Introduction
Let
(
N
,
g
)
be a compact Riemannian manifold, with no boundary, of dimension
n
+
1
, and
H
>
0
, be a fixed constant.
In this paper we investigate two classes,
𝔐
(
N
,
g
)
, and
ℭ
H
(
N
,
g
)
.
Here,
𝔐
(
N
,
g
)
is the class of smooth, closed, properly embedded, minimal hypersurfaces of N, with respect to the metric g, and
ℭ
H
(
N
,
g
)
is the class of smooth, closed, properly 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
Λ
>
0
, and
I
∈
ℤ
≥
0
, we define the subclasses,
𝔐
(
N
,
g
,
Λ
,
I
)
=
{
M
∈
𝔐
(
N
,
g
)
:
ℋ
g
n
(
M
)
≤
Λ
,
ind
(
M
)
≤
I
}
,
ℭ
H
(
N
,
g
,
Λ
,
I
)
=
{
M
∈
ℭ
H
(
N
,
g
)
:
ℋ
g
n
(
M
)
≤
Λ
,
ind
(
M
)
≤
I
}
.
Making use of the curvature estimates for stable minimal hypersurfaces by Schoen, Simon and Yau [32] and Schoen and Simon [31] (see also the recent proof by Bellettini [6]), for
2
≤
n
≤
6
, compactness properties have been proven for
𝔐
(
N
,
g
,
Λ
,
I
)
by Sharp [33], and for
ℭ
H
(
N
,
g
,
Λ
,
I
)
by Bourni, Sharp and Tinaglia [9].
In dimension
n
+
1
=
3
, it is worth noting that various other compactness results have been shown for minimal surfaces by Choi and Schoen [14], Anderson [4], Ros [28], and White [40], and for H-CMC surfaces by Sun [35].
We note that
𝔐
(
N
,
g
,
Λ
,
I
)
is sequentially compact (under the correct notion of convergence), whereas for
ℭ
H
(
N
,
g
,
Λ
,
I
)
, we must expand our class to quasi-embedded, H-CMC hypersurfaces (see Definition 1).
We denote this enlarged class by
ℭ
H
¯
(
N
,
g
,
Λ
,
I
)
.
We briefly describe this notion of convergence, with full details described in point (1) of Definition 2.
Consider a sequence
{
M
k
}
⊂
𝔐
(
N
,
g
,
Λ
,
I
)
(resp.
ℭ
H
(
N
,
g
,
Λ
,
I
)
), then after potentially taking a subsequence and renumerating, there is a smooth, closed, embedded minimal hypersurface (resp. H-CMC quasi-embedded hypersurface)
M
∞
, and a finite set of points
ℐ
⊂
M
∞
, where
|
ℐ
|
≤
I
such that on compact subsets of
N
∖
ℐ
,
M
k
will converge to
M
∞
, smoothly and locally graphically, with integer multiplicity potentially greater than 1.
It is then shown that
M
∞
∈
𝔐
(
N
,
g
,
Λ
,
I
)
(resp.
ℭ
H
¯
(
N
,
g
,
Λ
,
I
)
).
The set of points
ℐ
⊂
M
∞
, is defined by the condition that for each
y
∈
ℐ
, there exists a sequence points
{
x
k
y
∈
M
k
}
k
∈
ℕ
such that
x
k
y
→
y
, and the curvature
|
A
M
k
(
x
k
y
)
|
, blows up as
k
→
∞
.
Thus we call
ℐ
the singular set of the convergence.
In a bid to understand the formation of such singularities, a bubble analysis was carried out by Chodosh, Ketover and Maximo [12], Buzano and Sharp [10], and Bourni, Sharp and Tinaglia [9].
Zooming in at appropriate rates, along particular sequences of points converging onto
ℐ
, yields a complete, embedded, non-planar, minimal hypersurface in
ℝ
n
+
1
, of finite index, with Euclidean volume growth at infinity.
These minimal hypersurfaces in
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
∞
is referred to as the “base”.
This terminology is borrowed from other non-linear geometric problems.
See Figure 1 for a heuristic picture.
In the case of
n
=
2
, a bubble analysis has been carried out by Ros [28], in
ℝ
3
with the standard Euclidean metric, assuming uniform bounds on the total curvature instead of the Morse index, and by White [40], in general 3-manifolds, assuming uniform bounds on the genus instead of the Morse index.
One may be interested in certain information about the hypersurfaces along these sequences, for example; genus ([12]), index and total curvature ([10, 9]).
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
→
M
∞
.
For example, if we have our sequence
{
M
k
}
as above, and we know all our “bubbles” are given by
Σ
1
,
…
,
Σ
J
⊂
ℝ
n
+
1
, then we say that
M
k
→
(
M
∞
,
Σ
1
,
…
,
Σ
J
)
“bubble converges” (see Definition 2 for a detailed definition), and [10, 9],
(1)
ind
(
M
∞
)
+
∑
j
=
1
J
ind
(
Σ
j
)
≤
lim inf
k
→
∞
ind
(
M
k
)
.
This inequality gives a quantitative way of accounting for some of the index lost when taking the limit
M
k
→
M
∞
.
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 of connected, two-sided minimal hypersurfaces .
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
, where
3
≤
n
≤
6
.
Let
{
M
k
}
be a sequence of smooth, closed, embedded minimal hypersurfaces of
(
N
,
g
)
such that
M
k
→
(
M
∞
,
Σ
1
,
…
,
Σ
J
)
bubble converges as in Definition 2, with
M
∞
⊂
N
being a smooth, connected, two-sided, closed, embedded minimal hypersurface.
Then
lim sup
k
→
∞
(
ind
(
M
k
)
+
nul
(
M
k
)
)
≤
m
(
ind
(
M
∞
)
+
nul
(
M
∞
)
)
+
∑
j
=
1
J
ind
(
Σ
j
)
+
nul
ω
Σ
j
,
R
(
Σ
j
)
,
where
m
∈
Z
≥
1
is the multiplicity of the convergence onto
M
∞
.
Here,
ind
(
Σ
j
)
=
lim
S
→
∞
ind
B
S
n
+
1
(
0
)
(
Σ
j
)
,
and R may be chosen to be any finite positive real number greater than some
R
0
=
R
0
(
Σ
1
,
…
,
Σ
J
)
∈
[
1
,
∞
)
, and
ω
Σ
j
,
R
(
x
)
=
{
R
-
2
,
x
∈
B
R
n
+
1
(
0
)
∩
Σ
j
,
|
x
|
-
2
,
x
∈
Σ
j
∖
B
R
n
+
1
(
0
)
,
and
nul
ω
Σ
j
,
R
(
Σ
j
)
=
dim
{
f
∈
C
∞
(
Σ
j
)
:
Δ
f
+
|
A
Σ
j
|
2
f
=
0
,
f
2
ω
Σ
j
,
R
∈
L
1
(
Σ
j
)
,
|
∇
f
|
2
∈
L
1
(
Σ
j
)
}
.
The inequality in Corollary 1 can be further strengthened by noting that in this situation if
m
≥
2
, then
ind
(
M
∞
)
=
0
(see [33, Claim 6]), and
nul
(
M
∞
)
=
1
(as the first eigenvalue of the stability operator will be simple).
In order to conclude that the inequality in Theorem 1 (and Corollary 1) is non-trivial, we must also show that for each bubble
Σ
j
, of finite index,
nul
ω
Σ
j
,
R
(
Σ
j
)
<
+
∞
.
This is shown in Proposition 10.
We also remark that Theorem 1 (and hence Corollary 1) also hold for
n
≥
7
, under the additional assumption that the bubbles
Σ
1
,
…
,
Σ
J
, have finite total curvature.
Results on the lower semicontinuity of index along converging sequences (1), are common in the literature.
For certain classes of minimal hypersurfaces see Sharp [33], Buzano and Sharp [10], Ambrozio, Carlotto and Sharp [3] and Ambrozio, Buzano, Carlotto and Sharp [2], and for certain classes of CMC hypersurfaces see Bourni, Sharp and Tinaglia [9].
In the setting of Allen–Cahn solutions see Le [20], Hiesmayr [18], Gaspar [17] and Mantoulidis [23].
For the setting of bubble converging harmonic maps see Moore and Ream [27, Theorem 6.1], and Hirsch and Lamm [19, 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 and Mantoulidis [13, Theorem 1.9] and Mantoulidis [23, 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 and Rivière [16] and Hirsch and Lamm [19].
Note that in [16] and [19, 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). The method of Da Lio, Gianocca and Rivière, has also been extended to prove upper semicontinuity of index plus nullity in the setting of Willmore immersions by Michelat and Rivière [26], and to biharmonic maps by Michelat [25].
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 Chodosh, Ketover and Maximo [12] and Buzano and Sharp [10] for minimal hypersurfaces (in dimensions
3
≤
n
≤
6
)
.
We briefly remark on the strategy of the proof for Theorem 1, which is close to the strategy of Da Lio, Gianocca and Rivière [16, Section IV].
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
∞
, and the bubbles,
Σ
1
,
…
,
Σ
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
∞
(Section 2.2), and the convergence on the bubbles
Σ
1
,
…
,
Σ
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 [16].
One such difference is that in our case, the favourable dimension (
n
≥
3
) allows us to use a Lorentz–Sobolev inequality to deduce strict stability on the neck, whereas in [16] (where
n
=
2
) the authors must work significantly harder.
Another key difference is that in our setting the bubbles are non-compact.
This poses 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 subtle.
It is worth pointing out that the method used in [16], 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
≤
n
≤
6
, if we have a sequence
{
M
k
}
⊂
M
(
N
,
g
)
(
{
M
k
}
⊂
C
H
(
N
,
g
)
) such that
M
k
→
(
M
∞
,
Σ
1
,
…
,
Σ
J
)
bubble converges as in Definition 2, with
M
∞
=
⋃
i
=
1
l
M
i
, where each
M
i
is a closed minimal hypersurface (resp. closed, quasi-embedded H-CMC hypersurface, with
co
(
M
i
)
connected) and
θ
|
M
i
=
m
i
∈
Z
≥
1
(
θ
i
=
m
i
∈
Z
≥
1
), then
lim sup
k
→
∞
(
ind
(
M
k
)
+
nul
(
M
k
)
)
≤
∑
i
=
1
l
co
(
m
)
i
(
anl
-
ind
(
co
(
M
∞
i
)
)
+
anl
-
nul
(
co
(
M
∞
i
)
)
)
+
∑
j
=
1
J
ind
(
Σ
j
)
+
nul
ω
Σ
j
,
R
(
Σ
j
)
,
where for each
i
=
1
,
…
,
l
,
co
(
m
)
i
∈
Z
≥
1
, is such that
co
(
m
)
i
≤
m
i
if
M
i
is one-sided, and
co
(
m
)
i
=
m
i
if
M
i
is two-sided.
Here,
ind
(
Σ
j
)
=
lim
S
→
∞
ind
(
Σ
j
∩
B
S
n
+
1
(
0
)
)
,
and R may be chosen to be any finite positive real number greater than some
R
0
=
R
0
(
Σ
1
,
…
,
Σ
J
)
∈
[
1
,
∞
)
, and
ω
Σ
j
,
R
(
x
)
=
{
R
-
2
,
x
∈
B
R
n
+
1
(
0
)
∩
Σ
j
,
|
x
|
-
2
,
x
∈
Σ
j
∖
B
R
n
+
1
(
0
)
.
and
nul
ω
Σ
j
,
R
(
Σ
j
)
=
dim
{
f
∈
C
∞
(
Σ
j
)
:
Δ
f
+
|
A
Σ
j
|
2
f
=
0
,
f
2
ω
Σ
j
,
R
∈
L
1
(
Σ
j
)
,
|
∇
f
|
2
∈
L
1
(
Σ
j
)
}
.
The exact method of proof we employ does not extend to the case
n
=
2
(two-dimensional surfaces in 3-manifolds).
Two key reasons are the choice of weight (Remark 4), and the criticality of the Lorentz–Sobolev inequality (Proposition 4) for
n
=
p
=
2
.
We take a moment to comment on the terms
anl
-
ind
(
M
∞
i
)
and
anl
-
nul
(
M
∞
i
)
, that appear in the statement of Theorem 1.
These terms respectively stand for the analytic index and analytic nullity of
M
∞
i
.
This refers to the index and nullity of the stability operator acting on the function space
C
∞
(
co
(
M
∞
i
)
)
, where
co
(
M
∞
i
)
is a connected component of the two-sided double cover of
M
∞
i
⊂
N
.
We explain the reasoning behind this with the following example, which is also demonstrated in Figure 2.
Consider a unit hypersphere in
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
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
∞
, will not be equivalent to the Morse index and nullity of
M
∞
, which is customarily defined through ambient vector fields.
See Section 1.2 for further details.
Finally, we make note of another special case, by considering a sequence of H-CMC hypersurfaces
{
M
k
}
, which bubble converge
M
k
→
(
⋃
i
=
1
l
M
i
,
Σ
1
,
…
,
Σ
J
)
such that each
M
k
arises as the boundary of some open set
E
k
⊂
N
.
This particular setting has been analysed by Bourni, Sharp and Tinaglia [9], where they showed that for each i,
co
(
m
)
i
=
1
, and, by applying a uniqueness result of Schoen [30, Theorem 3], that each bubble
Σ
j
, will be given by a catenoid
𝒞
.
Thus, as catenoids have index 1, we have that
lim sup
k
→
∞
(
ind
(
M
k
)
+
nul
(
M
k
)
)
≤
∑
i
=
1
l
(
anl
-
ind
(
co
(
M
∞
i
)
)
+
anl
-
nul
(
co
(
M
∞
i
)
)
)
+
J
(
1
+
nul
ω
𝒞
,
R
(
𝒞
)
)
.
In Section 6 we investigate
nul
ω
𝒞
,
R
(
𝒞
)
, and show that it has a lower bound of n.
In particular, in Section 6 we analyse Jacobi fields on the n-dimensional catenoid
𝒞
⊂
ℝ
n
+
1
(for
n
≥
3
), which arise from rigid motions of
ℝ
n
+
1
(translations, rotations and scalings).
We show that the only non-trivial such Jacobi fields which lie in
W
1
,
2
(
𝒞
)
, or the weighted space
W
ω
𝒞
,
R
1
,
2
(
𝒞
)
, are those generated by translations which are parallel to the ends of
𝒞
.
1 Preliminaries
We briefly define some notation used throughout this paper.
Consider a Riemannian manifold
(
N
,
g
)
.
Throughout the paper, for
p
∈
N
, and
r
∈
(
0
,
inj
(
N
)
)
, we denote
B
r
N
(
p
)
to be the open geodesic ball of radius r centred at point p.
For
r
>
0
, and
p
∈
ℝ
m
, we denote,
B
r
m
(
p
)
as the open ball in
ℝ
m
of radius r centred at point p.
We denote the Ricci curvature of
(
N
,
g
)
by
Ric
N
.
For a proper immersion
ι
:
S
→
N
, we denote the second fundamental form of this immersion by
A
S
.
We denote an n-rectifiable varifold V in N, by a pair
V
=
(
Σ
,
θ
)
, where,
Σ
⊂
N
is an n-rectifiable set, and
θ
:
Σ
→
ℝ
≥
0
.
We refer the reader to [34, Chapter 4] for a more detailed discussion on n-rectifiable varifolds.
Definition 1.
For
H
>
0
, we say that a closed set
M
⊂
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, proper, two-sided immersion
ι
:
S
→
N
such that
M
=
ι
(
S
)
, and for every
y
∈
M
, there exists a
ρ
>
0
such that either:
B
ρ
N
(
y
)
∩
M
is a smooth embedded, n-dimensional disk, of constant mean curvature H, or
B
ρ
N
(
y
)
∩
M
is the union of two smooth embedded, n-dimensional 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
∈
M
which satisfy (i) will be referred to as embedded points of M, and we denote the set of such points by
e
(
M
)
.
Points in the set
t
(
M
)
=
M
∖
e
(
M
)
, which satisfy (ii), will be referred 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 referred to as almost embedded and in [9] they are referred to as effectively embedded.
We opt for the name quasi-embedded as in [8].
We define the set
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
In this subsection we give a precise definition of bubble convergence (Definition 2), and describe the structure of the neck regions in the bubble convergence (Remark 1), as well as the ends of the bubbles (Remark 2).
Definition 2.
Consider a Riemannian manifold
(
N
,
g
)
, of dimension
n
+
1
,
n
≥
2
, (and
H
>
0
) along with a sequence
{
M
k
}
k
∈
ℕ
⊂
𝔐
(
N
,
g
)
(
{
M
k
}
k
∈
ℕ
⊂
ℭ
H
(
N
,
g
)
), an
M
∞
∈
𝔐
(
N
,
g
)
(
M
∞
∈
ℭ
H
¯
(
N
,
g
)
), and a collection of non-planar, complete, properly embedded, minimal hypersurfaces
{
Σ
j
}
j
=
1
J
in
ℝ
n
+
1
, with
J
∈
ℤ
≥
1
.
Then we say that
M
k
→
(
M
∞
,
Σ
1
,
…
,
Σ
J
)
,
bubble converges if:
For the case of minimal hypersurfaces;
(
M
k
,
1
)
→
(
M
∞
,
θ
)
as varifolds, where
θ
:
M
∞
→
ℤ
≥
1
, and is constant on connected components of
M
∞
.
Moreover, there exists an at most finite collection of points
ℐ
⊂
M
∞
such that locally on
M
∞
∖
ℐ
,
M
k
converges smoothly and graphically, with multiplicity locally given by θ.
For the case of H-CMC hypersurfaces;
(
M
k
,
1
)
→
(
M
∞
,
θ
)
as varifolds, where
M
∞
=
⋃
i
=
1
a
M
∞
i
, and each
M
∞
i
is a distinct, closed, quasi-embedded H-CMC hypersurface such that for its respective immersion,
ι
i
:
S
i
→
M
∞
i
,
S
i
is connected, and there exists a
θ
i
∈
ℤ
≥
1
such that
θ
(
y
)
=
∑
i
=
1
a
(
|
(
ι
i
)
-
1
(
y
)
|
θ
i
)
.
