On stable constant mean curvature surfaces with free boundary

Abstract

Ros and Vergasta (Geom Dedicata 56:19–33, 1995) proved, among others interesting results, a theorem which states 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.