Multiplicity one for min-max theory in compact manifolds with boundary and its applications

TL;DR

This paper proves a multiplicity one theorem for free boundary minimal hypersurfaces in certain manifolds, develops new existence and regularity theories, and constructs new minimal hypersurfaces and self-shrinkers with large entropy.

Contribution

It introduces a generic min-max theory with Morse index bounds and develops regularity and compactness results for free boundary hypersurfaces with prescribed mean curvature.

Findings

Proves multiplicity one for free boundary minimal hypersurfaces in generic metrics.

Develops existence and regularity theory for hypersurfaces with prescribed mean curvature.

Constructs new minimal hypersurfaces and self-shrinkers with large entropy.

Abstract

We prove the multiplicity one theorem for min-max free boundary minimal hypersurfaces in compact manifolds with boundary of dimension between 3 and 7 for generic metrics. To approach this, we develop existence and regularity theory for free boundary hypersurface with prescribed mean curvature, which includes the regularity theory for minimizers, compactness theory, and a generic min-max theory with Morse index bounds. As applications, we construct new free boundary minimal hypersurfaces in the unit balls in Euclidean spaces and self-shrinkers of the mean curvature flows with arbitrarily large entropy.