Moreover, there exists an at most finite collection of points
ℐ
⊂
M
∞
such that locally on
M
∞
∖
ℐ
,
M
k
converges smoothly and graphically, with multiplicity locally given by θ.
For each
i
∈
{
1
,
…
,
J
}
, there exist point-scale sequences
{
(
p
k
i
,
r
k
i
)
}
k
∈
ℕ
such that for each
k
∈
ℤ
≥
1
,
p
k
i
∈
M
k
, and there exists a
y
i
∈
ℐ
such that
p
k
i
→
y
i
,
r
k
i
→
0
.
Moreover, for each
R
∈
(
0
,
∞
)
, and large enough k, the connected component of
M
k
∩
B
R
r
k
i
N
(
p
k
i
)
, through
p
k
i
, denoted
Σ
k
i
,
R
, is such that, if we rescale the geodesic ball
B
R
r
k
i
N
(
p
k
i
)
by
r
k
i
, and denote
Σ
~
k
i
,
R
∩
B
R
n
+
1
(
0
)
≔
1
r
k
i
exp
p
k
i
-
1
(
Σ
k
i
,
R
∩
B
R
r
k
i
N
(
p
k
i
)
)
⊂
B
R
n
+
1
(
0
)
⊂
ℝ
n
+
1
,
then
Σ
~
k
i
,
R
→
Σ
i
∩
B
R
(
0
)
smoothly and graphically, and hence with multiplicity one.
Furthermore, for
i
≠
j
, either
lim
k
→
∞
r
k
i
r
k
j
+
r
k
j
r
k
i
+
dist
g
N
(
p
k
i
,
p
k
j
)
r
k
i
+
r
k
j
=
∞
,
or for each
R
∈
(
0
,
∞
)
, and then large enough
k
∈
ℕ
,
p
k
j
∉
Σ
k
i
,
R
.
Defining
d
k
(
x
)
≔
min
i
=
1
,
…
,
J
dist
g
N
(
x
,
p
k
i
)
,
then
lim
δ
→
0
lim
R
→
∞
lim
k
→
∞
sup
x
∈
M
k
∩
(
⋃
y
∈
ℐ
B
δ
N
(
y
)
∖
⋃
i
=
1
J
Σ
k
i
,
R
)
∫
M
k
x
∩
B
d
k
(
x
)
/
2
N
(
x
)
|
A
k
|
n
=
0
,
where
M
k
x
∩
B
d
k
(
x
)
/
2
N
(
x
)
is the connected component of
M
k
∩
B
d
k
(
x
)
/
2
N
(
x
)
, that contains x.
Consider such a convergence
M
k
→
(
M
∞
,
Σ
1
,
…
,
Σ
J
)
, as in Definition 2. Then we may remark:
The convergence considered in the bubble analysis of Chodosh, Ketover and Maximo [12] and Buzano and Sharp [10] for minimal hypersurfaces, and Bourni, Sharp and Tinaglia [9] for CMC boundaries, satisfies Definition 2.
As the multiplicity function
θ
:
M
∞
→
ℤ
≥
1
is uniformly bounded, and
ℋ
n
(
M
∞
∩
U
)
<
+
∞
, for
U
⊂
N
compact, then we may deduce by the varifold convergence that
sup
k
ℋ
n
(
M
k
∩
U
)
<
+
∞
.
Applying this fact, and using the monotonicity formula for varifolds with bounded mean curvature ([34, Theorem 17.7]), we may deduce that each
Σ
j
must have Euclidean volume growth at infinity (see [10, Corollary 2.6]).
The final sentence of point (ii) in Definition 2 guarantees that all of these bubbles are distinct, in the sense that for all
R
∈
(
0
,
∞
)
, and
i
≠
j
, then for all large enough k,
Σ
k
i
,
R
∩
Σ
k
j
,
R
=
∅
.
For each
j
=
1
,
…
,
J
, as
Σ
j
is complete and properly embedded, by [29],
Σ
j
is two-sided in
ℝ
n
+
1
.
This is why in the statement of Theorem 1, we may refer to the index and (weighted) nullity of
Σ
j
, rather than the just analytic index and (weighted) nullity.
We also now assume that each
Σ
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.
Thus, for
3
≤
n
≤
6
, by a result of Tysk [37], the finite index and Euclidean volume growth at infinity, imply that each bubble
Σ
j
will have finite total curvature
∫
Σ
j
|
A
Σ
j
|
n
<
+
∞
.
The following curvature estimate, which may also be viewed as an ε-regularity statement, (Proposition 1) is important in our analysis, and the proof follows from a standard point-picking argument, which may be found in [39, Lecture 3].
We briefly list notation used in the statement; if
ι
:
M
→
N
is a proper immersion, and
S
⊂
N
,
x
∈
M
, and
ι
(
x
)
∈
S
, then we denote
ι
-
1
(
S
)
x
as the connected component of
ι
-
1
(
S
)
which contains x.
Proposition 1.
Consider
(
B
1
n
+
1
(
0
)
,
g
)
, where g is a Riemannian metric (a constant
H
>
0
), and a proper, g-minimal (g-H-CMC) immersion
ι
:
M
→
B
1
n
+
1
(
0
)
such that
ι
(
∂
M
)
⊂
∂
B
1
n
+
1
(
0
)
.
There exists an
ε
0
=
ε
0
(
g
)
(
=
ε
0
(
g
,
H
)
)
>
0
such that for
x
∈
ι
-
1
(
B
1
/
2
n
+
1
(
0
)
)
,
r
∈
(
0
,
1
4
)
, and
ε
∈
(
0
,
ε
0
)
, if
∫
ι
-
1
(
B
r
n
+
1
(
ι
(
x
)
)
)
x
|
A
|
n
≤
ε
,
then
sup
y
∈
ι
-
1
(
B
r
n
+
1
(
ι
(
x
)
)
)
x
dist
g
B
1
n
+
1
(
0
)
(
ι
(
y
)
,
∂
B
r
n
+
1
(
ι
(
x
)
)
)
|
A
|
(
y
)
≤
C
ε
,
with
C
ε
=
C
(
g
,
ε
)
(
=
C
(
g
,
H
,
ε
)
)
<
+
∞
.
Moreover,
C
ε
→
0
as
ε
→
0
.
Proposition 1 implies that if we take a sequence
δ
k
→
0
and
R
k
→
∞
such that
R
k
r
k
j
→
0
for all
j
=
1
,
…
,
J
, and
δ
k
≥
4
R
k
r
k
j
for all
k
∈
ℤ
≥
1
and
j
=
1
,
…
,
J
, and pick a sequence of points
x
k
∈
M
k
∩
(
⋃
y
∈
ℐ
B
δ
k
N
(
y
)
∖
⋃
i
=
1
J
Σ
k
i
,
R
k
)
,
and denote
s
k
=
1
2
d
k
(
x
k
)
, and
M
k
~
∩
B
1
n
+
1
(
0
)
≔
1
s
k
exp
x
k
-
1
(
M
k
∩
B
s
k
N
(
x
k
)
)
,
then, after potentially taking a subsequence and renumerating, the component of
M
~
k
through the origin must smoothly converge to a plane through the origin, on compact sets of
B
1
n
+
1
(
0
)
.
Remark 1 (Graphicality of the necks).
Following arguments in [10, Claim 1 of Lemma 4.1] for minimal hypersurfaces, and [9, Lemma 5.6] for CMC hypersurfaces, and replacing the use of stability with the combination of point (3) of Definition 2 and the curvature estimate in Proposition 1, we have that there exists positive constants
S
0
and
s
0
such that for each
y
∈
ℐ
,
s
∈
(
0
,
s
0
)
, and
S
∈
[
S
0
,
+
∞
)
, and taking large enough k, if we let
C
k
denote a connected component of
M
k
∩
(
B
s
N
(
y
)
∖
⋃
i
=
1
J
Σ
k
i
,
S
¯
)
, then there exists a non-empty open set
A
k
=
A
k
(
C
k
,
s
,
S
)
⊂
T
y
M
∞
, and a smooth function
u
k
:
A
k
→
ℝ
such that
C
k
=
{
exp
y
(
x
+
u
k
(
x
)
ν
(
y
)
)
:
x
∈
A
k
}
,
where
ν
(
y
)
is a choice of unit normal to
M
∞
at y.
Moreover, we have that
lim
s
→
0
lim
S
→
∞
lim
k
→
∞
∥
u
k
∥
C
1
=
0
.
Remark 2 (Graphicality of the ends of bubbles).
Consider
ι
:
Σ
→
ℝ
n
+
1
, a n-dimensional (
n
≥
3
), complete, connected, two-sided, proper minimal immersion, of finite total curvature,
∫
Σ
|
A
Σ
|
n
<
+
∞
,
and Euclidean volume growth at infinity,
lim
R
→
∞
ℋ
n
(
ι
(
Σ
)
∩
B
R
n
+
1
(
0
)
)
R
n
<
+
∞
.
Following arguments in [37, Lemma 3 and 4] and [7, Proposition 3], replacing any use of stability with the finite total curvature assumption and Proposition 1 (see also [21, Appendix A]), we may conclude that Σ will have finitely many ends
E
1
,
…
,
E
m
such that for each
i
=
1
,
…
,
m
, there exists a rotation
r
i
, centred at the origin, a compact set
B
i
⊂
ℝ
n
, and a smooth function
u
i
:
ℝ
n
∖
B
i
→
ℝ
such that
r
i
(
ι
(
E
i
)
)
=
graph
(
u
i
)
≔
{
(
x
,
u
i
(
x
)
)
:
x
∈
ℝ
n
∖
B
i
}
.
Moreover,
lim
R
→
∞
∥
∇
u
i
∥
L
∞
(
ℝ
n
∖
B
R
n
(
0
)
)
=
0
.
1.2 Stability operator, index and nullity
Consider a smooth, closed, connected, and properly embedded hypersurface
M
⊂
N
.
We denote its normal bundle by
T
M
⊥
, and its two-sided double cover by
o
(
M
)
, which may be given by,
o
(
M
)
=
{
(
x
,
ν
)
:
x
∈
M
,
ν
∈
T
x
⊥
M
and
|
ν
|
=
1
}
.
We then define the obvious projections
ι
:
o
(
M
)
→
M
, and
ν
:
o
(
M
)
→
T
M
⊥
.
Taking a connected component of
o
(
M
)
, which we denote by
co
(
M
)
, we have that either, M is two-sided (and therefore
T
M
⊥
is trivial), and thus
co
(
M
)
=
M
, or M is one-sided, and thus
co
(
M
)
=
o
(
M
)
.
Let M be a minimal hypersurface. Then we consider the following bilinear form (which arises as the second variation of the area functional with respect to ambient variations [15, Chapter 1 Section 8]) on sections of
T
M
⊥
,
B
L
[
v
,
v
]
≔
∫
M
|
∇
⊥
v
|
2
-
|
A
M
|
2
|
v
|
2
-
Ric
N
(
v
,
v
)
d
ℋ
n
.
In the case when M is two-sided, we see that this is equivalent to considering the following bilinear form, which can be extended to the function space
W
1
,
2
(
M
)
,
(1.1)
B
L
[
f
,
h
]
≔
∫
M
∇
f
⋅
∇
h
-
|
A
M
|
2
f
h
-
Ric
N
(
ν
,
ν
)
f
h
d
ℋ
n
,
where ν is a choice of unit normal to M.
In the case when M is one-sided, this is equivalent to considering the bilinear form
(1.2)
B
L
[
f
,
h
]
≔
∫
o
(
M
)
∇
f
⋅
∇
h
-
|
A
M
∘
ι
|
2
f
h
-
Ric
N
(
ν
(
⋅
)
,
ν
(
⋅
)
)
f
h
d
ℋ
n
,
but on the function space
W
1
,
2
(
o
(
M
)
)
-
≔
{
f
∈
W
1
,
2
(
o
(
M
)
)
:
for a.e.
(
x
,
ν
)
∈
o
(
M
)
,
f
(
(
x
,
ν
)
)
=
-
f
(
(
x
,
-
ν
)
)
}
.
In both scenarios, integrating by parts, we see that
B
L
, is the weak formulation of the well known stability (or Jacobi) operator,
(1.3)
L
=
Δ
+
|
A
M
|
2
+
Ric
(
ν
,
ν
)
.
In order to stop having to differentiate between the cases when M is two-sided and one-sided we introduce the notation
W
1
,
2
(
co
(
M
)
)
-
, which simply equals
W
1
,
2
(
M
)
, when M is two-sided, and
W
1
,
2
(
o
(
M
)
)
-
, when M is one-sided.
The Morse index of M (denoted
ind
(
M
)
) may then be defined to be the number of negative eigenvalues (counted with multiplicity) of the stability operator L, acting on the function space
W
1
,
2
(
co
(
M
)
)
-
.
The nullity of M (denoted
nul
(
M
)
), may then be defined as the number of zero eigenvalues of L (counted with multiplicity) acting on the function space
W
1
,
2
(
co
(
M
)
)
-
.
In this paper it is also necessary for us to also define the analytic index (denoted
anl
-
ind
(
M
)
) and analytic nullity (denoted
anl
-
nul
(
M
)
) of M, which is defined as the number of negative and zero eigenvalues of L, acting on the function space
W
1
,
2
(
co
(
M
)
)
.
We see that if M is two-sided, then the Morse index and nullity and analytic index and nullity are equal.
One should think of
W
1
,
2
(
co
(
M
)
)
-
, as “ambient variations”, and
W
1
,
2
(
co
(
M
)
)
as “intrinsic variations”.
If M is an embedded H-CMC hypersurface such that
M
=
∂
E
, for some non-empty, open set E, then one sees that the bilinear form
B
L
, defined on sections of
T
M
⊥
, arises as the second variation of the functional
ℱ
H
(
E
)
=
ℋ
n
(
∂
E
)
-
H
ℋ
n
+
1
(
E
)
with respect to ambient variations ([5, Proposition 2.5]).
However, this bilinear form
B
L
(and corresponding operator L), can be defined on M, even if it does not arise as the boundary, and thus (as embedded H-CMC hypersurfaces are two-sided for
H
≠
0
), we may similarly define the Morse index and nullity of M as the number of negative and zero eigenvalues of the operator L, acting on the space
W
1
,
2
(
M
)
.
In general, for a quasi-embedded H-CMC hypersurface M, given by the proper two-sided immersion
ι
:
S
→
N
, we denote
co
(
M
)
=
S
, and define the analytic index and analytic nullity of M, as the number of negative and zero eigenvalues of the operator
L
=
Δ
+
|
A
S
|
2
+
Ric
N
(
ν
,
ν
)
,
acting on the function space
W
1
,
2
(
co
(
M
)
)
unifying with the notation above we denote
W
1
,
2
(
co
(
M
)
)
-
=
W
1
,
2
(
co
(
M
)
)
.
Here,
A
S
is the second fundamental form of the immersion ι, and ν is a choice of unit normal for this two-sided immersion.
Again we see that if M is embedded then the Morse index and nullity match up with this analytic index and analytic nullity.
1.3 Lorentz spaces
Let
(
M
,
g
)
be a Riemannian manifold and μ be the volume measure associated to g.
For a μ-measurable function
f
:
M
→
ℝ
, we define the function
α
f
(
s
)
≔
μ
(
{
x
∈
M
:
|
f
(
x
)
|
>
s
}
)
.
We may then define the decreasing rearrangement
f
*
, of f by,
f
*
(
t
)
≔
{
inf
{
s
>
0
:
α
f
(
s
)
≤
t
}
,
t
>
0
,
ess
sup
|
f
|
,
t
=
0
.
For
p
∈
[
1
,
∞
)
, and
q
∈
[
1
,
∞
]
, and a μ-measurable function f on M, we define,
∥
f
∥
(
p
,
q
)
≔
{
(
∫
0
∞
t
q
p
f
*
(
t
)
q
d
t
t
)
1
q
,
1
≤
q
<
∞
,
sup
t
>
0
t
1
p
f
*
(
t
)
,
q
=
∞
.
The Lorentz space
L
(
p
,
q
)
(
M
,
g
)
is then defined to be the space of μ-measurable functions f such that
∥
f
∥
(
p
,
q
)
<
+
∞
.
It is worth noting that
∥
⋅
∥
(
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,
∥
⋅
∥
p
,
q
(see [11, Definition 2.10]), on the space
L
(
p
,
q
)
(
M
,
g
)
, so that the normed space
(
L
(
p
,
q
)
(
M
,
g
)
,
∥
⋅
∥
p
,
q
)
is a Banach space [11, Theorem 2.19].
Moreover, for
1
<
p
<
∞
, and
1
≤
q
≤
∞
, we have the following equivalence ([11, Proposition 2.14]):
(1.4)
∥
⋅
∥
(
p
,
q
)
≤
∥
⋅
∥
p
,
q
≤
p
p
-
1
∥
⋅
∥
(
p
,
q
)
.
Proposition 2 (Hölder–Lorentz inequality, [11, Theorem 2.9]).
Take
p
1
,
p
2
∈
(
1
,
∞
)
and
q
1
,
q
2
∈
[
1
,
∞
]
such that
1
p
1
+
1
p
2
=
1
q
1
+
1
q
2
=
1
.
Then for
f
∈
L
(
p
1
,
q
1
)
(
M
,
g
)
and
h
∈
L
(
p
2
,
q
2
)
(
M
,
g
)
we have
∫
M
|
f
h
|
𝑑
μ
≤
∥
f
∥
(
p
1
,
q
1
)
∥
h
∥
(
p
2
,
q
2
)
.
The following fact can be easily derived from the definition of
∥
⋅
∥
(
p
,
q
)
.
Proposition 3.
Take
1
<
p
<
+
∞
,
1
≤
q
≤
∞
, and
γ
>
0
. Then for
f
∈
L
(
p
,
q
)
(
M
,
g
)
we have
∥
|
f
|
γ
∥
(
p
γ
,
q
γ
)
=
∥
f
∥
(
p
,
q
)
γ
.
The following Lorentz–Sobolev inequality on
ℝ
n
is crucial in Section 2.4. For a proof see [1, Appendix].
Proposition 4 (Lorentz–Sobolev inequality on
R
n
).
