Upper Semicontinuity of Index Plus Nullity for Minimal and CMC Hypersurfaces
Abstract
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--Rivi\`{e}re. 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.
Keywords
Cite
@article{arxiv.2312.09227,
title = {Upper Semicontinuity of Index Plus Nullity for Minimal and CMC Hypersurfaces},
author = {Myles Workman},
journal= {arXiv preprint arXiv:2312.09227},
year = {2024}
}
Comments
Revised document, in which we have: corrected minor typos, added extra references, shortened preliminary section on bubble convergence, extended various statements on minimal hypersurfaces in Euclidean space to higher dimensions (under appropriate assumptions on their total curvature), and added a final section on Jacobi fields on the higher dimensional catenoid