Flow of pinched convex hypersurfaces by powers of curvature functions in hyperbolic space

Abstract

This paper concerns closed, h-convex hypersurfaces of dimension n≥2 in the hyperbolic space Hκn+1 of constant sectional curvature κ evolving in direction of its normal vector, where the speed equals a power β>1 of a curvature function F, which is monotone, symmetric, homogeneous of degree 1. It is shown that if the initial h-convex hypersurface is pinched, then this is maintained under the flow, and the hypersurfaces shrink to a round point in Hκn+1 in finite time. As a consequence, when rescaling appropriately, the evolving hypersurfaces converge smoothly and exponentially to the unit sphere of Rn+1.