Take
1
<
p
<
n
, and
p
*
=
n
p
n
-
p
. Then there exists a constant
C
=
C
(
n
,
p
)
such that for all
u
∈
C
c
∞
(
R
n
)
,
∥
u
∥
(
p
*
,
p
)
≤
C
∥
∇
u
∥
L
p
.
By standard covering and partitions of unity arguments, from Proposition 4 we may also obtain a Lorentz–Sobolev inequality on a bounded subset of a Riemannian manifold.
Proposition 5 (Lorentz–Sobolev inequality on manifolds).
Let
(
M
,
g
)
be a complete Riemannian manifold of dimension n.
Take
1
<
p
<
n
,
p
*
=
n
p
n
-
p
, and an open, bounded set
Ω
⊂
M
.
Then there exists a constant
C
=
C
(
Ω
,
g
,
n
,
p
)
<
+
∞
such that for all
u
∈
C
c
∞
(
Ω
)
,
∥
u
∥
(
p
*
,
p
)
≤
C
∥
u
∥
W
1
,
p
(
M
)
.
Remark 3.
We may extend the inequalities in Propositions 4 and 5 to
u
∈
W
1
,
p
(
ℝ
n
)
and
W
0
1
,
p
(
Ω
)
respectively, by using a standard density argument, the fact
(
L
(
p
,
q
)
(
M
,
g
)
,
∥
⋅
∥
p
,
q
)
is a Banach space, and the equivalence in (1.4).
Proposition 6.
Let
(
M
,
g
)
be a compact Riemannian manifold of dimension
n
≥
3
, with
ω
∈
L
(
n
2
,
∞
)
(
M
,
g
)
. Then for any
f
1
,
f
2
∈
W
1
,
2
(
M
)
, we have that there exists a
C
=
C
(
M
,
g
)
<
∞
such that
|
∫
M
f
1
f
2
ω
|
≤
C
∥
ω
∥
(
n
2
,
∞
)
∥
f
1
∥
W
1
,
2
(
M
)
∥
f
2
∥
W
1
,
2
(
M
)
.
Proof.
Using Propositions 2, 3 and 5,
|
∫
M
f
1
f
2
ω
|
≤
(
∫
M
|
ω
|
f
1
2
)
1
2
(
∫
M
|
ω
|
f
2
2
)
1
2
≤
∥
ω
∥
(
n
2
,
∞
)
∥
f
1
2
∥
(
2
*
2
,
1
)
1
2
∥
f
2
2
∥
(
2
*
2
,
1
)
1
2
≤
∥
ω
∥
(
n
2
,
∞
)
∥
f
1
∥
(
2
*
,
2
)
∥
f
2
∥
(
2
*
,
2
)
≤
C
∥
ω
∥
(
n
2
,
∞
)
∥
f
1
∥
W
1
,
2
(
M
)
∥
f
2
∥
W
1
,
2
(
M
)
.
∎
2 Weighted eigenfunctions
We proceed with the proof of Theorem 1, taking a sequence
M
k
→
(
M
∞
,
Σ
1
,
…
,
Σ
J
)
as in Definition 2.
We assume that for each
k
∈
ℤ
≥
1
,
M
k
is a single connected component.
Thus for the case of minimal hypersurfaces,
l
=
1
.
The general statement is proven by applying the argument to each individual connected component of
M
k
.
We also note that the reader may find it easier to follow the rest of this paper by only considering the case where each
M
k
, and
M
∞
, are two-sided minimal hypersurfaces.
By doing so much of the notation introduced in Section 1.2 can be ignored, and we can just consider
M
k
instead of
co
(
M
k
)
, and
W
1
,
2
(
M
k
)
instead of
W
1
,
2
(
co
(
M
k
)
)
-
.
2.1 The weight
Take
R
≥
4
and
δ
>
0
such that for large enough k, and all
j
=
1
,
…
,
J
,
4
R
r
k
j
<
δ
<
min
{
inj
(
N
)
8
,
min
y
1
,
y
2
∈
ℐ
,
y
1
≠
y
2
dist
g
N
(
y
1
,
y
2
)
8
}
.
We first define our weight, on
co
(
M
k
)
, about the point scale sequence
{
(
p
k
j
,
r
k
j
)
}
k
∈
ℕ
,
ω
k
,
δ
,
R
j
(
x
)
≔
{
max
{
δ
-
2
,
dist
g
N
(
ι
(
x
)
,
p
k
j
)
-
2
}
,
ι
k
(
x
)
∈
M
k
∖
B
R
r
k
j
N
(
p
k
j
)
,
(
R
r
k
j
)
-
2
,
ι
(
x
)
∈
B
R
r
k
j
(
p
k
j
)
∩
M
k
.
We consider the weight
ω
k
,
δ
,
R
(
x
)
≔
max
j
=
1
,
…
,
J
ω
k
,
δ
,
R
j
(
x
)
.
We also define
ω
δ
∈
W
loc
1
,
∞
(
co
(
M
∞
)
∖
ι
∞
-
1
(
ℐ
)
)
∩
L
(
n
2
,
∞
)
(
co
(
M
∞
)
)
, by
ω
δ
(
x
)
≔
max
{
δ
-
2
,
dist
g
N
(
ι
∞
(
x
)
,
ℐ
)
-
2
}
.
The fact that
ω
δ
∈
L
(
n
2
,
∞
)
(
co
(
M
∞
)
)
will follow from a similar calculation to that in Claim 1.
Recall the stability operator on
co
(
M
k
)
,
L
k
≔
Δ
+
|
A
k
|
2
+
R
k
,
where
A
k
denotes the second fundamental form of the immersion
ι
k
:
M
k
→
N
, and
R
k
(
x
)
=
Ric
N
(
ν
k
(
x
)
,
ν
k
(
x
)
)
(
ι
k
(
x
)
)
,
and the associated bilinear form,
B
k
, acting on
W
1
,
2
(
co
(
M
k
)
)
,
B
k
[
φ
,
ψ
]
≔
∫
M
k
∇
k
φ
⋅
∇
k
ψ
-
(
|
A
k
|
2
+
R
k
)
φ
ψ
.
We define the unweighted eigenspace for an eigenvalue λ of
L
k
by
(2.1)
ℰ
(
λ
;
L
k
,
W
1
,
2
(
co
(
M
k
)
)
-
)
≔
{
f
∈
W
1
,
2
(
co
(
M
k
)
)
-
:
B
k
[
f
,
ψ
]
=
λ
∫
M
k
f
ψ
for all
ψ
∈
W
1
,
2
(
co
(
M
k
)
)
-
}
,
and the weighted eigenspace for a weighted eigenvalue λ of
L
k
by
(2.2)
ℰ
ω
k
,
δ
,
R
(
λ
;
L
k
,
W
1
,
2
(
co
(
M
k
)
)
-
)
≔
{
f
∈
W
1
,
2
(
co
(
M
k
)
)
-
:
B
k
[
f
,
ψ
]
=
λ
∫
M
k
f
ψ
ω
k
,
δ
,
R
for all
ψ
∈
W
1
,
2
(
co
(
M
k
)
)
-
}
.
Identical definitions hold for
ℰ
(
λ
;
L
∞
,
W
1
,
2
(
co
(
M
∞
)
)
,
ℰ
ω
δ
(
λ
;
L
∞
,
W
1
,
2
(
co
(
M
∞
)
)
)
.
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”.
Lemma 1.
There exists a
C
=
C
(
N
,
g
,
M
∞
,
Σ
1
,
…
,
Σ
J
,
δ
,
R
)
<
+
∞
such that for all k,
(2.3)
|
A
k
|
2
+
|
R
k
|
ω
k
,
δ
,
R
≤
C
.
Proof.
The proof follows from a contradiction argument.
We outline the basic idea, and leave details to the reader.
If we had a sequence of points
{
x
k
∈
M
k
}
such that
|
A
k
|
2
(
x
k
)
+
|
R
k
|
(
x
k
)
ω
k
,
δ
,
R
(
x
k
)
→
∞
.
Then by smooth convergence on
M
∞
∖
ℐ
, and the bubbles
Σ
1
,
…
,
Σ
J
, and the scale invariance of the quantity, we see that no subsequences can concentrate on the base
M
∞
∖
ℐ
, or on the bubbles
Σ
1
,
…
,
Σ
J
.
Thus this sequence must concentrate on the intermediate neck regions of the bubble convergence, however this cannot happen by point (iii) in Definition 2 and Proposition 1.
∎
Claim 1.
We have that there exists a constant
C
=
C
(
N
,
g
,
M
∞
,
m
,
δ
,
J
)
(
=
C
(
N
,
g
,
M
∞
,
m
1
,
…
,
m
a
,
H
,
δ
,
J
)
)
<
+
∞
such that for large enough
k
∈
Z
≥
1
, and
R
≥
1
,
∥
ω
k
,
δ
,
R
∥
(
n
2
,
∞
)
≤
C
.
Proof.
For a
j
=
1
,
…
,
J
, consider
f
=
ω
k
,
δ
,
R
j
.
We have
α
f
(
s
)
=
{
ℋ
n
(
co
(
M
k
)
)
,
s
∈
[
0
,
δ
-
2
)
,
ℋ
n
(
ι
k
-
1
(
B
1
/
s
N
(
p
k
j
)
)
∩
co
(
M
k
)
)
,
δ
-
2
≤
s
<
(
R
r
k
j
)
-
2
,
0
,
s
≥
(
R
r
k
j
)
-
2
.
We may choose k large enough such that
sup
k
ℋ
n
(
M
k
)
≤
m
ℋ
n
(
M
∞
)
+
1
.
Then, by the monotonicity formula ([34, Theorem 17.7]), there exists a uniform
C
=
C
(
N
,
g
,
m
,
M
∞
)
(
=
C
(
N
,
g
,
M
∞
,
m
1
,
…
,
m
a
,
H
)
)
<
+
∞
such that for
r
∈
(
0
,
1
2
inj
(
N
)
)
,
ℋ
n
(
co
(
M
k
)
∩
ι
k
-
1
(
B
r
N
(
p
k
j
)
)
)
≤
C
r
n
.
Then
{
f
*
(
t
)
=
0
,
t
≥
ℋ
n
(
co
(
M
k
)
)
,
f
*
(
t
)
=
δ
-
2
,
ℋ
n
(
co
(
M
k
)
∩
ι
k
-
1
(
B
δ
N
(
p
k
j
)
)
)
≤
t
<
ℋ
n
(
co
(
M
k
)
)
,
f
*
(
t
)
≤
(
C
t
)
2
n
,
ℋ
n
(
co
(
M
k
)
∩
ι
k
-
1
(
B
R
r
k
j
N
(
p
k
j
)
)
)
≤
t
≤
ℋ
n
(
co
(
M
k
)
∩
ι
k
-
1
(
B
δ
N
(
p
k
j
)
)
)
,
f
*
(
t
)
=
(
R
r
k
j
)
-
2
,
0
≤
t
≤
ℋ
n
(
co
(
M
k
)
∩
ι
k
-
1
(
B
R
r
k
j
N
(
p
k
j
)
)
)
.
Thus,
∥
f
∥
(
n
2
,
∞
)
=
sup
t
>
0
t
2
n
f
*
(
t
)
≤
C
for
C
=
C
(
N
,
g
,
m
,
M
∞
,
δ
)
(
=
C
(
N
,
g
,
M
∞
,
m
1
,
…
,
m
a
,
H
,
δ
)
)
<
+
∞
.
This in turn implies
∥
ω
k
,
δ
,
R
∥
(
n
2
,
∞
)
≤
(
n
n
-
2
)
∑
j
=
1
J
∥
ω
k
,
δ
,
R
j
∥
(
n
2
,
∞
)
≤
C
for
C
=
C
(
N
,
g
,
m
,
M
∞
,
δ
,
J
)
(
=
C
(
N
,
g
,
M
∞
,
m
1
,
…
,
m
a
,
H
,
δ
,
J
)
)
<
+
∞
.
The
n
n
-
2
factor is coming from (1.4).
∎
Remark 4.
This choice of weight
ω
k
,
δ
,
R
fails to work for the case of
n
=
2
.
In [16] (in which
n
=
2
) the choice of weight is subtle, and relies on improved 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
The goal of this subsection is to show that (weighted) eigenfunctions with non-negative eigenvalues on
M
k
, exhibit appropriate convergence to (weighted) eigenfunctions with non-negative eigenvalues on the base
M
∞
.
The ideal strategy is to write
M
k
as a smooth graph over
M
∞
, define sequences of functions on
M
∞
by composing the eigenfunctions on
M
k
with this graphing function, and deduce uniform
W
1
,
2
(
M
∞
)
bounds along these sequences.
However, in general,
M
k
cannot be written as graphs over
M
∞
, but instead, away from the singular set of convergence
ℐ
, we can write a subset of
co
(
M
k
)
as a collection of m graphs over a subset of
co
(
M
∞
)
.
While this allows us to extract appropriate limiting eigenfunctions on
co
(
M
∞
)
, one cannot in general conclude that these are related to ambient variations, hence why in the statement of Theorem 1, we must consider the analytic index and nullity of
M
∞
.
Consider a sequence of functions
{
f
k
∈
W
1
,
2
(
co
(
M
k
)
)
-
}
k
∈
ℤ
≥
1
, which satisfy the following weighted eigenvalue problem:
(2.4)
∫
co
(
M
k
)
∇
f
k
⋅
∇
φ
-
(
|
A
k
|
2
+
R
k
)
f
k
φ
=
λ
k
∫
co
(
M
k
)
f
k
φ
ω
k
,
δ
,
R
for all
φ
∈
W
1
,
2
(
co
(
M
k
)
)
-
, with
λ
k
≤
0
, for all k.
We take
∫
co
(
M
k
)
f
k
2
ω
k
,
δ
,
R
=
1
,
and by Lemma 1,
∫
co
(
M
k
)
|
∇
f
k
|
2
≤
∫
co
(
M
k
)
(
|
A
k
|
2
+
|
R
k
|
)
f
k
2
≤
C
∫
co
(
M
k
)
f
k
2
ω
k
,
δ
,
R
=
C
.
Furthermore,
δ
-
2
∥
f
k
∥
L
2
(
co
(
M
k
)
)
2
≤
∫
co
(
M
k
)
f
k
2
ω
k
,
δ
,
R
=
1
.
Thus, for all
k
∈
ℕ
,
∥
f
k
∥
W
1
,
2
(
co
(
M
k
)
)
≤
C
=
C
(
N
,
g
,
δ
,
R
,
m
,
M
∞
,
Σ
1
,
…
,
Σ
J
)
<
+
∞
.
Using Lemma 1, we may also obtain a lower bound on the negative eigenvalues,
(2.5)
λ
k
=
λ
k
∫
co
(
M
k
)
f
k
2
ω
k
,
δ
,
R
=
∫
co
(
M
k
)
|
∇
f
k
|
2
-
(
|
A
k
|
2
+
R
k
)
f
k
2
≥
-
C
∫
co
(
M
k
)
f
k
2
ω
k
,
δ
,
R
=
-
C
.
Thus, after potentially taking a subsequence and renumerating, we may assume that
λ
k
→
λ
∞
≤
0
.
We define the map
F
:
co
(
M
∞
)
×
ℝ
→
N
,
(
x
,
t
)
↦
exp
ι
∞
(
x
)
(
t
ν
∞
(
x
)
)
.
Note, as
M
∞
is smooth, and properly embedded, there exists a
τ
=
τ
(
N
,
M
∞
,
g
)
>
0
such that
F
:
co
(
M
∞
)
×
(
-
τ
,
τ
)
→
F
(
co
(
M
∞
)
×
(
-
τ
,
τ
)
)
⊂
N
is a smooth, local diffeomorphism.
We define the metric
g
~
=
F
*
g
, on
co
(
M
∞
)
×
(
-
τ
,
τ
)
, and assume that for all
k
∈
ℤ
≥
1
,
M
k
⊂
F
(
co
(
M
∞
)
×
(
-
τ
,
τ
)
)
.
First we consider the case of minimal hypersurfaces.
As
M
∞
is properly embedded, and N is compact, we may take
τ
>
0
such that
F
-
1
(
M
∞
)
=
co
(
M
∞
)
×
{
0
}
.
For
r
>
0
, define the open set
Ω
r
⊂
co
(
M
∞
)
, by
Ω
r
≔
ι
∞
-
1
(
M
∞
∖
⋃
y
∈
ℐ
B
r
N
(
y
)
¯
)
⊂
co
(
M
∞
)
.
We define
M
k
r
≔
M
k
∩
F
(
Ω
r
×
(
-
τ
,
τ
)
)
, and
M
~
k
r
=
F
-
1
(
M
k
r
)
∩
(
Ω
r
×
(
-
τ
,
τ
)
)
.
By the convergence described in Definition 2, along with the fact that
co
(
M
∞
)
is two-sided in
co
(
M
∞
)
×
(
-
τ
,
τ
)
, and
θ
|
M
∞
≡
m
, for large enough k, there exists m smooth functions
u
k
i
,
r
:
Ω
r
→
(
-
τ
,
τ
)
such that
u
k
1
,
r
<
u
k
2
,
r
<
⋯
<
u
k
m
,
r
, and
M
~
k
r
=
⋃
i
=
1
m
{
(
x
,
u
k
i
,
r
(
x
)
)
:
x
∈
Ω
r
}
⊂
Ω
r
×
(
-
τ
,
τ
)
.
We also note that
u
k
i
,
r
→
0
in
C
l
(
Ω
r
)
for all
l
∈
ℤ
≥
1
, and for
0
<
r
<
s
,
u
k
i
,
r
=
u
k
i
,
s
on
Ω
s
⊂
Ω
r
.
Moreover, we define the metric
g
k
=
ι
k
*
(
g
|
M
k
)
on
co
(
M
k
)
, and the metric
g
∞
=
ι
∞
*
(
g
|
M
∞
)
on
co
(
M
∞
)
.
First we consider the case in which
M
k
is one-sided.
For each connected component of
M
~
k
r
,
M
~
k
i
,
r
≔
{
(
x
,
u
k
i
,
r
(
x
)
)
:
x
∈
Ω
r
}
,
we denote the
ν
k
i
,
r
to be the choice of unit normal to
M
~
k
i
,
r
(with respect to
g
~
) which points in the positive τ direction.
