Mean curvature rigidity of horospheres, hyperspheres and hyperplanes Rabah Souam

TL;DR

This paper proves that certain geometric surfaces in hyperbolic space cannot be locally deformed to increase their mean curvature, extending a known Euclidean result to hyperbolic geometry.

Contribution

It extends Gromov's Euclidean mean curvature rigidity result to hyperbolic spaces for horospheres, hyperspheres, and hyperplanes.

Findings

Horospheres, hyperspheres, and hyperplanes in hyperbolic space admit no compactly supported perturbations increasing mean curvature.

The result generalizes Euclidean mean curvature rigidity to hyperbolic geometry.

No local deformation can increase the mean curvature of these surfaces in hyperbolic space.

Abstract

We prove that horospheres, hyperspheres and hyperplanes in a hyperbolic space H n , n $\ge$ 3, admit no perturbations with compact support which increase their mean curvature. is is an extension of the analogous result in the Euclidean spaces, due to M. Gromov, which states that a hyperplane in a Euclidean space R n admits no compactly supported perturbations having mean curvature $\ge$ 0.