[title=Index of Notation]

Ars Inveniendi Analytica (2022), Paper No. 2, 27 pp.

DOI 10.15781/j4aj-kd66

ISSN: 2769-8505

00footnotetext: © Otis Chodosh, Yevgeny Liokumovich, and Luca
Spolaor00footnotetext:  Licensed under a Creative Commons Attribution License
(CC-BY).

Singular behavior and generic regularity of min-max minimal hypersurfaces

Otis Chodosh

Stanford University

Yevgeny Liokumovich

University of Toronto

Luca Spolaor

UC San Diego

Communicated by Francesco Maggi

Abstract. We show that for a generic $8$-dimensional Riemannian manifold with
positive Ricci curvature, there exists a smooth minimal hypersurface. Without
the curvature condition, we show that for a dense set of $8$-dimensional
Riemannian metrics there exists a minimal hypersurface with at most one
singular point. This extends previous work on generic regularity that only
dealt with area-minimizing hypersurfaces.

These results are a consequence of a more general estimate for a one-parameter
min-max minimal hypersurface $\Sigma\subset(M,g)$ (valid in any dimension):

$\mathcal{H}^{0}(\mathcal{S}_{\textnormal{nm}}(\Sigma))+\operatorname{Index}(\Sigma)\leqslant
1$

where $\mathcal{S}_{\textnormal{nm}}(\Sigma)$ denotes the set of singular
points of $\Sigma$ with a unique tangent cone non-area minimizing on either
side.

Keywords. Minimal surfaces, generic regularity, min-max.

## Introduction

It is well known that $7$-dimensional area minimizing hypersurfaces can have
isolated singularities. Using work of Hardt–Simon [HS85], Smale proved in
[Sma93] that in an $8$-dimensional manifold $M$ with
$H_{7}(M;\varmathbb{Z})\neq 0$, there exists a smooth embedded area minimizing
hypersurface for a generic choice of metric. In other words, he showed that
isolated singularities of an area-minimizing $7$-dimensional hypersurface can
generically be perturbed away.

One may thus seek to find a smooth embedded minimal hypersurface in all
$8$-manifolds $M$ equipped with a generic metric $g$, even when
$H_{7}(M;\varmathbb{Z})=0$. Here, we find such a hypersurface in the case of
positive Ricci curvature, and give a partial answer in general. We let
$\operatorname{Met}^{2,\alpha}(M)$ denote the space of Riemannian metrics of
regularity $C^{2,\alpha}$ on $M$ and
$\operatorname{Met}^{2,\alpha}_{\operatorname{Ric}>0}(M)\subset\operatorname{Met}^{2,\alpha}(M)$
denote the open subset of Ricci positive metrics.

######  Theorem 0.1 (Generic regularity with positive Ricci in dimension $8$).

Let $M^{8}$ be a compact smooth $8$-manifold. There is an open and dense set
$\mathcal{G}\subset\operatorname{Met}_{\operatorname{Ric}>0}^{2,\alpha}(M)$ so
that for $g\in\mathcal{G}$, there exists a smooth embedded minimal
hypersurface $\Sigma\subset M$.

Without the curvature condition, we have the following partial result.

######  Theorem 0.2 (Generic almost regularity in dimension $8$).

Let $M^{8}$ be a compact smooth $8$-manifold. There exists a dense set
$\mathcal{G}\subset\operatorname{Met}^{2,\alpha}(M)$ so that for
$g\in\mathcal{G}$, there exists a smooth embedded minimal hypersurface
$\Sigma\subset M$ with at most one singular point.

We actually prove more general results valid in all dimensions, see Theorem
0.3 below.

As mentioned above, the principal motivation for such results is to study
generic regularity of non-minimizing, high-dimensional minimal submanifolds.
This contrasts with previous works on generic regularity:

  * •

Hardt–Simon [HS85] (resp. Smale [Sma93]), cf. [Liu19], show that regular
singularities of (one-sided) minimizing hypersurfaces can be perturbed away by
perturbing the boundary (resp. metric).

  * •

White [Whi85, Whi19] shows that minimizing integral 2-cycles are smoothly
embedded surfaces for a generic metric.

  * •

Moore [Moo06, Moo17] shows that parametrized minimal (2-dimensional) surfaces
are free of branch points for a generic ambient metric.

In fact, our work proves that generically there exists a minimal hypersurface
of optimal regularity avoiding _certain_ singularities in ambient dimensions
beyond the singular dimension. Indeed, Theorem 0.2 is a consequence of a more
general result stated below.

######  Theorem 0.3 (Generic removability of isolated singularities).

Consider a compact smooth $(n+1)$-manifold, for $n\geqslant 7$. There is a
dense set $\mathcal{G}\subset\operatorname{Met}^{2,\alpha}(M)$ with the
following properties:

  * •

If $g\in\mathcal{G}$ then there exists a minimal hypersurface $\Sigma$, smooth
away from a closed singular set of Hausdorff dimension at most $n-7$, so that
for $\mathcal{S}_{0}\subset\operatorname{Sing}(\Sigma)$ the set of singular
points with regular tangent cones, we have $\mathcal{H}^{0}(S_{0})\leqslant
1$.

  * •

If
$g\in\mathcal{G}\cap\operatorname{Met}^{2,\alpha}_{\operatorname{Ric}>0}(M)$
then the same statement holds, except we can conclude that
$\mathcal{H}^{0}(\mathcal{S}_{0})=0$.

In order to remove the topological condition $H_{7}(M;\varmathbb{Z})\neq 0$ of
Smale, we will use the Almgren–Pitts min-max construction [Pit81], which
guarantees the existence of a minimal hypersurface $\Sigma^{n}$ in a closed
Riemannian manifold $(M^{n+1},g)$. As in the area-minimizing case, when the
dimension $n$ satisfies $2\leqslant n\leqslant 6$, the Almgren–Pitts minimal
hypersurface is smooth, but for larger values of $n$ there may be an at most
$(n-7)$-dimensional singular set (this follows from work of Schoen–Simon
[SS81]). However tangent cones to min-max hypersurfaces are _a priori_ only
stable, while only area-minimizing cones have complements that are foliated by
smooth minimal hypersurfaces (cf. [BDGG69, Law91]) and it seems that such a
foliation is needed (at least on one side) to perturb the singularity away by
adjusting the metric [HS85].

The key technical result of this paper is that (for one-parameter min-max) at
all points—except possibly one—of the singular set with a regular tangent
cone, the tangent cone is area minimizing on at least one side. Put another
way, we show that tangent cones that are not area minimizing on either side
“contribute to the Morse index” from the point of view of min-max (and these
are precisely the cones that we are unable to perturb away using Hardt–Simon
[HS85]).

###  0.1. Detailed description of results

Let $(M^{n+1},g)$ be a closed Riemannian manifold. By a _sweepout_ of $M$ we
will mean a family of (possibly singular) hypersurfaces
$\\{\Phi(x)=\partial\Omega(x)\\}_{x\in[0,1]}$, where each hypersurface
$\Phi(x)$ is the boundary of an open set $\Omega(x)$ with
$\Omega(0)=\emptyset$ and $\Omega(1)=M$, and we denote the family of such
sweepouts by $\mathcal{S}$ (see Section 2 for the precise definition). The
width, $W(M)$, is then defined by

$W(M)=\inf_{\Phi\in\mathcal{S}}\left\\{\sup_{x}\mathbf{M}(\Phi(x))\right\\}\,.$

Given a stationary integral varifold $V$, with $\operatorname{supp}V$ regular
outside of a set of $n-7$ Hausdorff dimension, we define

$\mathfrak{h}_{\textnormal{nm}}(V):=\left\\{p\in\operatorname{supp}(V)\,:\,\begin{gathered}\textrm{for
all $r>0$ small, $\operatorname{supp}V\cap B_{r}(p)$ is not one-sided}\\\
\text{homotopy area minimizing on either side (in
$B_{r}(p)$).}\end{gathered}\right\\}\,$

In other words, $p\in\mathfrak{h}_{\textrm{nm}}(V)$ implies that in any small
ball there are one-sided homotopies on both sides of $\operatorname{supp}V$
that strictly decrease area without ever increasing area. Let $\mathcal{R}$
denote the set of integral varifolds, whose support is a complete embedded
minimal hypersurface regular away from a closed singular set of Hausdorff
dimension $n-7$. Finally, we let $\operatorname{Index}(V)$ denote the Morse
index of the regular part of the support of $V$, that is

$\operatorname{Index}(V)=\operatorname{Index}(\operatorname{supp}(\operatorname{Reg}(V)))\,.$

Then the main technical estimate of this paper is the following result.

######  Theorem 0.4 (Index plus non-area minimizing singularities bound).

For $n\geqslant 7$, let $(M^{n+1},g)$ be a closed Riemannian manifold of class
$C^{2}$. There exists a stationary integral varifold $V\in\mathcal{R}$ such
that $|V|(M)=W$, which satisfies

(0.1)
$\mathcal{H}^{0}(\mathfrak{h}_{\textnormal{nm}}(V))+\operatorname{Index}(V)\leqslant
1\,.$

If equality holds in (0.1), then for any point
$p\in\operatorname{supp}V\setminus\mathfrak{h}_{\textnormal{nm}}(V)$ there is
$\varepsilon>0$ so that $\operatorname{supp}V$ is area-minimizing to one side
in $B_{\varepsilon}(p)$. Finally, we can write
$V=\sum_{i}\kappa_{i}\,|\Sigma_{i}|$, where $\Sigma_{i}$ are finitely many
disjoint embeddded minimal hypersursufaces smooth away from finitely many
points with $\kappa_{i}\leqslant 2$ for every $i$; if $\Sigma_{i}$ is one-
sided then $\kappa_{i}=2$ and if $\kappa_{j}=2$ for some $j$ then each
$\Sigma_{i}$ is stable.

The above bound is valid in all dimensions and can be seen as a generalization
of the work of Calabi–Cao concerning min-max on surfaces [CC92]. Indeed if we
define $\mathcal{S}_{\textnormal{nm}}(V)$ by111Here $\omega$ is a modulus of
continuity, and we could take it to be logarithmic, as suggested by the work
of [Sim83a]. Notice in fact that at all isolated singularities
$\mathcal{S}_{0}$, minimal surfaces have unique tangent cone and are locally
$C^{1,\log}$ deformation of the cone itself.

$\mathcal{S}_{\textnormal{nm}}(V):=\left\\{p\in\operatorname{supp}(V)\,:\,\begin{gathered}V\text{
is locally a $C^{1,\omega}$ graph over its \emph{unique} tangent cone
$\mathcal{C}$}\\\ \text{at $p$ and both sides of $\mathcal{C}$ are not one-
sided minimizing}\end{gathered}\right\\}\,$

then we will see that
$\mathcal{S}_{\textnormal{nm}}(V)\subset\mathfrak{h}_{\textrm{nm}}(V)$ in
Lemma 3.4. In particular, (0.1) implies that

$\mathcal{H}^{0}(\mathcal{S}_{\textnormal{nm}}(V))+\operatorname{Index}(V)\leqslant
1.$

Thus, if we are guaranteed to have $\operatorname{Index}(V)=1$ (e.g., in
positive curvature) we see that $\mathcal{S}_{\textnormal{nm}}(V)=\emptyset$.
This is precisely the higher dimensional analogue of the result of Calabi–Cao
(cf. Figure 1 and the remark below).

See also the more recent work of Mantoulidis [Man17] which makes a more
explicit connection with Morse index, using the Allen–Cahn approach (as
developed by Guaraco and Gaspar [Gua18, GG19]) rather than Almgren–Pitts; it
would be interesting to elucidate the relationship between Mantoulidis’s
Allen–Cahn techniques and our proof of Theorem 0.4.

Figure 1. The figure eight geodesic $c$ is an example of a min-max closed
geodesic that is stable and has one singularity with non-area minimizing
tangent cone.

######  Remark 0.5.

By the index bound in Theorem 0.4, any tangent cone to $V$ has stable regular
part. Moreover, we note that the Simons cones [Sim68] in $\varmathbb{R}^{8}$
(formed from products of two spheres) are all stable and area minimizing on
(at least) one side (cf. [Law91]). We particularly emphasize that the Simons
cone

$\mathbf{C}^{1,5}:=\\{(x,y)\in\varmathbb{R}^{2}\times\varmathbb{R}^{6}:5|x|^{2}=|y|^{2}\\}$

is one-sided minimizing (and stable), but is not minimizing on the other side.
It seems to be an open question whether or not there exists an $n$-dimensional
stable cone that does not minimize area on either side, for $n\geqslant 7$.

Even assuming the existence of a stable minimal cone which is not area
minimizing on either sides, it is hard to decide if the above bound is
optimal. In dimension $n=1$, such an example is provided by the classical
starfish example (cf. Figure 1), whose tangent cone at the singular point (the
union of two lines through the origin) is indeed stable non-area minimizing on
either sides (and the starfish fails to be one-sided homotopy minimizing on
either side).

