Flow by powers of the Gauss curvature

Abstract

We prove that convex hypersurfaces in Rn+1 contracting under the flow by any power α>1n+2 of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the initial body we prove that the limit is the round sphere for α≥1.