Surfaces of constant mean curvature bounded by convex curves (Q1364936)

Let \(\Sigma\) be a compact connected surface with smooth boundary \(\partial \Sigma\) and \(\phi : \Sigma \to \mathbb{R} ^{3}\) an immersion with constant mean curvature \(H \neq 0\), such that \(\phi \), restricted to \(\partial \Sigma\), is a diffeomorphism onto a Jordan curve \(\Gamma \) on a plane \(\pi \). To determine all such immersions is a problem that has received recent attention of several geometers. In the present paper the author proves that, in the case where \(\phi \) is an imbedding, \(\Gamma \) is a circle of radius 1 and \(\phi (\Sigma)\) is included in a solid slab with width \(1/ | H| \), then \(\phi (\Sigma)\) is a spherical cap of radius \(1/ | H| \). The principal tools in the proof are Alexandrov's reflection principle and the maximum principle.