# Upper Semicontinuity of Index Plus Nullity for Minimal and CMC Hypersurfaces

Myles Workman
University College London

###### Abstract

We consider a sequence, $\\{M_{k}\\}_{k\in\mathbb{N}}$, of bubble converging
minimal hypersurfaces, or $H$-CMC hypersurfaces, in compact Riemannian
manifolds without boundary, of dimension $4\leq n+1\leq 7$. We prove the
following upper semicontinuity of index plus nullity:

$\displaystyle\limsup_{k\rightarrow\infty}(\text{ind}\,(M_{k})+\text{nul}\,(M_{k}))$
$\displaystyle\leq$ $\displaystyle\sum_{i=1}^{\mu}co(m)_{i}(\text{anl-
ind}\,(co(M^{i}_{\infty}))+\text{anl-nul}\,(co(M^{i}_{\infty})))$
$\displaystyle+\sum_{j=1}^{J}\text{ind}\,(\Sigma^{j})+\text{nul}_{\omega_{\Sigma^{j},R}}\,(\Sigma^{j}),$

for such a bubble converging sequence
$M_{k}\rightarrow(\cup_{i=1}^{\mu}M_{\infty}^{i},\Sigma^{1},\ldots,\Sigma^{L})$,
where $co(m)_{i}\in\mathbb{Z}_{\geq 1}$ is a notion of multiplicity of the
convergence to the connected component $M_{\infty}^{i}$, and
$\Sigma^{1},\ldots,\Sigma^{L}$ are the bubbles. This complements the
previously known lower semicontinuity of index obtained by Buzano–Sharp [12],
and Bourni–Sharp–Tinaglia [11].

The strategy of our proof is to analyse a weighted eigenvalue problem along
our sequence of hypersurfaces, $\\{M_{k}\\}_{k\in\mathbb{N}}$. This strategy
is inspired by the recent work of Da Lio–Gianocca–Rivière [18]. A key aspect
of our proof is making use of a Lorentz–Sobolev inequality to study the
behaviour of these weighted eigenfunctions on the neck regions along the
sequence, as well as the index and nullity of our non-compact bubbles
$\Sigma^{1},\ldots,\Sigma^{J}$.

Let $(N,g)$ be a compact Riemannian manifold, with no boundary, of dimension
$n+1$ ($2\leq n\leq 6$), and $H>0$, be a fixed constant. In this work we
investigate two classes, $\mathfrak{M}(N,g)$, and $\mathfrak{C}_{H}(N,g)$.
Here, $\mathfrak{M}(N,g)$ is the class of smooth, closed, embedded, minimal
hypersurfaces of $N$, with respect to the metric $g$, and
$\mathfrak{C}_{H}(N,g)$ is the class of smooth, closed, embedded hypersurfaces
in $N$, of constant mean curvature $H>0$, with respect to the metric $g$.

As these hypersurfaces arise as critical points to appropriately chosen area-
type functionals, a natural property to study is their Morse index (with
respect to the associated functional). For two fixed numbers $\Lambda>0$, and
$I\in\mathbb{Z}_{\geq 0}$, we define the subclasses,

$\begin{split}\mathfrak{M}(N,g,\Lambda,I)=\\{M\in\mathfrak{M}(N,g)\colon\mathcal{H}^{n}_{g}(M)\leq\Lambda,\,\text{ind}\,(M)\leq
I\\},\\\
\mathfrak{C}_{H}(N,g,\Lambda,I)=\\{M\in\mathfrak{C}_{H}(N,g)\colon\mathcal{H}^{n}_{g}(M)\leq\Lambda,\,\text{ind}\,(M)\leq
I\\}.\\\ \end{split}$

Relying on the curvature estimates for stable minimal hypersurfaces by
Schoen–Simon–Yau [33] and Schoen–Simon [32] (see also the recent proof by
Bellettini [7]), compactness properties have been proven for
$\mathfrak{M}(N,g,\Lambda,I)$ by Sharp [34], and for
$\mathfrak{C}_{H}(N,g,\Lambda,I)$ by Bourni–Sharp–Tinaglia [11]. We note that
$\mathfrak{M}(N,g,\Lambda,I)$ is sequentially compact (under the correct
notion of convergence), whereas for $\mathfrak{C}_{H}(N,g,\Lambda,I)$, we must
expand our class to quasi-embedded, $H$-CMC hypersurfaces (see Definition 1).
We denote this enlarged class by $\overline{\mathfrak{C}_{H}}(N,g,\Lambda,I)$.
We briefly describe this notion of convergence, with full details described in
point 1 of Definition 2. Consider a sequence
$\\{M_{k}\\}\subset\mathfrak{M}(N,g,\Lambda,I)$ (resp.
$\mathfrak{C}_{H}(N,g,\Lambda,I)$), then after potentially taking a
subsequence and renumerating, there is a smooth, closed, embedded minimal
hypersurface (resp. $H$-CMC quasi-embedded hypersurface) $M_{\infty}$, and a
finite set of points $\mathcal{I}\subset M_{\infty}$, where $|\mathcal{I}|\leq
I$, such that on compact subsets of $N\setminus\mathcal{I}$, $M_{k}$ will
converge to $M_{\infty}$, smoothly and locally graphically, with integer
multiplicity potentially greater than $1$. It is then shown that
$M_{\infty}\in\mathfrak{M}(N,g,\Lambda,I)$ (resp.
$\overline{\mathfrak{C}_{H}}(N,g,\Lambda,I)$). The set of points
$\mathcal{I}\subset M_{\infty}$, is defined by the condition that for each
$y\in\mathcal{I}$, there exists a sequence points $\\{x_{k}^{y}\in
M_{k}\\}_{k\in\mathbb{N}}$, such that $x_{k}^{y}\rightarrow y$, and the
curvature $|A_{M_{k}}(x_{k}^{y})|$, blows up as $k\rightarrow\infty$. Thus we
call $\mathcal{I}$ the singular set of the convergence. We also direct the
reader to similar results for minimal and CMC surfaces in $3$-manifolds [16,
5, 31, 40, 36].

In a bid to understand the formation of these singularities, a bubble analysis
was carried out by Chodosh–Ketover–Maximo [14], Buzano–Sharp [12], and
Bourni–Sharp–Tinaglia [11] (See also the work of Ros [31] and White [40]).
Zooming in at appropriate rates, along particular sequences of points
converging onto $\mathcal{I}$, yields a complete, embedded, non-planar,
minimal hypersurface in $\mathbb{R}^{n+1}$, of finite index, with Euclidean
volume growth at infinity. These minimal hypersurfaces in $\mathbb{R}^{n+1}$
are referred to as the ‘bubbles’, and they are the singularity models at the
singular points of the convergence. The hypersurface $M_{\infty}$ is referred
to as the ‘base’. This terminology is borrowed from other non-linear geometric
problems. See Figure 1 for a heuristic picture.

$M_{\infty}$$y$$\Sigma$ Figure 1: The top row depicts a local picture of a
converging sequence about a point $y\in\mathcal{I}$, which converges with
multiplicity 2 on the base $M_{\infty}$. The second row then depicts a
dilation of the dotted circles in the top row, which when we take a limit, as
seen in the last column, allows us to see a catenoid as the bubble $\Sigma$.

One may be interested in certain information about the hypersurfaces along
these sequences, for example; index, topology and total curvature.
Understanding the formation of these singularities through this bubble
analysis allows us to track this information along the sequence, and how it
behaves when taking the limit $M_{k}\rightarrow M_{\infty}$. For example, if
we have our sequence $\\{M_{k}\\}$ as above, and we know all our ‘bubbles’ are
given by $\Sigma^{1},\ldots,\Sigma^{J}\subset\mathbb{R}^{n+1}$, then we say
that $M_{k}\rightarrow(M_{\infty},\Sigma^{1},\ldots,\Sigma^{J})$ ‘bubble
converges’ (see Definition 2 for a detailed definition), and [12, 11],

$\text{ind}\,(M_{\infty})+\sum_{j=1}^{J}\text{ind}\,(\Sigma^{j})\leq\liminf_{k\rightarrow\infty}\,\text{ind}\,(M_{k}).$
(1)

This inequality gives a quantitative way of accounting for some of the index
lost when taking the limit $M_{k}\rightarrow M_{\infty}$. In this work we are
interested in proving an opposite inequality, which will give a finer analysis
and description of the index along such a converging sequence, and shows that
in certain situations, this bubble analysis will account for all the index in
the limit.

Our main result is Theorem 1 below, however we first illustrate the
conclusions in a simplified setting with the following special case. We delay
the statement of Theorem 1 until the end of this section.

######  Corollary 1.

Consider a compact Riemannian manifold $(N,g)$, without boundary and of
dimension $n+1$, $3\leq n\leq 6$. Let $\\{M_{k}\\}$ be a sequence of smooth,
closed, embedded minimal hypersurfaces of $(N,g)$, such that
$M_{k}\rightarrow(M_{\infty},\Sigma^{1},\ldots,\Sigma^{J})$ bubble converges
as in Definition 2, with $M_{\infty}\subset N$ being a smooth, connected, two-
sided, closed, embedded minimal hypersurface. Then:

$\displaystyle\limsup_{k\rightarrow\infty}(\text{ind}\,(M_{k})+\text{nul}\,(M_{k}))$
$\displaystyle\leq$ $\displaystyle
m(\text{ind}\,(M_{\infty})+\text{nul}\,(M_{\infty}))$
$\displaystyle+\sum_{j=1}^{J}\text{ind}\,(\Sigma^{j})+\text{nul}_{\omega_{\Sigma^{j},R}}\,(\Sigma^{j}),$

where $m\in\mathbb{Z}_{\geq 1}$ is the multiplicity of the convergence onto
$M_{\infty}$. Here,

$\text{ind}\,(\Sigma^{j})=\lim_{S\rightarrow\infty}\text{ind}\,(\Sigma^{j}\cap
B_{S}^{n+1}(0)),$

and $R$ may be chosen to be any finite positive real number greater than some
$R_{0}=R_{0}(\Sigma^{1},\ldots,\Sigma^{L})\in[1,\infty)$, and

$\omega_{\Sigma^{j},R}(x)=\begin{cases}R^{-2},\,x\in
B_{R}^{n+1}(0)\cap\Sigma^{j},\\\ |x|^{-2},\,x\in\Sigma^{j}\setminus
B_{R}^{n+1}(0).\end{cases}$

and,

$\text{nul}_{\omega_{\Sigma^{j},R}}\,(\Sigma^{j})=\text{dim}\,\\{f\in
C^{\infty}(\Sigma^{j})\colon\Delta
f+|A_{\Sigma^{j}}|^{2}f=0,\,f^{2}\,\omega_{\Sigma^{j},R}\in
L^{1}(\Sigma^{j}),\,|\nabla f|^{2}\in L^{1}(\Sigma^{j})\\}.$

The inequality in Corollary 1 can be further strengthened by noting that in
this situation if $m\geq 2$, then $\text{ind}\,(M_{\infty})=0$ ([34, Claim
6]).

Results on the lower semicontinuity of index along converging sequences (1),
are common in the literature. For certain classes of minimal hypersurfaces see
Sharp [34], Buzano–Sharp [12], Ambrozio–Carlotto–Sharp [4] and
Ambrozio–Buzano–Carlotto–Sharp [3], and for certain classes of CMC
hypersurfaces see Bourni–Sharp–Tinaglia [11]. In the setting of Allen–Cahn
solutions see Le [24], Hiesmayr [22], Gaspar [21] and Mantoulidis [27]. For
the setting of of bubble converging harmonic see Moore–Ream [29, Theorem 6.1],
and Hirsch–Lamm [23, Theorem 1.1].

The opposite upper semicontinuity inequality (Theorem 1) is more intricate. We
recall a few examples of such results from the literature. When convergence
happens with multiplicity one for sequences of critical points of the
Allen–Cahn functional, upper semicontinuity of the index plus nullity has been
established by Chodosh–Mantoulidis [15, Theorem 1.9] and Mantoulidis [27,
Theorem 1 (c)]. In the case of bubble converging harmonic maps such an
inequality was first established by Yin [41, 42], and then by Da
Lio–Gianocca–Rivière [18] and Hirsch–Lamm [23]. Note that in [18] and [23,
Section 6] the proofs are for a bubble converging sequence of critical points
for a general class of conformally invariant lagrangians (fixed along the
sequence).

When combined with the lower semicontinuity of index, the inequality in
Theorem 1 shows that in the case of the limiting hypersurface being two-sided
and minimal (as is the case of Corollary 1), the index along the sequence can
be _fully_ accounted for in the limit. Thus we should view Theorem 1 as saying
that we cannot lose index to the neck, or index cannot merely just disappear
in the bubble convergence of Buzano–Sharp [12] for minimal hypersurfaces.

In order to conclude that the inequality in Theorem 1 is non-trivial, we must
show that for each bubble, $\Sigma^{j}$, of finite index,
$\text{nul}_{\omega_{\Sigma^{j},R}}\,(\Sigma^{j})<+\infty$. This is shown in
Proposition 9. Proposition 9 also has the following Corollary which may be of
interest to some readers (and may be compared with [25, Corollary 2]).

######  Corollary 2.

Let $3\leq n\leq 6$, and $\iota\colon\Sigma\rightarrow\mathbb{R}^{n+1}$ be a
complete, proper, minimal immersion, of codimension 1, with finite index and
Euclidean volume growth at infinity, i.e.

$\limsup_{r\rightarrow\infty}\frac{\mathcal{H}^{n}(\iota(\Sigma)\cap
B_{r}^{n+1}(0))}{r^{n}}<+\infty.$

Then:

$\text{nul}\,(\Sigma)\coloneqq\text{dim}\,\\{f\in W^{1,2}(\Sigma)\colon\Delta
f+|A_{\Sigma}|^{2}f=0\\}<+\infty.$

We note that as $\Sigma$ is not compact, Corollary 2 does not follow from
analysing the spectrum of a compact operator. In fact in the case when $n=2$,
and all the other assumptions in Corollary 2 hold, the spectrum of the
stability operator $L$ is not discrete, and in fact its essential spectrum is
equal to $[0,\infty)$ ([30, Theorem 5.3]).

We briefly remark on the strategy of the proof for Theorem 1, which is close
to the strategy of Da Lio–Gianocca–Rivière [18]. We prove Theorem 1 by
reframing the problem in terms of a weighted eigenvalue problem. The weight is
specifically chosen so that sequences of normalised weighted eigenfunctions
$\\{f_{k}\\}$, along the sequence $\\{M_{k}\\}$, with non-positive weighted
eigenvalues, exhibit good convergence on the base $M_{\infty}$, and the
bubbles, $\Sigma^{1},\ldots,\Sigma^{J}$. The key steps for the proof are
showing the equivalence of the weighted and unweighted eigenvalue problems
(Section 3), the convergence on the base $M_{\infty}$ (Section 2.2), and the
convergence on the bubbles $\Sigma^{1},\ldots,\Sigma^{J}$ (Section 2.3), along
with a Lorentz–Sobolev inequality on the neck, which shows that the normalised
weighted eigenfunctions cannot concentrate on the neck (Section 2.4).

Due to the different settings, there are several key differences between our
work and that of [18]. One such difference is our use of a Lorentz–Sobolev
inequality to deduce strict stability on the neck. Another major difference is
that in our setting the bubbles are non-compact. This poses significant
complications in the theory of the elliptic operator on the bubble. In
particular its spectrum may not be discrete, and thus effectively analysing
the index and nullity of these bubbles is intricate.

It is worth pointing out that the method used in [18], and in Theorem 1 is
rather general. In the proof of Theorem 1, only a few aspects rely
specifically on the mean curvature assumptions of the submanifolds. Thus it is
plausible that the ideas and techniques could be applied to a large range of
problems in which one wishes to study how an elliptic PDE behaves along a
sequence of (sub)manifolds which ‘bubble converge’ in an appropriate sense.

######  Theorem 1.

For a compact Riemannian manifold $(N,g)$ without boundary, of dimension
$n+1$, $3\leq n\leq 6$, if we have a sequence
$\\{M_{k}\\}\subset\mathfrak{M}(N,g)$
($\\{M_{k}\\}\subset\mathfrak{C}_{H}(N,g)$), such that
$M_{k}\rightarrow(M_{\infty},\Sigma_{1},\ldots,\Sigma_{J})$ bubble converges
as in Definition 2, with $M_{\infty}=\cup_{i=1}^{\mu}M^{i}$, where each
$M^{i}$ is a closed minimal hypersurface (resp. closed, quasi-embedded $H$-CMC
hypersurface, with $e(M^{i})$ connected) and $\theta_{|M^{i}}\equiv
m_{i}\in\mathbb{Z}_{\geq 1}$, then

$\displaystyle\limsup_{k\rightarrow\infty}(\text{ind}\,(M_{k})+\text{nul}\,(M_{k}))$
$\displaystyle\leq$ $\displaystyle\sum_{i=1}^{\mu}co(m)_{i}(\text{anl-
ind}\,(co(M^{i}_{\infty}))+\text{anl-nul}\,(co(M^{i}_{\infty})))$
$\displaystyle+\sum_{j=1}^{J}\text{ind}\,(\Sigma^{j})+\text{nul}_{\omega_{\Sigma^{j},R}}\,(\Sigma^{j}),$

where for each $i=1,\ldots,\mu$, $co(m)_{i}\in\mathbb{Z}_{\geq 1}$, is such
that $co(m)_{i}\leq m_{i}$ if $M^{i}$ is one-sided, and $co(m)_{i}=m_{i}$ if
$M^{i}$ is two-sided. Here,

$\text{ind}\,(\Sigma^{j})=\lim_{S\rightarrow\infty}\text{ind}\,(\Sigma^{j}\cap
B_{S}^{n+1}(0)),$

and $R$ may be chosen to be any finite positive real number greater than some
$R_{0}=R_{0}(\Sigma^{1},\ldots,\Sigma^{L})\in[1,\infty)$, and

$\omega_{\Sigma^{j},R}(x)=\begin{cases}R^{-2},\,x\in
B_{R}^{n+1}(0)\cap\Sigma^{j},\\\ |x|^{-2},\,x\in\Sigma^{j}\setminus
B_{R}^{n+1}(0).\end{cases}$

and,

$\text{nul}_{\omega_{\Sigma^{j},R}}\,(\Sigma^{j})=\text{dim}\,\\{f\in
C^{\infty}(\Sigma^{j})\colon\Delta
f+|A_{\Sigma^{j}}|^{2}f=0,\,f^{2}\,\omega_{\Sigma^{j},R}\in
L^{1}(\Sigma^{j}),\,|\nabla f|^{2}\in L^{1}(\Sigma^{j})\\}.$

The exact method of proof we implore does not extend to the case of $n=2$ ($2$
dimensional surfaces in $3$-manifolds). Two key reasons are the choice of
weight (Remark 3), and the criticality of the Lorentz–Sobolev inequality
(Proposition 3) for $n=p=2$.

We take a moment to comment on the terms $\text{anl-ind}\,(M_{\infty}^{i})$
and $\text{anl-nul}\,(M_{\infty}^{i})$, that appear in the statement of
Theorem 1. These terms respectively stand for the _analytic index_ and
_analytic nullity_ of $M_{\infty}^{i}$. This refers to the index and nullity
of the stability operator acting on the function space
$C^{\infty}(co(M_{\infty}^{i}))$, where $co(M_{\infty}^{i})$ is a connected
component of the oriented double cover of $M_{\infty}^{i}\subset N$. We
explain the reasoning behind this with the following example, which is also
demonstrated in Figure 2. Consider a unit hypersphere in $\mathbb{R}^{n+1}$
(this is a CMC hypersurface), then the function $f=1$ on the hypersphere, is
an eigenfunction of the stability operator, with negative eigenvalue, and
corresponds to shrinking the hypersphere. Now consider a sequence of two
disjoint unit hyperspheres in $\mathbb{R}^{n+1}$, such that in the limit they
touch at a point. In order to account for these eigenfunctions in the limit,
we must allow for variations that act on the hyperspheres independently, even
at the touching point. Thus we view the hyperspheres as immersions, and allow
variations which ‘shrink’ the hyperspheres separately. This type of variation
cannot arise through an ambient vector field due to the behaviour at the
touching point. Thus in general, the analytic index and analytical nullity of
$M_{\infty}$, will not be equivalent to the Morse index and nullity of
$M_{\infty}$, which is customarily defined through ambient vector fields. See
Section 1.2 for further details.

Figure 2: Sequence of two spheres coming together to touch at a point. The
arrows attached to the spheres demonstrate the ambient vector field needed to
give rise to the eigenfunction which corresponds to ‘shrinking’ of these
spheres. Notice that at the non-embedded point this vector field is not well
defined as an ambient one.

## Acknowledgements

First and foremost I am indebted to my supervisor Costante Bellettini for time
and support throughout this project. I also wish to thank Ben Sharp and
Giuseppe Tinaglia for informative discussions at the beginning of this
project. Moreover, I am grateful to Matilde Gianocca for taking the time to
answer questions related to [18], as well as Jonas Hirsch for an interesting
and insightful conversation on aspects related to this work, and [23]. I would
also like to thank Royu P.T. Wang for an interesting discussion on spectral
theory, and Louis Yudowitz for his interest in this work and valuable comments
on preliminary versions of the manuscript.

The author was supported by the Engineering and Physical Sciences Research
Council [EP/S021590/1]. The EPSRC Centre for Doctoral Training in Geometry and
Number Theory (The London School of Geometry and Number Theory), University
College London.

