On stable constant mean curvature surfaces with free boundary

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.