Mean curvature type flow and sharp Micheal-Simon inequalities

TL;DR

This paper introduces new mean curvature flows to prove sharp Michael-Simon inequalities and characterizes conditions for equality, demonstrating convergence to spheres for starshaped and convex hypersurfaces.

Contribution

It presents novel curvature flow methods to establish sharp inequalities and provides conditions for equality, extending the understanding of geometric inequalities in Euclidean space.

Findings

Flow (1.5) exists globally and converges to a sphere for starshaped hypersurfaces.

Flow (1.7) preserves convexity and converges exponentially to a sphere.

New sharp Michael-Simon inequalities for mean and kth mean curvature are derived.

Abstract

In this paper, we first investigate a new locally constrained mean curvature flow (1.5) and prove that if the initial hypersurface is of smoothly compact starshaped, then the solution of the flow (1.5) exists for all time and converges to a sphere in smooth topology. Following this flow argument, not only do we achieve a new proof of the celebrated sharp Michael-Simon inequality for mean curvature in (n+1) dimensional Euclidean space, but we also get the necessary and sufficient condition for the establishment of the equality. In the second part of this paper, we study a mean curvature type flow (1.7) of static convex hypersurfaces in (n+1) dimensional Euclidean space, and prove that the flow (1.7) has a unique smooth solution for all time t>0, and the static convexity of the hypersurface is preserved along the flow (1.7). Moreover, The solution of the flow (1.7) converges exponentially to a sphere of radius R in smooth topology as time tends to infinity. By exploiting the properties of this flow, we develop and present a new sharp Michael-Simon inequality for kth mean curvature.