We conjecture that if there is a regular stable minimal cone that is not area-
minimizing on either side, then it can arise as the tangent cone to a min-max
minimal hypersurface (possibly in a manifold geometrically similar to the
starfish); note that were this to occur, Theorem 0.4 would imply that the
resulting hypersurface would necessarily be stable.

Theorem 0.4 generalizes the index upper bound of Marques and Neves [MN16] for
Riemannian manifolds $M^{n+1}$, $3\leqslant n+1\leqslant 7$ (see also
[Zho15]). In recent years there has been tremendous progress in the
understanding of the geometry of minimal hypersurfaces constructed using min-
max methods in these dimensions (see [DLT13], [MN19], [CM20], [Zho19], [Son18]
and references therein).

For manifolds of dimension $n+1\geqslant 8$ much less is known. When Ricci
curvature is positive Zhou obtained index and multiplicity bounds for one-
parameter min-max minimal hypersurface [Zho17] (see also the work of Ramírez-
Luna [RL19] and Bellettini [Bel20]). Upper Morse index bounds are known to
hold in arbitrary manifolds of any dimensions for hypersurfaces constructed by
Allen–Cahn, as proven by Hiesmayr and Gaspar [Hie18, Gas20] (see also the
recent work of Dey showing that the Almgren–Pitts and Allen–Cahn approaches
are equivalent [Dey20]). Li proved [Li19] existence of infinitely many
distinct minimal hypersurfaces constructed via min-max methods for a generic
set of metrics, using the Weyl law of Liokumovich–Marques–Neves [LMN18].

###  0.2. Overview of the proof

The construction of a minimal hypersurface in Almgren-Pitts min-max theory
proceeds by considering a sequence of sweepouts $\\{\Phi_{i}(x)\\}$ with the
supremum of the mass $\sup_{x}\mathbf{M}(\Phi_{i}(x))\rightarrow W(M)$ as
$i\rightarrow\infty$. It is then proved that we can find a subsequence
$\\{i_{k}\\}$ and $\\{\Phi_{i_{k}}(x_{k})\\}$ with mass tending to $W$, so
that $|\Phi_{i_{k}}|(x_{k})$ converges to some $V\in\mathcal{R}$.

We outline the proof of Theorem 0.4. For the sake of simplicity, let’s focus
on the non-cancellation case, i.e., when all multiplicities of $V$ are one (in
the case of cancellation we must argue slightly differently but the main
strategy is the same). The main geometric idea is to show that there cannot be
two disjoint open sets $U_{1},U_{2}$ so that $\Sigma=\operatorname{supp}V$
fails to be one-sided homotopy minimizing on the same side in both $U_{1}$ and
$U_{2}$. This property is reminiscent of (but different from) almost
minimizing property introduced by Pitts to prove regularity of min-max minimal
hypersurfaces.

Granted this fact, it is easy to deduce the bound (0.1). For example, if
$\operatorname{Index}(\Sigma)=1$ and
$\operatorname{\mathfrak{h}_{\textnormal{nm}}}(\Sigma)=\\{p\\}$, then we can
localize the index in some $U$ disjoint from $p$. Because $\Sigma$ is unstable
in $U$, we can find area decreasing homotopies to both sides there, and we can
also find $B_{r}(p)$ disjoint from $U$ with area decreasing homotopies (by
definition). This contradicts the above fact.

As such, we want to show the one-sided homotopy minimizing property in pairs
by using the fact that $V$ is a min-max minimal hypersurface. However, this
leads us to a major difficulty. Indeed, the approximating currents
$\Phi_{i_{k}}(x_{k})$ might cross $\Sigma$ many times, making it difficult to
glue in one-sided homotopies to push down the mass.

At a technical level, the main tool used in this paper is that it is possible
to simplify the one-parameter case of min-max theory by constructing a nested
optimal sweepout $\Phi(x)$ with $\sup\mathbf{M}(\Phi(x))=W$. This allows us to
work with one sweepout $\Phi(x)$ instead of a sequence of sweepouts. The
nested property allows us to directly “glue in” the one-sided homotopies to
push down the mass.

The existence of a nested optimal sweepout follows from a monotonization
technique from [CL20]. There Chambers and Liokumovich proved that each
sweepout $\Phi_{i}(x)$ can be replaced by a nested sweepout $\Psi_{i}(x)$ with
$\sup\mathbf{M}(\Psi_{i}(x))\leqslant\sup\mathbf{M}(\Psi_{i}(x))+\frac{1}{i}$.
“Nested” here means that $\Psi_{i}(x)=\partial\Omega(x)$ for a family of open
sets with $\Omega(x)\subset\Omega(y)$ if $x<y$. The proof used ideas of
Chambers and Rotman [CR18] on existence of monotone homotopies of closed
curves on surfaces.

After we reparametrize $\Psi_{i}(x)$ by the volume swept out we obtain a
sequence of families that is uniformly Lipschitz in flat topology. By
Arzelà–Ascoli a subsequence will converge to an optimal sweepout.

In the Almgren–Pitts theory, a “pull-tight” procedure is used to find a
varifold achieving the width with good properties. We can apply this procedure
to our sweepout to deduce that one of the critical varifolds in the sweepout
is a smooth (up to small singular set) minimal hypersurface. We would then
like to prove that this hypersurface satisfies (0.1). However, this poses
another issue, namely that it could a priori be possible to push the mass of
the sweepout near this critical value down, while not decreasing the global
width. This could lead to an infinite sequence of “pushing down” operations
that could create extra critical points not dealt with previously.

As such, our second main technical tool is that we construct an optimal
sweepout $\Phi(x)$ with the following special property. For every point
$x_{0}$ with $\sup_{x\in U}\mathbf{M}(\Phi(x))=W$ for every neighbourhood $U$
of $x_{0}$, there _does not_ exist an open interval $I\ni x_{0}$ and a family
$\\{\Phi^{\prime}(x)\\}$ that coincides with $\Phi$ for $x\notin I$ and
satisfying $\sup_{x\in I^{\prime}}\mathbf{M}(\Phi^{\prime}(x))<W$ for every
closed interval $I^{\prime}\subset I$. (In fact, we will prove a somewhat
stronger property that holds for open, half-open and closed intervals $I$). In
other words, we can not make a small “dip” in the graph of $\mathbf{M}(x)$,
pushing it below $W$ in the neighbourhood of $x_{0}$.

This property of $\Phi$ allows us to easily prove the “homotopy minimizing to
one-side” property of $V$ discussed above (and thus Theorem 0.4). A diagram of
the procedure to prove this can be found in Figure 2.

$\Rightarrow$$\Rightarrow$ Figure 2. A diagram of how to use two disjoint
regions that are not one-sided homotopy minimizing to either side to push down
the mass. The “bumps” have smaller mass than the original surface and are
homotopic along one-sided homotopies that never increase mass. By doing one
homotopy fully and then the other, we can see that the mass is always strictly
decreased in this range. This can then be glued into a nested sweepout without
increasing mass of the adjacent leaves.

It thus remains to explain how we perturb remaining singularities. Hardt-Simon
(cf. [HS85]) proved that one sided perturbations of area minimizing cones with
isolated singularities are smooth, and Liu extended this result to one sided
stationary minimizers (cf. [Liu19]). The proofs of the bounds in Theorems 0.1,
0.2, and 0.3 are then obtained by combining a weak generalization of White
[Whi94], where “stability” is replaced by the stronger hypothesis that the
surface is homotopic minimizing to one side, with a simple surgery procedure
in an annulus around the singularity, to show that singular points with
regular tangent cones in
$\operatorname{Sing}(V)\setminus\operatorname{\mathfrak{h}_{\textnormal{nm}}}(V)$
are not generic. As such, the main results follow from (0.1) when combined
with a result of Simon on uniqueness of the blow up at certain singularities
[Sim83a]. The rest of the theorem follows from the index lower bound in
manifolds with positive Ricci curvature.

###  0.3. Organization of the paper

The paper is divided into four sections. The first section contains the basic
definitions and some useful geometric tools. The second section introduces the
notion of non-excessive optimal sweepouts, which is a key idea in the present
work. The third section proves Theorem 0.4, while the last section is
dedicated to the proofs of Theorems 0.2 and 0.3.

###  0.4. Remark

Since the first version of this paper appeared, Li–Wang have shown in a
remarkable work [LW20] that the results of this paper can be combined with
[Wan20] and some further geometric arguments to prove a version of Theorem 0.1
for generic metrics without any curvature assumption.

###  0.5. Acknowledgements

We are grateful to Zhihan Wang for pointing out an error in the original
version of this paper and for pointing out an interesting alternative approach
to prove Theorem 0.1. We are also grateful to Zhichao Wang for carefully
reading an earlier version of this paper and making several helpful
suggestions. We would like to thank the anonymous referees for numerous
corrections.

O.C. was partially supported by a Terman Fellowship, a Sloan Fellowship, and
an NSF grant DMS-1811059/2016403. Y. L. was partially supported by NSERC
Discovery grant. L. S. acknowledges the support of the NSF grant
DMS-$1951070$.

##  1\. Geometric preliminaries

In this section we introduce the main notations, we prove the existence of
optimal nested sweepouts and we recall some useful geometric tools.

###  1.1. Notations

Let $(M^{n+1},g)$ be a closed Riemannian manifold. By scaling, it suffices to
consider $\operatorname{Vol}(M,g)=1$, which we will always assume below. We
use $\mathcal{Z}_{n}(M;\varmathbb{Z}_{2})$ to denote the space of mod 2 flat
cycles in $M$. The topology on the space
$\mathcal{Z}_{n}(M;\varmathbb{Z}_{2})$ is induced by the usual flat norm
$\mathcal{F}$.

We will make extensive use of the notion of _Caccioppoli set_. A measurable
set $E\subset M$ is Caccioppoli if

$\operatorname{Per}(E):=\sup\left\\{\int_{M}\chi_{E}\,{\rm
div}\omega\,:\,\omega\in X(M)\,,\,\,\|\omega\|_{C^{0}}\leqslant
1\right\\}<\infty\,,$

where $\chi_{E}$ denotes the indicator function of $E$. By De Giorgi’s
strutcure theorem we have that the distributional derivative $D\chi_{E}$
(which is is a Radon measure) of a set of finite perimeter $E$ is given by
$D\chi_{E}=\nu_{E}\,\mathcal{H}^{n}\mathop{\hbox{\vrule
height=7.0pt,width=0.3pt,depth=0.0pt\vrule
height=0.3pt,width=5.0pt,depth=0.0pt}}\nolimits\partial^{*}E$, where
$\partial^{*}E$ is the reduced boundary of $E$, which is a $n$-rectifiable
set, and $\nu_{E}$ is the normal direction to $\partial^{*}E$ pointing outside
$E$ defined $\mathcal{H}^{n}$-a.e.. This allows us to identify $D\chi_{E}$
with an element of $\mathcal{Z}_{n}(M;\varmathbb{Z}_{2})$, which we will abuse
notation and denote by

$\partial E:=\nu_{E}\,\mathcal{H}^{n}\mathop{\hbox{\vrule
height=7.0pt,width=0.3pt,depth=0.0pt\vrule
height=0.3pt,width=5.0pt,depth=0.0pt}}\nolimits\partial^{*}E\,.$

In particular, with this identification we have

$\mathbf{M}(\partial
E)=\operatorname{Per}(E)\qquad\text{and}\qquad\mathcal{F}(\partial E,\partial
F)=\|\chi_{E}-\chi_{F}\|_{L^{1}}=\operatorname{Vol}(E\Delta F)\,,$

where $E\Delta F$ denotes the symmetric difference between two sets. As usual,
the perimeter of $E$ in an open set $U$, denoted by
$\operatorname{Per}(E\,|\,U)$, is the total variation of $D\chi_{E}$ in the
set $U$.

We let $\mathcal{V}$ denote the set of varifolds of $(M^{n+1},g)$. Given a
cycle $\Gamma\in\mathcal{Z}_{n}(M;\varmathbb{Z}_{2})$, we will write
$|\Gamma|$ for the associated integral varifold. In particular if
$\Gamma=\partial\Omega$, then

$|\Gamma|=\mathcal{H}^{n}\mathop{\hbox{\vrule
height=7.0pt,width=0.3pt,depth=0.0pt\vrule
height=0.3pt,width=5.0pt,depth=0.0pt}}\nolimits\partial^{*}\Omega\,\otimes\,\delta_{T\partial^{*}\Omega}\qquad\text{and}\qquad\mu_{\Gamma}:=\mathcal{H}^{n}\mathop{\hbox{\vrule
height=7.0pt,width=0.3pt,depth=0.0pt\vrule
height=0.3pt,width=5.0pt,depth=0.0pt}}\nolimits\partial^{*}\Omega$

is the total variation measure of the measure $D\chi_{\Omega}$. Finally, given
a set $\Sigma$ regular outside a set of dimension $n-7$, we will denote with
$|\Sigma|$ the associated integral varifold.

###  1.2. Optimal nested sweepouts

We start by recalling the notion of sweepouts that we will use in this paper.

