Volume preserving flow by powers of $k$-th mean curvature

Abstract

We consider the flow of closed convex hypersurfaces in Euclidean space with the speed given by any positive power of the $k$-th mean curvature plus a global term such that the volume of the domain enclosed by the flow hypersurface remains constant. We prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges smoothly to a round sphere as the time goes to infinity. No curvature pinching assumption is required on the initial hypersurface. The same conclusion is also true for the flow preserving a general non-degenerate increasing function of the volume and the mixed volume $V_{n+1-k}$. This is joint work with Ben Andrews.