Uniqueness of stable capillary hypersurfaces in a ball
Guofang Wang, Chao Xia
TL;DR
This paper proves that all immersed stable capillary hypersurfaces in a ball are totally umbilical, solving a long-standing open problem using a new Minkowski type formula and establishing related geometric inequalities.
Contribution
It introduces a new Minkowski type formula and proves the uniqueness of stable capillary hypersurfaces in a ball, resolving a major open problem in differential geometry.
Findings
All immersed stable capillary hypersurfaces in a ball are totally umbilical.
Established a Heintze-Karcher-Ros type inequality for hypersurfaces in a ball.
Provided a new proof of Alexandrov's Theorem for embedded CMC hypersurfaces with free boundary.
Abstract
In this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical. This solves completely a long-standing open problem. In the proof one of crucial ingredients is a new Minkowski type formula. We also prove a Heintze-Karcher-Ros type inequality for hypersurfaces in a ball, which, together with the new Minkowski formula, yields a new proof of Alexandrov's Theorem for embedded CMC hypersurfaces in a ball with free boundary.
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