Some dynamic properties of volume preserving curvature driven flows

Abstract

This paper is concerned with the following three types of geometric evolution equations: the volume preserving mean curvature flow, the intermediate surface diffusion flow, and the surface diffusion flow. Important common properties of these flows are the preservation of volume and the decrease of perimeter. It is shown in this paper that the intermediate surface diffusion flow can lose convexity. Hence the volume preserving mean curvature flow is the only flow among the evolution equations under consideration which preserves convexity, cf. [11, 16, 14, 17]. Moreover, several sufficient conditions are presented, which illustrate that each of the above mentioned flows can move smooth initial configurations into singularities in finite time.