Volume preserving non homogeneous mean curvature flow in hyperbolic space

TL;DR

This paper investigates a volume and area preserving curvature flow of convex hypersurfaces in hyperbolic space, proving exponential convergence to geodesic spheres under a broad class of speed functions.

Contribution

It introduces a new class of curvature flows with non-homogeneous speeds in hyperbolic space and proves their exponential convergence to geodesic spheres.

Findings

Flow preserves convexity by horospheres.

Hypersurfaces converge exponentially to geodesic spheres.

Uniform bounds on curvature are established.

Abstract

We study a volume/area preserving curvature flow of hypersurfaces that are convex by horospheres in the hyperbolic space, with velocity given by a generic positive, increasing function of the mean curvature, not necessarly homogeneous. For this class of speeds we prove the exponential convergence to a geodesic sphere. The proof is ispired by [10] and is based on the preserving of the convexity by horospheres that allows to bound the inner and outer radii and to give uniform bounds on the curvature by maximum principle arguments. In order to deduce the exponential trend, we study the behaviour of a suitable ratio associated to the hypersurface that converges exponentially in time to the value associated to a geodesic sphere.