the min-max procedure of [BW20a] which
contradicts the assumption that $M$ arose from this construction and allows us
to establish Theorem 3. In the following Subsections we construct the separate
pieces of our desired path, with maximum energy along these pieces bounded a
fixed amount below $\mathcal{F}_{\lambda}(E)$, independently of $\varepsilon$.

###  5.1 Paths with energy drop fixing a ball

We first construct the shifted functions described in Step 2 of Subsection 1.3
and compute upper energy bounds.

####  5.1.1 Choosing radii for local properties

We fix various radii, dependent only on the geometry of $M$ around an isolated
singularity, in order to later define functions for our path.

Let $p\in\mathrm{Sing}(M)$ be an isolated singularity of $M$, there then
exists an $R_{p}>0$ such that
$\overline{B_{R_{p}}(p)}\cap\mathrm{Sing}(M)=\\{p\\}$, i.e. such that
$M\cap\overline{B_{R_{p}}(p)}\setminus\\{p\\}$ is smooth.

The bounded mean curvature of $M$ (specifically as the monotonicity formula
holds at all points of $\overline{M}$) provides Euclidean volume growth,
namely that there exists two constants $C_{p},r_{p}>0$ both depending on the
point $p$ such that for all $r<r_{p}$ we have

$\mathcal{H}^{n}(M\cap B_{r_{p}}(p))\leq C_{p}r^{n}.$ (28)

We first fix $r_{0}\in(0,\min\\{r_{p},R_{p}\\})$ such that

$\mathcal{H}^{n}(M\setminus B_{r_{0}}(p))>\frac{3\mathcal{H}^{n}(M)}{4}.$ (29)

Set $\widehat{K}=M\cap(\overline{B_{R_{p}}(p)}\setminus B_{r_{0}}(p))$, which
is compact in $M$, so by Remark 8 we ensure that there exists a
$c_{\widehat{K}}>0$ such that
$F(\widehat{K}\times(-c_{\widehat{K}},c_{\widehat{K}}))\cap(\mathrm{Sing}(M)\cup\overline{S_{d_{\overline{M}}}})=\emptyset$.
Defining on $K\times(-c_{\widehat{K}},c_{\widehat{K}})\subset V_{M}$ the
function

$h(x,a)=\mathrm{dist}_{N}(F(\Pi_{V_{M}}(x,a)),p)$ (30)

we ensure that for $(x,a)\in K\times(-c_{\widehat{K}},c_{\widehat{K}})\subset
V_{M}$ we have, by Remark 9, that

$|\nabla h(x,a)|_{(x,a)}\leq C_{h},$ (31)

where here we are computing the gradient, $\nabla$, on $V_{M}$ with respect to
the pullback metric $F^{*}g$, for some constant $C_{h}>0$ dependent only on
$\widehat{K}$, $N$ and $g$.

We may now fix $r<\frac{1}{4}(\min\\{r_{p},R_{p}\\}-r_{0})$ sufficiently small
to ensure that on the annulus $\mathcal{A}=M\cap(B_{r_{0}+3r}(p)\setminus
B_{r_{0}+2r})$ we have

$\frac{\mathcal{H}^{n}(\mathcal{A})}{r^{2}}\leq\left(\frac{me^{-\frac{\lambda^{2}}{2m}}\mathcal{H}^{n}(M)}{8C^{2}_{h}}\right);$
(32)

where here we are using the Euclidean volume growth of the mass of $M$ (which
follows from the monotonicity formula) to ensure that
$\mathcal{H}^{n}(\mathcal{A})$ is of order $r^{n}$, combined with the
assumption that, from Remark 4, $n\geq 7$. The above choice of $r$ will become
clear when calculating the energy of the functions $v^{t,s}_{\varepsilon}$ in
Subsection 5.1.3.

We now define the balls $B_{i}=B_{r_{0}+ir}(p)$ for $i\in\\{1,2,3,4\\}$ which
are such that $B_{1}\subset\subset B_{2}\subset\subset B_{3}\subset\subset
B_{4}$; with this notation we have that
$\mathcal{A}=M\cap(\overline{B_{3}}\setminus B_{1})$.