###### Contents

  1. 1 Preliminaries
    1. 1.1 Bubble Convergence Preliminaries
    2. 1.2 Stability Operator, Index and Nullity
    3. 1.3 Lorentz Spaces
  2. 2 Weighted Eigenfunctions
    1. 2.1 The Weight
    2. 2.2 Convergence on the Base
    3. 2.3 Convergence on the Bubble
    4. 2.4 Strict Stability of the Neck
  3. 3 Equivalence of Weighted and Unweighted Eigenspaces
  4. 4 Proof of the Theorem
  5. 5 Finiteness of the Nullity

##  1 Preliminaries

Let $(N,g)$ be a Riemannian manifold of dimension $n+1$, with $n\geq 2$. We
define the set $\mathfrak{M}(N,g)$ to be the set of closed, smooth, embedded
minimal hypersurfaces $M$ in $N$, with respect to the metric $g$. Similary,
for a constant $H>0$, we define the set, $\mathfrak{C}_{H}(N,g)$, to be the
set of closed, smooth, embedded, $H$-CMC hypersurfaces in $N$, with respect to
the metric $g$.

Throughout the paper, for $r>0$, $m\in\mathbb{Z}_{\geq 1}$, and
$p\in\mathbb{R}^{m}$, we denote, $B_{r}^{m}(p)$ as the open ball in
$\mathbb{R}^{m}$ of radius $r$ centred at point $p$. For $p\in N$, and
$r\in(0,\text{inj}\,(N))$, we denote $B_{r}^{N}(p)$ to be the open geodesic
ball of radius $r$ centred at point $p$.

We denote $n$-rectifiable varifolds $V$, in $N$, by $V=(\Sigma,\theta)$, where
$\Sigma\subset N$ is an $n$-rectifiable set, and
$\theta\colon\Sigma\rightarrow\mathbb{R}_{\geq 0}$, is a
$\mathcal{H}^{n}_{g}$-measureable function on $\Sigma$. Here,
$\mathcal{H}^{n}_{g}$ denotes the $n$-dimensional Hausdorff measure on $N$,
with respect to the metric $g$. We will drop the subscript $g$ when the
implied metric on the manifold is clear. Denote $\mu_{V}$ to be the associated
Radon measure on $(N,g)$ [35, Section 15]. We note that the definition of an
$n$-rectifiable varifold is always up to an equivalence (see the first
paragraph of [35, chapter 4]), however we will always take
$\Sigma=\text{spt}\,\mu_{V}$, and for $p\in\Sigma$,

$\theta(p)=\liminf_{r\rightarrow
0}\frac{\mu_{V}(B_{r}^{N}(p))}{\omega_{n}r^{n}},$

where we are setting, $\omega_{n}=\mathcal{H}^{n}(B_{1}^{n}(0))$. We refer the
reader to [35, Chapter 4] for a more detailed discussion on rectifiable
$n$-varifolds.

######  Definition 1.

For $H>0$, we say that a closed set $M\subset N^{n+1}$ is a $H$-CMC quasi-
embedded hypersurface of $N$, if there exists a smooth manifold $S$ of
dimension $n$, and a smooth immersion $\iota\colon S\rightarrow N$, such that
$M=\iota(S)$, and for every $y\in M$, there exists a $\rho>0$, such that
either:

  1. 1.

$B_{\rho}^{N}(y)\cap M$ is a smooth embedded disk, of constant mean curvature
$H$,

  2. 2.

$B_{\rho}^{N}(y)\cap M$ is the union of two smooth embedded disks, both of
constant mean curvature $H$, and only intersecting tangentially along a set
containing $y$, with mean curvature vectors pointing in opposite directions.

Points $y\in M$ which satisfy 1 will be refered to as embedded points of $M$,
and we denote the set of such points by $e(M)$. Points in the set
$t(M)=M\setminus e(M)$, which satisfy 2, will be refered to as non-embedded
points of $M$.

It is worth noting that in the literature these hypersurfaces are defined
under different names. In [43] they are refered to as almost embedded and in
[11] they are refered to as effectively embedded. We opt for the name quasi-
embedded as in [10]. We define the set $\overline{\mathfrak{C}_{H}}(N,g)$ to
be the set of quasi-embedded, $H$-CMC hypersurfaces in $N$, with respect to
the metric $g$.

###  1.1 Bubble Convergence Preliminaries

######  Definition 2.

Consider a Riemannian manifold $(N,g)$, of dimension $n+1$, $n\geq 2$, (and
$H>0$) along with a sequence
$\\{M_{k}\\}_{k\in\mathbb{N}}\subset\mathfrak{M}(N,g)$
($\\{M_{k}\\}_{k\in\mathbb{N}}\subset\mathfrak{C}_{H}(N,g)$), an
$M_{\infty}\in\mathfrak{M}(N,g)$
($M_{\infty}\in\overline{\mathfrak{C}_{H}}(N,g)$), and a collection of non-
planar minimal hypersurfaces $\\{\Sigma_{j}\\}_{j=1}^{J}$ in
$\mathbb{R}^{n+1}$, with $J\in\mathbb{Z}_{\geq 1}$. Then, we say that

$M_{k}\rightarrow(M_{\infty},\Sigma_{1},\ldots,\Sigma_{J}),$

bubble converges, if:

  1. 1.

$(M_{k},1)\rightarrow(M_{\infty},\theta)$ as varifolds, where
$\theta:M_{\infty}\rightarrow\mathbb{Z}_{\geq 1}$, and is constant on
connected, components on $M_{\infty}$ (resp. constant on connected components
of $e(M_{\infty})$). Moreover, there exists an at most finite collection of
points $\mathcal{I}\subset M_{\infty}$, such that for $y\in
M_{\infty}\setminus\mathcal{I}$, there exists a
$\rho\in(0,\text{inj}\,(N)/2)$, such that $M_{\infty}\cap
B_{\rho}^{N}(y)=\cup_{i=1}^{\theta(y)}D_{i,y}$, where each $D_{i,y}$ is a
smooth embedded minimal ($H$-CMC) disk containing $y$, and convergence of
$M_{k}$ to $M_{\infty}$ in $B_{\rho}^{N}(y)$, happens as a collection of
$\theta(y)$ disconnected, smooth disks, $\\{D_{i,y,k}\\}_{i=1}^{\theta(y)}$,
where for each $i=1,\ldots,\theta(y)$, $D_{i,y,k}\rightarrow D_{i,y}$
smoothly, and graphically.

  2. 2.

For each $i\in\\{1,\ldots,J\\}$, there exist point-scale sequences
$\\{(p_{k}^{i},r_{k}^{i})\\}_{k\in\mathbb{N}}$, such that for each
$k\in\mathbb{N}$, $p_{k}^{i}\in M_{k}$, and there exists a
$y^{i}\in\mathcal{I}$, such that $p_{k}^{i}\rightarrow y^{i}$,
$r_{k}^{i}\rightarrow 0$. Moreover, for each $R\in(0,\infty)$, and large
enough $k$, the connected component of $M_{k}\cap
B_{Rr_{k}^{i}}^{N}(p_{k}^{i})$, through $p_{k}^{i}$, labelled
$\Sigma_{k}^{i,R}$, is such that, if we rescale the geodesic ball
$B^{N}_{Rr_{k}^{i}}(p_{k}^{i})$ by $r_{k}^{i}$, and denote

$\tilde{\Sigma}_{k}^{i,R}\cap
B_{R}^{n+1}(0)\coloneqq\frac{1}{r_{k}^{i}}\exp_{p_{k}^{i}}^{-1}(\Sigma_{k}^{i,R}\cap
B_{Rr_{k}^{i}}^{N}(p_{k}^{i}))\subset B_{R}^{n+1}(0)\subset\mathbb{R}^{n+1},$

then $\tilde{\Sigma}_{k}^{i,R}\rightarrow\Sigma^{i}\cap B_{R}(0)$ smoothly and
graphically, with multiplicity one. Furthermore, for $i\not=j$, either

$\lim_{k\rightarrow\infty}\frac{r_{k}^{i}}{r_{k}^{j}}+\frac{r_{k}^{j}}{r_{k}^{i}}+\frac{\text{dist}_{g}^{N}(p_{k}^{i},p_{k}^{j})}{r_{k}^{i}+r_{k}^{j}}=\infty,$

or for each $R\in(0,\infty)$, and then large enough $k\in\mathbb{N}$,
$p_{k}^{j}\not\in\Sigma_{k}^{i,R}$.

  3. 3.

Defining,

$d_{k}(x)\coloneqq\min_{i=1,\ldots,L}\text{dist}_{g}^{N}(x,p_{k}^{i}),$

then,

$\lim_{\delta\rightarrow
0}\lim_{R\rightarrow\infty}\lim_{k\rightarrow\infty}\sup_{x\in
M_{k}\cap(\cup_{y\in\mathcal{I}}B^{N}_{\delta}(y)\setminus\cup_{i=1}^{J}\Sigma_{k}^{i,R})}\int_{M^{x}_{k}\cap
B^{N}_{d_{k}(x)/2}(x)}|A_{k}|^{n}=0,$

where $M_{k}^{x}\cap B^{N}_{d_{k}(x)/2}(x)$ is the connected component of
$M_{k}\cap B^{N}_{d_{k}(x)/2}(x)$, that contains $x$.

Consider such a convergence
$M_{k}\rightarrow(M_{\infty},\Sigma^{1},\ldots,\Sigma^{J})$, as in Definition
2, then we may remark:

  * •

The convergence considered in the bubble analysis of Buzano–Sharp [12] for
minimal hypersurfaces, and Bourni–Sharp–Tinaglia [11] for separating CMC
hypersurfaces, satisfy Definition 2.

  * •

By Allard’s Compactness Theorem for Integral Varifolds [1, Theorem 6.4], and
the Constancy Theorem ([35, Theorem 41.1] for the case of $M_{\infty}$ being a
minimal hypersurface, and [9, Lemma A.1] for the case of $M_{\infty}$ being a
quasi-embedded $H$-CMC hypersurface), the assumption that values of $\theta$
lie in $\mathbb{Z}_{\geq 1}$, and are constant on connected components of
$M_{\infty}$, (resp. $e(M_{\infty})$) follows from the varifold convergence,
and they are not additional assumptions.

  * •

As the multiplicity function $\theta\colon
M_{\infty}\rightarrow\mathbb{Z}_{\geq 1}$ is uniformly bounded, then we may
deduce by the monotonicity formula ([35, Corollary 17.8]) that
$\sup_{k}\mathcal{H}^{n}(M_{k})<+\infty$. Applying this fact, and again using
the monotonicity formula, we may deduce that each $\Sigma^{j}$ must have
Euclidean volume growth at infinity ([12, Corollary 2.6]).

  * •

The final sentence of point 2 in Definition 2 guarantees that all of these
bubbles are distinct.

  * •

The fact that the convergence onto the bubble is of multiplicity one is not an
extra assumption. This follows from the fact that $\mathbb{R}^{n+1}$ is simply
connected, and the convergence onto the bubble is smooth.

  * •

Without loss of generality we may assume that $|A_{\Sigma^{j}}|(0)>0$, for
each $j=1,\ldots,J$.

We may also now assume that each $\Sigma^{j}$ has finite index. This is a
reasonable assumption for us to make, as if it does not hold then the result
that we are interested in (Theorem 1) would hold trivially.

The following curvature estimate is important in our analysis, and the proof
follows from a standard point-picking argument, which may be found in [39,
Lecture 3].

######  Proposition 1.

Given a Riemannian manifold $(N,g)$, of dimension $n+1\geq 3$, (and $H>0$)
there exists an $\varepsilon_{0}=\varepsilon_{0}(N,g)$
($=\varepsilon_{0}(N,g,H)$) and an $r_{0}=r_{0}(N,g)$ ($=r_{0}(N,g,H)$)
$\leq\text{inj}\,(N)$ such that if $M\in\mathfrak{M}(N,g)$
($M\in\mathfrak{C}_{H}(N,g)$), $x\in M$, $r\in(0,r_{0})$, and
$\varepsilon\in(0,\varepsilon_{0})$, such that

$\int_{B_{r}(x)\cap M}|A|^{n}\leq\varepsilon,$

then there is a $C_{\varepsilon}=C(N,g,\varepsilon)<+\infty$, such that

$\sup_{y\in B_{r}(x)\cap M}\text{dist}_{g}^{N}(y,\partial
B^{N}_{r}(x))|A|(y)\leq C_{\varepsilon},$

and moreover, $C_{\varepsilon}\rightarrow 0$ as $\varepsilon\rightarrow 0$.

Proposition 1 implies that if we take a sequences $\delta_{k}\rightarrow 0$
and $R_{k}\rightarrow\infty$, such that $R_{k}r_{k}^{j}\rightarrow 0$ for all
$j=1,\ldots,L$, and $\delta_{k}\geq 4R_{k}r_{k}^{j}$ for all $k\in\mathbb{N}$
and $j=1,\ldots,L$, and pick a sequence of points,

$x_{k}\in
M_{k}\cap(\cup_{y\in\mathcal{I}}B^{N}_{\delta}(y)\setminus\cup_{i=1}^{L}\Sigma_{k}^{i,R_{k}}),$

and denote, $s_{k}=d_{k}(x_{k})/2$, and

$\tilde{M_{k}}\cap
B_{1}^{n+1}(0)\coloneqq\frac{1}{s_{k}}\exp_{x_{k}}^{-1}(M_{k}\cap
B_{s_{k}}^{N}(x_{k})),$

then the component of $\tilde{M}_{k}$ through the origin must smoothly
converge to a plane through the origin, on compact sets of $B_{1}^{n+1}(0)$.

######  Lemma 1.

Consider a point-scale sequence $\\{(p_{k}^{i},r_{k}^{i})\\}$ from Definition
2, and a sequence $s_{k}\rightarrow 0$, such that
$s_{k}/r_{k}^{i}\rightarrow\infty$. Then if we denote
$\tilde{M}_{k}^{\prime}\subset B_{1}^{n+1}(0)$ to be the connected component
of

$\tilde{M}_{k}\cap
B_{1}^{n+1}(0)\coloneqq\frac{1}{s_{k}}\exp_{p_{k}^{i}}^{-1}(M_{k}\cap
B_{s_{k}}^{N}(p_{k}^{i})),$

that passes through the origin, we have that after potentially taking a
subsequence and renumerating, there exists a hyperplane
$E\subset\mathbb{R}^{n+1}$, and an $m\in\mathbb{Z}_{\geq 1}$, such that,

$(\tilde{M}_{k}^{\prime},1)\rightarrow(E,m),$

in $B_{1}^{n+1}(0)$ as varifolds, and smoothly, as $m$ disjoint graphs, on
open sets of $B_{1}^{n+1}(0)$ which are compactly contained away from finitely
many points in $\overline{B_{1}^{n+1}(0)}$.

###### Proof.

For any $R\in(2,\infty)$, and large enough $k$, consider the map,

$\displaystyle G_{k}^{R}\colon B_{R}^{n+1}(0)$ $\displaystyle\rightarrow$
$\displaystyle B_{Rs_{k}}^{N}(p_{k}^{i}),$ $\displaystyle x$
$\displaystyle\mapsto$ $\displaystyle\exp_{p_{k}^{i}}(s_{k}x),$

and the metric $g_{k}=(G^{R}_{k})^{*}\,g$, on $B_{R}^{n+1}(0)$. We have that
$(g_{k})_{\alpha,\beta}\rightarrow\delta_{\alpha,\beta}$, smoothly as
$k\rightarrow\infty$. Denote $\tilde{M}_{k}=(G_{k}^{R})^{-1}(M_{k}\cap
B_{Rs_{k}}^{N}(p_{k}^{i}))$. By the monotonicity formula, for large enough $k$
we have,

$\mathcal{H}^{n}_{g_{k}}(\tilde{M}_{k}\cap B_{R}^{n+1}(0))\leq
C\,R^{n}\,m\,(\mathcal{H}^{n}_{g}(M_{\infty})+1)$

with $C=C(N,g)<+\infty$. Denote $\tilde{M}_{k}^{\prime}$ to be the connected
component of $\tilde{M}_{k}$ that passes through the origin. Thus, by Allard’s
Compactness Theorem [1, Theorem 6.4] (we use the theorem as stated in [35,
Theorem 42.7 and Remark 42.8]) we know that, after potentially taking a
subsequence and renumerating, there exists an integral, $n$-rectifiable,
stationary varifold $V$ in $B_{R}^{n+1}(0)$, such that
$(\tilde{M}_{k}^{\prime},1)\rightarrow V$, as varifolds. By the monotonicity
formula, as for each $k$, $0\in\tilde{M}_{k}^{\prime}$, and
$\tilde{M}_{k}^{\prime}$ is connected, then,
$0\in\text{spt}\,V\not=\emptyset$, and $\text{spt}\,V$ is a closed connected
set. Thus, if we can show that $\text{spt}\,V$ is a plane, then by the
Constancy Theorem [35, Theorem 41.1] we may conclude that $\theta\equiv
m\in\mathbb{Z}_{\geq 1}$.

After potentially taking a subsequence and renumerating, we may assume that
for each $j=1,\ldots,L$, the following sequence has a well defined limit
(potentially $+\infty$),

$\frac{\text{dist}_{g}^{N}(p_{k}^{i},p_{k}^{j})}{s_{k}}.$

We denote the set,

$B_{i,\\{s_{k}\\}}=\\{j\colon\lim_{k\rightarrow\infty}\text{dist}_{g}^{N}(p_{k}^{i},p_{k}^{j})/s_{k}\leq
2R\\},$

and for each $j\in B_{i,\\{s_{k}\\}}$, and large enough $k$, we denote,

$q_{k}^{j}\coloneqq(G_{k}^{3R})^{-1}(p_{k}^{j})\in B_{3R}^{n+1}(0),$

and, again after potentially taking a subsequence and renumerating,

$q^{j}\coloneqq\lim_{k\rightarrow\infty}q_{k}^{j}\in\overline{B_{2R}^{n+1}(0)}.$

######  Claim 1.

For each $r>0$, there exists a sequence $R_{k}\rightarrow\infty$, such that
for large enough $k$,

$G_{k}^{R}((\tilde{M}_{k}^{\prime}\cap B_{R}^{n+1}(0))\setminus\cup_{j\in
B_{i,\\{s_{k}\\}}}B_{r}^{n+1}(q^{j}))\subset(M_{k}\setminus\cup_{j=1}^{L}\Sigma_{k}^{j,R_{k}})\cap
B_{Rs_{k}}^{N}(p_{k}^{i}).$

We prove this claim by contradiction. We assume there exists an $r>0$,
$S<+\infty$, and a subsequence, which we may renumerate by such that there is
an $l\in\\{1,\ldots,L\\}$, and points,

$z_{k}\in G_{k}^{R}((\tilde{M}_{k}^{\prime}\cap
B_{R}^{n+1}(0))\setminus\cup_{j\in
B_{i,\\{s_{k}\\}}}B_{r}^{n+1}(q^{j}))\cap\Sigma_{k}^{l,S}.$

Thus,

$\text{dist}_{g}^{N}(z_{k},p_{k}^{l})<Sr_{k}^{l},$

and,

$\text{dist}_{g}^{N}(z_{k},p_{k}^{i})<Rs_{k}.$

Moreover, for large enough $k$,

$\min_{j=1,\ldots,L}\text{dist}_{g}^{N}(z_{k},p_{k}^{j})\geq\frac{r}{2}s_{k},$

which implies that,

$s_{k}<\frac{2S}{r}r_{k}^{l},$

and thus,

$\text{dist}_{g}^{N}(p_{k}^{i},p_{k}^{l})<\left(\frac{2R}{r}+1\right)Sr_{k}^{l}.$

Then as $\Sigma_{k}^{l,2(r^{-1}+R^{-1})SR}$ and $\tilde{M}_{k}^{\prime}$ are
both connected and contain $z_{k}$, we have that,

$p_{k}^{i}\in\Sigma_{k}^{l,2(r^{-1}+R^{-1})SR}.$

However, by smooth convergence on the bubble,

$s_{k}|A_{k}|(p_{k}^{i})\leq\frac{2S}{r}r_{k}^{l}|A_{k}|(p_{k}^{i})\leq
4\frac{S}{r}\sup_{\Sigma^{l}\cap
B_{2(r^{-1}+R^{-1})SR}^{n+1}(0)}|A_{\Sigma^{l}}|<+\infty$

which contradicts fact that,

$s_{k}|A_{k}|(p_{k}^{i})=\frac{s_{k}}{r_{k}^{i}}r_{k}^{i}|A_{k}|(p_{k}^{i})\rightarrow\infty.$

Therefore we have proven the claim.