Through this choice of unit normal we identify
Ω
r
as a subset of
co
(
M
k
)
, by the map (which is a diffeomorphism onto its image)
(2.6)
F
k
i
,
r
:
Ω
r
→
co
(
M
k
)
,
x
↦
(
F
(
x
,
u
k
i
,
r
(
x
)
)
,
d
F
(
ν
k
i
,
r
)
)
,
and define
f
~
k
i
,
r
(
x
)
=
(
f
k
∘
F
k
i
,
r
)
(
x
)
.
We do note that
f
~
k
i
,
r
depends on the choice of unit normal to
M
~
k
i
,
r
that we pick, however as
f
k
∈
W
1
,
2
(
co
(
M
k
)
)
-
, this choice is only up to a sign.
For the case of
M
k
being two-sided, we simply define
F
k
i
,
r
:
Ω
r
→
co
(
M
k
)
=
M
k
,
x
↦
F
(
x
,
u
k
i
,
r
(
x
)
)
,
and define
f
~
k
i
,
r
(
x
)
=
(
f
k
∘
F
k
i
,
r
)
(
x
)
.
For the case of H-CMC hypersurfaces, we have that
co
(
M
∞
)
=
⊔
j
=
1
a
co
(
M
∞
j
)
,
where each
M
∞
j
is a distinct, closed, quasi-embedded H-CMC hypersurface such that
co
(
M
∞
j
)
is connected.
We have that
θ
j
=
m
j
∈
ℤ
, and we denote,
Ω
r
/
2
j
⊂
co
(
M
∞
j
)
as before.
Then similarly to before, for each
j
=
1
,
…
,
a
, and large enough k, there exist
m
j
smooth graphs (
i
=
1
,
…
,
m
j
),
u
k
i
,
j
,
r
:
Ω
r
2
j
→
(
-
τ
,
τ
)
such that
u
k
1
,
j
,
r
<
u
k
2
,
j
,
r
<
⋯
<
u
k
m
j
,
j
,
r
,
u
k
i
,
j
,
r
→
0
in
C
l
(
Ω
r
j
)
for all
l
∈
ℤ
≥
1
, and, for large enough k,
M
k
∖
⋃
y
∈
ℐ
B
r
N
(
y
)
⊂
⋃
j
=
1
a
⋃
i
=
1
m
i
{
F
(
x
,
u
k
i
,
j
,
r
2
(
x
)
)
:
x
∈
Ω
r
/
2
j
}
.
Define
M
~
k
i
,
j
,
r
=
{
(
x
,
u
k
i
,
j
,
r
(
x
)
)
:
x
∈
Ω
r
j
}
, and as before we identify this as a subset of
co
(
M
k
)
, and similarly define the map
F
k
i
,
j
,
r
, and the function
f
~
k
i
,
j
,
r
∈
W
1
,
2
(
Ω
r
j
)
.
For ease of notation we just consider the case of minimal hypersurfaces.
For an open set
Ω
⊂
⊂
co
(
M
∞
)
∖
ι
∞
-
1
(
ℐ
)
we may take
r
>
0
such that
Ω
⊂
⊂
Ω
r
, and then define, for large enough k,
f
~
k
i
(
x
)
=
f
~
k
i
,
r
(
x
)
,
x
∈
Ω
.
Note that this definition is independent of the choice of
0
<
r
<
r
0
, for
Ω
⊂
Ω
r
0
. When dealing with a fixed open set
Ω
⊂
⊂
co
(
M
∞
)
∖
ι
∞
-
1
(
ℐ
)
, for appropriate choices of r, we drop the superscript r in the notation of the maps
F
k
i
,
r
, and functions
u
k
i
,
r
.
Then, choosing k large enough (so that
∥
u
k
i
∥
C
1
(
Ω
r
)
is small enough), we have that
∥
f
~
k
i
∥
W
1
,
2
(
Ω
)
≤
2
∥
f
k
∥
W
1
,
2
(
co
(
M
k
)
)
≤
C
and thus for each
i
=
1
,
…
,
m
, there exists an
f
~
∞
i
∈
W
loc
1
,
2
(
co
(
M
∞
)
∖
ι
∞
-
1
(
ℐ
)
)
such that, after potentially picking a subsequence and renumerating,
(2.7)
{
f
~
k
i
⇀
f
~
∞
i
in
W
loc
1
,
2
(
co
(
M
∞
)
∖
ι
∞
-
1
(
ℐ
)
)
,
f
~
k
i
→
f
~
∞
i
in
L
loc
2
(
co
(
M
∞
)
∖
ι
∞
-
1
(
ℐ
)
)
.
Note that by lower semicontinuity of the
W
1
,
2
norm for (2.7), for all open
Ω
⊂
co
(
M
∞
)
∖
ι
∞
-
1
(
ℐ
)
, we have a uniform bound
∥
f
~
∞
i
∥
W
1
,
2
(
Ω
)
≤
C
.
Therefore, we may deduce that in fact
f
~
∞
i
∈
W
1
,
2
(
co
(
M
∞
)
)
.
Moreover, we have that
∫
co
(
M
∞
)
(
f
~
∞
i
)
2
(
ω
δ
∘
ι
∞
)
≤
1
.
For
i
=
1
,
…
,
m
, and large enough k, we define the metric,
g
~
k
i
≔
(
F
k
i
)
*
g
k
, and its associated gradient
∇
~
k
i
, on Ω.
Let
J
k
i
denote the Jacobian of the map
F
k
i
with respect to the metric
g
∞
on Ω.
For a point
x
0
∈
Ω
, we may choose
s
>
0
, small enough so that
B
s
co
(
M
∞
)
(
x
0
)
⊂
⊂
Ω
,
F
(
B
s
co
(
M
∞
)
(
x
0
)
×
(
-
τ
,
τ
)
)
⊂
B
inj
(
N
)
/
2
N
(
ι
(
x
0
)
)
.
Consider
φ
∈
C
c
∞
(
B
s
co
(
M
∞
)
(
x
0
)
)
, and for each
i
=
1
,
…
,
m
and
x
∈
B
s
co
(
M
∞
)
(
x
0
)
, we define the function
φ
k
i
(
F
k
i
(
x
)
)
=
φ
(
x
)
on
C
c
∞
(
F
k
i
(
B
s
co
(
M
∞
)
(
x
0
)
)
)
⊂
C
∞
(
co
(
M
k
)
)
.
As each
M
k
is properly embedded, by [12, Lemma C.1] (cf. [29]),
{
ι
k
(
F
k
i
(
x
)
)
:
x
∈
B
s
co
(
M
∞
)
(
x
0
)
}
⊂
M
k
∩
B
inj
(
N
)
/
2
N
(
ι
(
x
0
)
)
is two-sided, and thus we can extend
φ
k
i
to a vector field on N, and thus to a function in
C
∞
(
co
(
M
k
)
)
-
.
Thus we may plug
φ
k
i
into (2.4) and obtain
∫
Ω
g
k
i
(
∇
~
k
i
f
~
k
i
,
∇
~
k
i
φ
)
J
k
i
=
λ
k
∫
Ω
f
~
k
i
φ
(
ω
k
,
δ
,
R
∘
F
k
i
)
J
k
i
+
∫
Ω
(
(
|
A
k
|
2
+
R
k
)
∘
F
k
i
)
f
~
k
i
φ
J
k
i
.
Hence, by (2.7), convergence of
ω
k
,
δ
,
R
∘
F
k
i
→
ω
δ
∘
ι
∞
, in
L
∞
(
Ω
)
, and smooth convergence of,
F
k
i
→
id
, on Ω, we have that
(2.8)
∫
co
(
M
∞
)
∇
f
~
∞
i
⋅
∇
φ
-
(
(
|
A
∞
|
2
+
R
∞
)
∘
ι
∞
)
f
~
∞
i
φ
=
λ
∞
∫
co
(
M
∞
)
f
~
∞
i
φ
(
ω
δ
∘
ι
∞
)
holds for all
φ
∈
C
c
∞
(
B
s
co
(
M
∞
)
(
x
0
)
)
.
Thus by standard regularity theory for linear elliptic PDEs we have that
f
~
∞
i
∈
W
2
,
2
(
Ω
)
and
Δ
f
~
∞
i
+
(
(
|
A
∞
|
2
+
R
∞
)
∘
ι
∞
)
f
~
∞
i
+
λ
∞
f
~
∞
i
(
ω
δ
∘
ι
∞
)
=
0
a.e. on Ω.
This then implies that (2.8) holds for all
φ
∈
C
c
∞
(
co
(
M
∞
)
∖
ι
∞
-
1
(
ℐ
)
)
.
Proposition 7.
Let
(
M
,
g
)
be a compact, n-dimensional, Riemannian manifold, with
n
≥
3
.
Consider
V
∈
L
∞
(
M
)
, and
ω
∈
L
(
n
2
,
∞
)
(
M
)
.
Suppose that we have
u
∈
W
1
,
2
(
M
)
, and a finite set of points
J
⊂
M
such that, for all
φ
∈
C
c
∞
(
M
∖
J
)
,
(2.9)
∫
M
∇
u
⋅
∇
φ
-
V
u
φ
-
ω
u
φ
=
0
.
Then in fact (2.9) holds for all
φ
∈
W
1
,
2
(
M
)
.
Proof.
Consider a smooth function on
ℝ
with the following properties:
{
χ
(
t
)
=
1
,
t
≤
1
,
χ
(
t
)
=
0
,
t
≥
2
,
-
3
≤
χ
′
(
t
)
≤
0
.
Then for
ε
>
0
, chosen small enough we define the following smooth function on M:
χ
ε
(
x
)
=
χ
(
d
𝒥
(
x
)
ε
)
,
where
d
𝒥
(
x
)
=
dist
g
(
x
,
𝒥
)
.
Then, for
φ
=
χ
ε
φ
+
(
1
-
χ
ε
)
φ
∈
C
∞
(
M
)
, by Hölder inequality, Proposition 6, and small enough
ε
>
0
,
|
∫
M
∇
u
⋅
∇
φ
-
V
u
φ
-
ω
u
φ
|
≤
C
∥
φ
∥
C
1
(
M
)
∥
u
∥
W
1
,
2
(
M
)
(
1
+
∥
V
∥
L
∞
(
M
)
+
∥
ω
∥
(
n
2
,
∞
)
(
M
)
)
ε
n
-
2
2
Thus we see that (2.9) holds for all
φ
∈
C
∞
(
M
)
, and the Proposition may then be completed by a standard density argument.
∎
By Proposition 7 we have that (2.8) holds for all
φ
∈
W
1
,
2
(
co
(
M
∞
)
)
.
Thus,
f
~
∞
i
∈
ℰ
ω
δ
(
λ
∞
;
L
∞
,
W
1
,
2
(
co
(
M
∞
)
)
)
.
It is worth noting that we could have
f
~
∞
i
=
0
.
2.3 Convergence on the bubble
For
S
>
0
fixed, and
i
∈
{
1
,
…
,
J
}
, consider the bubble
Σ
k
i
,
S
, and its associated point-scale sequence,
{
(
p
k
i
,
r
k
i
)
}
k
∈
ℕ
.
Let
{
∂
1
,
…
,
∂
n
+
1
}
be an orthonormal basis for
T
p
k
i
N
, with respect to the metric g, and define the map
G
k
i
:
ℝ
n
+
1
=
span
{
∂
1
,
…
,
∂
n
+
1
}
→
N
,
x
↦
exp
p
k
i
(
r
k
i
x
)
.
Then on
B
2
S
n
+
1
(
0
)
, for large enough k we define the metric
g
~
k
=
(
r
k
i
)
-
2
(
G
k
i
)
*
g
, and we have that
(
g
~
k
)
α
,
β
(
x
)
≔
g
α
,
β
(
r
k
i
x
)
→
δ
α
,
β
,
and for our bubble,
Σ
~
k
i
,
2
S
≔
1
r
k
i
exp
p
k
i
-
1
(
Σ
k
i
,
S
∩
B
2
S
r
k
i
N
(
p
k
i
)
)
→
Σ
i
∩
B
2
S
n
+
1
(
0
)
≕
Σ
i
,
2
S
,
smoothly.
As
Σ
i
is two-sided, there is a choice of unit normal ν, and a
τ
>
0
such that the map
F
:
Σ
i
,
S
×
(
-
τ
,
τ
)
→
ℝ
n
+
1
,
(
x
,
t
)
↦
x
+
t
ν
(
x
)
is a diffeomorphism onto its image.
Then, for large enough k, there exists a smooth function
v
k
i
,
S
:
Σ
i
,
S
→
(
-
τ
,
τ
)
such that
Σ
~
k
i
,
2
S
∩
F
(
Σ
i
,
S
×
(
-
τ
,
τ
)
)
=
{
F
(
x
,
v
k
i
,
S
(
x
)
)
:
x
∈
Σ
i
,
S
}
.
As before, for large enough k, define the smooth map
F
k
i
,
S
:
Σ
i
,
S
→
ℝ
n
+
1
,
x
↦
F
(
x
,
v
k
i
,
S
(
x
)
)
.
From ν, we get a choice of unit normal
ν
k
i
(which points in the
d
F
(
∂
t
)
direction), to
Σ
~
k
i
,
2
S
∩
F
(
Σ
i
,
S
×
(
-
τ
,
τ
)
)
, with respect to
g
~
k
.
If
M
k
is one-sided, then we define the following functions on
Σ
i
,
S
(recalling our functions
f
k
from Section 2.2),
f
~
k
i
,
S
(
x
)
≔
(
r
k
i
)
n
2
-
1
f
k
(
(
G
k
i
∘
F
k
i
)
(
x
)
,
(
r
k
i
)
-
1
d
G
k
i
(
ν
k
i
(
F
k
i
(
x
)
)
)
)
and
ω
k
Σ
i
,
R
,
S
(
x
)
≔
(
r
k
i
)
2
ω
k
,
δ
,
R
(
(
G
k
i
∘
F
k
i
)
(
x
)
,
(
r
k
i
)
-
1
d
G
k
i
(
ν
k
i
(
F
k
i
(
x
)
)
)
)
.
We note that while
f
~
k
i
,
S
depends on our choice of unit normal ν to
Σ
i
, this dependence is only up to a choice in sign.
If
M
k
is two sided, then we define the following functions on
Σ
i
,
S
:
f
~
k
i
,
S
(
x
)
≔
(
r
k
i
)
n
2
-
1
f
k
(
(
G
k
i
∘
F
k
i
)
(
x
)
)
and
ω
k
Σ
i
,
R
,
S
(
x
)
≔
(
r
k
i
)
2
ω
k
,
δ
,
R
(
(
G
k
i
∘
F
k
i
)
(
x
)
)
.
We then have (after potentially taking further subsequences and renumerating) that there exists a function
ω
Σ
i
,
R
∈
W
1
,
∞
(
Σ
i
)
such that
ω
k
Σ
i
,
R
,
S
→
ω
Σ
i
,
R
in
L
∞
(
Σ
i
,
S
)
, and moreover satisfies the following: there exists a
T
<
+
∞
such that
ess
inf
ω
Σ
i
,
R
>
0
in
B
T
n
+
1
(
0
)
∩
Σ
i
, and there exists a
Λ
≥
1
such that
(2.10)
1
Λ
|
x
|
2
≤
ω
Σ
i
,
R
(
x
)
≤
Λ
|
x
|
2
on
Σ
i
∖
B
T
n
+
1
(
0
)
.
We have, for large k,
∫
Σ
i
,
S
(
f
~
k
i
,
S
)
2
ω
k
Σ
i
,
R
,
S
≤
2
∫
co
(
M
k
)
f
k
2
ω
k
,
δ
,
R
=
2
,
implying that, for large enough k,
∫
Σ
i
,
S
(
f
~
k
i
,
S
)
2
≤
1
min
Σ
i
,
S
ω
Σ
i
,
R
∫
Σ
i
,
S
(
f
~
k
i
,
S
)
2
ω
k
,
R
Σ
i
,
S
≤
C
(
S
,
R
,
Σ
1
,
…
,
Σ
J
)
<
+
∞
.
Moreover,
∫
Σ
i
,
S
|
∇
f
~
k
i
,
S
|
2
≤
2
∫
co
(
M
k
)
|
∇
f
k
|
2
≤
C
(
N
,
g
,
δ
,
R
,
M
∞
,
Σ
1
,
…
,
Σ
J
)
<
+
∞
.
Similar to previous, for
0
<
S
1
<
S
2
<
+
∞
, and large enough k, we have that
f
~
k
i
,
S
1
=
f
~
k
i
,
S
2
on
Σ
i
,
S
1
.
Thus, for any open, bounded set
Ω
⊂
Σ
i
, we may take any
S
>
0
such that
Ω
⊂
Σ
i
,
S
, then for large enough k, the function
f
~
k
i
=
f
~
k
i
,
S
is well defined on Ω, with
∫
Ω
(
f
~
k
i
)
2
+
∫
Ω
|
∇
f
~
k
i
|
2
≤
C
(
Ω
,
N
,
g
,
δ
,
R
,
M
∞
,
Σ
1
,
…
,
Σ
J
)
.
We may conclude that there exists an
f
~
∞
i
∈
W
loc
1
,
2
(
Σ
i
)
,
{
f
~
k
i
→
f
~
∞
i
in
L
loc
2
(
Σ
i
)
,
f
~
k
i
⇀
f
~
∞
i
in
W
loc
1
,
2
(
Σ
i
)
,
and, we have
f
~
∞
i
∈
W
ω
Σ
i
,
R
1
,
2
(
Σ
i
)
≔
{
f
∈
W
loc
1
,
2
(
Σ
i
)
:
∫
Σ
i
|
∇
f
|
2
<
+
∞
,
∫
Σ
i
ω
Σ
i
,
R
f
2
<
+
∞
}
.
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
φ
∈
C
c
∞
(
Σ
i
)
,
(2.11)
∫
Σ
i
∇
f
~
∞
i
⋅
∇
φ
-
|
A
Σ
i
|
2
f
~
∞
i
φ
-
λ
∞
ω
Σ
i
,
R
f
~
∞
i
φ
=
0
.
