Min-max theory for free boundary minimal hypersurfaces I - regularity theory

TL;DR

This paper completes the min-max theory for free boundary minimal hypersurfaces in compact manifolds with boundary, proving existence and multiplicity results for such hypersurfaces using advanced geometric analysis techniques.

Contribution

It extends the min-max theory to manifolds with boundary, establishing the existence of smooth embedded free boundary minimal hypersurfaces and their infinite multiplicity under certain curvature conditions.

Findings

Existence of smooth embedded free boundary minimal hypersurfaces in Euclidean domains.

Infinitely many such hypersurfaces in manifolds with nonnegative Ricci curvature.

Application of min-max methods to manifolds with boundary.

Abstract

In 1960s, Almgren initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts and Schoen-Simon in 1980s when the manifold has no boundary. In this paper, we finish this program for general compact manifold with nonempty boundary. As a result, we prove the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. An application of our general existence result combined with the work of Marques and Neves shows that for any compact Riemannian manifolds with nonnegative Ricci curvature and convex boundary, there exist infinitely many embedded minimal hypersurfaces with free boundary which are properly embedded.