Thus, for each $r>0$, small enough so that for each $k$,
$\tilde{M}_{k}^{\prime}\cap B_{R}^{n+1}(0)\setminus\cup_{j\in
B_{i,\\{s_{k}\\}}}B_{r}^{n+1}(q^{j})$ is non-empty, we have by Point 3 of
Definition 2, and Proposition 1, that $\text{spt}\,V\setminus\cup_{j\in
B_{i,\\{s_{k}\\}}}B_{r}^{n+1}(q^{j})$ must be a collection of hyperplanes.
Moreover, the convergence on $B_{R}^{n+1}(0)\setminus\cup_{j\in
B_{i,\\{s_{k}\\}}}B_{r}^{n+1}(q^{j})$, must be smooth, and thus by the
embeddedness assumption along the sequence, these hyperplanes cannot intersect
transversely on $B_{R}^{n+1}(0)\setminus\cup_{j\in
B_{i,\\{s_{k}\\}}}B_{r}^{n+1}(q^{j})$. Therefore, after potentially taking $R$
larger, and renumerating along a further subsequence, we see that these
hyperplanes must be parallel. Then taking $r\rightarrow 0$ we see that
$\text{spt}\,V$ must be this collection of parallel hyperplanes. To conclude
we note that as $\text{spt}\,V$ is connected, it must just be a single
hyperplane through the origin, which we converge to smoothly, away from the
finite set of points $\\{q^{j}\colon j\in B_{i,\\{s_{k}\\}}\\}$. ∎

######  Lemma 2.

Take $\rho>0$ and $2\leq n\leq 6$. Consider a sequence of smooth, embedded,
minimal ($H_{k}$-CMC) hypersurfaces, $\\{M_{k}\\}\subset B_{\rho}^{n+1}(0)$,
with respect to Riemannian metrics $\\{g_{k}\\}$, with $(B_{\rho}(0),g_{k})$
being geodesic normal coordinates about $0$. Further suppose that
$g_{k}\rightarrow g_{\mathbb{R}^{n+1}}$ smoothly, ($H_{k}\rightarrow 0$), and

$(M_{k},1)\rightarrow(\\{x_{n+1}=0\\},m),$

as varifolds, with $m\in\mathbb{Z}_{\geq 1}$, and that on open sets
$K\subset\subset B_{\rho}^{n+1}(0)\setminus\\{0\\}$, the convergence is as $m$
disjoint graphs over $K$, smoothly converging to 0. Moreover, suppose we have
a sequence $r_{k}\rightarrow 0$, such that for any other convergent sequence
$\\{t_{k}\\}\subset(0,\rho]$, (say $t_{k}\rightarrow t_{\infty}\in[0,\rho]$),
which satisfies,

$\frac{t_{k}}{r_{k}}\rightarrow\infty$

we have the condition that, after potentially taking a subsequence and
renumerating,

$\tilde{M}_{k}\coloneqq\frac{M_{k}}{t_{k}}\subset B_{1/t_{k}}^{n+1}(0),$

converges (potentially with multiplicity) to some plane $E_{\\{t_{k}\\}}$,
through the origin, with the convergence being smooth and graphical on compact
sets of $B_{1/t_{\infty}}^{n+1}(0)\setminus\\{0\\}$,
($\mathbb{R}^{n+1}\setminus\\{0\\}$ if $t_{\infty}=0$). Then for any sequence
$\\{t_{k}\\}$ as above, $E_{\\{t_{k}\\}}=\\{x_{n+1}=0\\}$, and there exists an
$R_{0}$, such that if $C_{k}$ is a connected component of
$M_{k}\cap(B_{\rho}^{n+1}(0)\setminus B_{R_{0}r_{k}}^{n+1}(0))$, then,
$\tilde{C}_{k}\coloneqq C_{k}/t_{k}$ converges to $\\{x_{n+1}=0\\}$ with
multiplicity one.

###### Proof.

The multiplicity one convergence of $\tilde{C}_{k}$ may be seen by following
an argument used in [8, Proposition 3]. The fact that $E=\\{x_{n+1}=0\\}$
follows from the foliation and maximum principle argument in [12, Claim 1 of
Lemma 4.1] and [11, Lemma 5.6], making use of the fact that for each
$t\in[R_{0}r_{k},1]$, $C_{k}\cap\partial B_{t}(0)$ is connected. ∎

######  Remark 1.

([3, Lemma 42]) We remark that combining point 3 in Definition 2, along with a
contradiction argument that makes use of Proposition 1 and Lemma 2, implies
that there exists positive constants $R_{0}$ and $\delta_{0}$, such that for
each $y\in\mathcal{I}$, $\delta\in(0,\delta_{0})$, $R\in[R_{0},+\infty)$, and
taking large enough $k$, if we let $C_{k}$ denote a connected component of
$M_{k}\cap(B_{\delta}^{N}(y)\setminus\cup_{i=1}^{L}\Sigma_{k}^{i,R})$, then
there exists positive constants $\tau_{k}$, $s^{1}_{k},\ldots,s^{l}_{k}$, and
points $q_{k}^{1},\ldots,q_{k}^{l}\in T_{y}M_{\infty}$, with $l\leq L$,
$B_{s_{k}^{j}}^{n+1}(q_{k}^{j})\subset\subset B_{\tau_{k}}^{n+1}(0)$ for each
$j=1,\ldots,l$, and a smooth function

$u_{k}\colon(B_{\tau_{k}}^{n+1}(0)\setminus\cup_{j=1}^{l}B_{s_{k}^{j}}^{n+1}(q_{k}^{j}))\cap
T_{y}M_{\infty}\rightarrow\mathbb{R},$

such that,

$C_{k}=\\{\exp_{y}(x+u_{k}(x)\nu(y))\colon
x\in(B_{\tau_{k}}^{n+1}(0)\setminus\cup_{j=1}^{l}B_{s_{k}^{j}}^{n+1}(q_{k}^{j}))\cap
T_{y}M_{\infty}\\},$

where $\nu(y)$ is a choice of unit normal to $M_{\infty}$ at $y$. Moreover, we
have that $\|u_{k}\|_{L^{\infty}}\rightarrow 0$, and

$\lim_{\delta\rightarrow
0}\lim_{R\rightarrow\infty}\lim_{k\rightarrow\infty}\|\nabla
u_{k}\|_{L^{\infty}}=0.$

###  1.2 Stability Operator, Index and Nullity

Consider $M\subset N$, a closed, embedded minimal hypersurface and the area
functional,

$\mathcal{A}(M)\coloneqq\mathcal{H}^{n}(M).$

Letting $\Gamma(TN)$ denote the space of smooth vector fields on $N$, and
taking $X\in\Gamma(TN)$, then for small $\varepsilon>0$, we may define a flow,

$\Phi\colon N\times(-\varepsilon,\varepsilon)\rightarrow N,$

such that

$\frac{\partial}{\partial t}\Phi(\,\cdot\,,0)=X(\,\cdot\,).$

Then for the case of $M$ being minimal, we take the first variation of
$\mathcal{A}$, at $M$, in direction $X$ by [17, Chapter 1, Section 1.3]

$\delta\mathcal{A}(M)(X)\coloneqq\frac{d}{dt}_{|t=0}\mathcal{A}(\Phi(M,t))=0.$

We may also determine the second variation in direction $X$ [17, Chapter 1,
Section 8.1],

$\delta^{2}\mathcal{A}(M)(X)\coloneqq\frac{d^{2}}{dt^{2}}_{|t=0}\mathcal{A}(\Phi(M,t))=-\int_{M}g(X,LX),$

Here we have the elliptic operator $L$, which we call the stability operator.
For a point $x\in M$, taking $\\{E_{1},\ldots,E_{n}\\}$ to be an orthonormal
basis of $T_{x}M$, we may define the operator at $x$ by

$LX=\Delta_{M}^{\perp}X+\sum_{i,j=1}^{n}g(A(E_{i},E_{j}),X)\,A(E_{i},E_{j})+\text{Tr}\,R_{N}(\,\cdot\,,X)\,\cdot$

where $\Delta_{M}^{\perp}$ is the normal Laplacian to $M$, $A$ is the second
fundamental form of $M$, and $R_{N}$ is the Riemann curvature tensor of $N$.

We say that $\lambda\in\mathbb{R}$ is an eigenvalue of $L$, if there exists an
$X\in\Gamma(TN)$, such that $X_{|M}\not=0$, and

$LX+\lambda X=0.$

Then we define the Morse index of $M$ ($\text{ind}\,(M)$) to be the number of
negative eigenvalues of $L$, counted with the multiplicity given by the
dimension of the linear space of eigensections with such eingenvalue. The
nullity ($\text{nul}\,(M)$) of $M$ is defined to be the number of zero
eigenvalues, again counted with the same notion of multiplicity. We refer the
reader to [17, Chapter 1, Section 8] for a more detailed discussion.

Alternatively, for $H>0$, if $M\subset N$ is a closed, embedded, $H$-CMC
hypersurface, such that there is an open set $E\subset N$ with $\partial E=M$,
and the mean curvature vector of $M$ pointing into $E$, we may define the
functional

$\mathcal{F}_{H}(M)\coloneqq\mathcal{H}^{n}(M)-H\,\mathcal{H}^{n+1}(E),$

and as before, taking $X\in\Gamma(TN)$, we may compute the first and second
variations of $F_{H}$, at $M$, in direction $X$, [6, Propositions 2.3 and 2.5]

$\displaystyle\delta\mathcal{F}_{H}(M)(X)$ $\displaystyle\coloneqq$
$\displaystyle\frac{d}{dt}_{|t=0}\mathcal{F}_{H}(\Phi(M,t))=0,$
$\displaystyle\delta^{2}\mathcal{F}_{H}(M)(X)$ $\displaystyle\coloneqq$
$\displaystyle\frac{d^{2}}{d^{2}t}_{|t=0}\mathcal{F}_{H}(\Phi(M,t))=-\int_{M}g(X,LX).$

We then similarly define the index and nullity of $M$. Note that $M$ does not
need to be seperating on $N$ for the operator $L$ to be defined. Thus we do
not require the existence of an open set $E$ to define the index and nullity
of a $H$-CMC hypersurface.

Consider an embedded minimal hypersurface $M\subset N$. We define $o(M)$ to be
the oriented double cover of $M$,

$o(M)\coloneqq\\{(x,\nu)\colon x\in M,\,\nu\in
T_{x}^{\perp}M,\,\text{and}\,|\nu|=1\\},$

and $co(M)$ to be a connected component of $o(M)$. Note that if $M$ is two-
sided in $N$, then $co(M)=M$, and if $M$ is one-sided in $N$, then
$co(M)=o(M)$. We define the obvious continuous projections, $\iota\colon
o(M)\rightarrow M$, and $\nu\colon o(M)\rightarrow T^{\perp}M$, and note that
$z=(\iota(z),\nu(z))\in o(M)$.

For the case of $M\subset N$ being an quasi-embedded $H$-CMC hypersurface we
define,

$\displaystyle co(M)\coloneqq\\{(x,\nu)$ $\displaystyle\colon$ $\displaystyle
x\in M,\,\nu\in T_{x}^{\perp}M,\,|\nu|=1,$ $\displaystyle\text{and points in
direction of the mean curvature vector}\\}.$

Thus at non-embedded point $x\in M$ we have $(x,\nu)$, and $(x,-\nu)\in
co(M)$.

For $X\in\Gamma(TN)$ we define a function $f_{X}\in C^{\infty}(co(M))$ by,

$f_{X}(z)=g(X(\iota(z)),\nu(z)).$

and note that if $z_{1},\,z_{2}\in co(M)$ such that $z_{1}\not=z_{2}$ but
$\iota(z_{1})=\iota(z_{2})$, then $f_{X}(z_{2})=-f_{X}(z_{1})$. Thus,

$\displaystyle f_{X}\in C^{\infty}(co(M))^{-}\coloneqq\\{f\in C^{\infty}(M)$
$\displaystyle\colon$ $\displaystyle
if\,(x,\nu)\,\text{and}\,(x,-\nu)\,\text{are both in}\,co(M),$
$\displaystyle\text{then}\,f((x,-\nu))=-f((x,\nu))\\}.$

Moreover, by a standard extension argument, for each $f\in
C^{\infty}(co(M))^{-}$, there exists an $X_{f}\in\Gamma(TN)$ such that

$f(z)=g(X_{f}(\iota(z)),\nu(z)).$

Elements of the function space $C^{\infty}(co(M))^{-}$ should be thought of as
‘ambient variations’.

Thus, instead of considering the stability operator on $\Gamma(TN)$, it is
equivalent to consider the scalar elliptic operator (which we also denote by
$L$ and refer to as the stability operator),

$Lf=\Delta f+(|A_{M}|^{2}\circ\iota+\text{Ric}_{N}\,(\nu,\nu))f,$

on the function space $C^{\infty}(co(M))^{-}$. We associate to $L$ the
bilinear form,

$B_{L}[f,h]\coloneqq\int_{co(M)}\nabla f\cdot\nabla
h-(|A_{M}|^{2}\circ\iota+\text{Ric}_{N}\,(\nu,\nu))\,f\,h,$

for $f,\,h\in C^{\infty}(co(M))^{-}$, which we may extend to the Sobolev
space,

$\displaystyle W^{1,2}(co(M))^{-}\coloneqq\\{f\in W^{1,2}(co(M))$
$\displaystyle\colon$ $\displaystyle\text{for a.e.}\,(x,\nu)\in co(M),$
$\displaystyle\text{if}\,(x,-\nu)\in co(M),\,f((x,-\nu))=-f((x,\nu))\\}$

Then it may be shown (an argument is contained in the proof of Proposition 7),

$\displaystyle\text{ind}\,(M)+\text{nul}\,(M)=\sup\\{\text{dim}\,\Pi$
$\displaystyle\colon$ $\displaystyle\Pi\subset W^{1,2}(co(M))^{-}\,\text{is a
linear space,}$ $\displaystyle\text{on which}\,B_{L}\,\text{is negative semi-
definite}\\}.$

As $\text{Ric}_{N}$ is bilinear, the function
$x\mapsto\text{Ric}_{N}(\nu,\nu)(x)$, for $\nu\in T_{x}^{\perp}M$, $|\nu|=1$,
is a well defined function on $M$, even if $M$ is one-sided. Thus we may
consider the operator $L$ on the function space $C^{\infty}(M)$, and its index
and nullity on this function space. We define this as the analytic index
($\text{anl-ind}\,(M)$) and analytic nullity ($\text{anl-nul}\,(M)$) of $M$.
We may also define the bilinear form $B_{L}$ as before, which can be extended
to the Sobolev space $W^{1,2}(M)$. Note that if $M$ is two-sided then these
two notions of index and nullity will coincide. We also refer to the analytic
index and analytic nullity of $co(M)$ as the index and nullity of the operator
$L$ acting on the function space $C^{\infty}(co(M))$. Elements of the function
space $C^{\infty}(co(M))$ are referred to as ‘analytic variations’.

###  1.3 Lorentz Spaces

Let $(M,g)$ be a Riemannian manifold and $\mu$ be the volume measure
associated to $g$. For a $\mu$-measureable function $f\colon
M\rightarrow\mathbb{R}$, we define the function

$\alpha_{f}(s)\coloneqq\mu(\\{x\in M\colon|f(x)|>s\\}).$

We may then define the decreasing rearrangement $f^{*}$, of $f$ by,

$f^{*}(t)\coloneqq\begin{cases}\inf\\{s>0\colon\alpha_{f}(s)\leq t\\},&t>0,\\\
\operatorname*{ess\,sup}|f|,&t=0.\end{cases}$

For $p\in[1,\infty)$, and $q\in[1,\infty]$, and a $\mu$-measureable function
$f$ on $M$, we define,

$\|f\|_{(p,q)}\coloneqq\begin{cases}\left(\int_{0}^{\infty}t^{q/p}f^{*}(t)^{q}\frac{dt}{t}\right)^{1/q},&1\leq
q<\infty,\\\ \sup_{t>0}t^{1/p}f^{*}(t),&q=\infty.\end{cases}$

The Lorentz space $L(p,q)(M,g)$ is then defined to be the space of
$\mu$-measureable functions $f$ such that $\|f\|_{(p,q)}<+\infty$. It is worth
noting that $\|\,\cdot\,\|_{(p,q)}$ is not a norm on $L(p,q)(M,g)$ as it does
not generally satisfy the triangle inequality. However it is possible to
define an appropriate norm, $\|\cdot\|_{p,q}$ (see [13, Definition 2.10]), on
the space $L(p,q)(M,g)$, so that the normed space
$(L(p,q)(M,g),\|\cdot\|_{p,q})$ is a Banach Space [13, Theorem 2.19].
Moreover, for $1<p\leq\infty$, and $1\leq q\leq\infty$, we have the following
equivalence ([13, Proposition 2.14]),

$\|\,\cdot\,\|_{(p,q)}\leq\|\,\cdot\,\|_{p,q}\leq\frac{p}{p-1}\|\,\cdot\,\|_{(p,q)}.$
(2)

######  Proposition 2.

(Hölder–Lorentz inequality, [13, Theorem 2.9]) Take
$p_{1},\,p_{2}\in(1,\infty)$ and $q_{1},\,q_{2}\in[1,\infty]$ such that
$1/p_{1}+1/p_{2}=1/q_{1}+1/q_{2}=1$. Then for $f\in L(p_{1},q_{1})(M,g)$ and
$g\in L(p_{2},q_{2})(M,g)$, we have,

$\int_{M}|f\,g|\,d\mu\leq\|f\|_{(p_{1},q_{1})}\,\|g\|_{(p_{2},q_{2})}.$

The following Lorentz–Sobolev inequality on $\mathbb{R}^{n}$ will prove
crucial in Section 2.4. For a proof see [2, Appendix]

######  Proposition 3.

(Lorentz–Sobolev inequality on $\mathbb{R}^{n}$) Take $1<p<n$, and
$p^{*}=np/(n-p)$, then there exists a constant $C=C(n,p)$ such that for all
$u\in C_{c}^{\infty}(\mathbb{R}^{n})$,

$\|u\|_{(p^{*},p)}\leq C\|\nabla u\|_{L^{p}}.$

By a standard covering argument, from Proposition 3 we may also obtain a
Lorentz–Sobolev inequality on a bounded subset of a Riemannian manifold.

######  Proposition 4.

(Lorentz–Sobolev inequality on manifolds) Let $(M,g)$ be a Riemannian manifold
of dimension $n$. Take $1<p<n$, $p^{*}=np/(n-p)$, and an open, bounded set
$\Omega\subset M$. Then there exists a constant $C=C(\Omega,g,n,p)<+\infty$,
such that for all $u\in C_{c}^{\infty}(\Omega)$,

$\|u\|_{(p^{*},p)}\leq C\|u\|_{W^{1,p}(M)}.$

######  Remark 2.

We may extend the inequalties in Propositions 3 and 4 to $u\in
W^{1,p}(\mathbb{R}^{n})$ and $W^{1,p}(\Omega)$ respectively, by using a
standard density argument, the fact $(L(p,q)(M,g),\|\,\cdot\,\|_{p,q})$ is a
Banach space, the equivalence in (2), and the inequality [13, Lemma 2.17].

##  2 Weighted Eigenfunctions

We proceed with the proof of Theorem 1, taking a sequence
$M_{k}\rightarrow(M_{\infty},\Sigma^{1},\ldots,\Sigma^{J})$ as in Definition
2. We assume that for each $k\in\mathbb{N}$, $M_{k}$ is a single connected
component. Thus for the case of minimal hypersurfaces, $\mu=1$. The general
statement is proven by applying the argument to each individual disconnected
component of $M_{k}$.

###  2.1 The Weight

Take, $R\geq 4$, and $\delta>0$, such that for all $j=1,\ldots,J$,
$4Rr_{k}^{j}<\delta$. We first define our weight, on $co(M_{k})$, about the
point scale sequence $\\{(p_{k}^{j},r_{k}^{j})\\}_{k\in\mathbb{N}}$,

$\omega_{k,\delta,R,j}(x)\coloneqq\begin{cases}\max\\{\delta^{-2},\text{dist}_{g}^{N}(\iota(x),p_{k}^{j})^{-2}\\},&\iota(x)\in
M_{k}\setminus B_{Rr_{k}^{j}}^{N}(p_{k}^{j}),\\\
(Rr_{k}^{j})^{-2},&\iota(x)\in B_{Rr_{k}^{j}}(p_{k}^{j})\cap
M_{k},\end{cases}$

We consider the weight,

$\omega_{k,\delta,R}(x)\coloneqq\max_{j=1,\ldots
J}\,\omega_{k,\delta,R,j}(x).$

We also define $\omega_{\delta}\in
W^{1,\infty}_{\text{loc}}(co(M_{\infty})\setminus\iota_{\infty}^{-1}(\mathcal{I}))\cap
L(n/2,\infty)(co(M_{\infty}))$, by

$\omega_{\delta}(x)\coloneqq\max\\{\delta^{-2},\text{dist}_{g}^{N}(\iota(x),\mathcal{I})^{-2}\\}.$

The fact that $\omega_{\delta}\in L(n/2,\infty)(co(M_{\infty}))$ will follow
from a similar calculation to that in Claim 2.

Recall the stability operator on $co(M_{k})$,

$L_{k}\coloneqq\Delta+|A_{k}|^{2}+R_{k},$

where $A_{k}(x)=A_{M_{k}}(\iota_{k}(x))$, and
$R_{k}(x)=\text{Ric}_{N}(\nu_{k}(x),\nu_{k}(x))(\iota_{k}(x))$, and the
associated bilinear form, $B_{k}$, acting on $W^{1,2}(co(M_{k}))$,

$B_{k}[\varphi,\psi]\coloneqq\int_{M_{k}}\nabla^{k}\varphi\cdot\nabla^{k}\psi-(|A_{k}|^{2}+R_{k})\,\varphi\,\psi.$

We define the unweighted eigenspace for an eigenvalue $\lambda$ of $L_{k}$ by

$\begin{split}\mathcal{E}(\lambda;L_{k},W^{1,2}(co(M_{k}))^{-})\coloneqq\\{f\in
W^{1,2}(co(M_{k}))^{-}\colon&B_{k}[f,\psi]=\lambda\int_{M_{k}}f\,\psi,\\\
&for\,all\,\psi\in W^{1,2}(co(M_{k}))^{-}\\},\end{split}$ (3)

and the weighted eigenspace for a weighted eignevalue $\lambda$ of $L_{k}$ by,

$\begin{split}\mathcal{E}_{\omega_{k,\delta,R}}(\lambda;L_{k},W^{1,2}(co(M_{k}))^{-})\coloneqq\\{f\in
W^{1,2}(co(M_{k}))^{-}\colon&B_{k}[f,\psi]=\lambda\int_{M_{k}}f\,\psi\,\omega_{k,\delta,R},\\\
&for\,all\,\psi\in W^{1,2}(co(M_{k}))^{-}\\}.\end{split}$ (4)

Identical definitions hold for
$\mathcal{E}(\lambda;L_{\infty},W^{1,2}(co(M_{\infty}))$ and
$\mathcal{E}_{\omega_{\delta}}(\lambda;L_{\infty},W^{1,2}(co(M_{\infty})))$.
Recall that when we refer to function space $W^{1,2}(co(M))^{-}$ we are
considering ‘ambient variations’, and when we consider the function space
$W^{1,2}(co(M))$ we are considering ‘analytic variations’. In general, for
$\lambda\in\mathbb{R}$, a Riemannian manifold $(M,g)$, with a second order,
linear, elliptic operator $L$, with associated bilinear form $B_{L}$, and
$\omega\colon M\rightarrow\mathbb{R}$, we define the function space,

$\mathcal{E}_{\omega}(\lambda;L,A,B)\coloneqq\\{f\in A\colon
B_{L}[f,\varphi]=\int_{M}f\,\varphi\,\omega,\,for\,all\,\varphi\in B\\},$

where $A$, and $B$ are some function spaces on $M$. If $A=B$, we write
$\mathcal{E}_{\omega}(\lambda;L,A)$, and if $\omega\equiv c$, for some
constant $c$, we write,
$\mathcal{E}_{c}(\lambda;L,A,B)=\mathcal{E}(c\lambda;L,A,B)$.

