Capillary surfaces in a cone

TL;DR

This paper investigates the geometric properties of capillary surfaces within a cone, demonstrating conditions under which such surfaces are spherical caps, planar discs, or radially graphical, using advanced reflection methods.

Contribution

It extends the Alexandrov reflection method with sphere inversions to analyze capillary surfaces in cones, providing new classification results.

Findings

Capillary surfaces with non-positive mean curvature are radially graphical.

In circular cones, such surfaces are spherical caps or planar discs.

The method involves an innovative extension of the Alexandrov reflection technique.

Abstract

We show that a capillary surface in a solid cone, that is, a surface that has constant mean curvature and the boundary of surface meets the boundary of the cone with a constant angle, is radially graphical if the mean curvature is non-positive with respect to the Gauss map pointing toward the domain bounded by the surface and the boundary of the cone. In the particular case that the cone is circular, we prove that the surface is a spherical cap or a planar disc. The proofs are based on an extension of the Alexandrov reflection method by using inversions about spheres.