Flow by powers of the Gauss curvature

Abstract

We prove that convex hypersurfaces in Rⁿ⁺¹ contracting under the flow by any power α > 1/n+2 source 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.