######  Lemma 3.

There exists a
$C=C(N,g,M_{\infty},\Sigma^{1},\ldots,\Sigma^{J},\delta,R)<+\infty$ such that
for all $k$,

$\frac{|A_{k}|^{2}+|R_{k}|}{\omega_{k,\delta,R}}\leq C.$ (5)

###### Proof.

We assume the statement does not hold and prove by contradiction. After
potentially choosing a subsequence and renumerating we have that there exists
a sequence of points $x_{k}\in co(M_{k})$, such that

$\omega_{k,\delta,R}^{-1}(|A_{k}|^{2}+|R_{k}|)(x_{k})\rightarrow\infty.$

Note that there exists a $C=C(N,g)$, such that $|R_{k}|\leq C$, for all $k$,
thus,

$\omega_{k,\delta}^{-1}(|A_{k}|^{2}+C)(x_{k})\rightarrow\infty.$

Again potentially after taking a subsequence and renumerating we may assume
that $\iota_{k}(x_{k})\rightarrow\iota_{\infty}(x)\in M_{\infty}$.

If $\iota_{\infty}(x)\in M_{\infty}\setminus\mathcal{I}$, then for large
enough $k$,

$\omega_{k,\delta}\geq(2\,\text{dist}_{g}^{N}(x,\mathcal{I}))^{-2},$

which implies that,

$(2\text{dist}_{g}^{N}(x,\mathcal{I}))^{2}(|A_{k}|^{2}(x_{k})+C)\rightarrow\infty.$

Thus, $|A_{k}|^{2}(x_{k})\rightarrow\infty$, which contradicts the smooth
convergence away from $\mathcal{I}$.

We have that $x\in\iota_{\infty}^{-1}(\mathcal{I})$, and thus
$\omega_{k,\delta,R}^{-1}(x_{k})\rightarrow 0$, thus from here we just
consider the term $\omega_{k,\delta,R}^{-1}(x_{k})|A_{k}|^{2}(x_{k})$. There
exists an $i=1,\ldots,J$, such that after potentially taking a subsequence and
renumerating

$\omega_{k,\delta,R}(x_{k})=\omega_{k,\delta,R}^{i}(x_{k}).$

We split into different cases.

Case 1: There exists a $j=1,\ldots,J$, an $S<+\infty$, and a subsequence we
can renumerate along to get that $\iota_{k}(x_{k})\in\Sigma_{k}^{S,j}$ for all
$k$.

Then we have,

$\omega_{k,\delta,R}^{-1}(x_{k})=(\omega_{k,\delta,R}^{i}(x_{k}))^{-1}\leq(\omega_{k,\delta,R}^{j}(x_{k}))^{-1}\leq\max\\{S,R\\}^{2}(r_{k}^{j})^{2}.$

By the smooth convergence,

$\omega_{k,\delta,R}^{-1}(x_{k})|A_{k}|^{2}(x_{k})\leq
2\max\\{S,R\\}^{2}\max_{B_{S}(0)\cap\Sigma^{j}}|A_{\Sigma^{j}}|^{2},$

which clearly contradicts the assumption that
$\omega_{k,\delta,R}^{-1}(x_{k})|A_{k}|^{2}(x_{k})\rightarrow\infty$.

Case 2: There are sequences $\delta_{k}\rightarrow 0$, and
$R_{k}\rightarrow\infty$, such that, for all $k\in\mathbb{N}$,

$\iota_{k}(x_{k})\in M_{k}\cap
B_{\delta_{k}}(x)\setminus\cup_{j=1}^{L}\Sigma_{k}^{j,R_{k}}.$

There exists a $j=1,\ldots,L$, such that after potentially picking a
subsequence and renumerating,

$d_{k}(x_{k})=\text{dist}_{g}^{N}(\iota(x_{k}),p_{k}^{j}).$

We have,

$\omega_{k,\delta,R}^{-1}(x_{k})\leq(\omega_{k,\delta,R}^{j}(x_{k}))^{-1}.$

Case 2.1: Potentially after picking a subsequence and renumerating,

$\omega_{k,\delta,R}^{j}(x_{k})=\text{dist}_{g}^{N}(\iota(x_{k}),p_{k}^{j})^{-2}.$

Then,

$\omega_{k,\delta,R}^{-1}(x_{k})|A_{k}|^{2}(x_{k})\leq
d_{k}^{2}(x_{k})|A_{k}|^{2}(x_{k}).$

By Point 3 of Definition 2 and Proposition 1, we have that the right hand side
converges to $0$. This clearly contradicts assumption that
$\omega_{k,\delta,R}^{-1}(x_{k})|A_{k}|^{2}(x_{k})\rightarrow\infty$.

Case 2.2: We have that,

$\omega_{k,\delta,R}^{j}(x_{k})=R^{2}(r_{k}^{j})^{2}.$

We claim there exists an $S=S(R)>0$, such that for large enough $k$,

$\frac{\text{dist}_{g}^{N}(\iota_{k}(x_{k}),p_{k}^{j})}{r_{k}^{j}}\geq S,$ (6)

Assuming (6), we have

$\omega_{k,\delta,R}^{-1}(x_{k})|A_{k}|^{2}(x_{k})\leq\frac{2R^{2}}{S^{2}}d_{k}^{2}(x_{k})|A_{k}|^{2}(x_{k}),$

which converges to zero by Point 3 of Definition 2 and Proposition 1, again
contradicting our assumption.

We now prove (6) by contradiction. If (6) does not hold then after potentially
taking a subsequence and renumerating,

$\frac{\text{dist}_{g}^{N}(\iota_{k}(x_{k}),p_{k}^{j})}{r_{k}^{j}}\rightarrow
0.$

Repeating arguments similar to those in Lemma 1, we may conclude that the
connected component of

$\frac{1}{r_{k}^{j}}\exp^{-1}_{p_{k}^{j}}(M_{k}\cap
B_{r_{k}^{j}}(p_{k}^{j}))\subset B_{1}^{n+1}(0),$

through $(r_{k}^{j})^{-1}\exp^{-1}_{p_{k}^{j}}(\iota_{k}(x_{k}))$ must
converge to a plane through the origin, with convergence as described in Lemma
1. As $\Sigma^{j}$ is a non-trivial minimal embedded hypersurface through the
origin, by the one sided maximum principle, this plane must cross $\Sigma^{j}$
transversely. However, by the smooth convergence away from finitely many
points, this transverse intersection will contradict the embeddedness along
the sequence. ∎

######  Claim 2.

We have that there exists a
$C=C(N,g,M_{\infty},m^{1},\ldots,m^{\mu},\Sigma^{1},\ldots,\Sigma^{J})<+\infty$
such that for all $k\in\mathbb{N}$,
$\delta\in[0,\max\\{1,\text{inj}\,(N)/2\\})$ and $R\geq 1$,

$\|\omega_{k,\delta,R}\|_{(n/2,\infty)}\leq C.$

###### Proof.

For a $j=1,\ldots,J$, consider $f=\omega_{k,\delta,R,j}$. We may then show,

$\alpha_{f}(s)=\begin{cases}\mathcal{H}^{n}(M_{k}),&s\in[0,\delta^{-2}),\\\
\mathcal{H}^{n}(B_{1/\sqrt{s}}^{N}(p_{k}^{j})\cap M_{k}),&\delta^{-2}\leq
s<(Rr_{k}^{j})^{-2},\\\ 0,&s\geq(Rr_{k}^{j})^{-2}.\end{cases}$

Which then implies,

$t^{2/n}f^{*}(t)=\begin{cases}\left((Rr_{k}^{j})^{-n}\,\mathcal{H}^{n}(B_{Rr_{k}^{j}}^{N}(p_{k}^{j}\cap
M_{k}))\right)^{2/n},&0\leq
t<\mathcal{H}^{n}(B_{Rr_{k}^{j}}^{N}(p_{k}^{j})\cap M_{k}),\\\
\left((1/\sqrt{s})^{-n}\,\mathcal{H}^{n}(B_{1/\sqrt{s}}^{N}(p_{k}^{j}\cap
M_{k}))\right)^{2/n},&t=\mathcal{H}^{n}(B_{1/\sqrt{s}}^{N}(p_{k}^{j})\cap
M_{k}),\,\delta^{-2}<s\leq(Rr_{k}^{j})^{-2},\\\
\left(\delta^{-n}\mathcal{H}^{n}(M_{k})\right)^{2/n},&\mu(B_{\delta}^{N}(p_{k}^{j})\cap
M_{k}\leq t<\mathcal{H}^{n}(M_{k})),\\\
0,&t\geq\mathcal{H}^{n}(M_{k}).\end{cases}$

Thus, by the monotonicity formula ([35, Corollary 17.8]) and fact that we may
assume $\sup_{k}\mathcal{H}^{n}(M_{k})\leq m\mathcal{H}^{n}(M_{\infty})+1$,
there exists a $C=C(N,g,M_{\infty},m,\Sigma^{j})<+\infty$, such that,

$\|\omega_{k,\delta,R,j}\|_{(n/2,\infty)}\leq C,$

which in turn implies that there exists a
$C=C(N,g,M_{\infty},m,\Sigma^{1},\ldots,\Sigma^{J})<+\infty$,

$\|\omega_{k,\delta,R}\|_{(n/2,\infty)}\leq
2^{2J/n}\sum_{j=1}^{J}\|\omega_{k,\delta,R,j}\|_{(n/2,\infty)}\leq C.$

The $2^{2J/n}$ factor is coming from [13, Theorem 2.4 (a)]. ∎

######  Remark 3.

The choice of weight $\omega_{k,\delta,R}$ fails to work for the case of
$n=2$. In [18] (in which $n=2$) the choice of weight is subtle, and relies on
intricate estimates on the neck region of the bubbles. We were unable to
derive appropriate corresponding estimates on the neck regions in the setting
of this paper.

###  2.2 Convergence on the Base

Consider a sequence of functions $\\{f_{k}\\}_{k\in\mathbb{N}}$, $f_{k}\in
W^{1,2}(co(M_{k}))^{-}$, which satisfy the following weighted eigenvalue
problem,

$\int_{co(M_{k})}\nabla
f_{k}\cdot\nabla\varphi-(|A_{k}|^{2}+R_{k})\,f_{k}\,\varphi=\lambda_{k}\int_{co(M_{k})}f_{k}\,\varphi\,\omega_{k,\delta,R},$

for all $\varphi\in W^{1,2}(co(M_{k}))^{-}$, with $\lambda_{k}\leq 0$, for all
$k$. We take

$\int_{co(M_{k})}f_{k}^{2}\,\omega_{k,\delta,R}=1,$

and by Lemma 3,

$\displaystyle\int_{co(M_{k})}|\nabla f_{k}|^{2}$ $\displaystyle\leq$
$\displaystyle\int_{co(M_{k})}(|A_{k}|^{2}+|R_{k}|)f_{k}^{2},$
$\displaystyle\leq$ $\displaystyle
C\int_{co(M_{k})}f_{k}^{2}\,\omega_{k,\delta,R},$ $\displaystyle=$
$\displaystyle C.$

Furthermore,

$\delta^{-2}\|f_{k}\|^{2}_{L^{2}(co(M_{k}))}\leq\int_{co(M_{k})}f_{k}^{2}\,\omega_{k,\delta,R}=1.$

Thus, for all $k\in\mathbb{N}$.,

$\|f_{k}\|_{W^{1,2}(co(M_{k}))}\leq
C=C(N,g,\delta,R,m,M_{\infty},\Sigma^{1},\ldots,\Sigma^{J})<+\infty.$

Using Lemma 3 we may also obtain a lower bound on the negative eigenvalues,

$\lambda_{k}=\lambda_{k}\int_{co(M_{k})}f_{k}^{2}\,\omega_{k,\delta,R}=\int_{co(M_{k})}|\nabla^{k}f_{k}|^{2}-(|A_{k}|^{2}+R_{k})f_{k}^{2}\geq-C\int_{co(M_{k})}f_{k}^{2}\,\omega_{k,\delta,R}=-C.$
(7)

Thus, after potentially taking a subsequence and renumerating, we may assume
that $\lambda_{k}\rightarrow\lambda_{\infty}\leq 0$.

We define the map,

$\displaystyle F\colon co(M_{\infty})\times\mathbb{R}$
$\displaystyle\rightarrow$ $\displaystyle N,$ $\displaystyle(x,t)$
$\displaystyle\mapsto$
$\displaystyle\exp_{\iota_{\infty}(x)}(t\nu_{\infty}(x)).$

Note, as $co(M_{\infty})$ is smooth, ([26, Proposition 4.2]) there exists a
$\tau=\tau(N,M_{\infty},g)>0$, such that,

$F\colon co(M_{\infty})\times(-\tau,\tau)\rightarrow
F(co(M_{\infty})\times(-\tau,\tau))\subset N,$

is a smooth, local diffeomorphism. We define the metric $\tilde{g}=F^{*}g$, on
$co(M_{\infty})\times(-\tau,\tau)$, and assume that for all $k\in\mathbb{N}$,
$M_{k}\subset F(co(M_{\infty})\times(-\tau,\tau))$.

First we consider the case of minimal hypersurfaces. As $M_{\infty}$ is
embedded, we may take $\tau>0$, such that,
$F^{-1}(M_{\infty})=co(M_{\infty})\times\\{0\\}$. For $r>0$, define the open
set $\Omega_{r}\subset co(M_{\infty})$, by,

$\Omega_{r}\coloneqq\iota_{\infty}^{-1}\left(M_{\infty}\setminus\bigcup_{y\in\mathcal{I}}\overline{B_{r}^{N}(y)}\right)\subset
co(M_{\infty}).$

We define, $M_{k}^{r}\coloneqq M_{k}\cap F(\Omega_{r}\times(-\tau,\tau))$, and
$\tilde{M}_{k}^{r}=F^{-1}(M_{k}^{r})\cap\Omega_{r}\times(-\tau,\tau)$. By the
convergence described in Definition 2, along with the the fact that
$co(M_{\infty})$ is two-sided in $co(M_{\infty})\times(-\tau,\tau)$, and
$\theta_{|M_{\infty}}\equiv m$, for large enough $k$, there exists $m$ smooth
functions,

$u_{k}^{i,r}\colon\Omega_{r}\rightarrow(-\tau,\tau),$

such that, $u_{k}^{1,r}<u_{k}^{2,r}<\cdots<u_{k}^{m,r}$, and

$\tilde{M}_{k}^{r}=\bigcup_{i=1}^{m}\\{(x,u_{k}^{i,r}(x))\colon
x\in\Omega_{r}\\}\subset\Omega_{r}\times(-\tau,\tau).$

We also note that $u_{k}^{i,r}\rightarrow 0$ in $C^{l}(\Omega_{r})$ for all
$l\in\mathbb{N}$, and for $0<r<s$, $u_{k}^{i,r}=u_{k}^{i,s}$ on
$\Omega_{s}\subset\Omega_{r}$. Further note that each connected component of
$\tilde{M}_{k}^{r}$, ($\tilde{M}_{k}^{i,r}\coloneqq\\{(x,u_{k}^{i,r}(x))\colon
x\in\Omega_{r}\\}$), is two-sided in $\Omega_{r}\times(-\tau,\tau)$, and thus
has trivial normal bundle. Thus we can identify each $\tilde{M}_{k}^{i,r}$ as
a subset of $co(M_{k})$.

Consider the case of $\lambda$-CMC hypersurfaces. Due to the presence of
potential non-embedded points in $M_{\infty}$, our description of $M_{k}$
using a normal tubular neighbourhood of $co(M_{\infty})$ requires us to be
slighlty more delicate. We have,

$e(M_{\infty})=\sqcup_{i=1}^{\mu}M_{\infty}^{i},$

and $\theta_{|M_{\infty}^{i}}\equiv m^{i}\in\mathbb{Z}$. Denote,
$\Omega_{r/2}^{i}\subset co(\overline{M_{\infty}^{i}})$ as before. Then
similarly, for each $i=1,\ldots,\mu$, and large enough $k$, there exists
$m^{i}$ smooth graphs ($j=1,\ldots,m^{i}$),

$u_{k}^{j,i,r}\colon\Omega_{r/2}^{i}\rightarrow(-\tau,\tau),$

such that, $u_{k}^{1,i,r}<u_{k}^{2,i,r}<\cdots<u_{k}^{m^{i},i,r}$,
$u_{k}^{j,i,r}\rightarrow 0$ in $C^{l}(\Omega_{r}^{i})$ for all
$l\in\mathbb{N}$, and

$M_{k}\setminus\bigcup_{y\in\mathcal{I}}B_{r}^{N}(y)\subset\bigcup_{i=1}^{\mu}\bigcup_{j=1}^{m^{i}}\\{F(x,u_{k}^{j,i,r/2}(x))\colon
x\in\Omega_{r/2}^{i}\\}.$

Define $\tilde{M}_{k}^{j,i,r}=\\{(x,u_{k}^{j,i,r}(x))\colon
x\in\Omega_{r}^{i}\\}$, and we may identify this as a subset of
$co(M_{k})=M_{k}$.

For ease of notation we just consider the case of minimal hypersurfaces.
Denote the following maps for $i=1,\ldots,m$,

$\begin{split}F_{k}^{i,r}\colon\Omega_{r}&\rightarrow M_{k},\\\ x&\mapsto
F(x,u_{k}^{i,r}(x)).\end{split}$ (8)

and define the functions on $\Omega_{r}$

$\tilde{f}_{k}^{i,r}(x)\coloneqq(f_{k}\circ F_{k}^{i,r})(x).$

For an open set $\Omega\subset\subset
co(M_{\infty})\setminus\iota_{\infty}^{-1}(\mathcal{I})$, we may take $r>0$,
such that $\Omega\subset\subset\Omega_{r}$, and then define, for large enough
$k$,

$\tilde{f}_{k}^{i}(x)=\tilde{f}_{k}^{i,r}(x),\,x\in\Omega.$

Note that this definition is independent of the choice of $0<r<r_{0}$, for
$\Omega\subset\Omega_{r_{0}}$. When dealing with a fixed open set
$\Omega\subset co(M_{\infty})\setminus\iota_{\infty}^{-1}(\mathcal{I})$, for
appropriate choices of $r$, we also drop the superscript $r$ in the notation
of the map (8), and functions $u_{k}^{i,r}$. Then, choosing $k$ large enough
(so that $\|u_{k}^{i}\|_{C^{1}(\Omega_{r})}$ is small enough), we have that,

$\|\tilde{f}_{k}^{i}\|_{W^{1,2}(\Omega)}\leq
2\|f_{k}\|_{W^{1,2}(co(M_{k}))}\leq C.$

and thus for each $i=1,\ldots,m$, there exists an $\tilde{f}^{i}_{\infty}\in
W^{1,2}_{\text{loc}}(co(M_{\infty})\setminus\iota_{\infty}^{-1}(\mathcal{I}))$,
such that, after potentially picking a subsequence and renumerating,

$\begin{cases}\tilde{f}_{k}^{i}\rightharpoonup\tilde{f}_{\infty}^{i},&W^{1,2}_{\text{loc}}(co(M_{\infty})\setminus\iota_{\infty}^{-1}(\mathcal{I})),\\\
\tilde{f}_{k}^{i}\rightarrow\tilde{f}_{\infty}^{i},&L^{2}_{\text{loc}}(co(M_{\infty})\setminus\iota_{\infty}^{-1}(\mathcal{I})),\\\
\end{cases}$ (9)

Note that by lower semicontinuity of the $W^{1,2}$ norm for (9), for all open
$\Omega\subset co(M_{\infty})\setminus\iota^{-1}(\mathcal{I})$, we have a
uniform bound $\|\tilde{f}_{\infty}\|_{W^{1,2}(\Omega)}\leq C$. Multiplying
$\tilde{f}_{\infty}$, and $|\nabla\tilde{f}_{\infty}|$, by sequences of
appropriately chosen characteristic functions on $co(M_{\infty})$, we may
apply monotone convergence theorem to deduce that $\tilde{f}_{\infty}$, and
$|\nabla\tilde{f}_{\infty}|$, lie in $L^{2}(co(M_{\infty}))$. Then by a
standard $2$-capacity argument we may show that $\nabla\tilde{f}_{\infty}$
extends to be the distributional gradient of $\tilde{f}_{\infty}$, on
$co(M_{\infty})$. Thus, $\tilde{f}_{\infty}\in W^{1,2}(co(M_{\infty}))$.
Moreover, we have that