As
Σ
i
has finite total curvature, we may deduce, by Proposition 1, that there exists a
C
=
C
(
Σ
i
)
<
+
∞
such that
(2.12)
|
A
Σ
i
|
2
≤
C
ω
Σ
i
,
R
.
Thus by Hölder’s inequality we have, for
φ
∈
L
2
(
Σ
i
)
,
|
∫
Σ
i
|
A
Σ
i
|
2
f
~
∞
i
φ
+
ω
Σ
i
,
R
f
~
∞
i
φ
|
≤
C
∥
ω
Σ
i
,
R
∥
L
∞
(
Σ
i
)
1
2
∥
φ
∥
L
2
(
Σ
i
)
.
This allows us to apply a standard density argument to deduce that (2.11) holds for all
φ
∈
W
1
,
2
(
Σ
i
)
.
We now look to extend (2.11) to
φ
∈
W
ω
Σ
i
,
R
1
,
2
(
Σ
i
)
.
First we note that, for
f
,
h
∈
W
ω
Σ
i
,
R
1
,
2
(
Σ
i
)
, using (2.12),
|
∫
Σ
i
f
h
ω
Σ
i
,
R
|
+
|
∫
Σ
i
|
A
Σ
i
|
2
f
h
|
<
(
1
+
C
)
(
∫
Σ
i
f
2
ω
Σ
i
,
R
)
1
2
(
∫
Σ
i
h
2
ω
Σ
i
,
R
)
1
2
<
+
∞
.
Therefore the quantities in (2.11) are finite and well defined for
φ
∈
W
ω
Σ
i
,
R
1
,
2
(
Σ
i
)
.
Now, denote the function
(2.13)
{
χ
∈
C
∞
(
ℝ
;
[
0
,
1
]
)
,
χ
(
t
)
=
1
,
t
∈
(
-
∞
,
1
]
,
χ
(
t
)
=
0
,
t
∈
[
2
,
∞
)
,
-
3
≤
χ
′
(
t
)
≤
0
,
and for large
S
>
0
, we define the following smooth function on
Σ
i
:
χ
S
(
x
)
=
χ
(
|
x
|
S
)
.
Then for
φ
∈
W
ω
Σ
i
,
R
1
,
2
(
Σ
i
)
, we define
φ
S
=
φ
χ
S
∈
W
0
1
,
2
(
Σ
i
∩
B
2
S
n
+
1
(
0
)
)
.
Using (2.10) and (2.12), we may deduce that for
φ
∈
W
ω
Σ
i
,
R
1
,
2
(
Σ
i
)
,
∫
Σ
i
∇
f
~
∞
i
⋅
∇
φ
-
|
A
Σ
i
|
2
f
~
∞
i
φ
-
λ
∞
ω
Σ
i
,
R
f
~
∞
i
φ
=
lim
S
→
∞
∫
Σ
i
∇
f
~
∞
i
⋅
∇
φ
S
-
|
A
Σ
i
|
2
f
~
∞
i
φ
S
-
λ
∞
ω
Σ
i
,
R
f
~
∞
i
φ
S
=
0
.
Thus we have that
f
~
∞
i
∈
ℰ
ω
Σ
i
,
R
(
λ
∞
;
L
Σ
i
,
W
ω
Σ
i
,
R
1
,
2
(
Σ
i
)
)
≔
{
f
∈
W
ω
Σ
i
,
R
1
,
2
(
Σ
i
)
:
B
L
Σ
i
[
f
,
φ
]
=
λ
∞
∫
Σ
i
f
φ
ω
Σ
i
,
R
for all
φ
∈
W
ω
Σ
i
,
R
1
,
2
(
Σ
i
)
}
,
where
L
Σ
i
=
Δ
+
|
A
Σ
i
|
2
, is the stability operator on
Σ
i
, and
B
L
Σ
i
is the associated bilinear form.
Again it is worth noting that we could have
f
~
∞
i
=
0
.
Remark 5.
By the Michael–Simon–Sobolev inequality ([24, Theorem 2.1]), one may deduce that
f
~
∞
∈
L
2
*
(
Σ
~
)
, for
2
*
=
2
n
n
-
2
.
2.4 Strict stability of the neck
For a Riemannian manifold
(
M
,
g
)
, we define
W
0
1
,
2
(
M
,
g
)
to be the closure of
C
c
1
(
M
)
, with respect to the standard norm on
W
1
,
2
(
M
,
g
)
.
Lemma 2.
For
n
≥
3
, consider the cylinder
(
A
×
R
,
g
)
, where
A
⊂
R
n
is a non-empty, open set, and g is a smooth Riemannian metric such that there exists a constant
K
∈
[
1
,
∞
)
such that for
x
∈
A
×
R
, and
X
∈
R
n
+
1
,
(2.14)
1
K
〈
X
,
X
〉
≤
g
x
(
X
,
X
)
≤
K
〈
X
,
X
〉
,
where
〈
⋅
,
⋅
〉
is the standard metric on
R
n
.
Now suppose we have a smooth function
u
:
A
→
(
-
T
,
T
)
such that
∥
∇
R
n
u
∥
L
∞
(
A
)
≤
1
2
(
∇
R
n
denotes the gradient on
R
n
with respect to the standard Euclidean metric), and denote
M
≔
graph
(
u
)
⊂
A
×
(
-
2
T
,
2
T
)
.
For fixed
W
∈
(
0
,
∞
)
suppose we have functions
ω
∈
L
(
n
2
,
∞
)
(
M
,
g
)
, and
V
∈
L
∞
(
M
)
such that
∥
ω
∥
L
(
n
2
,
∞
)
(
M
,
g
)
≤
W
, and
ess
inf
ω
>
0
.
Then there exists an
ε
=
ε
(
n
,
K
,
W
)
>
0
, and
C
=
C
(
n
,
K
)
∈
(
0
,
+
∞
)
such that if
|
V
|
≤
ε
ω
on M, then
0
<
1
C
W
≤
inf
{
∫
M
|
∇
f
|
2
-
V
f
2
:
f
∈
W
0
1
,
2
(
M
)
,
∫
M
f
2
ω
=
1
}
.
Proof.
We define the following map,
F
(
x
)
≔
(
x
,
u
(
x
)
)
, and the metric
g
~
=
F
*
g
on A.
For
x
∈
A
, and
X
∈
ℝ
n
, we have
g
~
x
(
X
,
X
)
≤
K
〈
X
,
X
〉
+
2
K
∥
∇
ℝ
n
u
∥
L
∞
(
A
)
〈
X
,
X
〉
+
K
∥
∇
ℝ
n
u
∥
L
∞
(
A
)
2
〈
X
,
X
〉
≤
C
〈
X
,
X
〉
,
with
C
=
C
(
n
,
K
)
, which from here on may be rechosen at each step.
Thus, for
x
∈
A
, and
f
∈
C
1
(
M
)
,
|
∇
ℝ
n
(
f
∘
F
)
(
x
)
|
=
sup
〈
X
,
X
〉
≤
1
|
d
(
f
∘
F
)
(
x
)
(
X
)
|
≤
sup
g
~
x
(
X
,
X
)
≤
C
|
d
(
f
∘
F
)
(
x
)
(
X
)
|
≤
C
|
∇
g
~
(
f
∘
F
)
|
(
x
)
.
Moreover, if we consider the metric on A,
g
1
=
F
*
〈
⋅
,
⋅
〉
, then
g
1
(
X
,
X
)
≤
K
g
~
(
X
,
X
)
≤
C
〈
X
,
X
〉
, and thus
1
≤
1
+
|
∇
ℝ
n
u
(
x
)
|
2
=
|
g
1
|
≤
K
n
|
g
~
|
≤
C
.
Now, take an
f
∈
C
c
1
(
M
)
that satisfies
∫
M
f
2
ω
𝑑
μ
=
1
,
where μ denotes the volume measure of
(
M
,
g
)
.
We have, by the above, and applying Propositions 2, 3 and 4 (see [38, Theorem 1.1] for a similar computation)
1
=
∫
A
(
f
∘
F
)
2
(
ω
∘
F
)
|
g
~
|
𝑑
x
≤
C
∫
A
(
f
∘
F
)
2
(
ω
∘
F
)
𝑑
x
≤
C
∥
ω
∘
F
∥
(
n
2
,
∞
)
(
A
,
〈
⋅
,
⋅
〉
)
∥
f
∘
F
∥
(
2
*
,
2
)
(
A
,
〈
⋅
,
⋅
〉
)
2
≤
C
∥
ω
∘
F
∥
(
n
2
,
∞
)
(
A
,
〈
⋅
,
⋅
〉
)
∫
A
|
∇
ℝ
n
(
f
∘
F
)
|
2
𝑑
x
≤
C
∥
ω
∘
F
∥
(
n
2
,
∞
)
(
A
,
〈
⋅
,
⋅
〉
)
∫
M
|
∇
f
|
2
-
V
f
2
d
μ
+
C
∥
ω
∘
F
∥
(
n
2
,
∞
)
(
A
,
〈
⋅
,
⋅
〉
)
∫
M
V
f
2
𝑑
μ
≤
C
∥
ω
∘
F
∥
(
n
2
,
∞
)
(
A
,
〈
⋅
,
⋅
〉
)
(
ε
+
∫
M
|
∇
f
|
2
-
V
f
2
d
μ
)
,
where again we are potentially rechoosing
C
=
C
(
n
,
K
)
at each line.
Moreover,
∥
ω
∘
F
∥
(
n
2
,
∞
)
(
A
,
〈
⋅
,
⋅
〉
)
≤
K
∥
ω
∥
(
n
2
,
∞
)
(
M
,
g
)
≤
K
W
,
and thus, again rechoosing
C
=
C
(
n
,
K
)
, and choosing
ε
=
(
2
C
W
)
-
1
,
∫
M
|
∇
f
|
2
-
V
f
2
≥
1
2
C
W
>
0
for all
f
∈
C
c
1
(
M
)
.
Now the lemma may be concluded by a standard density argument.
∎
As previously, for ease of notation we only consider the case of minimal hypersurfaces, however the argument for H-CMC hypersurfaces is identical.
In Sections 2.2 and 2.3 we showed that if we have a sequence
f
k
∈
ℰ
ω
k
,
δ
,
R
(
λ
k
;
L
k
,
W
1
,
2
(
co
(
M
k
)
)
-
)
,
with
λ
k
≤
0
, and
∫
co
(
M
k
)
f
k
2
ω
k
,
δ
,
R
=
1
for all k, then after potentially taking a subsequence and renumerating we have that
λ
k
→
λ
∞
≤
0
and
f
k
→
(
(
f
∞
1
,
…
,
f
∞
m
)
,
f
∞
Σ
1
,
…
,
f
∞
Σ
J
)
,
where, for
i
=
1
,
…
,
m
,
f
∞
i
∈
W
1
,
2
(
co
(
M
∞
)
)
,
and for
j
=
1
,
…
,
J
,
f
∞
Σ
j
∈
W
ω
Σ
j
,
R
1
,
2
(
Σ
j
)
⊂
L
2
*
(
Σ
j
)
.
By this convergence we mean, for
i
=
1
,
…
,
m
(recalling notation from Section 2.2),
f
~
k
i
=
(
f
k
∘
F
k
i
)
,
(2.15)
{
f
~
k
i
⇀
f
∞
i
in
W
loc
1
,
2
(
co
(
M
∞
)
∖
ι
∞
-
1
(
ℐ
)
)
)
,
f
~
k
i
→
f
∞
i
in
L
loc
2
(
co
(
M
∞
)
∖
ι
∞
-
1
(
ℐ
)
)
)
,
and for
j
=
1
,
…
,
J
, setting
f
~
k
Σ
j
=
f
~
k
j
(this time
f
~
k
j
as defined in Section 2.3),
(2.16)
{
f
~
k
Σ
j
⇀
f
∞
Σ
j
in
W
loc
1
,
2
(
Σ
j
)
,
f
~
k
Σ
j
→
f
∞
Σ
j
in
L
loc
2
(
Σ
j
)
.
Furthermore, we are able to deduce that for
i
=
1
,
…
,
m
,
f
∞
i
∈
ℰ
ω
δ
(
λ
∞
;
L
∞
,
W
1
,
2
(
co
(
M
∞
)
)
)
,
and
j
=
1
,
…
,
J
,
f
∞
Σ
j
∈
ℰ
ω
Σ
j
,
R
(
λ
∞
;
L
Σ
j
,
W
ω
Σ
j
,
R
1
,
2
(
Σ
j
)
)
.
Claim 2.
Suppose that
{
f
1
,
k
,
…
,
f
b
,
k
}
is an orthonormal collection, with respect to the
ω
k
,
δ
,
R
-weighted
L
2
norm, of
ω
k
,
δ
,
R
-weighted eigenfunctions, with non-positive eigenvalues, and
∑
i
=
1
a
i
2
=
1
. Then
h
k
≔
∑
i
=
1
b
a
i
f
i
,
k
→
∑
i
=
1
b
a
i
(
(
f
i
,
∞
1
,
…
,
f
i
,
∞
m
)
,
f
i
,
∞
Σ
1
,
…
,
f
i
,
∞
Σ
J
)
≠
(
(
0
,
…
,
0
)
,
0
,
…
,
0
)
.
Proof.
This claim is proven by a contradiction argument similar to that in [16, Claim 1 of Lemma IV.6].
By Remark 1, for all
η
∈
(
0
,
1
2
]
,
τ
∈
(
0
,
1
)
, and
y
∈
ℐ
, there exists an
r
0
∈
(
0
,
δ
4
)
, and
R
0
∈
(
4
R
,
∞
)
such that, taking geodesic normal coordinates about
y
∈
N
such that
T
y
M
∞
=
{
x
n
+
1
=
0
}
, for large enough k, if we denote
C
k
to be a connected component of
exp
y
-
1
(
M
k
∩
B
r
0
N
(
y
)
∖
⋃
j
=
1
J
Σ
k
j
,
R
0
¯
)
⊂
ℝ
n
+
1
,
then there exists a non-empty open set
A
(
C
k
)
⊂
{
x
n
+
1
=
0
}
, and a smooth function,
u
k
:
A
(
C
k
)
→
(
-
τ
,
τ
)
such that
C
k
=
graph
(
u
k
)
, and
∥
∇
ℝ
n
u
k
∥
L
∞
≤
η
.
Moreover, for each
ε
>
0
, we may make further choices of
r
0
and
R
0
such that for large enough k, on each
C
k
, if we denote
V
k
=
(
|
A
k
|
2
+
R
k
)
|
C
k
∈
L
∞
(
C
k
)
, and
ω
k
=
(
ω
k
,
δ
,
R
)
|
C
k
∈
L
(
n
2
,
∞
)
(
C
k
,
g
)
∩
L
∞
(
C
k
)
, then
∥
ω
k
∥
L
(
n
2
,
∞
)
(
C
k
,
g
)
≤
W
and
|
V
k
|
≤
ε
ω
k
,
with
W
=
W
(
N
,
g
,
M
∞
,
m
,
δ
,
J
)
<
+
∞
(coming from Claim 1).
Thus, fixing
ε
=
ε
(
N
,
g
,
M
∞
,
m
,
δ
,
J
)
>
0
small enough, by Lemma 2, for large enough k,
(2.17)
inf
{
∫
C
k
|
∇
f
|
2
-
V
k
f
2
:
f
∈
W
0
1
,
2
(
C
k
)
,
∫
C
k
f
2
ω
k
=
1
}
≥
γ
>
0
,
with
γ
=
γ
(
N
,
g
,
M
∞
,
m
,
δ
,
J
)
>
0
.
Now assuming Claim 2 does not hold, we will provide a contradiction to (2.17), on at least one connected component
C
k
of
M
k
∩
(
⋃
y
∈
ℐ
B
r
0
N
(
y
)
∖
⋃
j
=
1
J
Σ
k
j
,
R
0
¯
)
.
Note that there is a uniform bound,
m
|
ℐ
|
<
+
∞
, on the number of such connected components.
Recall the smooth cutoff χ from (2.13), and consider the following distance functions defined on
co
(
M
k
)
:
d
k
(
x
)
=
dist
g
N
(
ι
k
(
x
)
,
ℐ
)
,
and for
j
=
1
,
…
,
J
,
d
k
j
(
x
)
=
dist
g
N
(
ι
k
(
x
)
,
p
k
j
)
.
We then define the following function
H
k
(
x
)
=
{
0
,
x
∈
co
(
M
k
)
∖
⋃
y
∈
ℐ
ι
k
-
1
(
B
r
0
N
(
y
)
)
,
χ
(
2
r
0
-
1
d
k
(
x
)
)
h
k
,
x
∈
ι
k
-
1
(
⋃
y
∈
ℐ
B
r
0
N
(
y
)
∖
⋃
j
=
1
J
Σ
k
j
,
2
R
0
)
,
h
k
(
1
-
χ
(
(
R
0
r
k
j
)
-
1
d
k
j
(
x
)
)
)
,
x
∈
ι
k
-
1
(
Σ
k
j
,
2
R
0
)
,
j
=
1
,
…
,
J
.
We compute the gradient of
H
k
.
For
x
∈
co
(
M
k
)
∖
⋃
y
∈
ℐ
ι
k
-
1
(
B
r
0
N
(
y
)
)
, or
x
∈
ι
k
-
1
(
Σ
k
j
,
R
0
)
,
j
=
1
,
…
,
J
, we have
∇
H
k
=
0
.
For
x
∈
co
(
M
k
)
∩
ι
k
-
1
(
B
r
0
N
(
y
)
∖
⋃
j
=
1
J
Σ
k
j
,
2
R
0
)
, we have
∇
H
k
(
x
)
=
∇
h
k
(
x
)
χ
(
2
r
0
-
1
d
k
(
x
)
)
+
h
k
(
x
)
2
r
0
-
1
χ
′
(
2
r
0
-
1
d
k
(
x
)
)
∇
d
k
(
x
)
.
Finally, for
x
∈
ι
k
-
1
(
Σ
k
j
,
2
R
0
∖
Σ
k
j
,
R
0
)
,
j
=
1
,
…
,
J
, we have
∇
H
k
(
x
)
=
∇
h
k
(
x
)
(
1
-
χ
(
(
R
r
k
j
)
-
1
d
k
j
(
x
)
)
)
-
h
k
(
x
)
(
R
0
r
k
j
)
-
1
χ
′
(
(
R
0
r
k
j
)
-
1
d
k
j
(
x
)
)
∇
d
k
j
(
x
)
.