######  Definition 1.1 (Sweepout).

A sweepout of $M$ is a map
$\Phi:[0,1]\rightarrow\mathcal{Z}_{n}(M;\varmathbb{Z}_{2})$ continuous in
$\mathcal{F}$-topology such that $\Phi(x)=\partial\Omega(x)$, where
$\\{\Omega(x)\,:\,x\in[0,1]\\}$ is a family of Caccioppoli sets with
$\Omega(0)$ the $0$-cycle and $\Omega(1)=M$. We will denote with $\mathcal{S}$
the collection of all such sweepouts. Moreover, we define the width $W$ to be

$W=\inf_{\Phi\in\mathcal{S}}\,\sup_{x\in[0,1]}\mathbf{M}(\Phi(x))\,.$

It is a consequence of the isoperimetric inequality that $W>0$ [DLT13,
Proposition 0.5].

We will switch freely between the equivalent notation $\mathbf{M}(\Phi(x))$
and $\operatorname{Per}(\Omega(x))$. We now introduce the notion of optimal
nested sweepouts and prove their existence.

######  Definition 1.2 (Optimal nested volume parametrized (ONVP) sweepout).

A sweepout $\\{\Phi(x)=\partial\Omega(x)\,:\,x\in[0,1]\\}$ is called

  * •

optimal if $\sup_{x\in[0,1]}\mathbf{M}(\Phi(x))=W$;

  * •

nested if $\Omega(x_{1})\subset\Omega(x_{2})$, for all $0\leqslant
x_{1}\leqslant x_{2}\leqslant 1$;

  * •

volume parametrized if $\operatorname{Vol}(\Omega(x))=x$, for every
$x\in[0,1]$ (recall that we have assumed $\operatorname{Vol}(M,g)=1$).

Nested volume parametrized sweepouts enjoy nice compactness properties.

######  Lemma 1.3 (Compactness for nested volume parametrized sweepouts).

Let $\left(\Phi_{i}\right)_{i}$ be a sequence of nested volume-parametrized
sweepouts with mass uniformly bounded, that is

(1.1) $\sup_{i\in\varmathbb N}\sup_{x\in[0,1]}\mathbf{M}(\Phi_{i}(x))\leqslant
M<\infty\,.$

Then there exists a subsequence $\left(\Phi_{i_{k}}\right)_{k}$ converging
uniformly to a nested volume parametrized sweepout $\Psi$ such that

(1.2)
$\sup_{x}\mathbf{M}(\Psi(x))\leqslant\liminf_{k}\left(\sup_{x}\mathbf{M}(\Phi_{i_{k}}(x))\right).$

###### Proof.

The sequence of continuous functions
$\Phi_{i}\colon[0,1]\to\mathcal{Z}_{n-1}(M;\varmathbb{Z}_{2})$ is uniformly
Lispchitz continuous, since for every $0\leqslant x<y\leqslant 1$ we have

$\mathcal{F}(\Phi_{i}(x),\Phi_{i}(y))\leqslant\operatorname{Vol}(\Omega_{i}(y)\setminus\Omega_{i}(x))=\operatorname{Vol}(\Omega_{i}(y))-\operatorname{Vol}(\Omega_{i}(x))=y-x$

and $\Phi_{i}(0)=\emptyset$ for every $i$, so by Arzelà–Ascoli Theorem there
exists a subsequence $\Phi_{i_{k}}$ and a nested volume parametrized sweepout
$\Psi\colon[0,1]\to\mathcal{Z}_{n}(M;\varmathbb{Z}_{2})$ such that
$\Phi_{i_{k}}$ converges uniformly to $\Psi$. Then (1.2) follows from (1.1)
and the lower semi-continuity of $\mathbf{M}$ with respect to the flat
topology. ∎

Optimal nested volume parametrized sweepouts exist.

######  Theorem 1.4 (Existence of (ONVP) sweepouts).

For any closed Riemannian manifold $(M,g)$ there exists an optimal nested
volume-parametrized sweepout.

###### Proof.

Let $\\{\Psi_{i}\\}_{i}$ be a min-max sequence of sweepouts with
$\lim_{i\rightarrow\infty}\sup_{x}\mathbf{M}(\Psi_{i}(x))=W$. By [CL20,
Theorem 1.4] we can replace $\\{\Psi_{i}(x)\\}_{i}$ by a sequence of nested
sweepouts $\\{\Phi_{i}\\}_{i}$, such that $\Phi_{i}(x)=f_{i}^{-1}(x)$ for a
surjective Morse function $f_{i}:M\rightarrow[0,1]$ and
$\lim_{i\rightarrow\infty}\sup_{x}\mathbf{M}(\Phi_{i}(x))=W$. Let
$\phi_{i}(x)=\operatorname{Vol}(f_{i}^{-1}([0,x]))$. Since $f_{i}$ is a Morse
function we have that $\phi_{i}:[0,1]\rightarrow[0,1]$ is a continuous
strictly increasing function. Then
$\left(\Phi_{i}\circ\phi_{i}^{-1}\right)_{i}$ is a sequence of nested volume-
parametrized sweepouts. By Lemma 1.3, a subsequence of
$\\{\Phi_{i}\circ\phi_{i}^{-1}\\}_{i}$ converges to a (ONVP) sweepout. ∎

Finally we recall the definition of critical set for a sweepout.

######  Definition 1.5 (Critical set).

Given a sweepout $\Phi$, we define

$M(x)=\limsup_{r\rightarrow 0}\\{\mathbf{M}(\Phi(y)):|y-x|<r\\}.$

If $\Phi$ is an optimal sweepout, we define the _critical domain of $\Phi$_ to
be the set

${\bf m}(\Phi)=\\{x\in[0,1]:M(x)=W\\}\,.$

We will say that a sequence $x_{i}\rightarrow x\in{\bf m}(\Phi)$ is a _min-max
sequence_ if $|\Phi|(x_{i})$ converges in the varifold sense to a varifold $V$
of mass $W$, i.e. $|V|(M)=W$. We denote the set of such varifolds by ${\bf
C}(\Psi)$.

In fact, it is convenient to refine this definition somewhat.

######  Definition 1.6 (Left and right critical set).

Given a sweepout $\Phi$, we say that $x\in{\bf m}_{L}(\Phi)$ if there are
$x_{i}\nearrow x$ with

$\mathbf{M}(\Psi(x_{i}))\to W$

and similarly, $x\in{\bf m}_{R}(\Phi)$ if there are $x_{i}\searrow x$ with

$\mathbf{M}(\Psi(x_{i}))\to W.$

Note that ${\bf m}(\Phi)={\bf m}_{L}(\Phi)\cup{\bf m}_{R}(\Phi)$ (and ${\bf
m}_{L}(\Phi)\cap{\bf m}_{R}(\Phi)$ need not necessarily be empty).

######  Definition 1.7 (Varifolds of optimal regularity).

For an open set $U\subset M$ we say that a varifold $V$ is in $\mathcal{R}(U)$
if

$V\mathop{\hbox{\vrule height=7.0pt,width=0.3pt,depth=0.0pt\vrule
height=0.3pt,width=5.0pt,depth=0.0pt}}\nolimits
U=\sum_{k=i}^{K}\kappa_{i}\,|\Sigma_{i}|\,,$

for $\Sigma_{i}\subset U$ embedded minimal hypersurfaces that are regular away
from a closed singular set of Hausdorff dimension $n-7$, and
$\kappa_{i}\in\varmathbb{N}$ integer multiplicities. We set
$\mathcal{R}=\mathcal{R}(M)$.

Abusing notation, we will say that a (singular) hypersurface $\Sigma$ is in
$\mathcal{R}(U)$ if the associated varifold satisfies
$|\Sigma|\in\mathcal{R}(U)$.

As we will discuss later, Almgren–Pitts theory [Pit81] implies that given an
optimal sweepout $\Psi$, there is $V\in{\bf C}(\Psi)$ with $V\in\mathcal{R}$.

###  1.3. (Homotopic) one sided minimizers

The notions of one sided minimizers and homotopic minimizers will be useful
when dealing with our deformations theorems, as we will see in Section 3.

Let $U\subset M^{n+1}$ be an open set. Given two Caccioppoli sets
$E\subset\Omega$ with $E\Delta\Omega\subset U$ and $\varepsilon\geqslant 0$,
we denote the inner families of deformations between $E$ and $\Omega$ in $U$
which do not increase the volume more than $\varepsilon$ by

$\mathcal{I}_{\varepsilon}(\Omega,E\,|\,U):=\left\\{\\{\Omega(t)\\}_{t\in[0,1]}:\begin{gathered}\,\Omega(0)=E,\,\Omega(1)=\Omega,(\Omega(t)\Delta\Omega)\setminus
U=\emptyset,\\\ \Omega(t_{1})\subset\Omega(t_{2})\text{ for }t_{1}<t_{2},\\\
\operatorname{Per}(\Omega_{t})\leqslant\operatorname{Per}(\Omega)+\varepsilon\
\end{gathered}\right\\}\,,$

and analogously for $\Omega\subset E$ with $E\Delta\Omega\subset U$, we define
the outer families of deformations between $\Omega$ and $E$ in $U$ which do
not increase the volume more than $\varepsilon$ by

$\mathcal{O}_{\varepsilon}(\Omega,E\,|\,U):=\left\\{\\{\Omega(t)\\}_{t\in[0,1]}:\begin{gathered}\Omega(0)=\Omega,\,E=\Omega(1),(\Omega(t)\Delta\Omega)\setminus
U=\emptyset,\\\ \Omega(t_{1})\subset\Omega(t_{2})\text{ for }t_{1}<t_{2}\\\
\operatorname{Per}(\Omega_{t})\leqslant\operatorname{Per}(\Omega)+\varepsilon\end{gathered}\right\\}\,.$

Moreover, given $\Omega$ and $\varepsilon\geqslant 0$, we denote the
collections of inner and outer Caccioppoli sets that can be reached by an
inner or outer family of deformations by

$\displaystyle\mathcal{I}_{\varepsilon}(\Omega\,|\,U)=\\{E\subset\Omega\,:E\Delta\Omega\subset
U,\,\mathcal{I}_{\varepsilon}(\Omega,E\,|\,U)\neq\emptyset\\},$
$\displaystyle\mathcal{O}_{\varepsilon}(\Omega\,|\,U)=\\{E\supset\Omega\,:E\Delta\Omega\subset
U,\,\mathcal{O}_{\varepsilon}(\Omega,E\,|\,U)\neq\emptyset\\}\,.$

In both definitions, if we do not include an $\varepsilon$-subscript, it
should be understood that we are taking $\varepsilon=0$; this will happen most
of the time below, but we will crucially rely on the definition with
$\varepsilon>0$ to obtain _regular_ homotopic minimizers in certain
situations.

######  Definition 1.8 (Homotopic inner and outer minimizers).

Given a Caccioppoli set $\Omega$ we say that a Caccioppoli set
$L(\Omega\,|\,U)\in\mathcal{I}(\Omega\,|\,U)$ is a _homotopic inner minimizer
for $\Omega$ in $U$_, if

  1. (1)

$\operatorname{Per}(L(\Omega\,|\,U)\,|\,U)\leqslant\operatorname{Per}(\Omega^{\prime}\,|\,U)$,
for every $\Omega^{\prime}\in\mathcal{I}(\Omega\,|\,U)$ and

  2. (2)

if $E\in\mathcal{I}(\Omega\,|\,U)$ satisfies (1) and $L(\Omega\,|\,U)\subset
E$ then $E=L(\Omega\,|\,U)$.

Similarly, define $R(\Omega\,|\,U)\in\mathcal{O}(\Omega\,|\,U)$ to be a
_homotopic outer minimizer for $\Omega$ in $U$_, if

  1. (1)

$\operatorname{Per}(R(\Omega\,|\,U)\,|\,U)\leqslant\operatorname{Per}(\Omega^{\prime}\,|\,U)$,
for every $\Omega^{\prime}\in\mathcal{I}(\Omega\,|\,U)$;

  2. (2)

if $E\in\mathcal{I}(\Omega\,|\,U)$ satisfies (1) and $E\subset
R(\Omega\,|\,U)$ then $E=R(\Omega\,|\,U)$.

We say that a Caccioppoli set $\Omega$ is an _inner (resp. outer) homotopic
minimizer in $U$_ if $\Omega$ is a homotopic inner (resp. outer) minimizer
relative to itself.

It is easy to see that inner and outer homotopic minimizers for a fixed set
$\Omega$ always exist.

######  Lemma 1.9 (Existence of homotopic minimizers).

For any Caccioppoli set $\Omega$ and open set $U$ we can find a homotopic
inner (resp. outer) minimizer $L(\Omega\,|\,U)$ (resp. $R(\Omega\,|\,U)$) for
$\Omega$ in $U$.

###### Proof.

We consider only the case of inner minimizers as the outer minimizers are
handled identically.

