A Mean Curvature Type Flow in Space Forms

Abstract

In this article, we introduce a new type of mean curvature flow (1.3) for bounded star-shaped domains in space forms and prove its longtime existence, exponential convergence without any curvature assumption. Along this flow, the enclosed volume is a constant and the surface area evolves monotonically. Moreover, for a bounded convex domain in |${\mathbb R}^{n+1}$|⁠, the quermassintegrals evolve monotonically along the flow which allows us to prove a class of Alexandrov–Fenchel inequalities of quermassintegrals.