####  5.1.2 Defining the shifted functions

We define, for $t\in[-t_{1},t_{1}]$ where $t_{1}>0$ is to be chosen and all
$\varepsilon>0$ sufficiently small, functions $v^{t,s}_{\varepsilon}\in
W^{1,2}(N)$ for $s\in[0,1]$ with the paths $t\in[-t_{1},t_{1}]\rightarrow
v^{t,s}_{\varepsilon}$ and $s\in[0,1]\rightarrow v^{t,s}_{\varepsilon}$
continuous in $W^{1,2}(N)$. The functions $v^{t,s}_{\varepsilon}$ will be
defined so that the following properties hold for all $t\in[-t_{1},t_{1}]$ and
$s\in[0,1]$

$\begin{cases}v^{t,1}_{\varepsilon}=\overline{\mathbb{H}}^{\varepsilon}\circ(d^{\pm}_{\overline{M}}-t)\text{
on }N\\\ v^{0,s}_{\varepsilon}=v_{\varepsilon}\text{ on }N\\\
v^{t,0}_{\varepsilon}=v_{\varepsilon}\text{ in }B_{r_{0}}(p)\subset
B_{1}\end{cases},$ (33)

and in such a way that, for some $E(\varepsilon)\rightarrow 0$ as
$\varepsilon\rightarrow 0$, we have

$\mathcal{F}_{\varepsilon,\lambda}(v^{t,s}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})+E(\varepsilon).$

Furthermore, we will show in Lemma 4 that there exists some $0<t_{0}<t_{1}$
and $\eta>0$ such that for all $s\in[0,1]$ we have

$\mathcal{F}_{\varepsilon,\lambda}(v^{\pm
t_{0},s}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\eta+E(\varepsilon).$

Finally, by Lemma 5, for $t>t_{0}$ we will have

$\mathcal{F}_{\varepsilon,\lambda}(v^{\pm
t}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v^{\pm
t_{0},1}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\eta+E(\varepsilon).$

These upper bounds on the energy along the paths provided by the functions
$v^{t,s}_{\varepsilon}$ will be calculated explicitly in Subsection 5.1.3.

Thus, assuming the above, and combined with the path provided in Subsection
5.3, which changes the functions $v^{t,0}_{\varepsilon}$ only in $B_{1}$ to
decrease their energy, we construct a path from $a_{\varepsilon}$ to
$b_{\varepsilon}$ in $W^{1,2}(N)$ whose energy along the path remains bounded
strictly below $\mathcal{F}_{\lambda}(E)$. We now proceed to define the
$v^{t,s}_{\varepsilon}$ explicitly.

Consider $B_{1}\subset\subset B_{2}\subset\subset B_{3}\subset\subset B_{4}$
centred at $p\in M$ as specified in Subsection 5.1.1. Let
$K=M\cap\overline{B_{4}}\setminus B_{1}$, which is compact in $M$, and fix
$c_{K}=c_{M\cap\overline{B_{4}}\setminus B_{1}}>0$ as in Remark 8; we then
have that $K\times(-c_{K},c_{K})\subset V_{M}$ is a two-sided tubular
neighbourhood of $M\cap\overline{B_{4}}\setminus B_{1}$ in $N$ under the map
$F$. Note that we have
$F\left(K\times(-c_{K},c_{K})\right)\cap(\mathrm{Sing}(M)\cup\overline{S_{d_{\overline{M}}}})=\emptyset$
by the choice of $c_{K}$ and as $K\subset\widehat{K}$ we ensure that
$c_{K}\leq c_{\widehat{K}}$ as in Subsection 5.1.1.