$\int_{co(M_{\infty})}\tilde{f}_{\infty}^{2}\,\omega_{\delta}\leq 1,$

For $i=1,\ldots,m$, and large enough $k$, we define the metric,
$\tilde{g}^{i}_{k}\coloneqq(F^{i}_{k})^{*}\tilde{g}$, and its associated
gradient $\tilde{\nabla}^{k}$, on $\Omega$. Let $J_{k}^{i}$ denote the
Jacobian of the map $F_{k}^{i}$ with respect to the metric $\tilde{g}$ on
$\Omega$. For a point $x_{0}\in\Omega$, we may choose $s>0$, small enough so
that $B_{s}^{co(M_{\infty})}(x_{0})\subset\subset\Omega$, and for large $k$,

$M_{k}\cap F(B_{s}^{co(M_{\infty})}(x_{0})\times(-\tau,\tau))\subset
B_{\text{inj}\,(N)/2}^{N}(\iota(x_{0})),$

are all graphical over $T_{\iota(x_{0})}M_{\infty}$. Consider $\varphi\in
C_{c}^{\infty}(B_{s}^{co(M_{\infty})}(x_{0}))$, and for each $i=1,\ldots,m$
and $x\in B_{s}^{co(M_{\infty})}(x_{0})$, we define

$\varphi_{k}^{i}(x,u_{k}^{i}(x))=\varphi(x).$

As $F(\\{(x,u_{k}^{i}(x))\colon x\in B_{s}^{co(M_{\infty})}(x_{0})\\})\subset
B_{\text{inj}\,(N)/2}^{N}(\iota(x_{0}))$ is two-sided, we can extend
$\varphi_{k}^{i}$ to a vector field on $N$, and thus to a function in
$C^{\infty}(co(M_{k}))^{-}$. Thus we have,

$\int_{\Omega}\tilde{g}_{k}(\tilde{\nabla}^{k}\tilde{f}^{i}_{k},\tilde{\nabla}^{k}\varphi)J_{k}^{i}=\lambda_{k}\int_{\Omega}\tilde{f}_{k}^{i}\,\varphi\,(\omega_{k,\delta,R}\circ
F\circ F_{k}^{i})\,J_{k}^{i}+\int_{\Omega}((|A_{k}|^{2}+R_{k})\circ F\circ
F_{k}^{i})\,\tilde{f}_{k}^{i}\,\tilde{\varphi}\,J_{k}^{i}.$

Hence, by (9), convergence of $\omega_{k,\delta,R}\circ F\circ
F_{k}^{i}\rightarrow\omega_{\delta}\circ\iota_{\infty}$, in
$W^{1,\infty}(\Omega)$, and smooth convergence of, $F_{k}^{i}\rightarrow id$,
on $\Omega$, we have that,

$\int_{co(M_{\infty})}\nabla\tilde{f}^{i}_{\infty}\cdot\nabla\varphi-((|A_{\infty}|^{2}+R_{\infty})\circ\iota_{\infty})\,\tilde{f}^{i}_{\infty}\,\varphi=\lambda_{\infty}\int_{co(M_{\infty})}\tilde{f}^{i}_{\infty}\,\varphi\,(\omega_{\delta}\circ\iota_{\infty}),$
(10)

holds for all $\varphi\in C_{c}^{\infty}(B_{s}^{co(M_{\infty})}(x_{0}))$. Thus
by standard regularity theory for linear elliptic PDEs we have that
$\tilde{f}^{i}_{\infty}\in W^{2,2}_{\text{loc}}(\Omega)$ and
$\Delta\tilde{f}^{i}_{\infty}+((|A_{\infty}|^{2}+R_{\infty})\circ\iota_{\infty})\,\tilde{f}_{\infty}^{i}+\lambda_{\infty}\tilde{f}_{\infty}^{i}\,(\omega_{\delta}\circ\iota_{\infty})=0$
a.e. on $\Omega$. This then implies that (10) holds for all $\varphi\in
C^{\infty}_{c}(co(M_{\infty})\setminus\iota_{\infty}^{-1}(\mathcal{I}))$.

######  Proposition 5.

Let $(M,g)$ be a compact, $n$-dimensional, Riemannian manifold, with $n\geq
3$. Consider $V\in L^{\infty}(M)$, and $\omega\in L(n/2,\infty)(M)$. Suppose
that we have $u\in W^{1,2}(M)$, and a finite set of points $\mathcal{J}\subset
M$, such that, for all $\varphi\in C^{\infty}_{c}(M\setminus\mathcal{J})$,

$\int_{M}\nabla u\cdot\nabla\varphi-Vu\varphi-\omega u\varphi=0,$ (11)

then in fact (11) holds for all $\varphi\in W^{1,2}(M)$.

###### Proof.

As $\mathcal{H}^{n-2}(\mathcal{J})=0$, by [19, Theorem 3, Section 4.7.2], we
have that the $2$-capacity of $\mathcal{J}$ is $0$. Thus, for all
$\varepsilon>0$, and
$r<\min\\{\text{inj}(M),\min_{y_{1},y_{2}\in\mathcal{J}}\text{dist}(y_{1},y_{2})/2\\}$,
there exists a $0<\rho<r$, and function $\chi\in
C^{\infty}_{c}\left(\cup_{y\in\mathcal{J}}B_{\rho}(y)\right)$ such that,

$\begin{cases}0\leq\chi_{\varepsilon,r}(x)\leq 1,&x\in M,\\\
\chi_{\varepsilon,r}(x)=1,&x\in\cup_{y\in\mathcal{J}}B_{\rho}(y),\\\
\int_{M}|\nabla\chi_{\varepsilon,r}|^{2}<\varepsilon.\end{cases}$

Take $\varphi\in C^{\infty}(M)$, and we define,

$\displaystyle\varphi_{1,\varepsilon,r}$ $\displaystyle\coloneqq$
$\displaystyle(1-\chi_{\varepsilon,r})\varphi,$
$\displaystyle\varphi_{2,\varepsilon,r}$ $\displaystyle\coloneqq$
$\displaystyle\chi_{\varepsilon,r}\varphi.$

Then, $\varphi=\varphi_{1,\varepsilon,r}+\varphi_{2,\varepsilon,r}$, and
$\varphi_{1,\varepsilon,r}\in C_{c}^{\infty}(M\setminus\mathcal{J})$, which
implies that,

$\int_{M}\nabla u\cdot\nabla\varphi-Vu\varphi-\omega u\varphi=\int_{M}\nabla
u\cdot\nabla\varphi_{2,\varepsilon,r}-Vu\varphi_{2,\varepsilon,r}-\omega
u\varphi_{2,\varepsilon,r}.$

Thus, by Hölder inequality and Proposition 6,

$\displaystyle\left|\int_{M}\nabla u\cdot\nabla\varphi-Vu\varphi-\omega
u\varphi\right|$ $\displaystyle\leq\int_{M}|\nabla
u|\,(|\varphi|\,|\nabla\chi_{\varepsilon,r}|+|\nabla\varphi|\,|\chi_{\varepsilon,r}|)+|V|\,|u|\,|\chi_{\varepsilon,r}|\,|\varphi|+|\omega|\,|u|\,|\varphi|\,|\chi_{\varepsilon,r}|,$
$\displaystyle\leq
C\|\varphi\|_{C^{1}(M)}\|u\|_{W^{1,2}(M)}(1+\|V\|_{L^{\infty}(M)}+\|\omega\|_{(n/2,\infty)(M)})\left(\mathcal{H}^{n}(\cup_{y\in\mathcal{J}}B_{r}(y))+\varepsilon\right)^{1/2}.$

Therefore, letting $\varepsilon\rightarrow 0$, and $r\rightarrow 0$, we
conclude that (11) holds for all $\varphi\in C_{c}^{\infty}(M)$.

The fact that (11) holds for all $\varphi\in W^{1,2}(M)$, then follows from a
standard density argument, along with Hölder inequality and Proposition 6. ∎

By Proposition 5 we have that (10) holds for all $\varphi\in
W^{1,2}(co(M_{\infty}))$. Thus,
$\tilde{f}^{i}_{\infty}\in\mathcal{E}_{\omega_{\delta}}(\lambda_{\infty};L_{\infty},W^{1,2}(co(M_{\infty})))$.
It is worth noting that we could have $\tilde{f}^{i}_{\infty}=0$.

###  2.3 Convergence on the Bubble

For $S>0$ fixed, and $i\in\\{1,\ldots,J\\}$, consider the bubble
$\Sigma_{k}^{i,S}$, and its associated point-scale sequence,
$\\{(p_{k}^{i},r_{k}^{i})\\}_{k\in\mathbb{N}}$. For the metric
$\tilde{g}_{k}$, on $B_{S}^{n+1}(0)$, obtained by appropriately rescaling
geodesic normal coordinates about $p_{k}^{i}$, in
$B_{r_{k}^{i}S}^{N}(p_{k}^{i})$, we have,

$(\tilde{g}_{k})_{\alpha,\beta}(x)\coloneqq
g_{\alpha,\beta}(r_{k}^{i}x)\rightarrow\delta_{\alpha,\beta},$

and for our bubble,

$\tilde{\Sigma}^{i,S}_{k}\cap
B_{S}^{n+1}(0)\coloneqq\frac{1}{r_{k}^{i}}\exp_{p_{k}^{i}}^{-1}(\Sigma^{i,S}_{k}\cap
B_{Sr_{k}^{i}}^{N}(p_{k}^{i}))\rightarrow\Sigma^{i}\cap B_{S}^{n+1}(0),$

smoothly. As $B_{S}^{n+1}(0)$ is simply connected, we have a choice of unit
normal $\nu$, on $\Sigma^{i}\cap B_{S}(0)$. There is a $\tau>0$, such that the
map,

$\displaystyle F\colon\Sigma^{i,S}\times(-\tau,\tau)$
$\displaystyle\rightarrow$ $\displaystyle\mathbb{R}^{n+1},$
$\displaystyle(x,t)$ $\displaystyle\mapsto$ $\displaystyle x+t\nu(x),$

is a diffeomorphism onto its image. Then for large enough $k$, there exists a
smooth function,

$v_{k}^{i,S}\colon\Sigma^{i,S}\rightarrow(-\tau,\tau),$

such that,

$\tilde{\Sigma}^{i,S}_{k}\cap
F(\Sigma^{i,S}\times(-\tau,\tau))=\\{F(x,v_{k}^{i,S}(x))\colon
x\in\Sigma^{i,S}\\}.$

As before, for large enough $k$, define the smooth map,

$\displaystyle F_{k}^{i,S}\colon\Sigma^{i,S}$ $\displaystyle\rightarrow$
$\displaystyle\mathbb{R}^{n+1},$ $\displaystyle x$ $\displaystyle\mapsto$
$\displaystyle F(x,v_{k}^{i,S}(x)).$

######  Remark 4.

For $p\in N$ and $r\in(0,\text{inj}\,(N))$, the geodesic ball $B_{r}^{N}(p)$
is simply connected, and thus for any closed, embedded hypersurface $M\subset
N$, without boundary, we have that $M\cap B_{r}^{N}(p)$ will be two-sided.
Thus restricting any function $f\in C^{\infty}(co(M))^{-}$, to the domain
$\iota^{-1}(M\cap B_{r}^{N}(p))$, it is equivalent to consider this
restriction as a function in $C^{\infty}(M\cap B_{r}^{N}(p))$, up to a
potential choice in sign. Similarly for any function space on $co(M)$ and
$M\cap B_{r}^{N}(p)$.

We define the functions (if $M_{k}$ is one-sided then these functions inherit
the choice of sign from $\nu$ on $\Sigma^{i}\cap B_{S}^{n+1}(0)$)

$\tilde{f}_{k}^{i,S}(x)\coloneqq(r_{k}^{i})^{n/2-1}f_{k}(p_{k}^{i}+r_{k}^{i}F_{k}^{i,S}(x))$

and,

$\omega_{k}^{\Sigma^{i},R,S}(x)\coloneqq(r_{k}^{i})^{2}\omega_{k,\delta,R}(p_{k}^{i}+r_{k}^{i}F_{k}^{i,S}(x)),$

both defined on $\Sigma^{i,S}$. For each $j=1,\ldots,L$, such that
$\text{dist}_{g}^{N}(p_{k}^{j},p_{k}^{i})\rightarrow 0$, we denote,

$q_{k}^{j}=(r_{k}^{i})^{-1}\exp_{p_{k}^{i}}(p_{k}^{j}),$

and, after potentially taking a subsequence and renumerating, either
$|q_{k}^{j}|\rightarrow\infty$, or there exists a $q^{j}\in\mathbb{R}^{n+1}$,
such that $q_{k}^{j}\rightarrow q^{j}$. If $q_{k}^{j}\rightarrow q^{j}$, then
there exists $\alpha_{j}\in[0,\infty]$, such that

$\frac{r_{k}^{j}}{r_{k}^{i}}\rightarrow\alpha_{j}.$

If $\alpha_{j}=+\infty$, we may rescale about $p_{k}^{j}$ at rate $r_{k}^{j}$,
and use Lemma 1 on the component passing through the rescaled point of
$p_{k}^{j}$ to deduce a contradiction in a similar way in which we proved (6).
Thus $\alpha_{j}\in[0,+\infty)$.

For each such $j$ we define the following functions on $\mathbb{R}^{n+1}$. For
the case $\alpha_{j}\not=0$,

$\omega^{i,j,R}(x)=\begin{cases}|x-q^{j}|^{-2},&x\not\in
B_{R\alpha_{j}}^{n+1}(q^{j}),\\\ (R\alpha_{j})^{-2},&x\in
B_{R\alpha_{j}}^{n+1}(q^{j}),\end{cases}$

and for $\alpha_{j}=0$,

$\omega^{i,j,R}(x)=|x-q^{j}|^{-2}.$

For all other such $j=1,\ldots,L$, $j\not=i$, we simply set $\omega^{j,R}=0$.
For $j=i$ we set,

$\omega^{i,i,R}(x)=\begin{cases}|x|^{-2},&x\not\in B_{R}^{n+1}(0),\\\
R^{-2},&x\in B_{R}^{n+1}(0).\end{cases}$

If $\alpha_{j}=0$, then we may deduce that $q^{j}\not\in\Sigma^{i}$, again
using Lemma 1 and a similar contradiction argument to that used to show (6).

We now define the following function on $\Sigma^{i}$,

$\omega^{\Sigma^{i},R}(x)=\max_{j=1,\ldots,L}\omega^{i,j,R}(x),$ (12)

and note that, $\omega^{\Sigma^{i},R}\in W^{1,\infty}(\Sigma^{i})$, and
$\omega_{k}^{\Sigma^{i},R,S}\rightarrow\omega^{\Sigma^{i},R}$ in
$W^{1,\infty}(\Sigma^{i,S})$. We have that,

$\int_{\Sigma^{i,S}}(\tilde{f}_{k}^{i,S})^{2}\omega_{k}^{\Sigma^{i},R,S}\leq\int_{co(M_{k})}f_{k}^{2}\omega_{k,\delta,R}=1,$

implying that, for large enough $k$,

$\int_{\Sigma^{i,S}}(\tilde{f}_{k}^{i,S})^{2}\leq\frac{2}{\min_{\Sigma^{i,S}}\omega^{\Sigma^{i},R}}\int_{\Sigma^{i,S}}(\tilde{f}_{k}^{i,S})^{2}\omega_{k,R}^{\Sigma^{i},S}\leq
C(S,R,\Sigma^{1},\ldots,\Sigma^{J})<+\infty.$

Moreover,

$\int_{\Sigma^{i,S}}|\nabla\tilde{f}_{k}^{i,S}|^{2}\leq\int_{M_{k}}|\nabla
f_{k}|^{2}\leq
C(N,g,\delta,R,M_{\infty},\Sigma^{1},\ldots,\Sigma^{J})<+\infty.$

Similar to previous, for $0<S_{1}<S_{2}<+\infty$, and large enough $k$, we
have that $\tilde{f}_{k}^{i,S_{1}}=\tilde{f}_{k}^{i,S_{2}}$ on
$\Sigma^{i,S_{1}}$. Thus, for any open, bounded set $\Omega\subset\Sigma^{i}$,
we may take any $S>0$ such that $\Omega\subset\Sigma^{i,S}$, then for large
enough $k$, the function $\tilde{f}_{k}^{i}=\tilde{f}_{k}^{i,S}$ is well
defined on $\Omega$, with,

$\int_{\Omega}(\tilde{f}_{k}^{i})^{2}\leq
C(\Omega,R,M_{\infty},\Sigma^{1},\ldots,\Sigma^{J}),$

and,

$\int_{\Omega}|\nabla\tilde{f}_{k}^{i}|^{2}\leq
C(N,g,\delta,R,M_{\infty},\Sigma^{1},\ldots,\Sigma^{J}).$

We may conclude that there exists an $\tilde{f}_{\infty}^{i}\in
W^{1,2}_{\text{loc}}(\Sigma^{i})$,

$\begin{cases}\tilde{f}_{k}^{i}\rightarrow\tilde{f}_{\infty}^{i},&L^{2}_{\text{loc}}(\Sigma^{i}),\\\
\tilde{f}_{k}^{i}\rightharpoonup\tilde{f}_{\infty}^{i},&W^{1,2}_{\text{loc}}(\Sigma^{i}),\end{cases}$

and, we have

$\int_{\Sigma^{i}}|\nabla\tilde{f}_{\infty}^{i}|^{2}<+\infty,$

and,

$\int_{\Sigma^{i}}(\tilde{f}_{\infty}^{i})^{2}\omega^{\Sigma^{i},R}\leq 1.$

Thus,

$\tilde{f}_{\infty}^{i}\in
W^{1,2}_{\omega^{\Sigma^{i},R}}(\Sigma^{i})\coloneqq\left\\{f\in
L^{1}_{\text{loc}}(\Sigma^{i})\colon\int_{\Sigma^{i}}|\nabla
f|^{2}<+\infty,\,\int_{\Sigma^{i}}\omega^{\Sigma^{i},R}f^{2}<+\infty\right\\}$

Similar to before, and the fact that our metric on $B_{S}^{n+1}(0)$ converges
to the standard Euclidean one, we deduce that for all $\varphi\in
C^{\infty}_{c}(\Sigma^{i})$,

$\int_{\Sigma^{i}}\nabla\tilde{f}_{\infty}^{i}\cdot\nabla\varphi-|A_{\Sigma^{i}}|^{2}\,\tilde{f}_{\infty}^{i}\,\varphi-\lambda_{\infty}\,\omega^{\Sigma^{i},R}\,\tilde{f}_{\infty}^{i}\,\varphi=0.$
(13)

As $\Sigma^{i}$ has finite index and Euclidean volume growth at infinity, we
may deduce that there exists an $S>0$, such that $\Sigma^{i}\setminus
B_{S}^{n+1}(0)$ is stable [20, Proposition 1]. Thus, noting that $\Sigma^{i}$
is embedded, by [32, Theorem 3] (alternatively, [33, Theorem 3] for $2\leq
n\leq 5$, and [7, Theorem 2] for $2\leq n\leq 6$), for all
$x\in\Sigma^{i}\setminus B_{2S}^{n+1}(0)$, we have that there exists a
$C=C(\Sigma^{i})<+\infty$, such that,

$|A_{\Sigma^{i}}|^{2}(x)\leq\frac{C}{(|x|-S)^{2}}\leq\frac{C}{|x|^{2}}.$ (14)

Moreover, we may choose $S$ large enough so that, $|x|^{-2}\leq
2\omega^{\Sigma^{i},R}$ on $\Sigma^{i}\setminus B_{S}^{n+1}(0)$. Thus, for
$x\in\Sigma^{i}\setminus B_{2S}^{n+1}(0)$,

$|A_{\Sigma^{i}}|^{2}(x)\leq C\omega^{\Sigma^{i},R}.$

Thus by Hölder’s inequality we have, for $\varphi\in L^{2}(\Sigma^{i})$,

$\left|\int_{\Sigma^{i}}|A_{\Sigma^{i}}|^{2}\,\tilde{f}_{\infty}\,\varphi+\omega^{\Sigma^{i},R}\,\tilde{f}_{\infty}\,\varphi\right|\leq
C\|\omega^{\Sigma^{i},R}\|_{L^{\infty}(\Sigma^{i})}^{1/2}\,\|\varphi\|_{L^{2}(\Sigma^{i})}.$

This allows us to apply a standard density argument to deduce that (13) holds
for all $\varphi\in W^{1,2}(\Sigma^{i})$.