Noting that
H
k
2
≤
h
k
2
, we have
(2.18)
|
B
k
(
h
k
,
h
k
)
-
B
k
(
H
k
,
H
k
)
|
≤
|
∫
co
(
M
k
)
|
∇
h
k
|
2
-
|
∇
H
k
|
2
|
+
C
∫
co
(
M
k
)
ω
k
,
δ
,
R
(
h
k
2
-
H
k
2
)
,
with
C
=
C
(
N
,
g
,
M
∞
,
m
,
Σ
1
,
…
,
Σ
J
,
δ
,
R
)
<
+
∞
.
We split the first term on the right-hand side of (2.18) into separate domains
(2.19)
|
∫
co
(
M
k
)
|
∇
h
k
|
2
-
|
∇
H
k
|
2
|
≤
I
+
I
I
+
∑
y
∈
ℐ
I
I
I
y
+
∑
j
=
1
J
(
I
V
j
+
V
j
)
,
where
I
=
∫
co
(
M
k
)
∖
⋃
y
∈
ℐ
ι
k
-
1
(
B
r
0
N
(
y
)
)
|
∇
h
k
|
2
,
I
I
≤
∫
⋃
y
∈
ℐ
ι
k
-
1
(
B
r
0
N
(
y
)
∖
B
r
0
/
2
N
(
y
)
)
|
∇
h
k
|
2
+
12
r
0
-
1
|
h
k
|
|
∇
h
k
|
+
36
r
0
-
2
h
k
2
,
≤
C
∫
⋃
y
∈
ℐ
ι
k
-
1
(
B
r
0
N
(
y
)
∖
B
r
0
/
2
N
(
y
)
)
|
∇
h
k
|
2
+
r
0
-
2
h
k
2
.
For
y
∈
ℐ
,
I
I
I
y
=
|
∫
ι
k
-
1
(
B
r
0
/
2
N
(
y
)
∖
(
⋃
j
=
1
J
Σ
k
j
,
2
R
0
)
)
|
∇
h
k
|
2
-
|
∇
H
k
|
2
|
=
0
,
and
j
=
1
,
…
,
J
,
I
V
j
≤
∫
ι
k
-
1
(
Σ
k
j
,
2
R
0
∖
Σ
k
j
,
R
0
)
|
∇
h
k
|
2
+
6
(
R
0
r
k
j
)
-
1
|
h
k
|
|
∇
h
k
|
+
9
(
R
0
r
k
j
)
-
2
h
k
2
,
≤
C
(
∫
ι
k
-
1
(
Σ
k
j
,
2
R
0
∖
Σ
k
j
,
R
0
)
|
∇
h
k
|
2
+
(
R
0
r
k
j
)
-
2
h
k
2
)
,
V
j
=
∫
ι
k
-
1
(
Σ
k
j
,
R
0
)
|
∇
h
k
|
2
.
Along our sequence we have (in a weak sense)
Δ
h
k
+
(
|
A
k
|
2
+
R
k
)
h
k
=
-
(
∑
i
=
1
b
a
i
λ
i
,
k
f
i
,
k
)
ω
k
,
δ
,
R
=
-
P
k
ω
k
,
δ
,
R
,
and we may note that
∥
h
k
∥
L
2
(
co
(
M
k
)
)
+
∥
P
k
∥
L
2
(
co
(
M
k
)
)
≤
C
∑
i
=
1
b
∥
f
i
,
k
∥
L
2
(
co
(
M
k
)
)
≤
C
,
with
C
=
C
(
N
,
g
,
M
∞
,
m
,
Σ
1
,
…
,
Σ
J
,
δ
,
R
)
<
+
∞
.
Recalling notation from Section 2.2, for each
l
=
1
,
…
,
m
, we denote
h
~
k
l
≔
∑
i
=
1
b
a
i
f
~
i
,
k
l
and
P
~
k
l
≔
∑
i
=
1
b
a
i
λ
i
,
k
f
~
i
,
k
l
,
and then by standard interior estimates,
∥
h
~
k
l
∥
W
2
,
2
(
Ω
r
0
/
4
)
≤
C
~
(
∥
h
~
k
l
∥
L
2
(
Ω
r
0
/
8
)
+
r
0
-
2
∥
P
~
k
l
∥
L
2
(
Ω
r
0
/
8
)
)
≤
C
~
,
for
C
~
=
C
~
(
N
,
g
,
M
∞
,
m
,
Σ
1
,
…
,
Σ
J
,
δ
,
R
,
r
0
)
<
+
∞
.
Thus, by our assumption that
h
k
→
(
(
0
,
…
,
0
)
,
0
…
,
0
)
,
after potentially taking a subsequence and renumerating, we have
{
h
~
k
l
⇀
0
weakly in
W
2
,
2
(
Ω
r
0
/
4
)
,
h
~
k
l
→
0
strongly in
W
1
,
2
(
Ω
r
0
/
4
)
.
By identical arguments, for each
j
=
1
,
…
,
J
denoting
h
~
k
Σ
j
=
∑
i
=
1
b
a
i
f
~
i
,
k
Σ
j
,
we have that, after potentially taking a subsequence and renumerating,
{
h
~
k
Σ
j
⇀
0
weakly in
W
2
,
2
(
Σ
j
,
4
R
0
)
,
h
~
k
Σ
j
→
0
strongly in
W
1
,
2
(
Σ
j
,
4
R
0
)
.
Therefore, we have that for all
ζ
>
0
, and then large enough k,
|
∫
co
(
M
k
)
|
∇
h
k
|
2
-
|
∇
H
k
|
2
|
≤
C
^
(
∑
l
=
1
m
∥
h
~
k
l
∥
W
1
,
2
(
Ω
r
0
/
4
)
2
+
∑
j
=
1
J
∥
h
~
k
Σ
j
∥
W
1
,
2
(
Σ
j
,
4
R
0
)
2
)
<
ζ
,
with
C
^
=
C
^
(
N
,
g
,
M
∞
,
m
,
Σ
1
,
…
,
Σ
J
,
δ
,
R
,
r
0
,
R
0
)
<
+
∞
.
Similarly we have
∫
co
(
M
k
)
ω
k
,
δ
,
R
(
h
k
2
-
H
k
2
)
≤
C
^
(
∑
l
=
1
m
∥
h
~
k
∥
L
2
(
Ω
r
0
/
4
)
2
+
∑
j
=
1
J
∥
h
~
k
Σ
j
∥
L
2
(
Σ
j
,
4
R
0
)
2
)
<
ζ
.
Thus, for large k, there exists a connected component
C
k
⊂
M
k
∩
(
⋃
y
∈
ℐ
B
r
0
N
(
y
)
∖
(
⋃
j
=
1
J
Σ
k
j
,
R
0
)
)
such that, denoting
H
~
k
=
(
H
k
)
|
ι
k
-
1
(
C
k
)
, we have that
H
~
k
∈
W
0
1
,
2
(
ι
k
-
1
(
C
k
)
)
,
and
∫
ι
k
-
1
(
C
k
)
H
~
k
2
ω
k
,
δ
,
R
≥
1
-
ζ
m
|
ℐ
|
,
B
k
[
H
~
k
,
H
~
k
]
<
ζ
.
Thus, choosing
ζ
≤
γ
m
|
ℐ
|
+
γ
,
we derive a contradiction to (2.17) on
C
k
.
∎
3 Equivalence of weighted and unweighted eigenspaces
Proposition 8.
Assume that
(
M
,
g
)
is a compact Riemannian manifold of dimension
n
≥
3
, with
V
∈
L
∞
(
M
)
,
ω
∈
L
(
n
2
,
∞
)
(
M
,
g
)
, and
ess
inf
ω
>
0
.
Define the elliptic operator,
L
≔
Δ
+
V
.
Then
(3.1)
span
{
⋃
λ
≤
0
ℰ
(
λ
;
L
,
W
1
,
2
(
M
)
)
}
=
⊕
λ
≤
0
ℰ
(
λ
;
L
,
W
1
,
2
(
M
)
)
,
(3.2)
span
{
⋃
λ
≤
0
ℰ
ω
(
λ
;
L
,
W
1
,
2
(
M
)
)
}
=
⊕
λ
≤
0
ℰ
ω
(
λ
;
L
,
W
1
,
2
(
M
)
)
,
and
dim
(
span
{
⋃
λ
≤
0
ℰ
ω
(
λ
;
L
,
W
1
,
2
(
M
)
)
}
)
=
dim
(
span
{
⋃
λ
≤
0
ℰ
(
λ
;
L
,
W
1
,
2
(
M
)
)
}
)
.
Proof.
First we note by Proposition 6, and fact that
ess
inf
ω
>
0
,
〈
f
1
,
f
2
〉
ω
≔
∫
M
f
1
f
2
ω
is a well-defined inner product on
W
1
,
2
(
M
)
.
Consider the bilinear form on
f
1
,
f
2
∈
W
1
,
2
(
M
)
, corresponding to our elliptic operator L,
B
L
[
f
1
,
f
2
]
≔
∫
M
∇
f
1
⋅
∇
f
2
-
V
f
1
f
2
.
If
f
i
∈
ℰ
ω
(
λ
i
;
L
,
W
1
,
2
(
M
)
)
, for
i
=
1
,
2
, with
λ
1
≠
λ
2
, we have
λ
1
〈
f
1
,
f
2
〉
ω
=
B
L
[
f
1
,
f
2
]
=
λ
2
〈
f
1
,
f
2
〉
ω
.
Thus,
〈
f
1
,
f
2
〉
ω
=
0
, and so (3.2) follows.
An identical argument will also conclude (3.1).
We now show that
dim
(
⊕
λ
≤
0
ℰ
(
λ
;
L
,
W
1
,
2
(
M
)
)
)
=
sup
{
dim
Π
:
Π
≤
W
1
,
2
(
M
)
is a linear space such that
(
B
L
)
|
Π
≤
0
}
=
dim
(
⊕
λ
≤
0
ℰ
ω
(
λ
;
L
,
W
1
,
2
(
M
)
)
)
.
Indeed, we begin by showing that,
(3.3)
dim
(
⊕
λ
≤
0
ℰ
ω
(
λ
;
L
,
W
1
,
2
(
M
)
)
)
≤
sup
{
dim
Π
:
Π
≤
W
1
,
2
(
M
)
is linear space such that
(
B
L
)
|
Π
≤
0
}
.
If the left-hand side is equal to 0, then the inequality is trivial.
Thus, take
b
∈
ℤ
≥
1
such that
b
≤
dim
(
⊕
λ
≤
0
ℰ
ω
(
λ
;
L
,
W
1
,
2
(
M
)
)
)
.
Then we have b ω-weighted eigenvectors
{
f
1
,
…
,
f
b
}
⊂
W
1
,
2
(
M
)
,
with respective non-positive eigenvalues
{
λ
1
,
…
,
λ
b
}
.
By the argument at the beginning of the proof, we may take the set
{
f
1
,
…
,
f
b
}
, to be orthonormal with respect to
〈
⋅
,
⋅
〉
ω
.
Thus,
(3.4)
Π
=
span
{
f
1
,
…
,
f
b
}
is a b-dimensional vector space, and for
a
1
,
…
,
a
b
∈
ℝ
, we have
B
L
[
∑
i
a
i
f
i
,
∑
j
a
j
f
j
]
=
∑
i
,
j
a
i
a
j
λ
i
〈
f
i
,
f
j
〉
ω
=
∑
i
λ
i
a
i
2
≤
0
.
Thus, (3.3) holds.
Identical argument shows that same inequality holds for unweighted eigenspaces.
We look to show reverse inequality of (3.3).
As we know that (3.3) holds, if the left-hand side of (3.3) is unbounded, then equality holds trivially.
Thus, we consider
b
∈
ℤ
≥
0
,
b
=
dim
(
⊕
λ
≤
0
ℰ
(
λ
;
L
,
W
1
,
2
(
M
)
)
)
<
∞
,
and prove reverse of (3.3) by contradiction.
Indeed, assume that we have a linear subspace
Π
~
≤
W
1
,
2
(
M
)
, of dimension
b
+
1
such that
B
L
is non-positive on
Π
~
.
Consider the b-dimensional linear subspace Π, identically defined as (3.4).
Note that
W
1
,
2
(
M
)
=
Π
⊕
Π
⊥
ω
.
The projection map
P
Π
:
Π
~
→
Π
, must have a non-trivial kernel, implying that there exists a
v
∈
Π
~
∩
Π
⊥
ω
, with
〈
v
,
v
〉
ω
=
1
.
Thus,
λ
~
=
inf
{
B
L
[
f
,
f
]
:
f
∈
Π
⊥
ω
,
〈
f
,
f
〉
ω
=
1
}
≤
0
.
Take
f
~
k
∈
Π
⊥
ω
,
〈
f
~
k
,
f
~
k
〉
ω
=
1
such that
λ
~
=
lim
k
→
∞
B
L
[
f
~
k
,
f
~
k
]
.
Thus, noting that
ess
inf
ω
>
0
, and
|
V
|
<
+
∞
, we may deduce a lower bound on
λ
~
≥
-
∥
V
∥
L
∞
(
M
)
ess
inf
ω
,
and uniform
W
1
,
2
(
M
)
bounds for
{
f
~
k
}
, and conclude that there exists an
f
~
∈
W
1
,
2
(
M
)
such that, after potentially taking a subsequence and renumerating,
{
f
~
k
⇀
f
~
in
W
1
,
2
(
M
)
,
f
~
k
→
f
~
in
L
2
(
M
)
.
Denote
h
1
≔
P
Π
(
f
~
)
,
h
2
≔
f
~
-
h
1
∈
Π
⊥
ω
.
There exists
a
1
,
…
,
a
b
∈
ℝ
such that
h
1
=
∑
i
=
1
b
a
i
f
i
.
Then, as
f
~
k
∈
Π
⊥
ω
,
B
L
[
f
~
k
,
f
~
]
=
∑
i
=
1
m
a
i
λ
i
〈
f
~
k
,
f
i
〉
ω
+
B
L
[
f
~
k
,
h
2
]
=
B
L
[
f
~
k
,
h
2
]
.
Thus, by weak convergence,
B
L
[
f
~
,
f
~
]
=
lim
k
→
∞
B
L
[
f
~
k
,
f
~
]
=
lim
k
→
∞
B
L
[
f
~
k
,
h
2
]
=
B
L
[
h
1
+
h
2
,
h
2
]
=
∑
i
=
1
m
a
i
λ
i
〈
f
i
,
h
2
〉
ω
+
B
L
[
h
2
,
h
2
]
=
B
L
[
h
2
,
h
2
]
.
After potentially taking a further subsequence and renumerating so that
f
~
k
→
f
~
pointwise a.e., by Fatou’s Lemma we have the following:
0
≤
α
=
〈
h
2
,
h
2
〉
ω
≤
〈
h
2
,
h
2
〉
ω
+
〈
h
1
,
h
1
〉
ω
=
〈
f
~
,
f
~
〉
ω
≤
lim inf
k
→
∞
〈
f
~
k
,
f
~
k
〉
ω
=
1
,
and we reduce our argument to three cases.
First consider
α
=
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
[
f
~
,
f
~
]
≤
lim
k
→
∞
B
L
[
f
~
k
,
f
~
k
]
=
λ
~
≤
0
.
Therefore,
λ
~
=
0
, and recalling the function
v
∈
Π
~
∩
Π
⊥
ω
, with
〈
v
,
v
〉
ω
=
1
, we must have
B
L
[
v
,
v
]
=
inf
{
B
L
[
f
,
f
]
:
f
∈
Π
⊥
ω
,
〈
f
,
f
〉
ω
=
1
}
=
0
.
Standard variational arguments then show that
v
∈
ℰ
ω
(
0
;
L
,
W
1
,
2
(
M
)
)
, which contradicts the definition b.
If
α
=
1
, then
B
L
[
h
2
,
h
2
]
=
inf
{
B
L
[
f
,
f
]
:
f
∈
Π
⊥
ω
,
〈
f
,
f
〉
ω
=
1
}
=
λ
~
.
Again, standard variational arguments then show that
h
2
∈
ℰ
ω
(
λ
~
;
L
,
W
1
,
2
(
M
)
)
, which contradicts the definition of b.
The final case is
0
<
α
<
1
.
Note that if
B
L
[
h
2
,
h
2
]
=
0
, then we may apply a similar argument to that in case
α
=
0
.
Therefore, we may assume that
B
L
[
h
2
,
h
2
]
<
0
.
Define
h
≔
α
-
1
2
h
2
.
Thus,
B
L
[
h
,
h
]
=
α
-
1
B
L
[
h
2
,
h
2
]
<
B
L
[
h
2
,
h
2
]
≤
λ
~
.
However, as
h
∈
Π
⊥
ω
, and
〈
h
,
h
〉
ω
=
1
, this is a contradiction.
∎
Remark 6.
For an embedded hypersurface
M
⊂
N
, the method of proof in Proposition 8 may be applied to show that,
span
{
⋃
λ
≤
0
ℰ
ω
(
λ
;
L
,
W
1
,
2
(
co
(
M
)
)
-
)
}
=
⊕
λ
≤
0
ℰ
ω
(
λ
;
L
,
W
1
,
2
(
co
(
M
)
)
-
)
,
and,
dim
(
⊕
λ
≤
0
ℰ
ω
(
λ
;
L
,
W
1
,
2
(
co
(
M
)
)
-
)
)
=
dim
(
⊕
λ
≤
0
ℰ
(
λ
;
L
,
W
1
,
2
(
co
(
M
)
)
-
)
)
.
When reapplying this method the two major things to note is that
W
1
,
2
(
co
(
M
)
)
-
is a linear space, and that when applying Rellich–Kondrachov, we have that the limit will also lie
W
1
,
2
(
co
(
M
)
)
-
.
Remark 7.
We remark that in the case for
ω
∈
L
∞
(
M
)
(which is the case for our weights
ω
k
,
δ
,
R
, along our sequence of hypersurfaces
M
k
), the method of proof presented for Proposition 8 simplifies.