For $r>0$ as chosen we define the following function on $M$,

$f(y)=\begin{cases}1\text{ if }y\in M\setminus\overline{B_{3}}\\\ 0\text{ if
}y\in M\cap\overline{B_{2}}\\\ \frac{1}{r}(\mathrm{dist}_{N}(y,p)-r)\text{ if
}y\in\mathcal{A}.\end{cases}$ (34)

Then we have $0\leq f\leq 1$ and set

$f_{s}(x)=s+(1-s)f(x).$

Recall, from Subsection 3.2, the definition of the smooth projection, $\Pi$ to
$M$ defined on
$N\setminus(\mathrm{Sing}(M)\cup\overline{S_{d_{\overline{M}}}})$. Fix
$t_{1}>0$ and an $0<\varepsilon_{2}<\varepsilon_{1}$ to ensure that
$2\varepsilon\Lambda_{\varepsilon}+t_{1}<\min\\{c_{K},\frac{r}{2}\\}$, for all
$0<\varepsilon<\varepsilon_{2}$. We now define, for $t\in[-t_{1},t_{1}]$,
$s\in[0,1]$ and $0<\varepsilon<\varepsilon_{2}$, the functions

$v^{t,s}_{\varepsilon}(x)=\begin{cases}\overline{\mathbb{H}}^{\varepsilon}(d^{\pm}_{\overline{M}}(x)-tf_{s}(\Pi(x))))\text{
if }x\in F(K\times[-c_{K},c_{K}])\\\
\overline{\mathbb{H}}^{\varepsilon}(d^{\pm}_{\overline{M}}(x)-ts)\text{ if
}x\in B_{r_{0}+\frac{3r}{2}}(p)\\\
\overline{\mathbb{H}}^{\varepsilon}(d^{\pm}_{\overline{M}}(x)-t)\text{ if
}x\in N\setminus B_{r_{0}+\frac{7r}{2}}(p)\\\ 1\text{ if }x\in
E\cap\\{d^{\pm}_{\overline{M}}>2\varepsilon\Lambda_{\varepsilon}+t_{1}\\}\\\
-1\text{ if }x\in(N\setminus
E)\cap\\{d^{\pm}_{\overline{M}}<-2\varepsilon\Lambda_{\varepsilon}-t_{1}\\}\\\
\end{cases}.$

We now show that the $v^{t,s}_{\varepsilon}$ are well defined Lipschitz
functions on $N$ so that in particular, as
$F(\mathcal{A}\times[-2\varepsilon_{2}\Lambda_{\varepsilon_{2}}-t_{1},2\varepsilon_{2}\Lambda_{\varepsilon_{2}}+t_{1}])\subset\subset\overline{B_{r_{0}+\frac{7r}{2}}(p)}\setminus
B_{r_{0}+\frac{3r}{2}}(p)$ for
$2\varepsilon_{2}\Lambda_{\varepsilon_{2}}+t_{1}<\frac{r}{2}$, we have

$v^{t,s}_{\varepsilon}(x)=\overline{\mathbb{H}}^{\varepsilon}(d^{\pm}_{\overline{M}}(x)-tf_{s}(\Pi(x)))\text{
on }N\setminus(\mathrm{Sing}(M)\cup\overline{S_{d_{\overline{M}}}}).$

Note that if $x\in\overline{B_{r_{0}+\frac{7r}{2}}(p)}\setminus
B_{r_{0}+\frac{3r}{2}}(p)$ and $|d^{\pm}_{\overline{M}}|\leq
2\varepsilon\Lambda_{\varepsilon}+t_{1}$ then by the triangle inequality we
see that $x\in F(K\times[-c_{K},c_{K}])$. In
$F(K\times[2\varepsilon_{2}\Lambda_{\varepsilon_{2}}+t_{1},c_{K}])$ we have
that $v^{t,s}_{\varepsilon}\equiv 1$ and in
$F(K\times[-c_{K},-(2\varepsilon_{2}\Lambda_{\varepsilon_{2}}+t_{1})])$ we
have that $v^{t,s}_{\varepsilon}\equiv-1$. Combining these two facts we see
that the $v^{t,s}_{\varepsilon}$ are in fact Lipschitz functions in
$\overline{B_{r_{0}+\frac{7r}{2}}(p)}\setminus B_{r_{0}+\frac{3r}{2}}(p)$.

