Capillary surfaces in a cone

We show that a capillary surface in a solid cone, that is, a surface that has constant mean curvature and for which the surface boundary meets the boundary of the cone at a constant angle, is radially graphical if the mean curvature is non-positive with respect to the Gauss map pointing towards the domain bounded by the surface and the boundary of the cone. In the particular case in which 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 using inversions about spheres.