A Mountain-pass Theorem in Hyperbolic Space and its Application

TL;DR

This paper develops a min-max theory for minimal hypersurfaces in hyperbolic space, demonstrating the existence of new hypersurfaces between stable ones and classifying them under entropy conditions.

Contribution

It introduces a novel min-max approach in hyperbolic space and establishes isotopy results for minimal hypersurfaces with low entropy.

Findings

Existence of a new minimal hypersurface between two stable ones

All minimal hypersurfaces with the same asymptotic boundary are isotopic under low entropy

Development of a min-max theory tailored for hyperbolic space

Abstract

We develop a min-max theory for certain complete minimal hypersurfaces in hyperbolic space. In particular, we show that given two strictly stable minimal hypersurfaces that are both asymptotic to the same ideal boundary, there is a new one trapped between the two. As an application, we show that under a low entropy condition, all the minimal hypersurfaces asymptotic to the same ideal boundary are isotopic.