This is once again an application of Arzelà–Ascoli theorem. Indeed, notice
that $\mathcal{I}(\Omega\,|\,U)\not=\emptyset$, since
$\Omega\in\mathcal{I}(\Omega\,|\,U)$, so we can consider a minimizing sequence
$(E_{j})_{j}$, that is

$\lim_{j}\operatorname{Per}(E_{j}\,|\,U)=\inf\\{\operatorname{Per}(E\,|\,U)\,:\,E\in\mathcal{I}(\Omega\,|\,U)\\}$

and let $\\{E_{j}(x):x\in[0,1]\\}\in\mathcal{I}(\Omega,E_{j}\,|\,U)$ be the
corresponding inner volume non increasing sweepout between $E_{j}$ and
$\Omega$. We can assume that it is volume parametrized (being nested).
Moreover $\operatorname{Per}(E_{j}(x)\,|\,U)$ is uniformly bounded by
$\operatorname{Per}(\Omega\,|\,U)$, so by Arzelà–Ascoli there is a subsequence
converging to $\\{E_{\infty}(x)\\}\in\mathcal{I}(\Omega,E_{\infty}\,|\,U)$,
with $E_{\infty}$ satisfying the desired minimality property by lower semi-
continuity of the perimeter.

Finally, again by Arzelà–Ascoli, we can find $L(\Omega\,|\,U)\subset\Omega$ in
the set of minimizers, which infimizes the flat distance to $\partial\Omega$,
and so satisfies condition (2) (otherwise there would be a competitor closer
to $\Omega$ in flat norm). ∎

We recall the definition of one-sided minimizers, which will be useful in the
sequel when we perform cut and paste arguments.

######  Definition 1.10 (One sided minimizers).

Let $E$ be a Caccioppoli set. We say that $E$ is _locally one-sided inner
(resp. outer) area-minimizing_ in $U$ if for every $A\Subset U$ and $V$ with
$V\Delta E\subset A$, we have

$\operatorname{Per}(E\,|\,A)\leqslant\operatorname{Per}(V\,|\,A)$

whenever $V\subset E$ (resp. $E\subset V$). We say that $E$ is _strictly
locally one-sided inner (resp. outer) area-minimizing_ if the inequality holds
strictly except when $E=V$ as Caccioppoli sets.

We show that homotopic minimizers are in fact strict one sided minimizers into
the region they sweep out.

######  Lemma 1.11 (Homotopic minimizers are one sided minimizers in the swept
out region).

Suppose $L(\Omega\,|\,U)$ is an homotopic inner (resp. outer) minimizer for
$\Omega$ in $U$. Then $L(\Omega\,|\,U)$ (resp. $R(\Omega\,|\,U)$) is strict
locally outer (resp. inner) one-sided minimizing in $U\cap\Omega$ (resp.
$U\setminus\Omega$) .

###### Proof.

We consider homotopic inner minimizers; the case of outer minimizers is
similar.

If $L(\Omega\,|\,U)$ is not a strict outer minimizer in $U\cap\Omega$ then
there is $V^{\prime}$ with $L(\Omega\,|\,U)\subset V^{\prime}$ and
$L(\Omega\,|\,U)\Delta V^{\prime}\subset A\Subset U$ and

$\operatorname{Per}(V^{\prime}\,|\,A)\leqslant P(L(\Omega\,|\,U)\,|\,A).$

We can minimize perimeter in $A$ among all such $V^{\prime}$ to find $V$.
Namely,

(1.3) $\operatorname{Per}(V\,|\,A)\leqslant\operatorname{Per}(W\,|\,A)$

for all $W$ with $W\Delta V\subset A\setminus L(\Omega\,|\,U)$. Since
$L(\Omega\,|\,U)\in\mathcal{I}(\Omega\,|\,U)$, there is
$\\{U(x)\,:\,x\in[0,1]\\}\in\mathcal{I}(\Omega,L(\Omega\,|\,U)\,|\,U)$. Set
$\Omega(x)=U(x)\cup V$. Since $V$ satisfies (1.3), we have that

$\operatorname{Per}(\Omega_{t}\,|\,A)\leqslant\operatorname{Per}(U_{t}\,|\,A).$

This implies that $\Omega(1)=V$ satisfies (1) of Definition 1.8 and $V\Delta
L(\Omega\,|\,U)\subset A\setminus L(\Omega\,|\,U)$, therefore by (2) of
Definition 1.8, it follows that $V=L(\Omega\,|\,U)$. This completes the proof.
∎

We have the following lemma that will allow us to find bounded mass homotopies
in certain situations.

######  Lemma 1.12 (Interpolation lemma).

Fix $L>0$. For every $\varepsilon>0$ there exists $\delta>0$, such that the
following holds. If $\Omega_{0},\Omega_{1}$ are two sets of finite perimeter,
such that $\Omega_{0}\subset\Omega_{1}$,
$\operatorname{Per}(\Omega_{i})\leqslant L$, $i=0,1$, and
$\operatorname{Vol}(\Omega_{1}\setminus\Omega_{0})\leqslant\delta$, then there
exists a nested $\mathcal{F}$-continuous family
$\\{\partial\Omega_{t}\\}_{t\in[0,1]}$ with

$\operatorname{Per}(\Omega_{t})\leqslant\max\\{\operatorname{Per}(\Omega_{0}),\operatorname{Per}(\Omega_{1})\\}+\varepsilon$

for all $t\in[0,1]$

###### Proof.

Let $\Omega$ be a Caccioppoli set that minimizes perimeter among sets
$\Omega^{\prime}$ with $\Omega_{0}\subset\Omega^{\prime}\subset\Omega_{1}$.

Fix $r>0$ such that for every $x\in M$ the ball $B(x,2r)$ is $2$-bi-Lipschitz
diffeomorphic to the Euclidean ball of radius $2r$. Let
$\\{B(x_{i},r)\\}_{i=1}^{N}$ be a collection of balls covering $M$. By coarea
inequality we can find a radius $r_{i}\in[r,2r]$, so that $\mathbf{M}(\partial
B(x_{i},r_{i})\cap\Omega\setminus\Omega_{0})\leqslant\frac{\delta}{r}$.

Let $U_{1}=B(x_{1},r_{1})\cap\Omega\setminus\Omega_{0}$. By a result of
Falconer (see [Fal80], [Gut07, Appendix 6]) there exists a family of
hypersurfaces sweeping out $U_{1}$ of area bounded by
$c(n)\delta^{\frac{n}{n+1}}$. It follows (see [CL20, Lemma 5.3]) that there
exists a nested family $\\{\Xi^{1}(t)\\}$ of Caccioppoli sets with
$\Xi^{1}(0)=\Omega_{0}$ and $\Xi^{1}(1)=\Omega_{0}\cup U_{1}$ and satisfying

$\operatorname{Per}(\Xi^{1}(t))\leqslant\operatorname{Per}(\Omega_{0})+2c(n)\delta^{\frac{n}{n+1}}\,.$

Let $\Omega^{1}=\Omega_{0}\cup U_{1}$. Observe, that the minimality of
$\Omega$ implies that

$\operatorname{Per}(\Omega^{1})\leqslant\operatorname{Per}(\Omega_{0})+\frac{2\delta}{r}$

Inductively, we define $\Omega^{k}=\Omega^{k-1}\cup U_{k}$ and
$U_{k}=B(x_{k},r_{k})\cap\Omega\setminus\Omega^{k-1}$. As above we can
construct a nested homotopy of Caccioppoli sets $\Xi^{k}(t)$ from
$\Omega^{k-1}$ to $\Omega^{k}$, satisfying

$\operatorname{Per}(\Xi^{k}(t))\leqslant\operatorname{Per}(\Omega_{0})+2c(n)\delta^{\frac{n}{n+1}}+\frac{2N\delta}{r}$

We choose $\delta>0$ so small that
$\operatorname{Per}(\Xi^{k}(t))<\operatorname{Per}(\Omega_{0})+\varepsilon$.
It follows then that we have obtained a homotopy from $\Omega_{0}$ to $\Omega$
satisfying the desired perimeter bound. Similarly, we construct a homotopy
from $\Omega$ to $\Omega_{1}$. ∎

Finally, we have the following result. Recall that White [Whi94] proves that
strictly stable _smooth_ minimal hypersurfaces are locally area-minimizing. A
generalization of such a result to the case of hypersurfaces with
singularities (i.e., elements of $\mathcal{R}$) would be very interesting. The
following (weaker) result will suffice for our needs; it can be seen as a
result along these lines, except “stability” is replaced by a stronger
hypothesis: the surface is homotopic minimizing to one side.222Note that one
certainly needs a condition on the singularities rather than just a condition
on the regular part like strict stability, since as we show in Proposition
3.2, the existence of (regular) non-minimizing tangent cones implies that the
hypersurface is not homotopic minimizing irrespective of any stability
condition that might hold on the regular part.

######  Proposition 1.13 (Comparing the notions of minimizing vs. homotopic
minimizing for minimal surfaces).

Suppose that $\Omega$ is a Caccioppoli set and for some strictly convex open
set $U\subset M$ with smooth boundary, the associated varifold
$V=|\partial\Omega|$ satisfies $V\in\mathcal{R}(U)$. Assume that
$\operatorname{supp}V\cap U$ is connected.

Suppose that $\Omega$ is inner (resp. outer) homotopy minimizing in $U$. Then,
at least one of the following two situations holds:

  1. (1)

for all $p\in\operatorname{supp}V\cap U$, there is $\rho_{0}>0$ so that for
$\rho<\rho_{0}$, $B_{\rho}(p)\subset U$ and $\Omega$ is inner (resp. outer)
minimizing in $B_{\rho}(p)$, or

  2. (2)

there exists a sequence of Caccioppoli sets $E_{i}\neq\Omega$ with $|\partial
E_{i}|\in\mathcal{R}(U)$ so that $E\Delta\Omega\subset\Omega\cap U$ (resp.
$\Omega^{c}\cap U$), $|\partial E_{i}|$ has stable regular part, and $\partial
E_{i}\to\partial\Omega$ in the flat norm.

######  Remark 1.14.

It is interesting to ask if the second possibility occurs; it seems possible
that one could rule this out in the case where $V$ has regular tangent cones
that are all strictly minimizing in the sense of Hardt–Simon [HS85, §3].

###### Proof of Proposition 1.13.

We consider the “inner” case, as the “outer” case is similar. Let
$E^{\varepsilon}\in\mathcal{I}_{\varepsilon}(\Omega\,|\,U)$ minimize perimeter
among all sets in $\mathcal{I}_{\varepsilon}(\Omega\,|\,U)$ (as usual, the
existence of $E^{\varepsilon}$ follows from Arzelà–Ascoli). We claim that
$E^{\varepsilon}$ is area-minimizing to the inside of $\Omega$ in sufficiently
small balls.

More precisely, for $r>0$ sufficiently small, suppose there was a Caccioppoli
set $E^{\prime}$ so that $E^{\prime}\Delta E^{\varepsilon}\subset B_{r}(p)\cap
U\cap\Omega$ and
$\operatorname{Per}(E^{\prime}\,|\,U)<\operatorname{Per}(E^{\varepsilon}\,|\,U)$.
As long as $r$ was chosen sufficiently small, Lemma 1.12 guarantees that
$E^{\prime}\in\mathcal{I}_{\varepsilon}(\Omega\,|\,U)$. This is a
contradiction.

Now, consider $p\in\operatorname{Reg}V\cap U$. We note that $E^{\varepsilon}$
is almost minimizing (with no constraint coming from $\Omega$) in the sense of
[Tam84], and thus has $C^{1,\alpha}$ boundary in $B_{r}(p)\cap U$, thanks to
standard results on the obstacle problem; see [Tam84, §1.9, §1.14(iv)]. As
such, away from $\operatorname{Sing}V$ (which has Hausdorff dimension at most
$n-7$) we can thus conclude that $\partial^{*}E^{\varepsilon}$ is regular,
stationary and stable.333Cf. the proof of [Liu19, Proposition 2.1] for the
proof of stability. A capacity argument then implies that $|\partial
E^{\varepsilon}|\in\mathcal{R}(U)$ and $\partial^{*}E^{\varepsilon}$ is
stable. Therefore, the maximum principle for (possibly singular) hypersurfaces
[Ilm96] implies that either $E^{\varepsilon}=\Omega$ or
$\partial^{*}E^{\varepsilon}\cap\operatorname{supp}V=\emptyset$. In the first
case, we can conclude that $\Omega$ is inner minimizing in small balls (since
$E^{\varepsilon}$ is).

We can thus assume that the latter possibility holds for all $\varepsilon>0$
sufficiently small. Taking $\varepsilon_{j}\to 0$, there is
$E\in\mathcal{I}_{0}(\Omega|U)$ so that $E^{\varepsilon_{j}}\to E$ with
respect to the flat norm. If $E=\Omega$, then the second possibility in the
conclusion of the proposition holds for $E_{j}=E^{\varepsilon_{j}}$.

The final case to consider is $E\neq\Omega$. By curvature estimates for stable
minimal hypersurfaces [SS81], $|\partial E|\in\mathcal{R}(\Omega)$ and thus
$\partial^{*}E\cap\operatorname{supp}V=\emptyset$ again by the maximum
principle.

