Stable constant mean curvature surfaces with circular boundary

Abstract

In this paper we study stable constant mean curvature surfaces in the Euclidean space R 3 \mathbf {R}^3 with circular boundary. We show that in the case of genus zero, the only such surfaces are the spherical caps and the flat discs. We also extend this result to the case of surfaces in the other space forms, namely the sphere S 3 \mathbf {S}^3 and the hyperbolic space H 3 \mathbf {H}^3 .