Singular Structures and Generic Regularity for Minimal Hypersurfaces

Abstract

Regularity theory of minimal hypersurfaces has long been studied since last century. A satisfactory regularity result for area-minimizing and stable minimal hypersurfaces in a manifold of dimension $\leq 7$ has been established over 40 years ago. In higher dimensional spaces, area-minimizing or stable minimal hypersurfaces are known to potentially have singular sets, and the wild behavior of these singularities obstructs us from further exploration of global geometric behaviors of minimal hypersurfaces as well as applications in other fields. In this thesis, we will discuss how singular sets affect the global behavior of minimal hypersurfaces. First, we generalize a Theorem of Hardt-Simon \cite{HardtSimon85} to approximate any minimizing (resp. viscosity mean convex) hypercone by complete smooth minimizing (resp. strictly mean convex) hypersurfaces. This suggests that singularity models for minimizing hypersurfaces may all be smoothable. Second, we develop the spectral analysis of the Jacobi operator on a closed locally stable minimal hypersurface with strongly isolated singularities. This in particular applies to all min-max minimal hypersurfaces in an eight manifold. We include primary application of this linear theory to obtain a local exitence and a local minimizing property of minimal hypersurface under reasonable spectral assumptions. Finally, in an ambient manifold of dimension $8$, by further exploiting the linear analysis of the Jacobi operators, joint with Yangyang Li, we prove that under generic metric, every minimal hypersurface constructed by min-max approach are entirely smooth.