Volume preserving flow by powers of κ-TH mean curvature

Abstract

We consider the flow of closed convex hypersurfaces in Euclidean space R with speed given by a power of the k-th mean curvature E plus a global term chosen to impose a constraint involving the enclosed volume V and the mixed volume V of the evolving hypersurface. We prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges to a round sphere smoothly. No curvature pinching assumption is required on the initial hypersurface. Show more