Geometric Variational Problems for Mean Curvature

Abstract

This thesis investigates variational problems related to the concept of mean curvature on submanifolds. Our primary focus is on the area functional, whose critical points are the minimal submanifolds and whose gradient flow is the mean curvature flow. We study these geometric objects from the perspectives of existence, regularity and rigidity; their variational stability plays an essential role for each of these perspectives. As direct minimisation tends to produce stable critical points, the natural method to construct unstable critical points becomes the min-max procedure, first developed for the area functional by Almgren-Pitts. We extend the min-max theory to produce constrained critical points of the area functional - hypersurfaces with prescribed constant mean curvature. A key concern is the regularity of the resulting hypersurface, which derives from local stability properties. To study the mean curvature flow, for which singular behaviour is in fact inevitable, we study self-shrinking solitons, which model any singularities of the flow and are themselves minimal submanifolds with respect to a Gaussian metric. We study the dynamical stability of these self-shrinkers, particularly in the case that the soliton is itself singular, and in relation to the Colding-Minicozzi entropy functional for the flow in Euclidean space. We use similar techniques to analyse the principal eigenvalue of certain geometric operators on potentially singular minimal hypersurfaces. Furthermore, we prove several strong rigidity theorems for self-shrinkers assuming certain almost-stability conditions. Finally, we describe two monotonicity results: The first is a `moving-centre' monotonicity formula for the area of minimal submanifolds; the second is a new entropy functional for mean curvature flow in the spherical space form. Both yield a priori estimates for the area of minimal submanifolds.