Uniqueness of free-boundary minimal hypersurfaces in rotational domains

TL;DR

This paper studies the existence and properties of free-boundary minimal hypersurfaces in rotational domains, introducing new integral identities and gap theorems that advance understanding in geometric analysis.

Contribution

It introduces an original integral identity for free-boundary minimal hypersurfaces and establishes a new gap theorem in Euclidean and rotational ellipsoid domains.

Findings

Existence results for free-boundary minimal hypersurfaces in rotational domains

New integral identity applicable to hypersurfaces with boundary as a regular level set

A gap theorem for hypersurfaces in Euclidean ball and rotational ellipsoid

Abstract

In this work, we investigate the existence of compact free-boundary minimal hypersurfaces immersed in several domains. Using an original integral identity for compact free-boundary minimal hypersurfaces that are immersed in a domain whose boundary is a regular level set, we study the case where this domain is a quadric or, more generally, a rotational domain. This existence study is done without topological restrictions. We also obtain a new gap theorem for free boundary hypersurfaces immersed in an Euclidean ball and in a rotational ellipsoid.