Upper Semicontinuity of Index Plus Nullity for Minimal and CMC Hypersurfaces
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913434846625792 |
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| author | Workman, Myles |
| author_facet | Workman, Myles |
| contents | We consider a sequence of bubble converging minimal hypersurfaces, or H-CMC hypersurfaces, in compact Riemannian manifolds without boundary, of dimension 4, 5, 6 or 7, and prove upper semicontinuity of index plus nullity, for such a bubble converging sequence. This complements the previously known lower semicontinuity of index obtained by Buzano--Sharp, and Bourni--Sharp--Tinaglia.
The strategy of our proof is to analyse a weighted eigenvalue problem along our sequence of hypersurfaces. This strategy is inspired by the recent work of Da Lio--Gianocca--Rivière. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_09227 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Upper Semicontinuity of Index Plus Nullity for Minimal and CMC Hypersurfaces Workman, Myles Differential Geometry Analysis of PDEs We consider a sequence of bubble converging minimal hypersurfaces, or H-CMC hypersurfaces, in compact Riemannian manifolds without boundary, of dimension 4, 5, 6 or 7, and prove upper semicontinuity of index plus nullity, for such a bubble converging sequence. This complements the previously known lower semicontinuity of index obtained by Buzano--Sharp, and Bourni--Sharp--Tinaglia. The strategy of our proof is to analyse a weighted eigenvalue problem along our sequence of hypersurfaces. This strategy is inspired by the recent work of Da Lio--Gianocca--Rivière. 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. |
| title | Upper Semicontinuity of Index Plus Nullity for Minimal and CMC Hypersurfaces |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2312.09227 |