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

TL;DR

This paper develops monotonicity formulas for capillary minimal surfaces in spherical caps, showing they maximize a modified energy and providing a partial index-based characterization in the hemisphere, advancing understanding of their geometric properties.

Contribution

It introduces new monotonicity formulas and characterizes capillary minimal surfaces in spherical caps, extending Urbano's index results to this setting.

Findings

Capillary minimal surfaces maximize a modified energy in their conformal orbit.

Monotonicity formulas are established for these surfaces.

A partial index-based characterization of capillary minimal surfaces in the hemisphere is provided.

Abstract

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.