Mean curvature rigidity of horospheres, hyperspheres and hyperplanes Rabah Souam

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.