Contracting convex hypersurfaces by functions of the mean curvature

TL;DR

This paper studies the evolution of convex hypersurfaces in Euclidean space under curvature-dependent flows, showing they shrink to a point while preserving convexity, generalizing previous results to broader classes of curvature functions.

Contribution

It extends existing results by analyzing convex hypersurface contraction under more general mean curvature functions without homogeneity assumptions.

Findings

Flow exists on a finite maximal interval

Convexity is preserved during evolution

Hypersurfaces shrink to a point as time approaches final

Abstract

This paper concerns the evolution of a closed convex hypersurface in ${\mathbb{R}}^{n+1}$, in direction of its inner unit normal vector, where the speed is given by a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow exists on a finite maximal interval, convexity is preserved and the hypersurfaces shrink down to a single point as the final time is approached. This result covers and generalises the corresponding result of Schulze \cite{Sch05} for the positive power mean curvature flow to a much larger possible class of flows by the functions depending only on the mean curvature.