Noting that
$F(\mathcal{A}\times[-2\varepsilon_{2}\Lambda_{\varepsilon_{2}}-t_{1},2\varepsilon_{2}\Lambda_{\varepsilon_{2}}+t_{1}])\subset\subset\overline{B_{r_{0}+\frac{7r}{2}}(p)}\setminus
B_{r_{0}+\frac{3r}{2}}(p)$ as
$2\varepsilon_{2}\Lambda_{\varepsilon_{2}}+t_{1}<\frac{r}{2}$, we have by
definition of $f$ that if $x\in B_{r_{0}+\frac{3r}{2}}(p)\cap
F(K\times[-c_{K},c_{K}])$ then
$v^{t,s}_{\varepsilon}=\overline{\mathbb{H}}^{\varepsilon}(d^{\pm}_{\overline{M}}(x)-ts)$,
and if $x\in(N\setminus B_{r_{0}+\frac{7r}{2}}(p))\cap
F(K\times[-c_{K},c_{K}])$ then
$v^{t,s}_{\varepsilon}=\overline{\mathbb{H}}^{\varepsilon}(d^{\pm}_{\overline{M}}(x)-t)$.

Combining the above two paragraphs, we conclude that the
$v^{t,s}_{\varepsilon}$ are indeed well defined continuous functions on $N$.
Furthermore, as noted above, the $v^{t,s}_{\varepsilon}$ are Lipschitz in
$\overline{B_{r_{0}+\frac{7r}{2}}(p)}\setminus B_{r_{0}+\frac{3r}{2}}(p)$, and
in the sets $N\setminus B_{r_{0}+\frac{7r}{2}}(p)$ and
$B_{r_{0}+\frac{3r}{2}}(p)$ by the $v^{t,s}_{\varepsilon}$ are Lipschitz by
definiton (as on these sets
$v^{t,s}_{\varepsilon}=\overline{\mathbb{H}}^{\varepsilon}(d^{\pm}_{\overline{M}}(x)-t)$
and
$v^{t,s}_{\varepsilon}=\overline{\mathbb{H}}^{\varepsilon}(d^{\pm}_{\overline{M}}(x)-ts)$
respectively).

In conclusion we have that the $v^{t,s}_{\varepsilon}$ are continuous on $N$
and Lipschitz on $\overline{B_{r_{0}+\frac{7r}{2}}(p)}\setminus
B_{r_{0}+\frac{3r}{2}}(p)$, $N\setminus B_{r_{0}+\frac{7r}{2}}(p)$ and
$B_{r_{0}+\frac{3r}{2}}(p)$; hence we have that the $v^{t,s}_{\varepsilon}\in
W^{1,\infty}(N)\subset W^{1,2}(N)$. Note also that by the definition of the
$v^{t.s}_{\varepsilon}$ we have that (33) holds as desired.

####  5.1.3 Energy calculations and continuity of paths

We calculate an upper bound on the energy of the functions
$v^{t,s}_{\varepsilon}$ for $t\in[t_{0},t_{0}]$, for some $0<t_{0}\leq t_{1}$,
all $s\in[0,1]$ and $\varepsilon>0$ sufficiently small. The calculation method
here is similar to those in [BW22, Sections 4, 6.1 and 7.1].

######  Lemma 4.

There exists $t_{0}\in(0,t_{1})$ such that for all $s\in[0,1]$ and
$t\in[-t_{0},t_{0}]$ we have that

$\mathcal{F}_{\varepsilon,\lambda}(v^{t,s}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})+E(\varepsilon)$

where $E(\varepsilon)\rightarrow 0$ as $\varepsilon\rightarrow 0$.
Furthermore, we have that, for all $\varepsilon>0$ sufficiently small there
exists

$\eta=\frac{m\sigma}{8}\mathcal{H}^{n}(M)t_{0}^{2}>0$

such that for all $s\in[0,1]$ we have

$\mathcal{F}_{\varepsilon,\lambda}(v^{\pm
t_{0},s}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\eta.$

