Contracting convex hypersurfaces by functions of the mean curvature

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.