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

TL;DR

This paper studies a volume-preserving geometric flow of convex hypersurfaces driven by powers of the k-th mean curvature, proving long-time existence and smooth convergence to a sphere without curvature pinching conditions.

Contribution

It introduces a new volume-preserving flow based on powers of the k-th mean curvature and proves its long-time smooth convergence to a sphere for convex initial hypersurfaces.

Findings

Flow exists for all time for convex initial hypersurfaces.

Flow converges smoothly to a round sphere.

No curvature pinching assumptions are needed.

Abstract

We consider the flow of closed convex hypersurfaces in Euclidean space $\mathbb{R}^{n+1}$ 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 $V_{n+1}$ and the mixed volume $V_{n+1-k}$ 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.