###### Proof.

We define, for $(x,a)\in V_{M}$,
$\hat{v}^{t,s}_{\varepsilon}((x,a))=v^{t,s}_{\varepsilon}(F(x,a))$ so that, as
$\Pi\circ F=F\circ\Pi_{V_{M}}$, we have that

$\hat{v}^{t,s}_{\varepsilon}(x,a)=\overline{\mathbb{H}}^{\varepsilon}(a-tf_{s}(F(\Pi_{V_{M}}(x,a))))\text{
on }V_{M}.$ (35)

Note that by (30) and (34) we have for points $(x,a)\in V_{M}$ with
$x\in\mathcal{A}$ that

$\nabla\big{(}f_{s}(F(\Pi_{V_{M}}(x,a)))\big{)}=\frac{1-s}{r}\nabla h(x,a),$

and if $x\in M\setminus\mathcal{A}$ then $f$ is constant and so

$\nabla\big{(}f_{s}(F(\Pi_{V_{M}}(x,a)))\big{)}=0.$

We then compute by the generalised Gauss’ Lemma, [Gra04, Chapter 2.4], that we
have

$|\nabla\hat{v}^{t,s}_{\varepsilon}(x,a)|_{(x,a)}^{2}=\left((\overline{\mathbb{H}}^{\varepsilon})^{\prime}(a-tf_{s}(x)\right)^{2}\left(1+\frac{(1-s)^{2}t^{2}}{r^{2}}|\nabla
h(x,a)|_{(x,a)}^{2}\chi_{\mathcal{A}\times\mathbb{R}}(x,a)\right),$ (36)

where here $\chi_{\mathcal{A}\times\mathbb{R}}$ is the indicator function for
the set $\mathcal{A}\times\mathbb{R}$ on $V_{M}$.

Using the co-area formula (slicing with $a$) and Fubini’s Theorem we have

$\displaystyle
2\sigma\mathcal{F}_{\varepsilon,\lambda}(v^{t,s}_{\varepsilon})$
$\displaystyle=\int_{V_{M}}e_{\varepsilon}(\hat{v}^{t,s}_{\varepsilon})-\sigma\lambda\hat{v}^{t,s}_{\varepsilon}d\mathcal{H}^{n+1}_{F^{*}g}$
$\displaystyle=\int_{V_{M}}\frac{\varepsilon}{2}|\nabla\hat{v}^{t,s}_{\varepsilon}|^{2}+\frac{W(\hat{v}^{t,s}_{\varepsilon})}{\varepsilon}-\sigma\lambda\hat{v}^{t,s}_{\varepsilon}d\mathcal{H}^{n+1}_{F^{*}g}(x,a)$
$\displaystyle=\int_{V_{M}}\frac{\varepsilon}{2}\left((\overline{\mathbb{H}}^{\varepsilon})^{\prime}(a-tf_{s}(x))\right)^{2}+\frac{W(\overline{\mathbb{H}}^{\varepsilon}(a-tf_{s}(x))}{\varepsilon}-\sigma\lambda\overline{\mathbb{H}}^{\varepsilon}(a-tf_{s}(x))d\mathcal{H}^{n+1}_{F^{*}g}(x,a)$
$\displaystyle\quad\quad+\int_{V_{M}}\frac{\varepsilon}{2}\left((\overline{\mathbb{H}}^{\varepsilon})^{\prime}(a-tf_{s}(x))\right)^{2}\frac{(1-s)^{2}t^{2}}{r^{2}}|\nabla
h(x,a)|_{(x,a)}^{2}\chi_{\mathcal{A}\times\mathbb{R}}(x,a)d\mathcal{H}^{n+1}_{F^{*}g}(x,a)$
$\displaystyle=\int_{M}\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\left[e_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(a-tf_{s}(x))\right)-\sigma\lambda\overline{\mathbb{H}}^{\varepsilon}(a-tf_{s}(x))\right]\theta(x,a)da\,d\mathcal{H}^{n}(x)$