The halfspace theorem for minimal hypersurfaces in regions bounded by minimal cones

TL;DR

This paper proves that minimal and r-minimal hypersurfaces cannot be properly immersed in regions bounded by unstable minimal cones in Euclidean space, extending the understanding of geometric barriers.

Contribution

It establishes the non-existence of properly immersed minimal hypersurfaces in regions bounded by unstable minimal cones, including the case of r-minimal hypersurfaces.

Findings

No properly immersed minimal hypersurfaces in regions bounded by unstable minimal cones.

Extension of the non-existence result to r-minimal hypersurfaces.

Provides a geometric barrier result for minimal hypersurfaces.

Abstract

We prove that there are no minimal hypersurfaces properly immersed in any region of the Euclidean space bounded by unstable minimal cones. We also prove the analogous result for $r$-minimal hypersurfaces.