In fact, in this case one may also simply diagonalise the bilinear form
B
L
with respect to the inner product
〈
⋅
,
⋅
〉
ω
, and deduce the equivalence of weighted and unweighted index and nullity through Sylvester’s Law of Inertia (see [19, Appendix A]).
Let
(
M
,
g
)
be a complete but not necessarily compact Riemannian manifold.
Recall the following function space:
W
ω
1
,
2
(
M
)
≔
{
f
∈
L
loc
1
(
M
)
:
|
∇
f
|
∈
L
2
(
M
)
and
f
2
ω
∈
L
1
(
M
)
}
,
and, for a second order linear elliptic PDE L, the eigenspace
ℰ
ω
(
λ
;
L
,
W
ω
1
,
2
(
M
)
)
≔
{
f
∈
W
ω
1
,
2
(
M
)
:
B
L
[
f
,
φ
]
=
λ
∞
∫
M
f
φ
ω
for all
φ
∈
W
ω
1
,
2
(
M
)
}
,
where
B
L
is the associated bilinear form of L.
Proposition 9.
Let Σ be a connected, complete n-dimensional manifold,
n
≥
3
, and let
ι
:
Σ
→
R
n
+
1
be a two-sided, proper, minimal immersion, with finite total curvature
∫
Σ
|
A
Σ
|
n
<
+
∞
,
and Euclidean volume growth at infinity.
Consider a function
ω
∈
L
∞
(
Σ
)
such that there exists
Λ
∈
[
1
,
∞
)
, and
R
∈
(
0
,
∞
)
such that
ess
inf
ω
>
0
on
Σ
∩
ι
-
1
(
B
R
n
+
1
(
0
)
¯
)
, and
1
Λ
|
ι
(
x
)
|
2
≤
ω
(
x
)
≤
Λ
|
ι
(
x
)
|
2
,
for
x
∈
Σ
∖
ι
-
1
(
B
R
n
+
1
(
0
)
)
.
Then
(3.5)
span
{
⋃
λ
≤
0
ℰ
ω
(
λ
;
L
Σ
,
W
ω
1
,
2
(
Σ
)
)
}
=
⊕
λ
≤
0
ℰ
ω
(
λ
;
L
Σ
,
W
ω
1
,
2
(
Σ
)
)
and
dim
(
⊕
λ
<
0
ℰ
ω
(
λ
;
L
Σ
,
W
ω
1
,
2
(
Σ
)
)
)
=
anl
-
ind
(
Σ
)
≔
lim
S
→
∞
anl
-
ind
ι
-
1
(
B
S
n
+
1
(
0
)
)
(
Σ
)
≔
lim
S
→
∞
dim
(
⊕
λ
<
0
ℰ
(
λ
;
L
Σ
,
W
0
1
,
2
(
Σ
∩
ι
-
1
(
B
S
n
+
1
(
0
)
)
)
)
)
.
Remark 8.
In the literature, for a two-sided, properly immersed minimal hypersurface Σ, the analytic index (
anl
-
ind
(
Σ
)
) and analytic nullity (
anl
-
nul
(
Σ
)
), are just referred to as the index and nullity of Σ.
We choose to maintain the terms of analytic index and analytic nullity to keep the notation and definitions consistent throughout the paper.
Proof.
Denote the stability operator on Σ by
L
=
L
Σ
.
As the immersion is proper,
ess
inf
ω
>
0
, on compact sets of Σ, and thus
W
ω
1
,
2
(
Σ
)
⊂
W
loc
1
,
2
(
Σ
)
.
First we note that one may deduce (3.5) in an identical manner to (3.2).
We proceed similarly to Proposition 8.
First recall that as Σ has finite total curvature and Euclidean volume growth at infinity, its (analytic) index is finite [37, Section 3] (cf. [22]).
Thus we may pick an
S
0
>
0
such that
(3.6)
anl
-
ind
ι
-
1
(
B
S
0
n
+
1
(
0
)
)
(
Σ
)
=
anl
-
ind
(
Σ
)
.
First we show that
(3.7)
dim
(
⊕
λ
<
0
ℰ
ω
(
λ
;
L
Σ
,
W
ω
1
,
2
(
Σ
)
)
)
≤
anl
-
ind
(
Σ
)
=
I
.
If the left-hand side is equal to 0, then the inequality is trivial, thus assume we have
b
∈
ℤ
≥
1
such that
b
≤
dim
(
⊕
λ
<
0
ℰ
ω
(
λ
;
L
Σ
,
W
ω
1
,
2
(
Σ
)
)
)
.
Therefore, we may pick a set of eigenfunctions
{
f
1
,
…
,
f
b
}
⊂
⊕
λ
<
0
ℰ
ω
(
λ
;
L
,
W
ω
1
,
2
(
Σ
)
)
,
which are orthonormal with respect to
〈
⋅
,
⋅
〉
ω
.
Recall the function χ from (2.13), and similarly to before, for large enough
S
>
0
, we define the smooth function,
χ
S
(
x
)
=
χ
(
S
-
1
|
ι
(
x
)
|
)
on Σ, and for a function f on Σ, we define
f
S
=
χ
S
f
.
The inequality then follows noting that for large enough
S
>
S
0
,
span
{
(
f
1
)
S
,
…
,
(
f
b
)
S
}
⊂
W
0
1
,
2
(
Σ
∩
B
2
S
n
+
1
(
0
)
)
is a b-dimensional subspace on which
B
L
is negative definite.
For the reverse of (3.7), we take
I
≥
1
(note that if
I
=
0
then (3.7) implies equality), and an increasing sequence
S
k
→
∞
.
For large enough
S
k
, as
Σ
∩
ι
-
1
(
B
S
k
n
+
1
(
0
)
)
is compact with smooth boundary, and
ω
∈
L
∞
(
Σ
)
, one may deduce (see [19, Appendix A]) that there exists sequences
λ
1
k
≤
⋯
≤
λ
I
k
<
0
and a set
{
f
1
k
,
…
,
f
I
k
}
⊂
W
0
1
,
2
(
Σ
∩
ι
-
1
(
B
S
k
n
+
1
(
0
)
)
)
,
which is orthonormal with respect to
〈
⋅
,
⋅
〉
ω
such that for all
φ
∈
W
0
1
,
2
(
Σ
∩
ι
-
1
(
B
S
k
n
+
1
(
0
)
)
)
,
B
L
[
f
i
k
,
φ
]
=
λ
i
k
〈
f
i
k
,
φ
〉
ω
.
By standard theory (see [18, Lemma 3.7]), for each
i
=
1
,
…
,
I
,
λ
i
k
≥
λ
i
k
+
1
.
There exists a uniform bound
λ
i
k
≥
-
C
(this follows from arguments similar to (2.5)), and thus for each
i
=
1
,
…
,
I
, there exists a
λ
i
∞
∈
[
-
C
,
0
)
such that
λ
i
k
→
λ
i
∞
.
By similar arguments contained in Section 2.3, we may deduce that after potentially taking a subsequence and renumerating, for each
i
=
1
,
…
,
I
, there exists an
f
i
∞
∈
W
ω
1
,
2
(
Σ
)
such that
{
f
i
k
⇀
f
i
∞
in
W
loc
2
,
2
(
Σ
)
,
f
i
k
→
f
i
∞
in
W
loc
1
,
2
(
Σ
)
,
and
f
i
∞
∈
ℰ
ω
(
λ
i
∞
;
L
Σ
,
W
ω
1
,
2
(
Σ
)
)
.
We now look to show that
dim
(
span
{
f
1
∞
,
…
,
f
I
∞
}
)
=
I
,
which will complete the proof.
For a contradiction, assume not.
Then there exists
a
1
,
…
,
a
I
∈
ℝ
, with
∑
a
i
2
=
1
such that
h
≔
∑
i
=
1
I
a
i
f
i
∞
=
0
.
We then define
h
k
≔
∑
i
=
1
I
a
i
f
i
k
∈
W
0
1
,
2
(
Σ
∩
ι
-
1
(
B
S
k
n
+
1
(
0
)
)
)
,
and note that
〈
h
k
,
h
k
〉
ω
=
1
for all k,
h
k
→
0
in
W
loc
1
,
2
(
Σ
)
, and by the monotonicity of the eigenvalues, there exists an
α
>
0
such that for all large enough k,
B
L
[
h
k
,
h
k
]
≤
-
α
<
0
.
Thus,
(
1
-
χ
S
0
)
h
k
∈
W
0
1
,
2
(
Σ
∖
ι
-
1
(
B
S
0
n
+
1
(
0
)
¯
)
)
, and for large enough k,
B
L
[
(
1
-
χ
S
0
)
h
k
,
(
1
-
χ
S
0
)
h
k
]
<
0
.
However, this clearly contradicts the stability of
Σ
∖
ι
-
1
(
B
S
0
n
+
1
(
0
)
¯
)
.
∎
4 Proof of the theorem
Again, for ease of notation we only write the proof of Theorem 1 for the case of minimal hypersurfaces; however, the identical argument works for the case of H-CMC hypersurfaces.
We follow similar arguments to those in [16, Lemma IV.6] and [19, Theorem 1.2].
If
lim sup
k
→
∞
(
ind
(
M
k
)
+
nul
(
M
k
)
)
=
0
, then the conclusion of Theorem 1 is trivial.
Suppose for
b
∈
ℤ
≥
1
,
b
≤
lim sup
k
→
∞
(
ind
(
M
k
)
+
nul
(
M
k
)
)
=
lim sup
k
→
∞
(
dim
(
⊕
λ
≤
0
ℰ
ω
k
,
δ
,
R
(
λ
;
L
k
,
W
1
,
2
(
co
(
M
k
)
)
-
)
)
)
,
where the equality comes from equivalence of considering the weighted and unweighted eigenvalue problems along our sequence (Proposition 8 and Remark 6).
After potentially taking a subsequence and renumerating we have that for each k, there exists a linear subspace
W
k
≔
span
{
f
i
,
k
}
i
=
1
b
⊂
W
1
,
2
(
co
(
M
k
)
)
-
,
where, for each
i
=
1
,
…
,
b
, there is a
λ
i
,
k
≤
0
such that
f
i
,
k
∈
ℰ
ω
k
,
δ
,
R
(
λ
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
ω
k
,
δ
,
R
-weighted
L
2
inner product, i.e. for
i
,
j
=
1
,
…
,
b
,
∫
M
k
f
i
,
k
f
j
,
k
ω
k
,
δ
,
R
=
δ
i
j
.
We may assume this by the argument used to prove (3.2).
Thus, as outlined in Section 2.4, for each
i
=
1
,
…
,
b
, after potentially taking a subsequence and renumerating,
f
i
,
k
→
(
(
f
i
,
∞
1
,
…
,
f
i
,
∞
m
)
,
f
i
,
∞
Σ
1
,
…
,
f
i
,
∞
Σ
J
)
∈
E
∞
where we are defining
E
∞
≔
(
∏
j
=
1
m
(
⊕
λ
≤
0
ℰ
ω
δ
(
λ
;
L
∞
,
W
1
,
2
(
co
(
M
∞
)
)
)
)
)
×
(
∏
j
=
1
J
(
⊕
λ
≤
0
ℰ
ω
Σ
j
,
R
(
λ
;
L
Σ
j
,
W
ω
Σ
j
,
R
1
,
2
(
Σ
j
)
)
)
)
)
.
We define the linear map
Π
k
:
W
k
→
E
∞
,
f
i
,
k
↦
(
(
f
i
,
∞
1
,
…
,
f
i
,
∞
m
)
,
f
i
,
∞
Σ
1
,
…
,
f
i
,
∞
Σ
J
)
.
Thus
W
∞
≔
Π
k
(
W
k
)
is a linear subspace of
E
∞
.
Define the integer
co
(
m
)
=
lim inf
r
→
0
lim inf
k
→
∞
|
{
connected components of
M
k
∖
⋃
y
∈
ℐ
B
r
N
(
y
)
}
|
≤
m
.
If
M
∞
is two-sided, by the graphical convergence on sets, compactly contained away from the finite collection of points
ℐ
, we have that
co
(
m
)
=
m
.
If
M
∞
is one-sided, taking r small enough and k large enough such that
co
(
m
)
=
|
{
connected components of
M
k
∖
⋃
y
∈
ℐ
B
r
N
(
y
)
}
|
,
we recall notation from Section 2.2, and we have
M
k
r
=
⋃
l
=
1
co
(
m
)
M
k
l
,
r
,
where each
M
k
l
,
r
is a connected hypersurface.
Then
⋃
j
=
1
m
{
(
x
,
u
k
j
,
r
(
x
)
)
:
x
∈
Ω
r
}
is a double cover of
M
k
r
with trivial normal bundle, implying that we identify (as in Section 2.2)
⋃
j
=
1
m
{
(
x
,
u
k
j
,
r
(
x
)
)
:
x
∈
Ω
r
}
=
⋃
l
=
1
co
(
m
)
o
(
M
k
l
,
r
)
.
If there is an
l
∈
{
1
,
…
,
co
(
m
)
}
, and
j
≠
j
′
∈
{
1
,
…
,
m
}
such that for all large enough k,
o
(
M
k
l
,
r
)
=
{
(
x
,
u
k
j
,
r
(
x
)
)
:
x
∈
Ω
r
}
∪
{
(
x
,
u
k
j
′
,
r
(
x
)
)
:
x
∈
Ω
r
}
,
then (depending on the choice of unit normal in Section 2.2), for each
i
=
1
,
…
,
m
, either
f
i
,
∞
j
(
(
x
,
ν
)
)
=
-
f
i
,
∞
j
′
(
(
x
,
-
ν
)
)
for all
(
x
,
ν
)
∈
co
(
M
∞
)
∖
ι
-
1
(
ℐ
)
or
f
i
,
∞
j
(
(
x
,
ν
)
)
=
f
∞
j
′
(
(
x
,
-
ν
)
)
for all
(
x
,
ν
)
∈
co
(
M
∞
)
∖
ι
-
1
(
ℐ
)
.
Thus we may define an injective map
P
:
W
∞
→
F
∞
,
where
F
∞
=
(
∏
l
=
1
co
(
m
)
(
⊕
λ
≤
0
ℰ
ω
δ
(
λ
;
L
∞
,
W
1
,
2
(
co
(
M
∞
)
)
)
)
)
×
(
∏
j
=
1
J
(
⊕
λ
≤
0
ℰ
ω
Σ
j
,
R
(
λ
;
L
Σ
j
,
W
ω
Σ
j
,
R
1
,
2
(
Σ
j
)
)
)
)
.
By Claim 2 we have that
Π
k
is injective, and thus,
dim
W
∞
=
dim
W
k
=
b
,
which implies that,
b
≤
dim
F
∞
.
We conclude Theorem 1 by combining the results in Section 3 (Proposition 8 and Proposition 9), and noting that
ℰ
ω
Σ
j
,
R
(
0
;
L
Σ
j
,
W
ω
Σ
j
,
R
1
,
2
(
Σ
j
)
)
=
ℰ
ω
Σ
j
,
R
(
0
;
L
Σ
j
,
W
ω
Σ
j
,
R
1
,
2
(
Σ
j
)
)
,
for
ω
Σ
j
,
R
as defined in the statement of Theorem 1, and
ω
Σ
j
,
R
as in Section 2.3.
Lastly, by standard regularity theory for elliptic PDEs, we note that
nul
ω
Σ
j
,
R
(
Σ
j
)
≔
dim
(
{
ψ
∈
C
∞
(
Σ
)
∩
W
ω
Σ
j
,
R
1
,
2
(
Σ
j
)
:
L
Σ
j
ψ
=
0
}
)
=
dim
(
ℰ
ω
Σ
j
,
R
(
0
;
L
Σ
j
,
W
ω
Σ
j
,
R
1
,
2
(
Σ
j
)
)
)
.
5 Finiteness of the nullity
Proposition 10.
Let Σ be a complete, connected, n-dimensional manifold,
n
≥
3
, and let
ι
:
Σ
→
ℝ
n
+
1
be a proper, two-sided, minimal immersion, of finite total curvature
∫
Σ
|
A
Σ
|
n
<
+
∞
,
and Euclidean volume growth at infinity
lim sup
R
→
∞
ℋ
n
(
ι
(
Σ
)
∩
B
R
n
+
1
(
0
)
)
R
n
<
+
∞
.
Consider a function
ω
∈
L
∞
(
Σ
)
such that there exists an
R
>
0
, and
Λ
≥
1
such that
ess
inf
ω
>
0
in
ι
-
1
(
B
R
n
+
1
(
0
)
)
, and for
x
∈
Σ
∖
ι
-
1
(
B
R
n
+
1
(
0
)
)
,
1
Λ
|
ι
(
x
)
|
2
≤
ω
≤
Λ
|
ι
(
x
)
|
2
.
Then
anl-nul
ω
(
Σ
)
≔
dim
{
ψ
∈
W
ω
1
,
2
(
Σ
)
:
L
Σ
ψ
=
0
}
<
+
∞
.
Proof.
We assume the statement does not hold and prove by contradiction.
There exists a set
{
ψ
1
,
ψ
2
,
…
}
⊂
W
ω
1
,
2
(
Σ
)
∩
C
∞
(
Σ
)
such that for all
k
,
l
∈
ℤ
≥
1
,
Δ
ψ
k
+
|
A
Σ
|
2
ψ
k
=
0
and
∫
Σ
ψ
k
ψ
l
ω
=
δ
k
l
.
Claim.
For any
δ
>
0
, there exists
k
,
l
∈
Z
≥
1
,
k
≠
l
such that
∥
ψ
k
-
ψ
l
∥
W
1
,
2
(
Σ
∩
ι
-
1
(
B
2
S
n
+
1
(
0
)
)
)
2
<
δ
.
Proof of Claim.
Fix
S
>
0
, for all
k
∈
ℤ
≥
1
,
∫
Σ
∩
ι
-
1
(
B
3
S
n
+
1
(
0
)
)
ψ
k
2
≤
C
(
S
)
<
+
∞
.
