Uniqueness of free boundary minimal hypersurfaces in rotational domains

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.