Workman, Myles;
(2024)
Minimal and CMC hypersurfaces of classical or diffused type: convergence properties under Morse index bounds.
Doctoral thesis (Ph.D), UCL (University College London).
Preview |
Text
Myles Workman - PhD Thesis.pdf - Other Download (843kB) | Preview |
Abstract
In this thesis we investigate minimal and constant mean curvature (CMC) hypersurfaces as they arise (and converge), in two limiting procedures, of distinct type, and under a certain control for the (Morse) index. In the first part of this thesis, which is joint work with C. Bellettini, we investigate CMC hypersurfaces which arise as the limit interface of sequences of particular (minmax, hence with index at most 1) solutions to the inhomogeneous Allen–Cahn equation. We prove that on a compact Riemannian manifold of dimension 3 or higher, with positive Ricci curvature, the Allen–Cahn min-max scheme of C. Bellettini and N. Wickramasekera, with prescribing function taken to be a non-zero constant lambda, produces an embedded hypersurface of constant mean curvature lambda. More precisely, we prove that the limit interface arising from said min-max contains no even-multiplicity minimal hypersurface and no quasi-embedded points (both of these occurrences are in principle possible in the conclusions of the aforementioned work by the C. Bellettini and N. Wickramasekera). In the second part of this thesis, we investigate sequences of bubble converging minimal hypersurfaces, or CMC hypersurfaces, in compact Riemannian manifolds without boundary, of dimension 4, 5, 6 or 7, and prove upper semi-continuity of index plus nullity, for such bubble converging sequences. This complements the previously known lower semi-continuity results for the index. The strategy of the proof is to analyse an appropriate weighted eigenvalue problem along the bubble converging sequence of hypersurfaces.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Minimal and CMC hypersurfaces of classical or diffused type: convergence properties under Morse index bounds |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2024. Original content in this thesis is licensed under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) Licence (https://creativecommons.org/licenses/by/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10195524 |
Archive Staff Only
 |
View Item |
