Marshall-Stevens, Kobe; (2024) On the generic regularity of constant mean curvature hypersurfaces. Doctoral thesis (Ph.D), UCL (University College London).

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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.

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