On the mean curvature type flow for convex capillary hypersurfaces in the ball

TL;DR

This paper investigates a mean curvature flow for convex hypersurfaces with capillary boundary in a ball, proving long-term convergence to spherical caps and establishing new geometric inequalities.

Contribution

It introduces and analyzes a curvature flow for convex hypersurfaces with capillary boundary, proving convergence and deriving new Alexandrov-Fenchel inequalities.

Findings

Flow preserves convexity of hypersurfaces.

Solutions exist globally and converge smoothly to spherical caps.

New Alexandrov-Fenchel inequalities are established.

Abstract

In this paper, we study the mean curvature type flow for hypersurfaces in the unit Euclidean ball with capillary boundary, which was introduced by Wang-Xia and Wang-Weng. We show that if the initial hypersurface is strictly convex, then the solution of this flow is strictly convex for $t>0$, exists for all positive time and converges smoothly to a spherical cap. As an application, we prove a family of new Alexandrov-Fenchel inequalities for convex hypersurfaces in the unit Euclidean ball with capillary boundary.