eprintid: 10194357
rev_number: 14
eprint_status: archive
userid: 699
dir: disk0/10/19/43/57
datestamp: 2024-10-18 10:12:30
lastmod: 2024-10-18 10:12:30
status_changed: 2024-10-18 10:12:30
type: thesis
metadata_visibility: show
sword_depositor: 699
creators_name: Marshall-Stevens, Kobe
title: On the generic regularity of constant mean curvature hypersurfaces
ispublished: unpub
divisions: UCL
divisions: B04
divisions: C06
divisions: F59
note: Copyright © The Author 2024.  Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/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.
abstract: This thesis concerns the regularity properties of constant mean curvature hypersurfaces. These hypersurfaces arise naturally as boundaries to isoperimetric regions (regions with least boundary area for a fixed enclosed volume) and more generally as critical points of an area-type functional. Historically, constant (and other prescribed) mean curvature hypersurfaces have served as effective tools in developing our understand- ing of the interaction between the curvature and topology of Riemannian manifolds.

Constant mean curvature hypersurfaces may be produced by various minimisation and min-max procedures. Sharp regularity theory guarantees that, in both cases, the hypersurface produced will be smoothly immersed away from a closed singular set of codimension seven. In particular, when the ambient manifold is of dimension eight, one produces a constant mean curvature hypersurface which is smoothly immersed away from finitely many isolated singular points.

The presence of a singular set in high dimensional hypersurfaces of constant mean curvature means that, in general, they may fail to be an effective tool for geometric and topological application. One method to deal with the presence of a singular set is to show that generically one can remove it, resulting in an entirely smooth constant mean curvature hypersurface suitable for effective application. This generic regularity approach requires a finer understanding of the singularities that arise as well as the development of perturbation procedures to remove them.

This thesis utilises and develops techniques in the calculus of variations, elliptic partial differential equations and geometric measure theory in order to produce the first generic regularity results for the general class of constant mean curvature hypersurfaces when the ambient manifold is of dimension eight. The first part of this thesis establishes relevant background while the second part obtains generic regularity results under the assumption of positive Ricci curvature.
date: 2024-07-28
date_type: published
full_text_type: other
thesis_class: doctoral_embargoed
thesis_award: Ph.D
language: eng
verified: verified_manual
elements_id: 2293147
lyricists_name: Marshall-Stevens, Kobe
lyricists_id: KMARS54
actors_name: Marshall-Stevens, Kobe
actors_id: KMARS54
actors_role: owner
full_text_status: restricted
pages: 151
institution: UCL (University College London)
department: Mathematics
thesis_type: Doctoral
citation:        Marshall-Stevens, Kobe;      (2024)    On the generic regularity of constant mean curvature hypersurfaces.                   Doctoral thesis  (Ph.D), UCL (University College London).    
 
document_url: https://discovery.ucl.ac.uk/id/eprint/10194357/1/Thesis.pdf