Two regularity results in the theory of minimal hypersurfaces

Abstract

Questions about the existence of minimal submanifolds in compact Riemannian manifolds occupy a prominent place in the theory of geometric variational problems, and have been addressed using a variety of tools coming from topology, partial differential equations, and the calculus of variations.Geometric measure theory methods have proved particularly successful in providing a weak formulation of this variational problem, and proving the existence of weak solutions under very general assumptions. Thanks to geometric measure theory techniques, a variety of regularity results have also been obtained, showing that in many important situations these weak solutions are indeed smooth minimal submanifolds. In this thesis, we will discuss two regularity results with applications to the theory of minimal hypersurfaces. In Part I, which is based on joint work with Douglas Stryker, we develop a sharp regularity theory for minimisers of the prescribed mean curvature functional in isotopy classes of surfaces in 3-manifolds, which enables us to construct prescribed mean curvature embedded spheres in $S^3$. Minimisation over isotopies also plays a central role in min-max constructions of minimal surfaces with prescribed (or controlled) topology. In Part II, which is based on joint work with Martin Li and Davide Parise, we describe a regularity theory for singular limits of solutions of the Allen--Cahn equation with a homogeneous Neumann boundary condition on a compact manifold with boundary. In particular, we show that these limit-interfaces are measure-theoretic free boundary minimal hypersurfaces, opening up the possibility of developing a min-max construction of free boundary minimal hypersurfaces based on the Allen--Cahn equation.