  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.