######  Remark 5.

We may also deduce $\tilde{f}_{\infty}\in L^{2^{*}}(\tilde{\Sigma})$, for
$2^{*}=2n/(n-2)$. Indeed, take $S>R$, large enough so that $|x|^{-2}\leq
2\omega_{R}^{\Sigma^{i}}$, and define a function $\chi_{S}$ on $\Sigma^{i}$,

$\begin{cases}\chi_{S}(x)=1,&x\in\Sigma^{i,S},\\\
\chi_{S}(x)=0,&\iota(x)\in\Sigma^{i,2S},\\\ |\nabla\chi_{S}|\leq
2/S.\end{cases}$

We plug the function $\chi_{S}\tilde{f}_{\infty}$ into the
Michael–Simon–Sobolev inequality ([28, Theorem 2.1]) on $\Sigma^{i}$ (noting
that $\Sigma^{i}$ is an embedded, minimal hypersurface in $\mathbb{R}^{n+1}$),

$\displaystyle\left(\int_{\Sigma^{i,S}}|\tilde{f}_{\infty}|^{2^{*}}\right)^{1/2^{*}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\Sigma^{i}}|\chi_{S}\,\tilde{f}_{\infty}|^{2^{*}}\right)^{1/2^{*}},$
$\displaystyle\leq$ $\displaystyle
C(n)\biggl{(}\int_{\Sigma^{i}}|\nabla\tilde{f}_{\infty}|^{2}\,|\chi_{S}|^{2}+2|\nabla\tilde{f}_{\infty}|\,|\chi_{S}|\,|\tilde{f}_{\infty}|\,|\nabla\chi_{S}|$
$\displaystyle\hskip
56.9055pt+|\tilde{f}_{\infty}|^{2}\,|\nabla\chi_{S}|^{2}\biggr{)}^{1/2},$
$\displaystyle\leq$ $\displaystyle
C\left(\int_{\Sigma^{i}}|\nabla\tilde{f}_{\infty}|^{2}+\int_{\Sigma^{i}}\omega^{\Sigma^{i},R}\tilde{f}_{\infty}^{2}\right)^{1/2}.$

The bound is independent of $S$, and thus we may deduce that
$\tilde{f}_{\infty}\in L^{2^{*}}(\Sigma^{i})$.

###  2.4 Strict Stability of the Neck

######  Lemma 4.

Consider the cylinder $(A\times\mathbb{R},g)$, where $A\subset\mathbb{R}^{n}$
is a non-empty, open set, and $g$ is a smooth Riemannian metric, such that
there exists a constant $K\in[1,\infty)$ such that as bilinear forms

$\begin{split}\frac{1}{K}\langle\,\cdot\,,\,\cdot\,\rangle&\leq g\leq
K\langle\,\cdot\,,\,\cdot\,\rangle\\\
\frac{1}{K}\langle\,\cdot\,,\,\cdot\,\rangle&\leq g^{-1}\leq
K\langle\,\cdot\,,\,\cdot\,\rangle,\end{split}$ (15)

where $\langle\,\cdot\,,\,\cdot\,\rangle$ is the standard metric on
$\mathbb{R}^{n}$. Now suppose we have a smooth function

$u\colon A\rightarrow(-T,T),$

such that $\|\nabla^{\mathbb{R}^{n}}u\|_{L^{\infty}(A)}\leq 1/2$, and denote
$M\coloneqq\text{graph}\,(u)\subset A\times(-2\,T,2\,T)$. For fixed
$W\in(0,\infty)$ suppose we have functions, $\omega\in L(n/2,\infty)(M)$, and
$V\in L^{\infty}(M)$, such that $\|\omega\|_{L(n/2,\infty)(M)}\leq W$, and
$\operatorname*{ess\,inf}\omega>0$. Then, there exists an
$\varepsilon=\varepsilon(n,K,W)\in(0,1)$, and $C=C(n,K)\in(0,+\infty)$ such
that if $|V|\leq\varepsilon\omega$ on $M$, then

$0<\frac{1}{C\,W}\leq\inf\Biggl{\\{}\int_{M}|\nabla f|^{2}-Vf^{2}\colon f\in
W_{0}^{1,2}(M),\,\int_{M}f^{2}\omega=1\Biggr{\\}}.$

###### Proof.

First, we recall that positive-definite real symmetric matrices are always
diagonalisable with positive eigenvalues, thus (15) implies that,

$K^{-(n+1)/2}\leq\sqrt{|g|}\leq K^{(n+1)/2}.$

We define the following map, $F(x)\coloneqq(x,u(x))$, and the metric
$\tilde{g}=F^{*}g$ on $A$. As metrics on $A$ we have that there exists a
$C=C(n,K)\in[1,\infty)$, such that,

$C^{-1}\langle\,\cdot\,,\,\cdot\,\rangle\leq\tilde{g}\leq
C\langle\,\cdot\,,\,\cdot\,\rangle.$

Thus, as bilinear forms,

$C^{-1}\langle\,\cdot\,,\,\cdot\,\rangle\leq\tilde{g}^{-1}\leq
C\langle\,\cdot\,,\,\cdot\,\rangle,$

and

$C^{-1}\leq\sqrt{|\tilde{g}|}\leq C.$

Taking an $f\in W^{1,2}_{0}(M)$, that satisfies,

$\int_{M}f^{2}\,\omega=1,$

we have, by applying Propositions 2 and 3, for large enough $k$ (see [38,
Theorem 1.1] for a similar computation)

$\displaystyle 1$ $\displaystyle=$ $\displaystyle\int_{A}(f\circ
F)^{2}\,(\omega\circ F)\sqrt{|\tilde{g}|}\,dx,$ $\displaystyle\leq$
$\displaystyle C\int_{A}(f\circ F)^{2}\,(\omega\circ F)\,dx,$
$\displaystyle\leq$ $\displaystyle C\|\omega\circ
F\|_{(n/2,\infty)(A,\langle\,\cdot\,,\,\cdot\,\rangle)}\|f\circ
F\|_{(2^{*},2)(A)}^{2},$ $\displaystyle\leq$ $\displaystyle C\|\omega\circ
F\|_{(n/2,\infty)(A,\langle\,\cdot\,,\,\cdot\,\rangle)}\int_{A}|\nabla^{\mathbb{R}^{n}}(f\circ
F)|^{2}\,dx,$ $\displaystyle\leq$ $\displaystyle C\|\omega\circ
F\|_{(n/2,\infty)(A,\langle\,\cdot\,,\,\cdot\,\rangle)}\int_{M}|\nabla
f|^{2}-Vf^{2}\,d\mu$ $\displaystyle\hskip 28.45274pt+C\|\omega\circ
F\|_{(n/2,\infty)(A,\langle\,\cdot\,,\,\cdot\,\rangle)}\int_{M}Vf^{2}\,d\mu,$
$\displaystyle\leq$ $\displaystyle C\|\omega\circ
F\|_{(n/2,\infty)(A,\langle\,\cdot\,,\,\cdot\,\rangle)}\left(\varepsilon+\int_{M}|\nabla
f|^{2}-Vf^{2}\,d\mu\right),$

where $C=C(n,K)$, and $\mu$ denotes the volume measure of $(M,g)$. Moreover,
we may make a further choice of $C=C(n,K)<+\infty$, such that,

$\|\omega\circ F\|_{(n/2,\infty)(A,\langle\,\cdot\,,\,\cdot\,\rangle)}\leq
C\|\omega\|_{(n/2,\infty)(M,g)}\leq C\,W.$

Thus,

$\int_{M}|\nabla f|^{2}-Vf^{2}\geq\frac{1-\varepsilon\,C\,W}{C\,W}>0,$

and for an appropriate choice of $\varepsilon\in(0,1)$, we may conclude the
result. ∎

In Sections 2.2 and 2.3 we showed that if we have a sequence,

$f_{k}\in\mathcal{E}_{\omega_{k,\delta,R}}(\lambda_{k};L_{k},W^{1,2}(co(M_{k}))),$

with $\lambda_{k}\leq 0$, and

$\int_{co(M_{k})}f_{k}^{2}\,\omega_{k,\delta,R}=1,$

for all $k$, then after potentially taking a subsequence and renumerating we
have that $\lambda_{k}\rightarrow\lambda_{\infty}\leq 0$, and,

$f_{k}\rightarrow((f_{\infty}^{1},\ldots,f_{\infty}^{m}),f_{\infty}^{\Sigma^{1}},\ldots,f_{\infty}^{\Sigma^{J}}),$

where, for $i=1,\ldots,m$, $f_{\infty}^{i}\in W^{1,2}(co(M_{\infty}))$, and
for $j=1,\ldots,L$, $f_{\infty}^{\Sigma^{j}}\in
W_{\omega^{\Sigma^{j},R}}^{1,2}(\Sigma^{j})\subset L^{2^{*}}(\Sigma^{j})$. By
this convergence we mean, for $i=1,\ldots,m$,

$\begin{cases}\tilde{f}_{k}^{i}\rightharpoonup
f_{\infty},&W^{1,2}_{\text{loc}}(co(M_{\infty})\setminus\iota^{-1}(\mathcal{I}))),\\\
\tilde{f}_{k}^{i}\rightarrow
f_{\infty},&L^{2}_{\text{loc}}(co(M_{\infty})\setminus\iota^{-1}(\mathcal{I}))),\end{cases}$
(16)

and for $j=1,\ldots,L$,

$\begin{cases}\tilde{f}_{k}^{\Sigma^{j}}\rightharpoonup
f_{\infty}^{j},&W^{1,2}_{\text{loc}}(\Sigma^{j}),\\\
\tilde{f}_{k}^{\Sigma^{j}}\rightarrow
f_{\infty},&L^{2}_{\text{loc}}(\Sigma^{j}).\end{cases}$ (17)

Furthermore, we are able to deduce that for $i=1,\ldots,m$,

$f_{\infty}^{i}\in\mathcal{E}_{\omega_{\delta}}(\lambda_{\infty};L_{\infty},W^{1,2}(co(M_{\infty}))),$

and $j=1,\ldots,J$,

$f_{\infty}^{\Sigma^{j}}\in\mathcal{E}_{\omega^{\Sigma^{j},R}}(\lambda_{\infty};L_{\Sigma^{j}},W^{1,2}_{\omega^{\Sigma^{j},R}}(\Sigma^{j}),W^{1,2}(\Sigma^{j})).$

######  Claim 3.

If $\\{f_{1,k},\ldots,f_{b,k}\\}$, is an orthonormal collection, with respect
to $\omega_{k,\delta,R}$-weighted $L^{2}$ norm, of eigenfunctions, with non-
positive eigenvalues for $M_{k}$, and $\sum_{i=1}a_{i}^{2}=1$, then,

$h_{k}\coloneqq\sum_{i=1}^{b}a_{i}f_{1,k}\rightarrow\sum_{i=1}^{b}a_{i}((\tilde{f}_{i,\infty}^{1},\ldots,\tilde{f}_{i,\infty}^{m}),\tilde{f}_{i,\infty}^{\Sigma^{1}},\ldots,\tilde{f}_{i,\infty}^{\Sigma^{J}})\not=((0,\ldots,0),0,\ldots,0).$

###### Proof.

This claim is proven by a contradiction argument similar to that in [18, Claim
1 of Lemma<EMAIL_ADDRESS>

By Remark 1, for all $\eta\in(0,1)$, there exists an $r_{0}\in(0,\delta/4)$,
and $R_{0}\in(4R,\infty)$, such that, for large enough $k$, each connected
component of

$M_{k}\cap B_{r_{0}}(y)\setminus\cup_{j=1}^{L}\Sigma_{k}^{j,R_{0}},$

may be written as the graph of a smooth function over an open set of
$T_{y}M_{\infty}$, with gradient bounded by $\eta$. More explicitly, taking
geodesic normal coordinates about $y\in N$, such that
$T_{y}M_{\infty}=\\{x_{n+1}=0\\}$, then for large enough $k$, if we denote
$C_{k}$ to be a connected component

$\exp_{y}^{-1}(M_{k}\cap
B_{\delta}^{N}(y)\setminus\cup_{j=1}^{J}\Sigma_{k}^{j,R_{0}})\cap(B_{r_{0}}^{n}(0)\times(-\tau,\tau)),$

then there exists a non-empty open set $A(C_{k})\subset\\{x_{n+1}=0\\}$, and a
smooth function,

$u_{k}\colon A(C_{k})\rightarrow(-\tau,\tau),$

such that, $C_{k}=\text{graph}\,(u_{k})$, and $\|\nabla
u_{k}\|_{L^{\infty}}\leq\eta$. Moreover, we may make further choices of
$r_{0}$ and $R_{0}$, such that on $M_{k}\cap
B_{r_{0}}^{n}(y)\times(-\tau,\tau)\setminus\cup_{j=1}^{J}\Sigma_{k}^{j,R_{0}}$,
$V=|A_{k}|^{2}+R_{k}$, and $\omega=\omega_{k,\delta,R}$, will satisfy the
hypothesis of Lemma 4 (recall our uniform bound on
$\|\omega_{k,\delta,R}\|_{(n/2,\infty)}$ from Claim 2). Assuming Claim 3 does
not hold, we will provide a contradiction to Lemma 4.

Consider the following function,

$\begin{cases}\chi\in C^{\infty}(\mathbb{R};[0,1]),\\\
\chi(t)=1,&t\in(-\infty,1],\\\ \chi(t)=0,&t\in[2,\infty),\\\
\chi^{\prime}(t)\leq 0,\\\ |\chi^{\prime}(t)|\leq 3.\end{cases}$

We then define the following function

$H_{k}(x)=\begin{cases}0,&x\in
M_{k}\setminus\cup_{y\in\mathcal{I}}B^{N}_{r_{0}}(y),\\\
\chi(2r_{0}^{-1}\text{dist}_{g}^{N}(x,y))h_{k},&x\in M_{k}\cap
B^{N}_{r_{0}}(y)\setminus\cup_{j=1}^{L}\Sigma_{k}^{j,2R_{0}},\,y\in\mathcal{I},\\\
h_{k}(1-\chi((R_{0}r_{k}^{j})^{-1}\text{dist}_{g}^{N}(x,p_{k}^{j}))),&x\in\Sigma_{k}^{j,2R_{0}},\,j=1,\ldots,L.\end{cases}$

We compute the gradient of $H_{k}$. For $x\in
M_{k}\setminus\cup_{y\in\mathcal{I}}B^{N}_{r_{0}}(y)$, or
$x\in\Sigma_{k}^{j,R_{0}}$, $j=1,\ldots,L$, we have, $\nabla H_{k}=0$. For
$x\in M_{k}\cap B^{N}_{r_{0}}(y)\setminus\cup_{j=1}^{L}\Sigma_{k}^{j,2R_{0}}$,
we have,

$\nabla H_{k}(x)=\nabla
h_{k}(x)\chi(2r_{0}^{-1}\text{dist}_{g}^{N}(x,y))+h_{k}(x)2r_{0}^{-1}\chi^{\prime}(2r_{0}^{-1}\text{dist}_{g}^{N}(x,y))\nabla\text{dist}_{g}^{N}(x,y).$

Finally, for $x\in\Sigma_{k}^{j,2R_{0}}\setminus\Sigma_{k}^{j,R_{0}}$,
$j=1,\ldots,L$, we have,

$\begin{split}\nabla H_{k}(x)=&\nabla
h_{k}(x)(1-\chi((Rr_{k}^{j,y})^{-1}\text{dist}_{g}^{N}(x,p_{k}^{j,y})))\\\
&\hskip 28.45274pt-
h_{k}(x)(R_{0}r_{k}^{j,y})^{-1}\chi^{\prime}((R_{0}r_{k}^{j,y})^{-1}\text{dist}_{g}^{N}(x,p_{k}^{j,y}))\nabla\text{dist}_{g}^{N}(x,p_{k}^{j,y})\end{split}$

We wish to derive a contradiction of Lemma 4, by proving that for any $\tau>0$
there exists a small enough $r_{0}$, and large enough $R_{0}$, such that for
large enough $k$,

$B_{k}(H_{k},H_{k})<\zeta,$ (18)

and

$\int_{M_{k}}\omega_{k,\delta,R}H_{k}^{2}>1-\zeta.$ (19)

The contradiction immediately follows from the fact that for large enough $k$,
the number of connected components of
$M_{k}\cap\cup_{y\in\mathcal{I}}B_{r_{0}}^{N}(y)\setminus\cup_{j=1}^{L}\Sigma_{k}^{j,R_{0}}$
is bounded $m\cdot|\mathcal{I}|<+\infty$.

Consider,

$|B_{k}(h_{k},h_{k})-B_{k}(H_{k},H_{k})|\leq\left|\int_{M_{k}}|\nabla
h_{k}|^{2}-|\nabla
H_{k}|^{2}\right|+C\left|\int_{M_{k}}\omega_{k,\delta,R}(h_{k}^{2}-H_{k}^{2})\right|.$
(20)

We split the first term on the right hand side of (20) into seperate domains,

$\left|\int_{M_{k}}|\nabla h_{k}|^{2}-|\nabla
H_{k}|^{2}\right|\leq\@slowromancap i@+\@slowromancap
ii@+\sum_{y\in\mathcal{I}}\@slowromancap
iii@^{y}+\sum_{j=1}^{J}(\@slowromancap iv@^{j}+\@slowromancap v@^{j}),$ (21)

where,

$\@slowromancap
i@=\int_{M_{k}\setminus\cup_{y\in\mathcal{I}}B_{r_{0}}^{N}(y)}|\nabla
h_{k}|^{2},$ (22)

$\begin{split}\@slowromancap
ii@\leq&\int_{M_{k}\cap(\cup_{y\in\mathcal{I}}B^{N}_{r_{0}}(y)\setminus
B^{N}_{r_{0}/2}(y))}|\nabla
h_{k}|^{2}(1-\chi(2r^{-1}\text{dist}_{g}^{N}(\,\cdot\,,y))^{2})\\\
&+\int_{M_{k}\cap(\cup_{y\in\mathcal{I}}B^{N}_{r}(y)\setminus
B^{N}_{r_{0}/2}(y))}4|h_{k}|r_{0}^{-1}\chi^{\prime}(2r_{0}^{-1}\text{dist}_{g}^{N}(\,\cdot\,,y))|g_{M_{k}}(\nabla
h_{k},\nabla\text{dist}_{g}^{N}(\,\cdot\,,y))|\\\
&+\int_{M_{k}\cap(\cup_{y\in\mathcal{I}}B^{N}_{r_{0}}(y)\setminus
B^{N}_{r_{0}/2}(y))}4r_{0}^{-2}h_{k}^{2}|\chi^{\prime}(2r_{0}^{-1}\text{dist}_{g}^{N}(\,\cdot\,,y))|^{2},\\\
&\leq C\left(\int_{M_{k}\cap(\cup_{y\in\mathcal{I}}B^{N}_{r_{0}}(y)\setminus
B^{N}_{r_{0}/2}(y))}|\nabla
h_{k}|^{2}+\int_{M_{k}\cap(\cup_{y\in\mathcal{I}}B^{N}_{r_{0}}(y)\setminus
B^{N}_{r_{0}/2}(y))}r_{0}^{-2}h_{k}^{2}\right)\end{split}$ (23)

For $y\in\mathcal{I}$,

$\@slowromancap iii@^{y}=\left|\int_{x\in M_{k}\cap
B^{N}_{r_{0}/2}(y)\setminus(\cup_{j=1}^{J}\Sigma_{k}^{j,2R_{0}})}|\nabla
h_{k}|^{2}-|\nabla H_{k}|^{2}\right|=0,$ (24)

and $j=1,\ldots,J$,

$\begin{split}\@slowromancap
iv@^{j}\leq&\int_{\Sigma_{k}^{j,2R_{0}}\setminus\Sigma_{k}^{j,R_{0}}}|\nabla
h_{k}|^{2}(1-(1-\chi((R_{0}r_{k}^{j})^{-1}\text{dist}_{g}^{N}(\,\cdot\,,p_{k}^{j})))^{2})\\\
&+\int_{\Sigma_{k}^{j,2R_{0}}\setminus\Sigma_{k}^{j,R_{0}}}2|h_{k}|(R_{0}r_{k}^{j,y})^{-1}\chi^{\prime}((R_{0}r_{k}^{j})^{-1}\text{dist}_{g}^{N}(\,\cdot\,,p_{k}^{j}))\\\
&\hskip
56.9055pt(1-\chi((R_{0}r_{k}^{j,y})^{-1}\text{dist}_{g}^{N}(\,\cdot\,,p_{k}^{j})))|g_{M_{k}}(\,\nabla
h_{k},\nabla\text{dist}_{g}^{N}(\,\cdot\,,p_{k}^{j})\,)|\\\
&+\int_{\Sigma_{k}^{j,2R_{0}}\setminus\Sigma_{k}^{j,R}}(R_{0}r_{k}^{j})^{-2}h_{k}^{2}|\chi^{\prime}((R_{0}r_{k}^{j})^{-1}\text{dist}_{g}^{N}(\,\cdot\,,p_{k}^{j}))|^{2},\\\
&\leq C\left(\int_{\Sigma_{k}^{j,2R_{0}}\setminus\Sigma_{k}^{j,R_{0}}}|\nabla
h_{k}|^{2}+\int_{\Sigma_{k}^{j,2R_{0}}\setminus\Sigma_{k}^{j,R_{0}}}(R_{0}r_{k}^{j})^{-2}h_{k}^{2}\right)\end{split}$
(25)

$\@slowromancap v@^{j}=\int_{\Sigma_{k}^{j,R_{0}}}|\nabla h_{k}|^{2}.$ (26)

Recalling notation from Section 2.2, we have that as $\tilde{h}_{k}\coloneqq
h_{k}\circ F_{k}$ solves a second order, linear, elliptic PDE on
$\Omega_{r_{0}/4}\subset\subset
co(M_{\infty})\setminus\iota_{\infty}^{-1}(\mathcal{I})$, by interior
estimates,

$\|\tilde{h}_{k}\|_{W^{2,2}(\Omega_{r_{0}/2})}\leq
C\|\tilde{h}_{k}\|_{W^{1,2}(\Omega_{r_{0}/4})},$

for $C=C(N,g,\delta,r_{0},M_{\infty},\Sigma^{1},\ldots,\Sigma^{J})<+\infty$.
Thus, by our assumption that $h_{k}\rightarrow((0,\ldots,0),0\ldots,0)$, after
potentially taking a subsequence and renumerating we have,

$\begin{cases}\tilde{h}_{k}\rightharpoonup 0,&W^{2,2}(\Omega_{r_{0}/2}),\\\
\tilde{h}_{k}\rightarrow 0,&W^{1,2}(\Omega_{r_{0}/2}).\end{cases}$

By identical arguments, we have that, after potentially taking a subsequence
and renumerating, for all $j=1,\ldots,J$,

$\begin{cases}\tilde{h}_{k}^{\Sigma^{j}}\rightharpoonup
0,&W^{2,2}(\Sigma^{j,2R_{0}}),\\\ \tilde{h}_{k}^{\Sigma^{j}}\rightarrow
0,&W^{1,2}(\Sigma^{j,2R_{0}}).\end{cases}$

Thus we have that for all $\tau>0$, and large enough $k$, there exists a
$C=C(N,g,\delta,R_{0},r_{0},M_{\infty},\Sigma^{1},\ldots,\Sigma^{J})<+\infty$

$\begin{split}\left|\int_{M_{k}}|\nabla h_{k}|^{2}-|\nabla
H_{k}|^{2}\right|\leq
C\biggl{(}&\|\nabla\tilde{h}_{k}\|_{L^{2}(\Omega_{r_{0}/2})}^{2}\\\
&+r_{0}^{-2}\|\tilde{h}_{k}\|_{L^{2}(\Omega_{r_{0}/2})}^{2}+\sum_{j=1}^{J}\|\tilde{h}^{\Sigma^{j}}_{k}\|^{2}_{W^{1,2}(\Sigma^{j,2R_{0}})}\biggr{)}<\frac{\zeta}{2}.\end{split}$

Similarly we have,

$\left|\int_{M_{k}}\omega_{k,\delta,R}(h_{k}^{2}-H_{k}^{2})\right|\leq
C\left(r_{0}^{-2}\int_{\Omega_{r_{0}/2}}\tilde{h}_{k}^{2}+\int_{\Sigma^{j,2R_{0}}}(\tilde{h}^{\Sigma^{j}}_{k})^{2}\right)<\frac{\zeta}{2C}.$

Thus there exists a
$\gamma=\gamma(N,g,m,|\mathcal{I}|,M_{\infty},\Sigma^{1},\ldots,\Sigma^{J})\in(0,1)$,
such that picking $\zeta\in(0,\gamma)$, will imply a contradiction to Lemma 4
on at least one of the finitely many connected components of

$\cup_{y\in\mathcal{I}}\,M_{k}\cap
B_{r_{0}}^{N}(y)\setminus\cup_{j=1}^{J}\Sigma_{k}^{j,R_{0}}).$

Note that there is a uniform upper bound on the number of such connected
components.

