VOLUME PRESERVING FLOW BY POWERS OF k-TH MEAN CURVATURE

We consider the flow of closed convex hypersurfaces in Euclidean space Rn+1 with speed given by a power of the k-th mean curvature E-k plus a global term chosen to impose a constraint involving the enclosed volume Vn+1 and the mixed volume Vn+1-k 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.