2 research outputs found
Uniqueness of stable capillary hypersurfaces in a ball
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.Comment: Final version, Math. Ann., to appea
Low index capillary minimal surfaces in Riemannian -manifolds
We prove a local rigidity result for infinitesimally rigid capillary surfaces
in some Riemannian -manifolds with mean convex boundary. We also derive
bounds on the genus, number of boundary components and area of any compact
two-sided capillary minimal surface with low index under certain assumptions on
the curvature of the ambient manifold and of its boundary.Comment: 16 pages, 2 figure