On the geometry of CMC-hypersurfaces embedded in a manifold of dimension 4 or 5.

Abstract

The intent of this thesis is to survey properties of constant mean curvature (CMC) hypersurfaces embedded in ambient manifold of dimension 4 and 5. We shall prove that certain complete, non-compact CMC-hypersurfaces with bounded second fundamental form that are embedded in homogeneously regular manifold of dimensions n = 4, 5 (bounded injectivity radius from below and bounded sectional curvature) must be properly embedded. We are aiming to generalise a result given by Elbert, Nelli and Rosenberg to show the non-existence of certain complete, almost-stable, open CMC-hypersurfaces immersed in dimensions n = 4, 5. We will also use their original result to prove our theorem, based upon the embedded Calabi-Yau problem, which is inspired by results given by Colding and Minicozzi for minimal surfaces and also Meeks and Tinaglia for CMC-surfaces.

Date of Award	1 Aug 2024
Original language	English
Awarding Institution	King's College London
Supervisor	Giuseppe Tinaglia (Supervisor)