We know that
$\operatorname{Per}(E_{i}|U)\leqslant\operatorname{Per}(\Omega|U)$, so in the
limit we get $\operatorname{Per}(E|U)\leqslant\operatorname{Per}(\Omega|U)$ By
Arzelà-Ascoli nested homotopies from $\Omega$ to $E_{i}$ will converge to a
nested homotopy $E(t)$ from $\Omega$ to $E$ that does not increase volume. By
the inner homotopy minimizing property of $\Omega$ we have

$\operatorname{Per}(E\,|\,U)=\operatorname{Per}(\Omega\,|\,U).$

Suppose we minimize perimeter among Caccioppoli sets $A^{\prime}$ sandwiched
between $E$ and $\Omega$, $E\subset A^{\prime}\subset\Omega$. We claim that
the minimizer $A$ has perimeter equal to that of $\Omega$. Indeed, if $A$ has
strictly smaller perimeter, then family $E(t)\cup A$ is an area decreasing
nested homotopy between $\Omega$ and $A$, contradicting that $\Omega$ is inner
homotopic minimizing.

We thus see that $\Omega$ is minimizing in $\Omega\cap E^{c}\cap U$, which
implies that it is inner minimizing in small balls, as asserted. ∎

##  2\. Non-excessive sweepouts

In this section we introduce the concept of excessive intervals and excessive
points for a sweepout and prove that there is a sweepout, such that every
point in the critical domain is not excessive.

######  Definition 2.1 (Excessive points and intervals).

Suppose $\\{\Phi(x)=\partial\Omega(x)\\}$ is a sweepout. Given a connected
interval $I$ (we allow $I$ to be open, closed, or half-open) we will say that
$\\{\Phi^{I}(x)=\partial\Omega^{I}(x)\\}_{x\in\bar{I}}$ is an _$I$
-replacement family for $\Phi$_ if $\Omega^{I}(a)=\Omega(a)$,
$\Omega^{I}(b)=\Omega(b)$ and for all $x\in I$,

$\limsup_{I\ni y\to x}\mathbf{M}(\Phi^{I}(y))<W.$

We say that a connected interval $I$ is an _excessive interval for $\Phi$_ if
there is an $I$-replacement family for $\Phi$. We say that a point $x$ is
_left (resp. right) excessive for $\Phi$_ if there is an excessive interval
$I$ for $\Phi$ so that $(x-\varepsilon,x]\subset I$ (resp.
$[x,x+\varepsilon)\subset I$) for some $\varepsilon>0$.

The goal of this section is to prove the following result.

######  Theorem 2.2 (Existence of non-excessive min-max hypersurface).

There exists a (ONVP) sweepout $\Psi$ such that every $x\in{\bf m}_{L}(\Psi)$
is not left excessive and every $x\in{\bf m}_{R}(\Psi)$ is not right
excessive.

###  2.1. Preliminary results

We establish several results that will be used in the proof of Theorem 2.2.

######  Lemma 2.3 (Extension lemma I).

If $I,J$ are excessive for $\Phi$ and $I\cap J\not=\emptyset$, then $I\cup J$
is excessive for $\Phi$.

###### Proof.

Let $\\{\partial\Omega^{I}(x)\\}_{x\in I}$ and
$\\{\partial\Omega^{J}(x)\\}_{x\in J}$ be $I$ and $J$ replacement families for
$\Phi$.

Let $a_{1}=\inf\\{x\in I\\}$, $a_{2}=\inf\\{x\in J\\}$ and $b_{1}=\sup\\{x\in
I\\}$, $b_{2}=\sup\\{x\in J\\}$. Assume without any loss of generality that
$a_{1}\leqslant a_{2}$ and $b_{1}\leqslant b_{2}$ and at least one of the two
inequalities is strict.

Let $K=I\cap J$; let $a,b$ denote, respectively, left and right boundary
points of $K$ and $c=\frac{a+b}{2}\in K$. Let $\tilde{\Omega}$ be a
Caccioppoli set minimizing perimeter among all $\Omega^{\prime}$ with
$\Omega(a)\subset\Omega^{\prime}\subset\Omega(b)$. Define
$\phi_{1}:[a_{1},c]\rightarrow[a_{1},b_{1}]$ and
$\phi_{2}:[c,b_{2}]\rightarrow[a_{2},b_{2}]$ given by
$\phi_{1}(x)=a_{1}+\frac{b_{1}-a_{1}}{c-a_{1}}(x-a_{1})$ and
$\phi_{2}(x)=a_{2}+\frac{b_{2}-a_{2}}{b_{2}-c}(x-c)$. We define an $I\cup J$
replacement family for $\Phi$ by setting

$\Phi^{I\cup
J}(x)=\begin{cases}\partial(\Omega^{I}(\phi_{1}(x))\cap\tilde{\Omega})&x\in[a_{1},c]\\\
\partial(\Omega^{J}(\phi_{2}(x))\cup\tilde{\Omega})&x\in[c,b_{2}]\end{cases}$

Observe that $\Phi^{I\cup J}$ is continuous since $\Phi^{I\cup
J}(c)=\partial\tilde{\Omega}$. It follows from our choice of $\tilde{\Omega}$
that $\mathbf{M}(\Phi^{I\cup
J}(x))\leqslant\mathbf{M}(\Phi^{I}(\phi_{1}^{-1}(x)))<W$ for $x\in
I\cap(-\infty,c]$ and $\mathbf{M}(\Phi^{I\cup
J}(x))\leqslant\mathbf{M}(\Phi^{J}(\phi_{2}^{-1}(x)))<W$ for $x\in
J\cap[c,\infty)$. ∎

######  Lemma 2.4 (Extension lemma II).

If $I$ is excessive for $\Phi$ and $J$ has $J\cap I\neq\emptyset$ and is
excessive for

$\Psi(x):=\begin{cases}\Phi^{I}(x)&x\in I\\\ \Phi(x)&x\not\in I\end{cases}$

then $J\cup I$ is excessive for $\Phi$.

###### Proof.

Define an $I\cup J$-replacement family $\Phi^{I\cup J}$ for $\Phi$ by

$\Phi^{I\cup J}(x)=\begin{cases}\Phi(x)&x\in[0,1]\setminus(I\cup J)\\\
\Phi^{I}(x)&x\in I\setminus J\\\ \Psi^{J}(x)&x\in J\end{cases}$

where $\Psi^{J}$ is a $J$-replacement family for $\Psi$. ∎

The following is the technical core of the proof of Theorem 2.2.

######  Proposition 2.5 (Existence of maximal excessive intervals).

Given an (ONVP) sweepout $\Phi$, if $\hat{J}$ is excessive for
$\\{\Phi(x)=\partial\Omega(x)\\}$, then there exists an excessive interval
$J\supset\hat{J}$ so that $J$ is maximal in the sense that if $\tilde{J}$ is
excessive with $\tilde{J}\cap J\not=\emptyset$, then $\tilde{J}\subset J$.

###### Proof.

Let

$\alpha:=\sup\\{|\tilde{J}|:\tilde{J}\textrm{ excessive
},\hat{J}\cap\tilde{J}\not=\emptyset\\}.$

Choose excessive intervals $\tilde{J}_{n}$ with
$\tilde{J}_{n}\cap\hat{J}\neq\emptyset$ and $|\tilde{J}_{n}|\to\alpha$. By
Lemma 2.3, we can replace $\tilde{J}_{n}$ by $\tilde{J}_{n}\cup\hat{J}$, and
thus assume that $\hat{J}\subset\tilde{J}_{n}$. In particular
$\tilde{J}_{n}\cap\tilde{J}_{m}\neq\emptyset$ for all $m,n$. Using Lemma 2.3
again, we can replace $\tilde{J}_{n}$ by

$\bigcup_{m=1}^{n}\tilde{J}_{m}$

so that the $\tilde{J}_{n}$ form an increasing sequence of excessive intervals
(still with $|\tilde{J}_{n}|\to\alpha$). Note that the interior of an
excessive interval is still excessive, so we can consider
$J_{n}:=(\tilde{J}_{n})^{\circ}$. Note that $|J_{n}|\to\alpha$ and the $J_{n}$
are increasing.

We will show below that

$J^{\prime}:=\bigcup_{n}J_{n}$

is excessive. Write $J^{\prime}=(a,b)$. Granted the fact that $J^{\prime}$ is
excessive, we claim that one of the intervals $(a,b),(a,b],[a,b)$, or $[a,b]$
is the desired maximal excessive interval. Note that by Lemma 2.3, if $[a,b)$
and $(a,b]$ are excessive, then so is $[a,b]$, so we can choose the largest
excessive interval out of these four choices and call it $J$. Suppose that
$\tilde{J}$ is excessive with $\tilde{J}\cap J\not=\emptyset$. Then,
$J\cup\tilde{J}$ is excessive by Lemma 2.3 and $\hat{J}\subset
J\cup\tilde{J}$. Thus,

$|J\cup\tilde{J}|\leqslant\alpha,$

so $\tilde{J}\subset\bar{J}$ (where $\bar{J}$ is the closure of $J$). Now,
$J\cup\tilde{J}$ is excessive, but strictly larger than $\tilde{J}$ (by
assumption). This contradicts the choice of $J$ as the largest excessive
interval out of $(a,b),(a,b],[a,b)$, and $[a,b]$. This shows that $J$ is
maximal, as desired.

It thus remains to prove that $J^{\prime}=\cup_{n}J_{n}$ is excessive for a
nested sequence of open excessive intervals $J_{n}$. Write
$J_{n}=(a_{n},b_{n})$ and set
$a^{\prime}_{n}=a_{n}+\frac{1}{n},b_{n}^{\prime}=b_{n}-\frac{1}{n}$.

Fix $i=0,1,\dots$ and assume we have real numbers $0<A_{1},\dots,A_{i}<W$ and
integers $n_{i}\geqslant i$ (with $n_{1}<n_{2}<\dots<n_{i}$) so that for
$n\geqslant n_{i}$, there is a $J_{n}$-replacement
$\\{\Phi^{n}_{i}(x)=\partial\Omega^{n}_{i}(x)\\}$ so that

$\operatorname{Per}(\Omega^{n}_{i}(x))\leqslant A_{j}$

for $x\in[a_{j}^{\prime},b_{j}^{\prime}]$ and $1\leqslant j\leqslant i$. (Note
that for $i=0$, we can find such objects because the $J_{n}$ are excessive.)

We will choose $0<A_{i+1}<W$, and $n_{i+1}>\max\\{n_{i},i+1\\}$ so that we can
construct $J_{n}$-replacements
$\\{\Phi^{n}_{i+1}(x)=\partial\Omega^{n}_{i+1}(x)\\}$ for $n\geqslant n_{i+1}$
with

$\operatorname{Per}(\Omega^{n}_{i+1}(x))\leqslant A_{j}$

for $x\in[a_{j}^{\prime},b_{j}^{\prime}]$ and $1\leqslant j\leqslant i+1$.
Granted this, we can easily (inductively) complete the proof by passing
$\Phi^{n_{i+1}}_{i+1}$ to a subsequential limit (using Arzelà–Ascoli).

It is useful to introduce the following notation, used in the construction of
$\Phi^{n}_{i+1}$. Given two nested sets of finite perimeter $V\subset W$, we
let

  * •

$\mathcal{M}_{V,W}$ an outermost Caccioppoli set minimizing perimeter among
all the Caccioppoli sets $\Omega$ with $V\subset\Omega\subset W$;

  * •

$\\{\mathcal{V}_{(V,W)}(x)\\}_{x}$ the optimal nested homotopy from $V$ to
$W$.

For $n\geqslant n_{i}$, we set

$L_{n}:=\mathcal{M}_{\Omega(a_{n}),\Omega^{n_{i}}_{i}(a_{i+1}^{\prime})},\qquad
U_{n}:=\mathcal{M}_{\Omega^{n_{i}}_{i}(b_{i+1}^{\prime}),\Omega(b_{n})}$

Note that for $n\leqslant m$, $L_{m}\subset L_{n}$ and $U_{n}\subset U_{m}$.
Hence, $L_{n}$ and $U_{n}$ have $\mathcal{F}$-limits as $n\to\infty$. For
$\varepsilon>0$ fixed so that

$\max\left\\{\operatorname{Per}\left(\Omega^{n_{i}}(a_{i+1}^{\prime})\right),\operatorname{Per}\left(\Omega^{n_{i}}_{i}(b_{i+1}^{\prime})\right)\right\\}+\varepsilon<W,$

Lemma 1.12 thus guarantees that there is $n_{i+1}\geqslant i+1$ sufficiently
large so that for $n\geqslant n_{i+1}$,

$\sup_{t}\operatorname{Per}\left(\mathcal{V}_{(L_{n},L_{n_{i+1}})}(t)\right)<W,\qquad\sup_{t}\operatorname{Per}\left(\mathcal{V}_{(U_{n_{i+1}},U_{n})}(t)\right)<W.$

For $n\geqslant n_{i+1}$, we define

$\tilde{\Phi}^{n}_{i+1}(x)=\begin{cases}\partial\left(\Omega^{n}_{i}(x+1)\cap
L_{n}\right)&x\in[a_{n}-1,b_{n}-1]\\\
\partial\tilde{\mathcal{V}}_{L_{n},L_{n_{i+1}}}(x)&x\in[b_{n}-1,a_{n_{i+1}}]\\\
\partial\left(\Omega^{n_{i+1}}_{i}(x)\cup L_{n_{i+1}}\cap
U_{n_{i+1}}\right)&x\in[a_{n_{i+1}},b_{n_{i+1}}]\\\
\partial\tilde{\mathcal{V}}_{U_{n_{i+1}},U_{n}}(x)&x\in[b_{n_{i+1}},a_{n}+1]\\\
\partial\left(\Omega^{n}_{i}(x-1)\cup
U_{n}\right)&x\in[a_{n}+1,b_{n}+1].\end{cases}$

Here, the $\tilde{\mathcal{V}}$ are the homotopies $\mathcal{V}$
reparametrized to be defined on the given intervals (the exact parametrization
is immaterial). It is easy to check that $\tilde{\Phi}^{n}_{i+1}$ is
continuous.

