Mean curvature type flow and sharp Micheal-Simon inequalities

Abstract

In this paper, we first investigate a new locally constrained mean curvature flow (1.5) and prove that if the initial hypersurface Σ0 is of smoothly closed starshaped, then the solution Σt of the flow (1.5) exists for all time and converges to a sphere in C∞-topology. Following this flow argument, we obtain a new proof of the celebrated sharp Michael-Simon inequality (1.2) for mean curvatures on smooth, compact, and starshaped hypersurface M0 (possibly with boundary). Specifically, if M0 is closed, we find a necessary and sufficient condition for the equality of (1.2 ′) holds.In the second part of this paper, by exploiting the properties of the inverse mean curvature type flow (1.6) obtained from the expanding geometric flow in [7] by rescaling, we develop and present a new sharp Michael-Simon inequality (1.7) for the k-th mean curvatures. When M0 is closed, starshaped, (k−1)-convex and g is of constant, the inequality (1.7) is equivalent to the Alexandrov-Fenchel type inequality (1.8).