Min-max minimal hypersurfaces in non-compact manifolds

TL;DR

This paper develops a modified min-max theory to prove the existence of embedded closed minimal hypersurfaces in certain non-compact manifolds with specific boundary and geometric conditions.

Contribution

It introduces a new min-max approach tailored for non-compact manifolds to produce minimal hypersurfaces with intersecting properties.

Findings

Existence of embedded closed minimal hypersurfaces in specified non-compact manifolds.

Development of a modified min-max theory for the area functional.

Application to manifolds with mean-concave boundary and controlled geometry at infinity.

Abstract

In this work we prove the existence of embedded closed minimal hypersurfaces in non-compact manifolds containing a bounded open subset with smooth and strictly mean-concave boundary and a natural behavior on the geometry at infinity. For doing this, we develop a modified min-max theory for the area functional following Almgren and Pitts' setting, to produce minimal surfaces with intersecting properties.