Let $\Phi_{i+1}^{n}$ denote the reparametrization of $\tilde{\Phi}_{i+1}^{n}$
by volume. We have arranged that $\Phi_{i+1}^{n}$ is a $J_{n}$-replacement.
Moreover, for $x\in[a_{i+1}^{\prime},b_{i+1}^{\prime}]$, we have that
$\Phi^{n}_{i+1}(x)=\Phi^{n_{i+1}}_{i}(x)$, so

$\mathbf{M}(\Phi^{n}_{i+1}(x))\leqslant A_{j}$

for $x\in[a_{j}^{\prime},b_{j}^{\prime}]$ and $1\leqslant j\leqslant i$.
Finally, we can set

$A_{i+1}:=\sup_{x\in[a_{i+1}^{\prime},b_{i+1}^{\prime}]}\mathbf{M}(\Phi^{n_{i+1}}_{i}(x))<W$

(which is independent of $n$). This completes the proof. ∎

###  2.2. Proof Theorem 2.2

We are now able to complete proof of Theorem 2.2

Let $\Phi$ be a nested optimal sweepout. Consider the collection $\mathcal{A}$
of the maximal (with respect to inclusion) excessive intervals for $\Phi$,
that is $I\in\mathcal{A}$ if for every excessive interval $I^{\prime}$ such
that $I^{\prime}\cap I\neq\emptyset$, we have $I\supset I^{\prime}$. The
existence of maximal intervals follows from Proposition 2.5 proven above.

Notice that by definition $I\neq J\in\mathcal{A}$ implies that $I\cap
J=\emptyset$, so we can define a new sweepout $\Psi$ in the following way

$\Psi(x)=\begin{cases}\Phi^{I}(x)&\quad\text{if }x\in I\in\mathcal{A}\\\
\Phi(x)&\quad\text{otherwise}\,.\end{cases}$

Note that $\Psi$ is a nested optimal sweepout, so up to reparametrization we
can assume it is (ONVP), and moreover by construction ${\bf
m}(\Psi)\subset{\bf m}(\Phi)$. Suppose that $x\in{\bf m}_{L}(\Psi)$ is left
excessive. Then, there is a $\Psi$-excessive interval $J$ with
$(x-\varepsilon,x]\subset J$. We claim that there is $I\in\mathcal{A}$ with
$J\subset I$. Indeed, if $J\cap I=\emptyset$ for all $I\in\mathcal{A}$, then
$J$ is a $\Phi$-excessive interval, contradicting the definition of
$\mathcal{A}$. On the other hand, if there is $I\in\mathcal{A}$ with $J\cap
I\not=\emptyset$, then $J\cup I$ is excessive by Lemma 2.4. Thus, $J\subset I$
by definition of $\mathcal{A}$ again. Thus, for $y\in(x-\varepsilon,x]\subset
I$, $\Psi(y)=\Psi^{I}(y)$. By the definition of replacement family, we know
that if $x_{i}\in(x-\varepsilon,x]$ has $x_{i}\to x$, then

$\limsup_{i\to\infty}\mathbf{M}(\Psi^{I}(x_{i}))<W.$

However, this contradicts the assumption that $x\in{\bf m}_{L}(\Psi)$. The
same proof works to prove that $x\in{\bf m}_{R}(\Psi)$ is not right excessive.
This finishes the proof. ∎

##  3\. Deformation Theorems and Proof of Theorem 0.4

In this section we conclude the proof of Theorem 0.4. By Theorem 2.2, there
exists an (ONVP) sweepout $\Phi$ so that every $x\in{\bf m}_{L}(\Phi)$ is not
left excessive and every $x\in{\bf m}_{R}(\Phi)$ is not right excessive. By
Almgren–Pitts pull-tight and regularity theory [Pit81], we find that for some
$x_{0}\in{\bf m}(\Phi)$, there is a min-max sequence $x_{i}\to x_{0}$ so that
$|\Phi(x_{i})|$ converges to some $V\in\mathcal{R}$. Indeed, we can pull-tight
$\Phi$ to find a sweepout (in the sense of Almgren–Pitts, not in the (ONVP)
sense considered in this paper) $\tilde{\Phi}$; we have that ${\bf
C}(\tilde{\Phi})\subset{\bf C}(\Phi)$ and some $V\in{\bf C}(\tilde{\Phi})$ is
in $\mathcal{R}$. By replacing $\Phi(x)$ by $\Phi(1-x)$ if necessary, we can
then assume for the rest of this section that:

(3.1) $\begin{gathered}\text{there is a (ONVP) sweepout
$\\{\Phi(x)=\partial\Omega(x)\\}$ and $x_{i}\nearrow x_{0}\in{\bf
m}_{L}(\Phi)$, so that}\\\ \text{$|\Phi(x_{i})|\to V\in\mathcal{R}$ and $\Phi$
is not left excessive at $x_{0}$}\end{gathered}$

We then consider two cases: $\mathbf{M}(\Phi(x_{0}))=W$ (no cancellation) and
$\mathbf{M}(\Phi(x_{0}))<W$ (cancellation). We analyze the geometric
properties of $V$ in both cases separately, proving deformation theorems
reminiscent of those in [MN16].

###  3.1. No cancellation

Throughout this subsection we will assume the no cancellation condition

$\mathbf{M}(\Phi(x_{0}))=W\,.$

In this case we have that $|\Phi(x_{i})|\to|\partial\Omega|$, see for instance
[DLT13, Proposition A.1], so we can rephrase our assumption (3.1) as

(3.2) $\begin{gathered}\text{there is a (ONVP) sweepout
$\\{\Phi(x)=\partial\Omega(x)\\}$ and $x_{i}\nearrow x_{0}\in{\bf
m}_{L}(\Phi)$, so that}\\\
\text{$|\Phi(x_{i})|\to|\Sigma|:=|\partial\Omega|\in\mathcal{R}$ and $\Phi$ is
not left excessive at $x_{0}$.}\end{gathered}$

In particular, in this case the multiplicity bound of Theorem 0.4 follows
immediately.

######  Proposition 3.1.

Let $\Sigma$ be as in (3.2). Suppose $\Sigma$ is not homotopic minimizing to
either side in some open set $U$. Then the following holds:

  1. (1)

for every $x\not\in\overline{U}$ there exists $r>0$, such that $\Sigma$ is
minimizing to one side in $B_{r}(x)$;

  2. (2)

for every open set $U^{\prime}$ disjoint from $U$, we have that $\Sigma$ is
homotopic minimizing to one side in $U^{\prime}$.

###### Proof.

We prove statement (1). There is $\delta>0$ and Caccioppoli sets
$E^{-}_{1}\in\mathcal{I}(\Omega\,|\,U)$ and
$E^{+}_{1}\in\mathcal{O}(\Omega\,|\,U)$ with

(3.3)
$\operatorname{Per}(E^{\pm}_{1}\,|\,U)\leqslant\operatorname{Per}(\Omega\,|\,U)-\delta\,,$

and nested families
$\\{\Omega^{-}_{1}(x)\,:x\in[0,1]\\}\in\mathcal{I}(\Omega,E_{1}^{-}\,|\,U)$
and
$\\{\Omega^{+}_{1}(x)\,:x\in[0,1]\\}\in\mathcal{O}(\Omega,E_{1}^{+}\,|\,U)$.
Furthermore, by Lemma 1.9, we can assume that $E_{1}^{+}$ are inner and
$E_{1}^{-}$ are outer homotopic minimizers in $U$.

Let $x\in\Sigma\setminus\overline{U}$ and assume, for contradiction, that
$\Sigma$ is not area minimizing on both sides in every ball $B_{r}(x)$,
$r<{\rm dist}(x,U)$. Let $E^{-}_{2}\subset\Omega$, with $\Omega\setminus
E^{-}_{2}\subset B_{r}(x)$, denote a Caccioppoli set that is a strict outer
minimizer in $\Omega\cap B_{r}(x)$. Similarly, let $\Omega\subset E^{+}_{2}$,
with $E^{+}_{2}\setminus\Omega\subset B_{r}(x)$, denote a Caccioppoli set that
is a strict inner minimizer in $\Omega\cap B_{r}(x)$. We have

$\operatorname{Per}(\Omega)>\max\\{\operatorname{Per}(E^{\pm}_{2})\\}.$

If we choose $r>0$ sufficiently small, then, by Lemma 1.12, there exist nested
families $\\{\Omega^{-}_{2}(x)\,:x\in[0,1]\\}$ and
$\\{\Omega^{+}_{2}(x)\,:x\in[0,1]\\}$ that interpolate between $E^{-}_{2}$ and
$\Omega$ and between $\Omega$ and $E^{+}_{2}$ and satisfying

(3.4)
$\operatorname{Per}(\Omega^{\pm}_{2}(x))\leqslant\operatorname{Per}(\Omega)+\frac{\delta}{2}$

Let $(x_{l},x_{r})\neq\emptyset$ be the interval (since $\Phi$ is nested) such
that

$\Phi(x)\cap(\cup_{i}E_{i}^{+}\setminus\cup_{i}E_{i}^{-})\neq\emptyset\,.$

Then we define a family
$\bar{\Psi}\colon[x_{l}-2,x_{r}+2]\to\mathcal{Z}_{n}(M;\varmathbb{Z}_{2})$ by
setting

$\bar{\Psi}(x):=\begin{cases}\partial\left(\Omega(x+2)\cap E_{1}^{-}\cap
E_{2}^{-}\right)&\text{if }x\in(x_{l}-2,x_{0}-2]\\\
\partial\left(\Omega_{1}^{-}(x-x_{0}+2)\cap E_{2}^{-}\right)&\text{if
}x\in[x_{0}-2,x_{0}-1]\\\ \partial\left(\Omega_{1}^{+}(x-x_{0}+1)\cap
E_{2}^{-}\right)&\text{if }x\in[x_{0}-1,x_{0}]\\\
\partial\left(\Omega_{2}^{-}(x-x_{0})\cup E_{1}^{+}\right)&\text{if
}x\in[x_{0},x_{0}+1]\\\ \partial\left(\Omega_{2}^{+}(x-x_{0}-1)\cup
E_{1}^{+}\right)&\text{if }x\in[x_{0}+1,x_{0}+2]\\\
\partial\left(\Omega(x-2)\cup E_{1}^{+}\cup E_{2}^{+}\right)&\text{if
}x\in[x_{0}+2,x_{r}+2)\end{cases}$

It is easy to see that $\bar{\Psi}$ is continuous, and moreover notice that,
since by Lemma 1.11 $E_{1}^{+}$ is a strict inner minimizer in $U$ and
$E_{1}^{-}$ strict outer minimizers in $U$, we have that

$\limsup_{y\to x}\mathbf{M}(\bar{\Psi}(y))<\limsup_{y\to
x}\mathbf{M}(\bar{\Phi}(y))\leqslant W$

for $x\in(x_{l}-2,x_{0}-2]\cup[x_{0}-2,x_{r}-2)$. Since the families
$\Omega_{1}^{\pm}(x)$ do not increase the volume of $\Sigma$ in $U_{i}$ and
using (3.3) and (3.4), we also have

$\mathbf{M}(\bar{\Psi}(x))\leqslant
W-\frac{\delta}{2}\qquad\forall\,x\in[x_{0}-2,x_{0}+2]\,.$

We let $\Psi$ be the volume reparametrization of the nested sweepout
$\bar{\Psi}$, then $\Psi$ is a $(x_{l},x_{r})$-replacement for $\Phi$, thus
giving a contradiction with the fact that $x_{0}\in(x_{l},x_{r})$ and
$x_{0}\in{\bf m}_{L}(\Phi)$.

