Volume-preserving flow by powers of the m-th mean curvature

TL;DR

This paper studies a volume-preserving curvature flow for convex hypersurfaces, showing that under certain conditions, the shape evolves smoothly and converges to a perfect sphere, extending previous results to powers of mean and Gauss curvature.

Contribution

It introduces a new curvature flow involving powers of the m-th mean curvature with volume preservation, proving long-term existence and convergence to a sphere.

Findings

Flow exists for all time under pinching conditions

Hypersurfaces converge to round spheres

Results include powers of mean and Gauss curvature

Abstract

We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the m-th mean curvature plus a volume preserving term, including the case of powers of the mean curvature or of the Gauss curvature. We prove that if the initial hypersurface satisfies a suitable pinching condition, the solution exists for all times and converges to a round sphere.