By standard interior estimates for linear elliptic PDEs we have
∥
ψ
k
∥
W
2
,
2
(
Σ
∩
ι
-
1
(
B
2
S
n
+
1
(
0
)
)
≤
C
(
S
)
<
+
∞
.
Therefore, there exists a subsequence
{
ψ
k
′
}
⊂
{
ψ
k
}
, and a function
ψ
∞
∈
W
2
,
2
(
Σ
∩
ι
-
1
(
B
2
S
n
+
1
(
0
)
)
)
such that,
{
ψ
k
′
⇀
ψ
∞
weakly in
W
2
,
2
(
Σ
∩
ι
-
1
(
B
2
S
n
+
1
(
0
)
)
)
,
ψ
k
′
→
ψ
∞
strongly in
W
1
,
2
(
Σ
∩
ι
-
1
(
B
2
S
n
+
1
(
0
)
)
)
.
Thus this subsequence is Cauchy, so for any
δ
>
0
, there exists
l
,
k
∈
ℤ
≥
1
,
l
≠
k
such that
∥
ψ
k
-
ψ
l
∥
W
1
,
2
(
Σ
∩
ι
-
1
(
B
2
S
n
+
1
(
0
)
)
)
2
<
δ
.
∎
For
δ
>
0
fixed we denote
ψ
δ
=
ψ
l
-
ψ
k
∈
W
ω
1
,
2
(
Σ
)
,
and note that
Δ
ψ
δ
+
|
A
Σ
|
2
ψ
δ
=
0
and
∫
Σ
ψ
δ
2
ω
=
2
.
Recall from Remark 2 that Σ has finitely many ends (say
m
∈
ℤ
≥
1
), which, for large enough
S
≥
R
, may be denoted by
⊔
i
=
1
m
E
i
=
Σ
∖
ι
-
1
(
B
S
n
+
1
(
0
)
)
.
Moreover, each end
E
i
is graphical over some hyperplane minus a compact set
B
i
(with the graphing function having small gradient), and for each
ε
>
0
, we may further choose
S
=
S
(
Σ
,
ι
,
R
,
Λ
,
ε
)
<
+
∞
such that
|
A
Σ
|
2
(
x
)
≤
ε
ω
,
for
x
∈
Σ
∖
ι
-
1
(
B
¯
S
n
+
1
(
0
)
)
.
We also remark that
ω
∈
L
(
n
2
,
∞
)
(
Σ
)
(this can be shown similarly to Claim 1).
Fixing a choice of
S
=
S
(
Σ
,
ι
,
Λ
,
R
,
ε
)
<
+
∞
, for any
ζ
>
0
, we may pick
δ
=
δ
(
S
,
ζ
)
>
0
. Then
T
=
T
(
δ
)
>
4
S
such that
∥
ψ
δ
∥
W
1
,
2
(
Σ
∩
ι
-
1
(
B
2
S
n
+
1
(
0
)
)
)
<
ζ
and
∥
∇
ψ
δ
∥
L
2
(
Σ
∖
ι
-
1
(
B
T
n
+
1
(
0
)
)
)
+
(
∫
Σ
∖
ι
-
1
(
B
T
n
+
1
(
0
)
)
ψ
δ
2
ω
)
1
2
<
ζ
.
Recalling definition of functions
χ
S
and
χ
T
from (2.13), we define
Ψ
δ
=
χ
T
(
1
-
χ
S
)
ψ
δ
∈
W
0
1
,
2
(
ι
-
1
(
B
2
T
n
+
1
(
0
)
∖
B
S
n
+
1
(
0
)
¯
)
)
.
Performing similar computations to those in Claim 2, we deduce
∫
Σ
(
ψ
δ
2
-
Ψ
δ
2
)
ω
≤
C
(
∫
Σ
∩
ι
-
1
(
B
2
S
n
+
1
(
0
)
)
ψ
δ
2
+
∫
Σ
∖
ι
-
1
(
B
T
n
+
1
(
0
)
)
ψ
δ
2
ω
)
<
C
ζ
and
|
B
L
[
ψ
δ
,
ψ
δ
]
-
B
L
[
Ψ
δ
,
Ψ
δ
]
|
≤
C
(
∥
ψ
δ
∥
W
1
,
2
(
Σ
∩
ι
-
1
(
B
2
S
n
+
1
(
0
)
)
)
2
+
∥
∇
ψ
δ
∥
L
2
(
Σ
∖
ι
-
1
(
B
T
n
+
1
(
0
)
)
)
2
+
∫
Σ
∖
ι
-
1
(
B
T
n
+
1
(
0
)
)
ψ
δ
2
ω
)
<
C
ζ
with
C
=
C
(
Σ
,
ι
,
ω
,
R
)
<
+
∞
.
Now, choosing small enough
ε
=
ε
(
ω
)
>
0
, large enough
S
=
S
(
Σ
,
ι
,
Λ
,
R
,
ε
)
, then small enough
ζ
=
ζ
(
Σ
,
ι
,
ω
,
R
,
ε
)
>
0
,
δ
=
δ
(
S
,
ζ
)
>
0
, and large enough
T
=
T
(
δ
)
>
4
S
,
Ψ
δ
will derive a contradiction to Lemma 2 (in a similar fashion to Claim 2) on at least one of the ends
E
1
,
…
,
E
m
.
∎
6 Jacobi fields on the higher-dimensional catenoid
In this section we analyse Jacobi fields on the n-dimensional catenoid,
n
≥
3
.
First, we briefly recall the definition of the n-dimensional catenoid for
n
≥
3
(as in [36, Section 2]).
For
h
0
>
0
, consider the following integral, for
n
≥
3
:
(6.1)
s
(
h
)
=
∫
h
0
h
d
τ
(
a
τ
2
(
n
-
1
)
-
1
)
1
2
,
with
a
=
h
0
-
2
(
n
-
1
)
.
Then the function
s
(
h
)
is increasing and maps
[
h
0
,
+
∞
)
to
[
0
,
s
∞
)
, with
s
∞
=
∫
h
0
∞
d
τ
(
a
τ
2
(
n
-
1
)
-
1
)
1
2
<
∞
.
Thus, the inverse of s,
h
:
[
0
,
s
∞
)
→
[
h
0
,
+
∞
)
is well defined, with
h
(
0
)
=
h
0
, and
h
′
(
0
)
=
0
.
We then smoothly extend h as an even function across
(
-
s
∞
,
s
∞
)
.
Now, letting
S
n
-
1
denote the unit sphere in
ℝ
n
, we define the catenoid,
𝒞
, in
ℝ
n
+
1
,
n
≥
3
, by the embedding,
F
:
(
-
s
∞
,
s
∞
)
×
S
n
-
1
→
ℝ
n
+
1
,
(
s
,
w
)
↦
(
h
(
s
)
w
,
s
)
.
For a point
y
=
F
(
s
,
w
)
, the unit normal to
𝒞
at y is given by
ν
(
y
)
=
(
w
,
-
h
′
(
s
)
)
(
1
+
(
h
′
)
2
)
1
2
.
It was shown by Schoen [30, Theorem 3], up to rotations, translations and scalings, the catenoid
𝒞
, is the unique complete, non-flat minimal hypersurface in
ℝ
n
+
1
, with two ends, which is regular at infinity (for a definition of regular at infinity see [30, p. 800]).
Recall our weight,
ω
𝒞
,
R
, which is given by
ω
𝒞
,
R
(
s
,
w
)
=
{
(
h
(
s
1
)
2
+
s
1
2
)
-
1
,
s
∈
(
-
s
1
,
s
1
)
,
(
h
(
s
)
2
+
s
2
)
-
1
,
s
∈
(
-
s
∞
,
s
1
]
∪
[
s
1
,
s
∞
)
,
for
s
1
∈
(
0
,
s
∞
)
, given by
R
2
=
h
(
s
1
)
2
+
s
1
2
.
We now look at Jacobi fields on
𝒞
,
JF
𝒞
≔
{
f
∈
C
∞
(
𝒞
)
:
Δ
𝒞
f
+
|
A
𝒞
|
2
f
=
0
}
.
In particular, we will focus on elements of
JF
𝒞
which are generated through rigid motions in
ℝ
n
+
1
(translations, scalings and rotations of
𝒞
), and look to see which of them lie in
W
1
,
2
(
𝒞
)
and
W
ω
𝒞
,
R
1
,
2
(
𝒞
)
.
We note that it is still an open question whether these Jacobi fields generated through rigid motions account for all the Jacobi fields on
𝒞
.
We will denote the Jacobi fields on
𝒞
defined through rigid motions by
RMJF
𝒞
.
Those arising through translations are generated by the span of
f
1
,
…
,
f
n
+
1
, where
(6.2)
f
i
(
y
)
≔
〈
ν
(
y
)
,
e
i
〉
=
{
〈
(
w
,
0
)
,
e
i
〉
(
1
+
(
h
′
)
2
)
-
1
2
,
i
=
1
,
…
,
n
,
-
h
′
(
1
+
(
h
′
)
2
)
-
1
2
,
i
=
n
+
1
.
The 1-dimensional subspace of
RMJF
𝒞
generated through scaling has a basis element given by
(6.3)
f
d
(
y
)
≔
〈
ν
(
y
)
,
y
〉
=
h
-
s
h
′
(
1
+
(
h
′
)
2
)
1
2
.
The last group to consider are those generated through rotations.
Rotations of
ℝ
n
+
1
about the origin are given by the special orthogonal group
SO
(
n
+
1
)
=
{
R
∈
M
(
n
+
1
)
:
det
R
=
1
,
R
-
1
=
R
T
}
,
where
M
(
n
+
1
)
denotes the set of all
(
n
+
1
)
×
(
n
+
1
)
real matrices.
Then a smooth rotation of
ℝ
n
+
1
is given by a smooth curve
γ
:
[
0
,
T
]
→
SO
(
n
+
1
)
,
with
γ
(
0
)
=
Id
(the identity).
Then the Jacobi field on
𝒞
generated by the 1-parameter family of catenoids,
γ
(
t
)
(
𝒞
)
, is
f
γ
(
y
)
=
〈
ν
(
y
)
,
γ
′
(
0
)
y
〉
,
where
γ
′
(
0
)
∈
T
Id
SO
(
n
+
1
)
.
One may show that
T
Id
SO
(
n
+
1
)
=
{
R
∈
M
(
n
+
1
)
:
R
T
=
-
R
}
.
Taking
R
=
(
R
i
j
)
i
j
∈
T
Id
SO
(
n
+
1
)
, we have that
R
y
=
R
(
h
w
,
s
)
=
∑
i
=
1
n
+
1
(
∑
j
=
1
n
h
R
i
j
w
j
+
R
i
,
n
+
1
s
)
e
i
and
〈
ν
(
y
)
,
γ
′
(
0
)
y
〉
=
(
1
+
(
h
′
)
2
)
-
1
2
(
h
∑
i
,
j
=
1
n
R
i
j
w
i
w
j
+
∑
i
=
1
n
R
i
,
n
+
1
s
w
i
-
h
h
′
∑
j
=
1
n
R
n
+
1
,
j
w
j
-
R
n
+
1
,
n
+
1
h
′
s
)
.
As
R
T
=
-
R
, this implies
R
i
i
=
0
and
∑
i
,
j
=
1
n
R
i
j
w
i
w
j
=
0
.
Thus, for
y
=
F
(
s
,
w
)
we have
f
γ
(
y
)
=
(
1
+
(
h
′
)
2
)
-
1
2
(
(
s
+
h
h
′
)
∑
i
=
1
n
R
i
,
n
+
1
w
i
)
.
We now look to see which elements of
RMJF
𝒞
lie in either
W
1
,
2
(
𝒞
)
, or
W
ω
𝒞
,
R
1
,
2
(
𝒞
)
.
The following tells us that to check if an element of
JF
𝒞
lies in
W
1
,
2
(
𝒞
)
(resp.
W
ω
𝒞
,
R
1
,
2
(
𝒞
)
), we only need to check if it lies in
L
2
(
𝒞
)
(resp.
L
ω
𝒞
,
R
2
(
𝒞
)
).
As previously discussed, there exists a
C
<
+
∞
such that
|
A
𝒞
|
2
≤
C
ω
𝒞
,
R
.
Now fix any
S
>
R
>
0
, and recall the function
χ
S
from Proposition 9.
Then we have that, for
f
∈
JF
𝒞
∩
L
ω
𝒞
,
R
2
(
𝒞
)
,
∫
χ
S
2
|
∇
f
|
2
=
∫
|
A
|
2
χ
S
2
f
2
-
∫
2
(
χ
S
∇
f
)
⋅
(
f
∇
χ
S
)
≤
C
∫
f
2
ω
𝒞
,
R
+
1
2
∫
χ
S
2
|
∇
f
|
2
+
2
∫
f
2
|
∇
χ
S
|
2
.
Thus,
∫
B
S
n
+
1
(
0
)
∩
𝒞
|
∇
f
|
2
≤
C
∫
f
2
ω
𝒞
,
R
.
As the upper bound is finite and independent of S, we have that
|
∇
f
|
∈
L
2
(
𝒞
)
.
Consider the pull back of the Euclidean metric to
(
-
s
∞
,
s
∞
)
×
S
n
-
1
by F,
g
=
(
1
+
(
h
′
)
2
)
d
s
2
+
h
2
g
S
n
-
1
,
where
g
S
n
-
1
is the standard round metric on
S
n
-
1
.
We have
|
g
|
=
h
n
-
1
1
+
(
h
′
)
2
.
We now compute the
L
2
-norm of our elements of
RMJF
𝒞
.
Starting with the translations (6.2), for
i
=
1
,
…
,
n
,
∫
f
i
2
=
∫
-
s
∞
s
∞
∫
S
n
-
1
f
i
2
|
g
|
𝑑
w
𝑑
s
=
(
∫
-
s
∞
s
∞
h
n
-
1
(
1
+
(
h
′
)
2
)
1
2
𝑑
s
)
(
∫
S
n
-
1
w
i
𝑑
w
)
.
Differentiating (6.1), we have that
(6.4)
h
′
=
(
a
h
2
(
n
-
1
)
-
1
)
1
2
,
which implies that
∫
f
i
2
=
2
s
∞
a
1
2
∫
S
n
-
1
w
i
2
𝑑
w
<
+
∞
.
Thus we have that
f
i
∈
W
1
,
2
(
𝒞
)
⊂
W
ω
𝒞
,
R
1
,
2
(
𝒞
)
.
For
f
n
+
1
, we have that
|
f
n
+
1
|
(
s
)
→
1
as
|
s
|
→
s
∞
.
Thus as
ω
𝒞
,
R
∉
L
1
(
𝒞
)
, this implies that
f
n
+
1
∉
L
ω
𝒞
,
R
2
(
𝒞
)
, and thus
f
n
+
1
∉
W
ω
𝒞
,
R
1
,
2
(
𝒞
)
⊂
W
1
,
2
(
𝒞
)
.
For the Jacobi field
f
d
, generated by scaling, we have
|
f
d
|
=
|
h
-
s
h
′
|
(
1
+
(
h
′
)
2
)
1
2
=
|
h
(
h
′
)
-
1
-
s
|
(
(
h
′
)
-
2
+
1
)
1
2
.
Thus as
|
s
|
→
s
∞
, we have that
|
h
′
|
→
∞
,
h
→
∞
, and recalling (6.4),
(6.5)
h
|
h
′
|
=
h
(
a
h
2
(
n
-
1
)
-
1
)
1
2
→
0
,
which implies that
|
f
d
|
→
s
∞
.
Thus, similar to above,
f
d
does not lie in
W
1
,
2
(
𝒞
)
or
W
ω
𝒞
,
R
1
,
2
(
𝒞
)
.
Finally, we look at elements of
RMJF
𝒞
which are generated by rotations.
Recall from above that all such Jacobi fields are of the form
f
(
s
,
w
)
=
(
1
+
(
h
′
)
2
)
-
1
2
(
(
s
+
h
h
′
)
∑
i
=
1
n
R
i
,
n
+
1
w
i
)
.
If
R
i
,
n
+
1
=
0
for all
i
=
1
,
…
,
n
, we have that
f
=
0
.
Assuming that
R
i
,
n
+
1
≠
0
for some
i
=
1
,
…
,
n
, then as
∑
i
=
1
n
R
i
,
n
+
1
w
i
is a smooth function on
S
n
-
1
, and is non-zero at
w
f
=
(
∑
i
=
1
n
R
i
,
n
+
1
2
)
-
1
(
R
1
,
n
+
1
,
…
,
R
n
,
n
+
1
)
∈
S
n
-
1
,
we may conclude that there exists an
α
>
0
, and set
U
α
⊂
S
n
-
1
of positive
ℋ
n
-
1
measure such that
|
∑
i
=
1
n
R
i
,
n
+
1
w
i
|
≥
α
.
Thus for
w
∈
U
α
, we have that
|
f
|
2
≥
α
2
(
s
+
h
h
′
)
2
1
+
(
h
′
)
2
=
α
2
(
s
(
h
′
)
-
1
+
h
)
2
(
h
′
)
-
2
+
1
,
implying that
|
f
|
2
becomes unbounded as
|
s
|
→
s
∞
, on a set of unbounded measure.
Thus, f does not lie in
W
1
,
2
(
𝒞
)
, or
W
ω
𝒞
,
R
1
,
2
(
𝒞
)
.
Therefore, we have shown that the only non-trivial Jacobi fields on
𝒞
, which are generated through rigid motions and lie in
W
1
,
2
(
𝒞
)
or
W
ω
𝒞
,
R
1
,
2
(
𝒞
)
are those spanned by the translations in the
{
x
n
+
1
=
0
}
hyperplane, i.e. Jacobi fields in
span
(
{
f
1
,
…
,
f
n
}
)
.
This implies that
n
≤
nul
(
𝒞
)
≤
nul
ω
𝒞
,
R
(
𝒞
)
.
We note that, as far as we know, this lower bound on
nul
(
𝒞
)
is larger than what has been previously recorded in the literature (for example, this bound is larger than if one evaluates the inequality [21, Theorem 1.1] for the case of the catenoid)
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 [16], as well as Jonas Hirsch for an interesting and insightful conversation on aspects related to this work, and [19].
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.
Finally, I wish to thank both anonymous referees for their careful reading and comments on the manuscript.
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