Uniqueness of stable capillary hypersurfaces in a ball

Abstract

In this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical. Our result also provides a proof of a conjecture proposed by Sternberg and Zumbrun (J Reine Angew Math 503:63–85, 1998). We also prove a Heintze–Karcher–Ros type inequality for hypersurfaces with free boundary 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.