  We consider a capillary drop that contacts several planar bounding walls so
as to produce singularities (vertices) in the boundary of its free surface. It
is shown under various conditions that when the number of vertices is less than
or equal to three, then the free surface must be a portion of a sphere. These
results extend the classical theorem of H. Hopf on constant mean curvature
immersions of the sphere. The conclusion of sphericity cannot be extended to
more than three vertices, as we show by examples.