The proof of statement (2) is completely analogous. ∎

######  Proposition 3.2.

Let $\Sigma$ be as in (3.2), then the following holds

  1. (1)

$\operatorname{Index}(\Sigma)\leqslant 1$;

  2. (2)

If $\operatorname{Index}(\Sigma)=1$, then for every point $x\in\Sigma$ there
exists $r>0$, such that $\Sigma$ is minimizing to one side in $B_{r}(x)$;

  3. (3)

If $\operatorname{\mathfrak{h}_{\textnormal{nm}}}(\Sigma)$ is non-empty, then
$\Sigma$ is stable,
$\mathcal{H}^{0}(\operatorname{\mathfrak{h}_{\textnormal{nm}}}(\Sigma))=1$ and
for every point
$x\in\Sigma\setminus\operatorname{\mathfrak{h}_{\textnormal{nm}}}(\Sigma)$
there exists $r>0$, such that $\Sigma$ is minimizing to one side in
$B_{r}(x)$.

In particular, Theorem 0.4 holds in the case of no cancellations.

###### Proof.

Note that if $U\cap\Sigma$ is smooth and unstable, it is easy to see that
$\Sigma$ is not homotopic minimizing to either side in $U$ (just consider the
normal flow generated by a compactly supported unstable variation of fixed
sign). Statements (2) and (3) of the Proposition now immediately follow from
Proposition 3.1. The upper bound on the index (1) follows from (2) of
Proposition 3.1 and Lemma 3.3 below. ∎

######  Lemma 3.3 (Localizing the index).

Suppose that $\Sigma\in\mathcal{R}$ is two-sided and has
$\operatorname{Index}(\Sigma)\geqslant 2$. Then, there is
$\Sigma^{*}_{1},\Sigma^{*}_{2}\subset\Sigma$ smooth hypersurfaces with
boundary so that the $\Sigma^{*}_{i}$ are both unstable (for variations fixing
the boundary).

###### Proof.

A standard capacity argument implies that there is a subset
$\Sigma^{\prime}\subset\Sigma$ where $\Sigma^{\prime}$ is a smooth minimal
surface with smooth boundary and
$\operatorname{Index}(\Sigma^{\prime})\geqslant 2$ (with Dirichlet boundary
conditions). Let $u$ denote the second (Dirichlet) eigenfunction (with
eigenvalue $\lambda<0$) for the stability operator for $\Sigma$. Because $u$
must change sign, there are at least two nodal domains
$\Sigma_{1},\Sigma_{2}\subset\Sigma$. One can find subsets with smooth
boundary $\Sigma^{*}_{i}\subset\Sigma_{i}$ so that $\Sigma^{*}_{i}$ are
unstable. This follows from the argument in [Cha84, p. 21] (namely, by
considering $(u|_{\Sigma_{i}}-\varepsilon)_{+}$ in the stability operator for
$\varepsilon\to 0$ chosen so that $\\{u|_{\Sigma_{i}}>\varepsilon\\}$ has
smooth boundary). ∎

######  Lemma 3.4.

$\mathcal{S}_{\textnormal{nm}}(\Sigma)\subset\mathfrak{h}_{\textrm{nm}}(\Sigma)$.

###### Proof.

Suppose that $p\in\mathcal{S}_{\textnormal{nm}}(V)$, we claim that $\Sigma$ is
not homotopic minimizing to either side in $B_{\varepsilon}(p)$ for any
$\varepsilon>0$ sufficiently small. Indeed, by assumption, the unique tangent
cone $\mathbf{C}=\partial\Omega_{\mathbf{C}}$ to $\Sigma$ at $p$ is not
minimizing to either side. This implies that there are Caccioppoli sets
$E_{\mathbf{C}}^{-}\subset\Omega_{\mathbf{C}}\subset E_{\mathbf{C}}^{+}$ so
that $E_{\mathbf{C}}^{\pm}\Delta\Omega_{\mathbf{C}}\subset
B_{1}\subset\varmathbb{R}^{n+1}$ and so that

$\operatorname{Per}_{\varmathbb{R}^{n+1}}(E_{\mathbf{C}}^{\pm}\,|\,B_{1})\leqslant\operatorname{Per}_{\varmathbb{R}^{n+1}}(\Omega_{\mathbf{C}}\,|\,B_{1})-\delta.$

Choose $C^{1,\omega}$ coordinates on $M$ around $p$ so that
$\Omega=\Omega_{\mathbf{C}}$ in $B_{\varepsilon}(p)$ and so that
$g_{ij}(p)=\delta_{ij}$, which we can do since $g\in C^{2}$ and $\Sigma$ is a
$C^{1,\omega}$ deformation of $\mathbf{C}$ near $p$ by assumption. Then, set

$E(x):=\begin{cases}(\Omega\setminus
B_{\varepsilon})\cup(|x|E_{\mathbf{C}}^{-}\cap B_{\varepsilon})&x<0\\\
\Omega&x=0\\\ (\Omega\setminus B_{\varepsilon})\cup(|x|E_{\mathbf{C}}^{+}\cap
B_{\varepsilon})&x>0\\\ \end{cases}$

We have that

$\operatorname{Per}_{g}(E(x))-\operatorname{Per}_{g}(\Omega)=-|x|^{n}\delta(1+o(1))$

as $x\to 0$ (since the metric $g_{ij}$ converges to the flat metric
$\delta_{ij}$ after rescaling $|x|\to 1$, by the $C^{1,\omega}$ regularity of
the chart). This shows that $\Sigma$ is not homotopic minimizing to either
side in $B_{\varepsilon}(p)$, so $p\in\mathfrak{h}_{\textrm{nm}}(\Sigma)$ as
claimed. ∎

###  3.2. Cancellation

We will assume the cancellation condition

$\mathbf{M}(\Phi(x_{0}))<W$

throughout this subsection. In particular, we can find
$q\in\operatorname{Reg}V$ so that for all $\varepsilon>0$ sufficiently small,

$\operatorname{Per}(\Omega\,|\,B_{\varepsilon}(q))<|V|(B_{\varepsilon}(q))$

where $\partial\Omega=\Phi(x_{0})$. Like in the previous section we set
$\Sigma:=\operatorname{supp}V$.

Furthermore we set $V=\sum_{i}\kappa_{i}\,|\Sigma_{i}|$, where each
$\Sigma_{i}$ is a minimal hypersurface with optimal regularity and
$\kappa_{i}\in\varmathbb{N}$ are constant multiplicities, by the constancy
theorem [Sim83b, Theorem 41.1]. So (3.1) becomes

(3.5) $\begin{gathered}\text{there is a (ONVP) sweepout
$\\{\Phi(x)=\partial\Omega(x)\\}$ and $x_{i}\nearrow x_{0}\in{\bf
m}_{L}(\Phi)$, so that}\\\ \text{$|\Phi(x_{i})|\to
V=\sum_{i}\kappa_{i}\,|\Sigma_{i}|\in\mathcal{R}$, $\Phi$ is not left
excessive at $x_{0}$ and}\\\ \text{there is $q\in\Sigma$ such that
}\,\operatorname{Per}(\Omega\,|\,B_{\varepsilon}(q))\leqslant|V|(B_{\varepsilon}(q))-\delta(\varepsilon)\quad\text{for
all }\varepsilon>0\,.\end{gathered}$

We write $\Omega=\Omega(x_{0})$ and observe that
$\Sigma\subset\overline{\Omega}$. We would like to claim that $\Sigma$ is
homotopically minimizing, but this condition might not make sense if $\Sigma$
is one-sided. However, thanks to the cancellation we can actually prove that
$\Sigma$ is area-minimizing in its neighborhood in $\Omega$ away from a small
ball around $q$.

######  Definition 3.5.

We will call a set $\Omega^{\prime}$ a
$(q,\varepsilon,\tau,\Sigma,\Omega)$-competitor if

$\big{(}\Omega\setminus
B_{\tau}(\Sigma)\big{)}\cup\big{(}B_{\varepsilon}(q)\setminus\Sigma\big{)}\subset\Omega^{\prime}\subsetneqq\Omega\setminus\Sigma\,.$

An $(q,\varepsilon,\tau,\Sigma,\Omega)$-competitor $\Omega^{\prime}$ will be
called a minimizing competitor if its perimeter is strictly less than
perimeter of any $(q,\varepsilon,\tau,\Sigma,\Omega)$-competitor
$\Omega^{\prime\prime}$ with $\Omega^{\prime}\subset\Omega^{\prime\prime}$.
(Note that we do not require $\operatorname{Per}(\Omega^{\prime})$ to be less
that the perimeter of all competitors, but only those that contain
$\Omega^{\prime}$).

######  Proposition 3.6.

Suppose (3.5) holds, then for every $\varepsilon>0$ there is $\tau>0$, such
that minimizing $(q,\varepsilon,\tau,\Sigma,\Omega)$-competitor does not
exist.

###### Proof.

For contradiction suppose there exists a minimizing
$(q,\varepsilon,\tau,\Sigma,\Omega)$-competitor $U$. Observe that by the
cancellation assumption for every $\eta>0$ we can find
$(q,\varepsilon,\tau,\Sigma,\Omega)$-competitors $\Omega^{\prime}$ with
$\operatorname{Per}(\Omega^{\prime})\leqslant W+\eta-\delta(\epsilon)$, where
$\delta(\epsilon)$ is from (3.5). It follows that

$\operatorname{Per}(U)\leqslant\operatorname{Per}(\Omega^{\prime})<W\,.$

If we choose $\tau>0$ sufficiently small, then by Lemma 1.12 there exists a
nested family $\\{E(x)\,:\,x\in[0,1]\\}$ with $E(0)=U$, $E(1)=\Omega$ and

$\operatorname{Per}(E(x))<W\,.$

Let $(x_{l},x_{0}]$ be the connected interval such that $\Omega(x)\setminus
U\neq\emptyset$, where $\\{\Phi(x)=\partial\Omega(x)\\}$, and define family
$\Psi\colon(x_{l},x_{0}+1]\to\mathcal{Z}_{n}(M,\varmathbb Z_{2})$ by

$\Psi(x):=\begin{cases}\partial(\Omega(x)\cap U)&\text{if
}x\in(x_{l},x_{0}]\\\ \partial E(x-x_{0})&\text{if
}x\in[x_{0},x_{0}+1]\end{cases}$

Clearly $\Psi$ is continuous, since $\Omega=\Omega(x_{0})$ and moreover we
have that

$\limsup_{y\to x}\mathbf{M}(\Psi(y))<\limsup_{y\to
x}\mathbf{M}(\Phi(x))\leqslant W$

for every $x\in(x_{l},x_{0})$ by strict minimality condition in Definition
3.5. For every $x\in[x_{0},x_{0}+1]$ we also have
$\mathbf{M}(\Psi(x))=\mathbf{M}(\partial E(x))<W$. This implies that $x_{0}$
is left excessive for $\Phi$ which is a contradiction. ∎

######  Proposition 3.7.

Suppose $V=\sum_{i}\kappa_{i}\,|\Sigma_{i}|$ is as in (3.5), then each
$\Sigma_{i}$ has stable regular part and
$\operatorname{\mathfrak{h}_{\textnormal{nm}}}(V)=\emptyset$. Moreover, for
every point $x\in support(V)$ there exists $r>0$, such that the support of $V$
is minimizing to one side in $B_{r}(x)$

###### Proof.

First we observe that we can find two points $q_{1}$ and $q_{2}$ in
$\operatorname{Reg}V$, such that for all $\varepsilon>0$ sufficiently small,

$\operatorname{Per}(\Omega\,|\,B_{\varepsilon}(q_{j}))<|V|(B_{\varepsilon}(q))\,.$

By Proposition 3.6 we have non-existence of minimizing
$(q_{j},\varepsilon,\tau,\Sigma,\Omega)$-competitors for $j=1,2$. This implies
that $\Sigma_{i}$ is area minimizing to one side in a small ball around every
point of $V$. In particular, we have
$\mathcal{H}^{0}(\operatorname{\mathfrak{h}_{\textnormal{nm}}}(V)))=0$.

The stability of the regular part of each $\Sigma_{i}$ also follows from the
non-existence of minimizing $(q,\varepsilon,\tau,\Sigma,\Omega)$-competitors.
Indeed, if a component $\Sigma_{i}$ has index $\geqslant 1$, then for
$\varepsilon>0$ sufficiently small, the minimal hypersurface
$\Sigma_{i}\setminus B_{\varepsilon}(q)$ with fixed boundary will be unstable
by a standard capacity argument. If $\Sigma_{i}$ is two-sided, then by
considering a minimization problem to one side of $\Sigma_{i}$ in
$B_{\tau}(\Sigma_{i})\setminus B_{\varepsilon}(q)$ we can find open set
$U\subset\Omega$, such that $\Omega\setminus U$ is a minimizing
$(q,\varepsilon,\tau,\Sigma,\Omega)$-competitor.

