Volume preserving non homogeneous mean curvature flow of convex hypersurfaces

TL;DR

This paper studies convex hypersurfaces evolving under volume or area-preserving flows driven by nonhomogeneous functions of mean curvature, proving convergence to a sphere for a broad class of speeds.

Contribution

It introduces a general framework for volume-preserving nonhomogeneous mean curvature flows and proves convergence to a sphere for convex hypersurfaces.

Findings

Convex hypersurfaces converge to a sphere under the flow.

Monotonicity of the isoperimetric ratio is key to the proof.

Uniform curvature bounds are established.

Abstract

We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed convex hypersurface converges to a round sphere. The proof is based on the monotonicity of the isoperimetric ratio, which allows to control the inner and outer radius of the hypersurface and to deduce uniform bounds on the curvature by maximum principle arguments.