Curvature flow in hyperbolic spaces

Abstract

Abstract We study the evolution of compact convex hypersurfaces in hyperbolic space ℍ n + 1 {\mathbb{H}^{n+1}} , with normal speed governed by the curvature. We concentrate mostly on the case of surfaces, and show that under a large class of natural flows, any compact initial surface with Gauss curvature greater than 1 produces a solution which converges to a point in finite time, and becomes spherical as the final time is approached. We also consider the higher-dimensional case, and show that under the mean curvature flow a similar result holds if the initial hypersurface is compact with positive Ricci curvature.