On stable constant mean curvature surfaces with free boundary

TL;DR

This paper extends the classification of stable constant mean curvature surfaces with free boundary in a ball, showing that genus 1 surfaces cannot occur, thus refining previous results.

Contribution

It completes the classification by proving that stable CMC surfaces with free boundary in a ball cannot have genus 1, using a modified Hersch type balancing argument.

Findings

Genus 1 stable CMC surfaces with free boundary do not exist.

Classified stable CMC surfaces as planar equators or spherical caps.

Extended previous results by excluding genus 1 cases.

Abstract

In [20], Ros and Vergasta proved that an immersed orientable compact stable constant mean curvature surface $\Sigma$ with free boundary in a closed ball $B\subset\mathbb{R}^3$ must be a planar equator, a spherical cap or a surface of genus 1 with at most two boundary components. In this article, by using a modified Hersch type balancing argument, we complete their work by proving that $\Sigma$ cannot have genus 1.