DOI: 10.1515/acv-2024-0082 ISSN: 1864-8258
Upper semicontinuity of index plus nullity for minimal and CMC hypersurfaces
Myles Workman Abstract
We consider a sequence,
{
M
k
}
k
∈
ℕ
{\{M_{k}\}_{k\in\mathbb{N}}}
, of bubble converging minimal hypersurfaces, or H-CMC hypersurfaces, in compact Riemannian manifolds without boundary, of dimension
4
≤
n
+
1
≤
7
{4\leq n+1\leq 7}
.
We prove the following upper semicontinuity of index plus nullity:
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
)
\limsup_{k\rightarrow\infty}(\operatorname{ind}(M_{k})+\operatorname{nul}(M_{k%
}))\leq\sum_{i=1}^{l}\operatorname{co}(m)_{i}(\operatorname{anl-ind}(%
\operatorname{co}(M^{i}_{\infty}))+\operatorname{anl-nul}(\operatorname{co}(M^%
{i}_{\infty})))+\sum_{j=1}^{J}\operatorname{ind}(\Sigma^{j})+\operatorname{nul%
}_{\omega_{\Sigma^{j},R}}(\Sigma^{j})
for such a bubble converging sequence
M
k
→
(
⋃
i
=
1
l
M
∞
i
,
Σ
1
,
…
,
Σ
J
)
{M_{k}\rightarrow(\bigcup_{i=1}^{l}M_{\infty}^{i},\Sigma^{1},\ldots,\Sigma^{J})}
, where
co
(
m
)
i
∈
ℤ
≥
1
{\operatorname{co}(m)_{i}\in\mathbb{Z}_{\geq 1}}
is a notion of multiplicity of the convergence to the connected component
M
∞
i
{M_{\infty}^{i}}
, and
Σ
1
,
…
,
Σ
J
{\Sigma^{1},\ldots,\Sigma^{J}}
are the bubbles.
This complements the previously known lower semicontinuity of index obtained in [R. Buzano and B. Sharp,
Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area,
Trans. Amer. Math. Soc. 370 2018, 6, 4373–4399] and [T. Bourni, B. Sharp and G. Tinaglia,
CMC hypersurfaces with bounded Morse index,
J. Reine Angew. Math. 786 2022, 175–203].
The strategy of our proof is to analyse a weighted eigenvalue problem along our sequence of degenerating hypersurfaces,
{
M
k
}
k
∈
ℕ
{\{M_{k}\}_{k\in\mathbb{N}}}
. This strategy is inspired by the recent work [F. Da Lio, M. Gianocca and T. Rivière,
Morse index stability for critical points to conformally invariant Lagrangians,
preprint 2022, https://arxiv.org/abs/2212.03124].
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
Σ
1
,
…
,
Σ
J
{\Sigma^{1},\ldots,\Sigma^{J}}
.