Constant mean curvature surfaces with boundary in Euclidean three-space (Q1306536)

This paper deals with constant mean curvature surfaces inscribed in a convex planar closed curve. In the first part certain characterizations of planar discs and hemispheres as the only cmc surfaces with certain boundary behavior are given. This generalizes previous results by E. Heinz and L. Barbosa. In the second part of the paper the main result states the following: Let \(H\) be a constant and let \(\Gamma=\partial\Omega\) be a convex planar closed curve with curvature \(\kappa>|H|\). Then there exists a graph defined on \(\Omega\) of constant mean curvature \(H\). Moreover, up to reflection with respect to the boundary plane, there is only one such \(H\)-cmc surface with boundary \(\Gamma\) which is small, i.e., contained in an open ball of radius less than \(1/|H|\). The main tools for the proof are the maximum principle on the one hand and a balancing formula on the other hand.