∎

##  3 Equivalence of Weighted and Unweighted Eigenspaces

######  Proposition 6.

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$, with
$\omega\in L(n/2,\infty)(M,g)$. Then for any $f_{1},\,f_{2}\in W^{1,2}(M)$, we
have that there exists a $C=C(M,g)<\infty$, such that,

$\left|\int_{M}f_{1}f_{2}\omega\right|\leq
C\|\omega\|_{(n/2,\infty)}\|f_{1}\|_{W^{1,2}(M)}\|f_{2}\|_{W^{1,2}(M)}.$

###### Proof.

Using Propositions 2 and 4,

$\displaystyle\left|\int_{M}f_{1}f_{2}\omega\right|$ $\displaystyle\leq$
$\displaystyle\left(\int_{M}\omega\,f_{1}^{2}\right)^{1/2}\,\left(\int_{M}\omega\,f_{2}^{2}\right)^{1/2},$
$\displaystyle\leq$
$\displaystyle\|\omega\|_{(n/2,\infty)}\,\|f_{1}^{2}\|_{(2^{*}/2,1)}^{1/2}\,\|f_{2}^{2}\|_{(2^{*}/2,1)}^{1/2},$
$\displaystyle\leq$
$\displaystyle\|\omega\|_{(n/2,\infty)}\,\|f_{1}\|_{(2^{*},2)}\,\|f_{2}\|_{(2^{*},2)},$
$\displaystyle\leq$ $\displaystyle
C\|\omega\|_{(n/2,\infty)}\,\|f_{1}\|_{W^{1,2}(M)}\,\|f_{2}\|_{W^{1,2}(M)}.$

∎

######  Proposition 7.

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$, with
$V\in L^{\infty}(M)$, $\omega\in L(n/2,\infty)(M,g)$, and
$\operatorname*{ess\,inf}\,\omega>0$. Define the elliptic operator,

$L\coloneqq\Delta+V.$

Then,

$\displaystyle\text{span}\,\left\\{\cup_{\lambda\leq
0}\,\mathcal{E}(\lambda;L,W^{1,2}(M))\right\\}$ $\displaystyle=$
$\displaystyle\oplus_{\lambda\leq 0}\,\mathcal{E}(\lambda;L,W^{1,2}(M)),$ (27)
$\displaystyle\text{span}\,\left\\{\cup_{\lambda\leq
0}\,\mathcal{E}_{\omega}(\lambda;L,W^{1,2}(M))\right\\}$ $\displaystyle=$
$\displaystyle\oplus_{\lambda\leq
0}\,\mathcal{E}_{\omega}(\lambda;L,W^{1,2}(M)),$ (28)

and

$\text{dim}\,\left(\text{span}\,\left\\{\cup_{\lambda\leq
0}\,\mathcal{E}_{\omega}(\lambda;L,W^{1,2}(M))\right\\}\right)=\text{dim}\,\left(\text{span}\,\left\\{\cup_{\lambda\leq
0}\,\mathcal{E}(\lambda;L,W^{1,2}(M))\right\\}\right).$

###### Proof.

First we note by Proposition 6, and fact that
$\operatorname*{ess\,inf}\omega>0$,

$\langle f_{1},f_{2}\rangle_{\omega}\coloneqq\int_{M}f_{1}\,f_{2}\,\omega,$

is a well-defined inner product on $W^{1,2}(M)$. Consider the bilinear form on
$f_{1},\,f_{2}\in W^{1,2}(M)$, corresponding to our elliptic operator $L$,

$B_{L}[f_{1},f_{2}]\coloneqq\int_{M}\nabla f_{1}\cdot\nabla
f_{2}-Vf_{1}f_{2}.$

If $f_{i}\in\mathcal{E}_{\omega}(\lambda_{i};L,W^{1,2}(M))$, for $i=1,\,2$,
with $\lambda_{1}\not=\lambda_{2}$, we have,

$\lambda_{1}\langle
f_{1},f_{2}\rangle_{\omega}=B_{L}[f_{1},f_{2}]=\lambda_{2}\langle
f_{1},f_{2}\rangle_{\omega}.$

Thus, $\langle f_{1},f_{2}\rangle_{\omega}=0$, and so (28) follows. An
identical argument will also conclude (27).

We show that,

$\displaystyle\text{dim}\,(\oplus_{\lambda\leq
0}\,\mathcal{E}(\lambda;L,W^{1,2}(M)))$ $\displaystyle=$
$\displaystyle\sup\\{\text{dim}\,\Pi\colon\Pi\leq W^{1,2}(M),\text{ is a
linear space such that, }(B_{L})_{|\Pi}\leq 0\\}$ $\displaystyle=$
$\displaystyle\text{dim}\,(\oplus_{\lambda\leq
0}\,\mathcal{E}_{\omega}(\lambda;L,W^{1,2}(M))).$

Indeed, we begin by showing that,

$\begin{split}\text{dim}\,(\oplus_{\lambda\leq
0}\,\mathcal{E}_{\omega}(\lambda;L,W^{1,2}(M)))\leq\sup\\{\text{dim}\,\Pi\colon&\Pi\leq
W^{1,2}(M),\\\ &\text{ is linear space such that, }(B_{L})_{|\Pi}\leq
0\\}.\end{split}$ (29)

If the left-hand side is equal to $0$ then the inequality is trivial. Thus,
take $b\in\mathbb{Z}_{\geq 1}$, such that

$b\leq\text{dim}\,(\oplus_{\lambda\leq
0}\,\mathcal{E}_{\omega}(\lambda;L,W^{1,2}(M))),$

then we have $b$, $\omega$-weighted eigenvectors,

$\\{f_{1},\ldots,f_{b}\\}\subset W^{1,2}(M),$

with respective non-positive eigenvalues
$\\{\lambda_{1},\ldots,\lambda_{b}\\}$. By the argument at the begining of the
proof, we may take the set $\\{f_{1},\ldots,f_{b}\\}$, to be orthonormal with
respect to $\langle\,\cdot\,,\,\cdot\,\rangle_{\omega}$. Thus,

$\Pi=\text{span}\\{f_{1},\ldots,f_{b}\\},$ (30)

is a $b$-dimensional vector space, and for $a_{1},\ldots,a_{b}\in\mathbb{R}$,
we have,

$\displaystyle B_{L}\left[\sum_{i}a_{i}f_{i},\sum_{j}a_{j}f_{j}\right]$
$\displaystyle=$ $\displaystyle\sum_{i,j}a_{i}a_{j}\lambda_{i}\langle
f_{i},f_{j}\rangle_{\omega},$ $\displaystyle=$
$\displaystyle\sum_{i}\lambda_{i}a_{i}^{2},$ $\displaystyle\leq$
$\displaystyle 0.$

Thus, (29) holds. Identical argument shows that same inequality holds for
unweighted eigenspaces.

We look to show reverse inequality of (29). As we know that (29) holds, if the
left-hand side of (29) is unbounded, then equality holds trivially. Thus, we
consider, $b\in\mathbb{Z}_{\geq 0}$,

$b=\text{dim}\,(\oplus_{\lambda\leq
0}\,\mathcal{E}(\lambda;L,W^{1,2}(M)))<\infty,$

and prove reverse of (29) by contradiction. Indeed, assume that we have a
linear subspace $\tilde{\Pi}\leq W^{1,2}(M)$, of dimension $b+1$, such that
$B_{L}$ is non-positive on $\tilde{\Pi}$. Consider the $b$ dimensional linear
subspace $\Pi$, identically defined as (30). Note that,

$W^{1,2}(M)=\Pi\oplus\Pi^{\perp_{\omega}}.$

The projection map $P_{\Pi}\colon\tilde{\Pi}\rightarrow\Pi$, must have a non-
trivial kernel, implying that there exists a
$v\in\tilde{\Pi}\cap\Pi^{\perp_{\omega}}$, with $\langle
v,v\rangle_{\omega}=1$. Thus,

$\tilde{\lambda}=\inf\\{B_{L}[f,f]\colon f\in\Pi^{\perp_{\omega}},\,\langle
f,f\rangle_{\omega}=1\\}\leq 0.$

Take $\tilde{f}_{k}\in\Pi^{\perp_{\omega}}$,
$\langle\tilde{f}_{k},\tilde{f}_{k}\rangle_{\omega}=1$, such that

$\tilde{\lambda}=\lim_{k\rightarrow\infty}B_{L}[\tilde{f}_{k},\tilde{f}_{k}].$

Thus, also noting that $\operatorname*{ess\,inf}\,\omega>0$, and
$|V|<+\infty$, we may apply Rellich–Kondrachov, to conclude that there exists
an $\tilde{f}\in W^{1,2}(M)$, such that after potentially taking a subsequence
and renumerating,

$\begin{cases}\tilde{f}_{k}\rightharpoonup\tilde{f},\,in\,W^{1,2}(M),\\\
\tilde{f}_{k}\rightarrow\tilde{f},\,in\,L^{2}(M).\end{cases}$

Denote,

$\displaystyle h_{1}$ $\displaystyle\coloneqq$ $\displaystyle
P_{\Pi}(\tilde{f}),$ $\displaystyle h_{2}$ $\displaystyle\coloneqq$
$\displaystyle\tilde{f}-h_{1}\in\Pi^{\perp_{\omega}}.$

There exists $a_{1},\ldots,a_{b}\in\mathbb{R}$, such that,

$h_{1}=\sum_{i=1}^{b}a_{i}f_{i}.$

Then, as $\tilde{f}_{k}\in\Pi^{\perp_{\omega}}$,

$\displaystyle B_{L}[\tilde{f}_{k},\tilde{f}]$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}a_{i}B_{L}[\tilde{f}_{k},f_{i}]+B_{L}[\tilde{f}_{k},h_{2}],$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{m}a_{i}\lambda_{i}\langle\tilde{f}_{k},f_{i}\rangle_{\omega}+B_{L}[\tilde{f}_{k},h_{2}],$
$\displaystyle=$ $\displaystyle B_{L}[\tilde{f}_{k},h_{2}].$

Thus, by weak convergence,

$\displaystyle B_{L}[\tilde{f},\tilde{f}]$ $\displaystyle=$
$\displaystyle\lim_{k\rightarrow\infty}B_{L}[\tilde{f}_{k},\tilde{f}],$
$\displaystyle=$
$\displaystyle\lim_{k\rightarrow\infty}B_{L}[\tilde{f}_{k},h_{2}],$
$\displaystyle=$ $\displaystyle B_{L}[h_{1}+h_{2},h_{2}],$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}a_{i}\lambda_{i}\langle
f_{i},h_{2}\rangle_{\omega}+B_{L}[h_{2},h_{2}],$ $\displaystyle=$
$\displaystyle B_{L}[h_{2},h_{2}].$

By Fatou’s Lemma we have the following,

$0\leq\alpha=\langle h_{2},h_{2}\rangle_{\omega}\leq\langle
h_{2},h_{2}\rangle_{\omega}+\langle
h_{1},h_{1}\rangle_{\omega}=\langle\tilde{f},\tilde{f}\rangle_{\omega}\leq\liminf_{k\rightarrow\infty}\langle\tilde{f}_{k},\tilde{f}_{k}\rangle_{\omega}=1,$

and we reduce our argument to three cases.

First consider, $\alpha=0$. Thus, $h_{2}=0$, and by lower semicontinuity of
the $W^{1,2}(M)$-norm under weak convergence,

$0=B_{L}[h_{2},h_{2}]=B_{L}[\tilde{f},\tilde{f}]\leq\lim_{k\rightarrow\infty}B_{L}[\tilde{f}_{k},\tilde{f}_{k}]=\tilde{\lambda}\leq
0.$

Therefore, $\tilde{\lambda}=0$, and recalling the function
$v\in\tilde{\Pi}\cap\Pi^{\perp_{\omega}}$, with $\langle
v,v\rangle_{\omega}=1$, we must have

$B_{L}[v,v]=\inf\\{B_{L}[f,f]\colon f\in\Pi^{\perp_{\omega}},\,\langle
f,f\rangle_{\omega}=1\\}.$

Standard variational arguments will then show that
$v\in\mathcal{E}_{\omega}(0;L,W^{1,2}(M))$, which contradicts the definition
$b$.

If $\alpha=1$, then,

$B_{L}[h_{2},h_{2}]=\inf\\{B_{L}[f,f]\colon f\in\Pi^{\perp_{\omega}},\,\langle
f,f\rangle_{\omega}=1\\}.$

Again standard variational arguments may then show that
$h_{2}\in\mathcal{E}_{\omega}(\tilde{\lambda};L,W^{1,2}(M)$, which contradicts
the definition of $b$.

The final case is $0<\alpha<1$. Note that if $B_{L}[h_{2},h_{2}]=0$, then we
may apply a similar argument to that in case $\alpha=0$. Therefore, we may
assume that $B_{L}[h_{2},h_{2}]<0$. Define,

$h\coloneqq\alpha^{-1/2}h_{2}.$

Thus,

$B_{L}[h,h]=\alpha^{-1}B_{L}[h_{2},h_{2}]<B_{L}[h_{2},h_{2}]\leq\tilde{\lambda}.$

However, as $h\in\Pi^{\perp_{\omega}}$, and $\langle h,h\rangle_{\omega}=1$,
this is a contradiction. ∎

######  Remark 6.

The method of proof in Proposition 7 may be applied to prove that for all
$k\in\mathbb{N}$,

$\text{span}\,\left\\{\cup_{\lambda\leq
0}\,\mathcal{E}_{\omega}(\lambda;L,W^{1,2}(co(M_{k}))^{-})\right\\}=\oplus_{\lambda\leq
0}\,\mathcal{E}_{\omega}(\lambda;L,W^{1,2}(co(M_{k}))^{-}),$

and,

$\text{dim}\,(\oplus_{\lambda\leq
0}\,\mathcal{E}_{\omega_{k,\delta,R}}(\lambda;L_{k},W^{1,2}(co(M_{k}))^{-}))=\text{dim}\,(\oplus_{\lambda\leq
0}\,\mathcal{E}(\lambda;L_{k},W^{1,2}(co(M_{k}))^{-})).$

When reapplying this method the two major things to note is that
$W^{1,2}(co(M_{k}))^{-}$ is a linear space, and that when applying
Rellich–Kondrachov, due to the strong convergence in $L^{2}(co(M_{k}))$, we
have that the limit will also lie $W^{1,2}(co(M_{k}))^{-}$.

Let $(M,g)$ be a complete but not necessarily compact Riemannian manifold.
Recall the following function space,

$W_{\omega}^{1,2}(M)\coloneqq\\{f\in L_{\text{loc}}^{1}(M)\colon|\nabla f|\in
L^{2}(M),\,and\,f^{2}\omega\in L^{1}(M)\\}.$

######  Proposition 8.

Let $\Sigma$ be a complete, properly and minimally immersed hypersurface in
$\mathbb{R}^{n+1}$, $3\leq n\leq 6$, with finite index and Euclidean volume
growth at infinity. For any $R>0$, define the function $\omega^{\Sigma,R}\in
W^{1,\infty}(\Sigma)$, as in the statment of Theorem 1, then

$\text{span}\,\\{\cup_{\lambda\leq
0}\,\mathcal{E}_{\omega^{\Sigma,R}}(\lambda;L_{\Sigma},W_{\omega^{\Sigma,R}}^{1,2}(\Sigma),W^{1,2}(\Sigma))\\}=\oplus_{\lambda\leq
0}\,\mathcal{E}_{\omega^{\Sigma,R}}(\lambda;L_{\Sigma},W_{\omega^{\Sigma,R}}^{1,2}(\Sigma),W^{1,2}(\Sigma)),$
(31)

and,

$\text{dim}\,(\oplus_{\lambda<0}\,\mathcal{E}_{\omega^{\Sigma,R}}(\lambda;L_{\Sigma},W_{\omega^{\Sigma,R}}^{1,2}(\Sigma),W^{1,2}(\Sigma)))=\text{ind}\,(\Sigma)\coloneqq\lim_{S\rightarrow\infty}\text{ind}\,(\Sigma\cap
B_{S}^{n+1}(0)).$

###### Proof.

Denote our immersion $\iota\colon\Sigma\rightarrow\mathbb{R}^{n+1}$, and
$\omega=\omega^{\Sigma,R}$. As $\operatorname*{ess\,inf}\omega>0$ on compact
sets of $\Sigma$, we have that $W_{\omega}^{1,2}(\Sigma)\subset
W^{1,2}_{\text{loc}}(\Sigma)$. As $\Sigma$ has finite index and Euclidean
volume growth at infinity, we may deduce that there exists an $S>0$, such that
$\Sigma\setminus\iota^{-1}(B_{S}^{n+1}(0))$ is stable [20, Proposition 1].
Thus, by the curvature estimates for stable minimal immersions, [33, Theorem
3] for $2\leq n\leq 5$, and [7, Theorem 2] for $2\leq n\leq 6$, for all
$x\in\Sigma\setminus\iota^{-1}(B_{2S}^{n+1}(0))$, we have that there exists a
$C=C(\Sigma)<+\infty$, such that,

$|A_{\Sigma}|^{2}(x)\leq\frac{C}{(|\iota(x)|-S)^{2}}\leq\frac{C}{|\iota(x)|^{2}}.$
(32)

Moreover, we may choose $S$ large enough so that, $|\iota(x)|^{-2}\leq
2\omega$ on $\Sigma^{i}\setminus\iota^{-1}(B_{S}^{n+1}(0))$. Thus, for
$x\in\Sigma\setminus\iota^{-1}(B_{2S}^{n+1}(0))$,

$|A_{\Sigma}|^{2}(x)\leq C\omega.$

Thus, for $f,\,h\in W_{\omega}^{1,2}(\Sigma)$,

$\left|\int_{\Sigma}f\,h\,\omega\right|+\left|\int_{\Sigma}|A_{\Sigma}|^{2}\,f\,h\right|<(1+C)\left(\int_{\Sigma}f^{2}\,\omega\right)^{1/2}\left(\int_{\Sigma}h^{2}\,\omega\right)^{1/2}<+\infty.$

Therefore the quantities, $B_{L}[f,h]$, and $\langle f,h\rangle_{\omega}$, are
finite and well defined. We also note that for $\lambda\in\mathbb{R}$, such
that

$\text{dim}\,(\mathcal{E}_{\omega^{\Sigma,R}}(\lambda;L_{\Sigma},W_{\omega}^{1,2}(\Sigma),W^{1,2}(\Sigma)))\not=0,$

then, similar to (7) we may deduce $\lambda\geq-C$, with
$C=C(\Sigma)<+\infty$.

Take $S\geq 1$, large enough, we may define the following function,

$\begin{cases}\chi_{S}(x)=1,&x\in\iota^{-1}(B_{S}^{n+1}(0))\cap\Sigma,\\\
\chi_{S}(x)=0,&x\in\Sigma\setminus\iota^{-1}(B_{2S}^{n+1}(0)),\\\
|\nabla\chi_{S}|\leq 4S^{-1}.\end{cases}$

For $f\in W^{1,2}_{\omega}(\Sigma)$, we define $f_{S}=f\chi_{S}\in
W_{0}^{1,2}(\Sigma\cap\iota^{-1}(B_{2S}^{n+1}(0)))$. Again using the estimate
(32), we may deduce that for $f,\,h\in W_{\omega}^{1,2}(\Sigma)$,

$\displaystyle\lim_{R\rightarrow\infty}B_{L}[f,h_{R}]$ $\displaystyle=$
$\displaystyle B_{L}[f,h],$ $\displaystyle\lim_{R\rightarrow\infty}\langle
f,h_{R}\rangle_{\omega}$ $\displaystyle=$ $\displaystyle\langle
f,h\rangle_{\omega}.$

and thus if
$f\in(\mathcal{E}_{\omega^{\Sigma,R}}(\lambda;L_{\Sigma},W_{\omega}^{1,2}(\Sigma),W^{1,2}(\Sigma)))$,
then in fact, for all $h\in W^{1,2}_{\omega}(\Sigma)$,
$B_{L_{\Sigma}}[f,h]=\lambda\langle f,h\rangle_{\omega}$. This allows us to
conclude (31) in an identical way to (28).

