A Heintze-Karcher type inequality for hypersurfaces with capillary boundary

TL;DR

This paper proves a new geometric inequality for hypersurfaces with capillary boundary conditions and uses it to establish a uniqueness theorem for constant mean curvature hypersurfaces with such boundaries.

Contribution

It introduces a Heintze-Karcher type inequality for hypersurfaces with capillary boundary and provides a new proof of Alexandrov's theorem in this setting.

Findings

Established a Heintze-Karcher inequality for capillary hypersurfaces.

Provided a new proof of Alexandrov's theorem for these hypersurfaces.

Extended geometric inequalities to hypersurfaces with contact angle in a half space or ball.

Abstract

In this paper, we establish a Heintze-Karcher type inequality for hypersurfaces with capillary boundary of contact angle $\theta\in (0,\frac{\pi}{2})$ in a half space or a half ball, by using solution to a mixed boundary value problem in Reilly type formula. Consequently, we give a new proof of Alexandrov type theorem for embedded capillary constant mean curvature hypersurfaces with contact angle $\theta\in (0,\frac{\pi}{2})$ in a half space or a half ball.