Free boundary and capillary minimal surfaces in spherical caps II: Low energy

This is the second of two articles in which we investigate the geometry of free boundary and capillary minimal surfaces in balls $B_R\subset {\mathbb{S}}^3$. In this article, we find monotonicity formulae which imply that capillary minimal surfaces maximise a certain modified energy in their conformal orbit (preserving $B_R$). In the hemisphere, this energy is precisely the capillary energy. We also prove a partial characterisation by index for capillary minimal surfaces in the hemisphere, analogous to Urbano’s characterisation of the Clifford torus.