Suppose $\Sigma_{i}$ is one-sided. Since $\Sigma_{i}\subset\overline{\Omega}$
we have that $B_{\tau}(\Sigma_{i})\setminus\Sigma_{i}\subset\Omega$ for all
sufficiently small $\tau>0$. In particular, for small $\tau<\varepsilon$ we
can minimize in the class of hypersurfaces $\\{S\subset
B_{\tau}(\Sigma_{i}):S\cap B_{\varepsilon}(q)=\Sigma_{i}\cap
B_{\varepsilon}(q)\\}$ to obtain a minimizer $\Sigma_{i}^{\prime}$ in the same
homology class and open set $U\subset\Omega$ with $\partial
U=\Sigma_{i}\cup\Sigma_{i}^{\prime}$. Then $\Omega\setminus U$ is a minimizing
$(q,\varepsilon,\tau,\Sigma,\Omega)$-competitor. ∎

###  3.3. Multiplicity $2$ bound

In this subsection we show that if $\kappa_{i}>2$ for some $i$, then $x_{0}$
is excessive, by using simple comparisons with disks. Notice that if any
multiplicity satisfies $\kappa_{i}\geqslant 2$ then we must be in the
cancellation case considered above.

######  Lemma 3.8 (Multiplicity $2$ bound).

Let $V=\sum_{i}\kappa_{i}\,|\Sigma_{i}|$ be as in (3.1). Then
$\kappa_{i}\leqslant 2$ for every $i$.

###### Proof.

Suppose by contradiction $\kappa_{i}\geqslant 3$ for some $i$. Then let
$p\in\operatorname{Reg}(\Sigma_{i})$, $p\neq q$ (where $q$ is the cancellation
point considered above). Consider a ball $B_{r}(p)$, $r<\frac{1}{2}dist(p,q)$,
sufficiently small so that $\Sigma_{i}\cap B_{r}(p)$ is two-sided. Let
$\tau(r)>0$ be a small constant to be chosen later and set $U=B_{r}(p)\cap
B_{\tau}(\Sigma_{i})$.

Consider sequence $x_{j}\nearrow x_{0}$ with
$|\partial\Omega(x_{j})|\rightarrow V$. We can assume that the radius $r$ was
chosen sufficiently small, so that

(3.6) $\mathbf{M}(\partial\Omega(x_{j})\cap
U)\geqslant\left(\kappa_{i}-\frac{1}{10}\right)\omega_{n}r^{n}\,,$

for all $j$ large enough, where $\omega_{n}$ denotes the measure of the
$n$-dimensional ball of radius one.

Let $\Omega_{j}^{\prime}\subset\Omega(x_{j})$, $\Omega_{j}^{\prime}\setminus
U=\Omega(x_{j})\setminus U$, be a strict one-sided outer area minimizer in
$\Omega(x_{j})\cap U$. Observe that if $\Omega^{\prime}_{j}$ does not converge
to $\Omega(x_{0})$, then $\lim\Omega_{j}^{\prime}$ is a
$(q,\frac{1}{2}dist(p,q),\tau,\Sigma,\Omega(x_{0}))$-competitor, which
contradicts Proposition 3.6.

We conclude that $\lim\Omega_{j}^{\prime}=\Omega(x_{0})$. On the other hand,
by comparing $\Omega(x_{j})\setminus U$ to $\Omega_{j}^{\prime}$ and assuming
that $\tau(r)$ was chosen sufficiently small, we have that one-sided area
minimizing property of $\Omega_{j}^{\prime}$ implies

$\mathbf{M}(\partial\Omega_{j}^{\prime}\cap
U)\leqslant\operatorname{Per}(U)\leqslant\left(2+\frac{1}{10}\right)\omega_{n}r^{n}\,,$

For $\tau(r)$ sufficiently small and $j$ large we can apply Lemma 1.12 to find
a nested family $E(x)$ interpolating between $\Omega_{j}^{\prime}$ and
$\Omega$, such that

$\displaystyle\operatorname{Per}(E(x))$
$\displaystyle\leqslant\max\\{\mathbf{M}(\partial\Omega_{j}^{\prime}\setminus
U),\mathbf{M}(\partial\Omega(x_{0})\setminus
U)\\}+\left(2+\frac{2}{10}\right)\omega_{n}r^{n}$ $\displaystyle\leqslant
W-\left(1-\frac{3}{10}\right)\omega_{n}r^{n}.$

By combining families $\Omega(x)\cap\Omega_{j}^{\prime}$ and $E(x)$ we obtain
that $x_{0}$ is left-excessive. ∎

###  3.4. Proof of Theorem 0.4

The result follow immediately by combining Corollary 2.2 with Propositions
3.1, 3.2, 3.7 and Lemma 3.8. ∎

##  4\. Proof of Theorems 0.1, 0.2, and 0.3

In this section we prove Theorem 0.3 (Theorems 0.1 and 0.2 follow immediately
from Theorem 0.3 when combined with the facts that when $n=8$ all
singularities are regular and that the set of bumpy metrics is open and dense
[Whi91, Whi17]). Theorem 0.3 will follow from Theorem 0.4 and Proposition
1.13, together with a simple surgery procedure.

###  4.1. Surgery procedure

We show here how to regularize minimal hypersurfaces with regular
singularities under the assumption that the hypersurface minimizes area in a
small ball around each singularity.

######  Proposition 4.1 (Perturbing away regular singularities of locally area
minimizing surfaces).

For $(M^{n+1},g)$ a compact $C^{2,\alpha}$-Riemannian metric and
$\Sigma\in\mathcal{R}$ a minimal hypersurface, recall that
$\mathcal{S}_{0}(\Sigma)\subset\operatorname{Sing}\Sigma$ is defined to be the
set of singular points with a regular tangent cone. There is
$\tilde{g}\in\operatorname{Met}^{2,\alpha}(M)$ arbitrarily close to $g$ and
$\tilde{\Sigma}$ arbitrarily close in the Hausdorff sense to $\Sigma$ so that
$\tilde{\Sigma}$ is minimal with respect to $\Sigma$ and
$\mathcal{S}_{0}(\tilde{\Sigma})\subset\operatorname{\mathfrak{h}_{\textnormal{nm}}}(\tilde{\Sigma})=\operatorname{\mathfrak{h}_{\textnormal{nm}}}(\Sigma)$.

###### Proof.

For every
$p\in\mathcal{S}_{0}(\Sigma)\setminus\operatorname{\mathfrak{h}_{\textnormal{nm}}}(\Sigma)$,
and $\varepsilon_{0}=\varepsilon_{0}(p)$ so that
$\Sigma\cap(B_{\varepsilon_{0}}(p)\setminus p)$ is regular, we will show how
to perturb $g$ and $\Sigma$ so that $p$ becomes regular. We will do this by
making an arbitrarily small change to $g$, $\Sigma$ supported in
$B_{\varepsilon_{0}}(p)$. Because $\mathcal{S}_{0}$ is discrete (but not
necessarily closed when $n\geqslant 9$) it is easy to enumerate the elements
of
$\mathcal{S}_{0}(\Sigma)\setminus\operatorname{\mathfrak{h}_{\textnormal{nm}}}(\Sigma)$
and make a summably small change around each point. As such, it suffices to
consider just the perturbation near $p$.

By definition, taking $\varepsilon<\varepsilon_{0}$ sufficiently small,
$\Sigma\cap B_{\varepsilon}(p)$ is one-sided homotopy area-minimizing. For
concreteness write $\Sigma\cap B_{\varepsilon}(p)=\partial\Omega$ in
$B_{\varepsilon}(p)$ and assume that $\Omega$ is inner homotopy minimizing. By
Lemma 3.4, the tangent cone at $p$ is area-minimizing (to the same side).

We claim that (after taking $\varepsilon>0$ smaller if necessary) there is a
sequence of $\Sigma_{i}\in\mathcal{R}(B_{\varepsilon}(p))$ with stable regular
part, with $\Sigma_{i}\subset\Omega$, $\Sigma_{i}$ disjoint from $\Sigma$, and
$\Sigma_{i}\to\Sigma$. Indeed, we can apply Proposition 1.13 to conclude that
either (after shrinking $\varepsilon>0$), $\Omega$ is area-minimizing to the
inside, or there are $\Sigma_{i}$ as asserted.

In the case that $\Omega$ is area-minimizing to the inside, we can still
construct the $\Sigma_{i}$ by shrinking $\varepsilon>0$ even further so that
$\Omega$ is strictly area-minimizing to the inside and then minimizing area
with respect to a boundary $\Sigma\cap\partial B_{\varepsilon}(p)+\delta_{i}$,
for a sequence $\delta_{i}\to 0$; i.e., the boundary of $\Sigma\cap
B_{\varepsilon}(p)$ pushed slightly into $\Omega$. By the unique minimizing
property, the minimizers will converge back to $\Sigma$ in
$B_{\varepsilon}(p)$.

For $i$ sufficiently large we can write the intersection of $\Sigma_{i}$ with
the annulus $A(p,\varepsilon/5,\varepsilon)$ as a graph of function $u_{i}$
over $\Sigma$.

Reasoning as in Hardt–Simon [HS85, Theorem 5.6] (cf. [Liu19, Theorem 3.1]),
for $i$ sufficiently large, $\Sigma_{i}$ will be regular in
$B_{\varepsilon/2}(p)$. We now set

$\tilde{\Sigma}_{i}=(\Sigma_{i}\cap B_{\varepsilon/5})\cup(\Sigma\setminus
B_{\varepsilon}(p))\cup((\Sigma+\chi u_{i})\cap
A(p,\varepsilon/5,\varepsilon))$

where $\chi$ is a smooth cutoff function with $\chi\equiv 1$ on
$B_{\varepsilon/5}$ and $\chi\equiv 0$ on $B_{3\varepsilon/5}$. Note that

$H_{g}(\tilde{\Sigma}_{i})\quad\text{is supported in
$B_{4\varepsilon/5}(p)\setminus B_{\varepsilon/5}(p)$}$

and $\|H_{g}(\tilde{\Sigma}_{i})\|_{C^{2,\alpha}}=o(1)$ as $i\to\infty$.

Now, define $\tilde{g}=e^{2f}g$, in this new metric, since $\tilde{\Sigma}$ is
smooth, we have the transformation

$H_{\tilde{g}}(\tilde{\Sigma})=e^{-f}\left(H_{g}(\tilde{\Sigma})+\frac{\partial
f}{\partial\nu}\right)\,,$

where $\nu$ is the normal direction to $\Sigma$. Setting
$H_{\tilde{g}}(\tilde{\Sigma})=0$, this reduces to the equation

$H_{g}(\tilde{\Sigma})+\frac{\partial f}{\partial\nu}=0$

which implies that $f=-H_{g}(\tilde{\Sigma})\zeta(\nu)$, for a function
$\zeta(t)$ such that $\zeta^{\prime}(0)=1$ and $\zeta\equiv 0$ for
$|t|\geqslant\varepsilon/100$ is a solution. Since, as observed,
$H_{g}(\tilde{\Sigma})$ is supported in $A(p,\varepsilon/5,4\varepsilon/5)$,
so is the metric change, and since $\|u\|_{C^{4,\alpha}}\leqslant o(1)$ and
$\chi$ is smooth, we have

$\|g-\tilde{g}\|_{2,\alpha}=\|e^{f}-1\|_{C^{2,\alpha}}\|g\|_{2,\alpha}\leqslant
C\,\|u\|_{C^{4,\alpha}}\,\|g\|_{2,\alpha}=o(1)$

as $i\to\infty$. This completes the proof. ∎

###  4.2. Proof of Theorem 0.3

For $g\in\operatorname{Met}^{2,\alpha}(M)$, apply Theorem 0.4 to find
$V\in\mathcal{R}$ with

$\mathcal{H}^{0}(\operatorname{\mathfrak{h}_{\textnormal{nm}}}(V))+\operatorname{Index}(V)\leqslant
1.$

We can apply Proposition 4.1 to $\Sigma=\operatorname{supp}V$ to find a metric
$\tilde{g}$ that is arbitrarily $C^{2,\alpha}$-close to $g$ and a $\tilde{g}$
minimal hypersurface $\tilde{\Sigma}\in\mathcal{R}$ so that
$\mathcal{S}_{0}(\Sigma)\subset\operatorname{\mathfrak{h}_{\textnormal{nm}}}(\Sigma)$.
(Note that if $\operatorname{Index}(V)=1$, then
$\operatorname{\mathfrak{h}_{\textnormal{nm}}}(\Sigma)=\emptyset$, so
$\mathcal{S}_{0}(\Sigma)=\emptyset$.) This completes the first part of the
proof.

We now consider
$g\in\operatorname{Met}^{2,\alpha}_{\operatorname{Ric}>0}(M)$.444The idea is
that positive Ricci curvature rules out stable hypersurfaces but this requires
the hypersurface to be two-sided. As such, we must consider two cases,
depending on whether $\Sigma$ is one or two-sided. If $\Sigma$ is two-sided,
then $\operatorname{Index}(\Sigma)\geqslant 1$, so we can argue as above. On
the other hand if $\Sigma$ is one-sided, then $[\Sigma]\neq 0\in
H_{n}(M,\varmathbb{Z}_{2})$. We can then find $\hat{\Sigma}\in[\Sigma]$ by
minimizing area in the homology class. The surface $\hat{\Sigma}$ may have
singularities, but they are all locally area minimizing. Thus, we can apply
Proposition 4.1 to $\Sigma$ yielding $\tilde{\Sigma}$ and $\tilde{g}$ with
$\mathcal{S}_{0}(\tilde{\Sigma})=\emptyset$. ∎

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