Index Estimates for CMC and Minimal Surfaces with Capillary Boundary

TL;DR

This paper establishes bounds on the index of constant mean curvature surfaces with capillary boundary conditions, linking it to topological and energetic properties, and provides comprehensive second variation formulas.

Contribution

It introduces new bounds on the index of CMC surfaces with capillary boundary based on topological and energetic factors, and derives detailed second variation formulas.

Findings

Index is bounded linearly by genus, boundary components, and branching order.

Derived second variation formulas for area, volume, and wetting functionals.

Provided auxiliary theorems comparing second variations at branched conformal maps.

Abstract

We prove that the index of a CMC surface with capillary boundary is bounded from above linearly by its genus, number of boundary components, and branching order, and also by some Willmore-type energy involving the area, mean curvature, contact angle, and ambient curvature. The main auxiliary theorem of more general interest is a comparison of the second variations of area and energy at a branched conformal map with boundary. In the appendix we derive the various second variation formulae for area, enclosed-volume, and wetting functionals away from critical points and for non-admissible variations, the purpose of which is to rather comprehensively fill a gap in the literature.