Constant mean curvature surfaces with boundary on a sphere

TL;DR

This paper investigates the geometric conditions under which a compact constant mean curvature surface with boundary on a sphere must be spherical, relevant for physical models like liquid drops on spherical surfaces.

Contribution

It characterizes when such surfaces are necessarily spherical based on boundary conditions and contact angles, extending understanding of capillary surfaces.

Findings

Surfaces with boundary on a sphere are spherical under certain geometric conditions.

Results apply to physical models of liquid drops on spherical surfaces.

Provides criteria for when constant mean curvature surfaces are spherical.

Abstract

In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere with a constant angle. We study under what geometric conditions the surface must be spherical. Our results apply in many scenarios in physics where in absence of gravity a liquid drop is deposited on a round solid ball and the air-liquid interface is a critical point for area under all variations that preserve the enclosed volume.