A Minkowski-type inequality for capillary hypersurfaces in a half-space

TL;DR

This paper introduces a flow method for capillary hypersurfaces in a half-space, proving convergence to spherical caps and deriving a new Minkowski-type inequality applicable to various contact angles.

Contribution

It establishes a global existence and convergence result for an inverse mean curvature flow for capillary hypersurfaces, leading to a novel Minkowski-type inequality.

Findings

Flow converges smoothly to spherical caps

New Minkowski-type inequality derived for star-shaped, mean convex hypersurfaces

Applicable for all contact angles in (0, π)

Abstract

In this article, we investigate a flow of inverse mean curvature type for capillary hypersurfaces in the half-space. We establish the global existence of solutions for this flow and demonstrate that it converges smoothly to a spherical cap as time tends to infinity. As a result, we derive a new Minkowski-type inequality for star-shaped and mean convex capillary hypersurfaces for the whole range of contact angle $\theta\in (0,\pi)$.