We proceed similarly to Proposition 7. Pick $S_{0}>0$, such that
$\text{ind}\,(\Sigma\cap\iota^{-1}(B_{S}^{n+1}(0)))=\text{ind}\,(\Sigma)$.
First we show that,

$\text{dim}\,(\oplus_{\lambda<0}\,\mathcal{E}_{\omega^{\Sigma,R}}(\lambda;L_{\Sigma},W_{\omega}^{1,2}(\Sigma),W^{1,2}(\Sigma)))\leq\text{ind}\,(\Sigma)=I.$
(33)

If the left hand side is equal to $0$, then the inequality is trivial, thus
assume we have $b\in\mathbb{Z}_{\geq 1}$, such that,

$b\leq\text{dim}\,(\oplus_{\lambda<0}\,\mathcal{E}_{\omega}(\lambda;L_{\Sigma},W_{\omega}^{1,2}(\Sigma),W^{1,2}(\Sigma))).$

Therefore, as in Proposition 7, we may pick a set of eigenfunctions,

$\\{f_{1},\ldots,f_{b}\\}\subset\oplus_{\lambda<0}\,\mathcal{E}_{\omega}(\lambda;L,\dot{W}_{\omega}^{1,2}(\Sigma),W^{1,2}(\Sigma)),$

which are orthonormal with respect to
$\langle\,\cdot,\cdot\,\rangle_{\omega}$. The inequality then follows noting
that for large enough $S>S_{0}$,

$\text{span}\\{(f_{1})_{S},\ldots,(f_{b})_{S}\\}\subset W_{0}^{1,2}(\Sigma\cap
B_{2S}^{n+1}(0)),$

is a $b$-dimensional subspace on which $B_{L_{\Sigma}}$ is negative definite.

For the reverse of (33), we take $I\geq 1$ (note that if $I=0$ then (33)
implies equality), and an increasing sequence $S_{k}\rightarrow\infty$. For
large enough $S_{k}$, by identical arguments to those contained in Proposition
7, there exists a sequence $\lambda_{i}^{k}\leq\cdots\leq\lambda_{I}^{k}<0$
and a set,

$\\{f_{1}^{k},\ldots,f_{I}^{k}\\}\subset
W_{0}^{1,2}(\Sigma\cap\iota^{-1}(B_{S_{k}}^{n+1}(0))),$

which is orthonormal with respect to $\langle\,\cdot,\cdot\,\rangle_{\omega}$,
such that for all $\varphi\in
W^{1,2}_{0}(\Sigma\cap\iota^{-1}(B_{S_{k}}^{n+1}(0)))$,

$B_{L_{\Sigma}}[f_{i}^{k},\varphi]=\lambda_{i}^{k}\langle
f_{i}^{k},\varphi\rangle_{\omega}.$

By standard theory (see [22, Lemma 3.7]), for each $i=1,\ldots,I$,
$\lambda_{i}^{k}\geq\lambda_{i}^{k+1}$. Then recalling our uniform bound
$\lambda_{i}^{k}\geq-C$, for each $i=1,\ldots,I$, there exists a
$\lambda_{i}^{\infty}\in(0,-C]$, such that
$\lambda_{i}^{k}\rightarrow\lambda_{i}^{\infty}$. By similar arguments
contained in Sections 2.2 and 2.3, we may deduce that after potentially taking
a subsequence and renumerating, for each $i=1,\ldots,I$, there exists an
$f_{i}^{\infty}\in W_{\omega}^{1,2}(\Sigma)$, such that,

$\begin{cases}f_{i}^{k}\rightharpoonup
f_{i}^{\infty},&W^{2,2}_{\text{loc}}(\Sigma),\\\ f_{i}^{k}\rightarrow
f_{i}^{\infty},&W^{1,2}_{\text{loc}}(\Sigma),\end{cases}$

and thus
$f_{i}^{\infty}\in\mathcal{E}_{\omega}(\lambda_{i}^{\infty};L_{\Sigma},W_{\omega}^{1,2}(\Sigma),W^{1,2}(\Sigma))$.

Assume that there exists $a_{1},\ldots,a_{I}\in\mathbb{R}$, with $\sum
a_{i}^{2}=1$, such that,

$h\coloneqq\sum_{i=1}^{I}a_{i}\,f_{i}^{\infty}=0.$

We then define,

$h_{k}\coloneqq\sum_{i=1}^{I}a_{i}\,f_{i}^{k}\in
W_{0}^{1,2}(\Sigma\cap\iota^{-1}(B_{S_{k}}^{n+1}(0))),$

and note that $\langle h_{k},h_{k}\rangle_{\omega}=1$ for all $k$, and
$h_{k}\rightarrow 0$ in $W^{1,2}_{\text{loc}}(\Sigma)$.

As $\Sigma$ has Euclidean volume growth at infinity, it will have finitely
many ends, $E^{1},\ldots,E^{m}$, which for large enough $S^{\prime}<+\infty$,
we may represent by,

$\sqcup_{i=1}^{m}E^{i}=\Sigma\setminus\iota^{-1}(B_{S^{\prime}}^{n+1}(0)).$

Furthermore, as $\Sigma$ has finite index, by applying the arguments in [37,
Lemma 4 and Section 4] with the curvature estimates of [33, Theorem 3] for
$2\leq n\leq 5$, and [7, Theorem 2] for $2\leq n\leq 6$, we may further choose
$S^{\prime}<+\infty$ so that the following holds for all $i=1,\ldots,m$: For
each end $E^{i}$, there exists a rotation $r_{i}$, and function
$u_{i}\colon\mathbb{R}^{n}\setminus B_{S}^{n}(0)\rightarrow\mathbb{R}$, such
that

$r_{i}(\iota(E^{i}))\setminus(B_{S^{\prime}}^{n}(0)\times\mathbb{R})=\text{graph}\,u_{i}=\\{(x,u_{i}(x))\colon
x\in\mathbb{R}^{n}\setminus B_{S^{\prime}}^{n}(0)\\}.$

and, if we fix $\eta>0$, and $\varepsilon>0$, we may make a further choice of
$S^{\prime}$ such that, $|\nabla u_{i}(x)|\leq\eta$, and
$|A_{\Sigma}|^{2}(x)\leq\varepsilon\omega(x)$, for all
$x\in\Sigma\setminus\iota^{-1}(B_{S^{\prime}}^{n+1}(0))$.

Then for large enough $S^{\prime\prime}\geq S^{\prime}>S_{0}$, and then large
enough $k$, a near identical calculation to that in Claim 3, will show that
the function

$H_{k}=(1-\chi_{S^{\prime\prime}})h_{k},$

will provide a contradiction to Lemma 4, on at least one of the ends of
$\Sigma$. Therefore

$\text{dim}\,(\oplus_{\lambda<0}\,\mathcal{E}_{\omega}(\lambda;L_{\Sigma},W_{\omega^{\Sigma,R}}^{1,2}(\Sigma),W^{1,2}(\Sigma)))\geq\text{dim}\,\text{span}\,\\{f_{1}^{\infty},\ldots,f_{I}^{\infty}\\}=I,$

and thus the proof of the Proposition is complete. ∎

##  4 Proof of the Theorem

We follow the arguments in proofs of [18, Lemma<EMAIL_ADDRESS>and [23, Theorem 1.2].

If $\limsup_{k\rightarrow\infty}(\text{ind}(M_{k})+\text{nul}(M_{k}))=0$, then
the conclusion of Theorem 1 is trivial. Suppose for $b\in\mathbb{Z}_{\geq 1}$,

$b\leq\limsup_{k\rightarrow\infty}(\text{ind}(M_{k})+\text{nul}(M_{k}))=\limsup_{k\rightarrow\infty}\left(\text{dim}\,(\oplus_{\lambda\leq
0}\,\mathcal{E}_{\omega_{k,\delta,R}}(\lambda;L_{k},W^{1,2}(co(M_{k}))^{-}))\right),$

where the equality comes from Remark 6. After potentially taking a subsequence
and renumerating we have that for each $k$, there exists a linear subspace,

$V_{k}\coloneqq\text{span}\\{f_{i,k}\\}_{i=1}^{b}\subset
W^{1,2}(co(M_{k}))^{-},$

where, for each $i=1,\ldots,b$, there is a $\lambda_{i,k}\leq 0$, such that,

$f_{i,k}\in\mathcal{E}_{\omega_{k,\delta,R}}(\lambda_{i,k};L_{k},W^{1,2}(co(M_{k}))^{-}),$

and the set $\\{f_{i,k}\\}_{i=1}^{b}$, is orthonormal with respect to the
inner product $\omega_{k,\delta,R}$-weighted $L^{2}$ inner product, i.e. for
$i,\,j=1,\ldots,b$,

$\int_{M_{k}}f_{i,k}\,f_{j,k}\,\omega_{k,\delta,R}=\delta_{ij}.$

We may assume this by the argument used to prove (28). Thus, as outlined in
Section 2.4, for each $i=1,\ldots,b$, after potentially taking a subsequence
and renumerating,

$f_{i,k}\rightarrow((f_{i,\infty}^{1},\ldots,f_{i,\infty}^{m}),f_{i,\infty}^{\Sigma^{1}},\ldots,f_{i,\infty}^{\Sigma^{J}})\in
E_{\infty}$

where we are defining,

$\displaystyle E_{\infty}$ $\displaystyle\coloneqq$
$\displaystyle\left(\oplus_{j=1}^{m}\,\text{span}\,\\{\cup_{\lambda\leq
0}\mathcal{E}_{\omega_{\delta}}(\lambda;L_{\infty},W^{1,2}(co(M_{\infty})))\\}\right)$
$\displaystyle\oplus\left(\oplus_{j=1}^{J}\,\text{span}\,\\{\cup_{\lambda\leq
0}\mathcal{E}_{\omega_{\Sigma^{j},R}}(\lambda;L_{\Sigma^{j}},\dot{W}_{\omega_{\Sigma^{j},R}}^{1,2}(\Sigma^{j}),W^{1,2}(\Sigma^{j}))\\}\right),$
$\displaystyle=$ $\displaystyle\left(\oplus_{j=1}^{m}\,\oplus_{\lambda\leq
0}\mathcal{E}_{\omega_{\delta}}(\lambda;L_{\infty},W^{1,2}(co(M_{\infty})))\right)$
$\displaystyle\oplus\left(\oplus_{j=1}^{J}\,\oplus_{\lambda\leq
0}\mathcal{E}_{\omega_{\Sigma^{j},R}}(\lambda;L_{\Sigma^{j}},\dot{W}_{\omega_{\Sigma^{j},R}}^{1,2}(\Sigma^{j}),W^{1,2}(\Sigma^{j}))\right),$

where the equality is coming from (28) and (31).

We define the linear map,

$\displaystyle\Pi_{k}\colon V_{k}$ $\displaystyle\rightarrow$ $\displaystyle
E_{\infty},$ $\displaystyle f_{i,k}$ $\displaystyle\mapsto$
$\displaystyle((f_{i,\infty}^{1},\ldots,f_{i,\infty}^{m}),f_{i,\infty}^{\Sigma^{1}},\ldots,f_{i,\infty}^{\Sigma^{J}}).$

Thus $W_{k}\coloneqq\Pi_{k}(V_{k})$ is a linear subspace of $E_{\infty}$.

Define the integer

$co(m)=\liminf_{r\rightarrow 0}\liminf_{k\rightarrow\infty}|\\{\text{connected
components of }M_{k}\setminus\cup_{y\in\mathcal{I}}B_{r}^{N}(y)\\}|\leq m.$

If $M_{\infty}$ is two-sided, by the graphical convergence on open sets,
compactly supported away from the finite collection of points $\mathcal{I}$,
we have that $co(m)=m$. If $M_{\infty}$ is one-sided, taking $r$ small enough
and $k$ large enough such that,

$co(m)=|\\{\text{connected components of
}M_{k}\setminus\cup_{y\in\mathcal{I}}B_{r}^{N}(y)\\}|,$

we may recall notation from Section 2.2, and we have

$M_{k}^{r}=\bigcup_{l=1}^{co(m)}M_{k}^{l,r},$

where each $M_{k}^{l,r}$ is a connected hypersurface. Then,
$\cup_{i=1}^{m}\\{(x,u_{k}^{i,r}(x))\colon x\in\Omega_{r}\\}$ is a double
cover of $M_{k}^{r}$ with trivial normal bundle, implying

$\cup_{i=1}^{m}\\{(x,u_{k}^{i,r}(x))\colon
x\in\Omega_{r}\\}=\cup_{l=1}^{co(m)}o(M_{k}^{l,r}).$

If there is an $l\in\\{1,\ldots,co(m)\\}$, and $i\not=j\in\\{1,\ldots,m\\}$,
such that for all large enough $k$,

$o(M_{k}^{l,r})=\\{(x,u_{k}^{i,r}(x))\colon
x\in\Omega_{r}\\}\cup\\{(x,u_{k}^{j,r}(x))\colon x\in\Omega_{r}\\},$

then we have that for $(x,\nu)\in
co(M_{\infty})\setminus\iota^{-1}(\mathcal{I})$,

$f_{\infty}^{i}((x,\nu))=f_{\infty}^{j}((x,-\nu)).$

Thus we may define the injective map

$P_{k}\colon W_{k}\rightarrow F_{\infty},$

where,

$\displaystyle F_{\infty}$ $\displaystyle=$
$\displaystyle\left(\oplus_{l=1}^{co(m)}\oplus_{\lambda\leq
0}\mathcal{E}_{\omega_{\delta}}(\lambda;L_{\infty},W^{1,2}(co(M_{\infty})))\right)$
$\displaystyle\oplus\left(\oplus_{j=1}^{L}\oplus_{\lambda\leq
0}\mathcal{E}_{\omega^{\Sigma^{j},R}}(\lambda;L_{\Sigma^{j}},W_{\omega^{\Sigma^{j},R}}^{1,2}(\tilde{\Sigma}^{j}),W^{1,2}(\Sigma^{j}))\right)$

######  Claim 4.

$\Pi_{k}$ is injective

###### Proof.

We prove by contradiction. Assume we have an $h_{k}$,

$h_{k}=\sum_{i=1}^{b}a^{i}f_{i,k},$

with $\sum_{i=1}^{b}|a^{i}|^{2}=1$, and $\Pi_{k}(h_{k})=0$. This implies that
for all $k$,

$\int_{co(M_{k})}h_{k}^{2}\,\omega_{k,\delta,R}=1,$

and,

$h_{k}\rightarrow((0,\ldots,0),0,\ldots,0),$

which contradicts Claim 3. ∎

Thus,

$\text{dim}\,W_{k}=\text{dim}\,V_{k}=b,$

which implies that, $b\leq\text{dim}\,F_{\infty}$. We conclude Theorem 1 by
combining Proposition 7, Proposition 8, and noting that

$\mathcal{E}_{\omega_{\Sigma^{i},R}}(0;L_{\Sigma^{i}},W_{\omega_{\Sigma^{i},R}}^{1,2}(\Sigma^{j}),W^{1,2}(\Sigma^{j}))=\mathcal{E}_{\omega^{\Sigma^{i},R}}(0;L_{\Sigma^{i}},W_{\omega^{\Sigma^{i},R}}^{1,2}(\Sigma^{j}),W^{1,2}(\Sigma^{j})).$

for $\omega_{\Sigma^{j},R}$ as defined in the statement of Theorem 1, and
$\omega^{\Sigma^{j},R}$ as defined in (12), and by standard regularity theory
for elliptic PDEs,

$\text{nul}_{\omega_{\Sigma^{j},R}}(\Sigma^{j})=\mathcal{E}_{\omega^{\Sigma^{i},R}}(0;L_{\Sigma^{i}},W_{\omega^{\Sigma^{i},R}}^{1,2}(\Sigma^{j}),W^{1,2}(\Sigma^{j}))$

##  5 Finiteness of the Nullity

######  Proposition 9.

Let $\Sigma$ be a complete, properly and minimally immersed hypersurface in
$\mathbb{R}^{n+1}$, $3\leq n\leq 6$, with finite index and Euclidean volume
growth at infinity. For any $R>0$, define the function $\omega_{\Sigma,R}\in
L^{\infty}(\Sigma)$, as in the statement of Theorem 1, then

$\text{dim}\,(\text{nul}_{\omega_{\Sigma,R}}(\Sigma))<+\infty.$

###### Proof.

We assume the statement does not hold and prove by contradiction. Denote our
proper immersion $\iota\colon\Sigma\rightarrow\mathbb{R}^{n+1}$. There exists
a set, $\\{\psi_{1},\psi_{2},\ldots\\}\subset
W^{1,2}_{\omega_{\Sigma,R}}(\Sigma)\cap C^{\infty}(\Sigma)$, such that for all
$k\in\mathbb{N}$,

$\int_{\Sigma}\psi_{k}^{2}\,\omega_{\Sigma,R}=1,$

and,

$\Delta\psi_{k}+|A_{\Sigma}|^{2}\psi_{k}=0,$

and, for $k\not=l$,

$\int_{\Sigma}\psi_{k}\,\psi_{l}\,\omega_{\Sigma,R}=0.$

We choose $S>0$ large enough so that there exists a $C<+\infty$ such that

$\frac{1}{|\iota(x)|^{2}}\leq C\omega_{\Sigma,R}(x),$

for all $x\in\Sigma\setminus\iota^{-1}(B_{S}^{n+1}(0))$. Recall the setup at
the end of Proposition 8, that we have finitely many ends,

$\sqcup_{i=1}^{m}E^{i}=\Sigma\setminus\iota^{-1}(B_{S}^{n+1}(0)),$

and for each $i=1,\ldots,m$,

$r_{i}(\iota(E^{i}))\setminus(B_{S}^{n}(0)\times\mathbb{R})=\text{graph}\,u_{i}=\\{(x,u_{i}(x))\colon
x\in\mathbb{R}^{n}\setminus B_{S}^{n}(0)\\}.$

Claim. For any $\delta>0$, there exists $k,\,l\in\mathbb{N}$, $k\not=l$, such
that

$\|\psi_{k}-\psi_{l}\|^{2}_{W^{1,2}(\Sigma\cap\iota^{-1}(B_{2S}^{n+1}(0)))}<\delta.$

###### Proof.

(of Claim) Fix $S>0$,

$\int_{B_{3S}^{n+1}(0)\cap\Sigma}\psi_{k}^{2}\leq C(S)<+\infty.$

By standard interior estimates for linear elliptic PDEs we have,

$\|\psi_{k}\|_{W^{2,2}(B_{2S}^{n+1}(0)\cap\Sigma)}\leq C(S)<+\infty.$

Therefore, there exists a subsequence
$\\{\psi_{k^{\prime}}\\}\subset\\{\psi_{k}\\}$, and a function
$\psi_{\infty}\in W^{2,2}(B_{2S}^{n+1}(0)\cap\Sigma)$ such that,

$\begin{cases}\psi_{k^{\prime}}\rightharpoonup\psi_{\infty},&\text{weakly
in}\,W^{2,2}(B_{2S}^{n+1}(0)\cap\Sigma),\\\
\psi_{k^{\prime}}\rightarrow\psi_{\infty},&\text{strongly
in}\,W^{1,2}(B_{2S}^{n+1}(0)\cap\Sigma).\end{cases}$

Thus this subsequence is Cauchy, so for any $\delta>0$, there exists
$l,\,k\in\mathbb{N}$, $l\not=k$, such that

$\|\psi_{k}-\psi_{l}\|^{2}_{W^{1,2}(B_{2S}(0)\cap\Sigma)}<\delta$

∎

For $\delta>0$ fixed we denote,

$\psi_{\delta}=\psi_{l}-\psi_{k}\in W^{1,2}_{\omega_{\Sigma,R}}(\Sigma),$

and note that, $\Delta\psi_{\delta}+|A_{\Sigma}|^{2}\psi_{\delta}=0$, and,

$\int_{\Sigma}\psi_{\delta}^{2}\,\omega_{\Sigma,R}=2.$

Recall the function $\chi$ defined in Claim 3, and for $T>4S$, define,

$\Psi_{\delta}(x)=\chi(|\iota(x)|/T)(1-\chi(|\iota(x)|/S))\psi_{\delta}(x).$

for $x\in\Sigma$. For large enough $S\in(0,\infty)$, small enough
$\delta\in(0,1)$, and then large enough $T\in(4\,S,\infty)$, as in the end of
Proposition 8, $\Psi_{\delta}$ will provide a contradiction to Lemma 4, on
atleast one of our ends. ∎

Corollary 2 may then be concluded by noting that,

$\text{nul}\,(\Sigma)\coloneqq\text{dim}\,\mathcal{E}(0;L_{\Sigma},W^{1,2}(\Sigma))\leq\text{dim}\,\mathcal{E}_{\omega_{\Sigma,R}}(0;L_{\Sigma},W_{\omega_{\Sigma,R}}^{1,2}(\Sigma),W^{1,2}(\Sigma))<+\infty.$

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