Surfaces of constant mean curvature with boundary in a sphere
Let \(\Sigma\) be a compact, connected and embedded surface in the Euclidean space \(\mathbb{R} ^{3}\) with constant mean curvature \(H \neq 0\), whose boundary \(\partial \Sigma\) is a curve \(C\). \textit{M. Koiso} [Math. Z. 191, 567-574 (1986; Zbl 0563.53007)] proved that, when \(C\) is a Jordan curve included in a plane \(\pi\) and \(\Sigma\) does not intersect the plane \(\pi \) outside of the region bounded by \(C\), then \(\Sigma\) is included in one of the two halfspaces determined by \(\pi \). In this paper the author obtains a similar result, but in the case where \(C\) is a Jordan curve included in an open hemisphere \(S_{+}\) of the unit sphere \(S^{2}(1)\). More precisely, denoting by \(\Omega \) the domain bounded by \(C\) in \(S_{+}\), he proves: if \(\Sigma\) has constant mean curvature 1 and does not intersect \(S^{2}(1)- {\overline \Omega }\), then \(\Sigma = {\overline \Omega }\) or \((\Sigma - \partial \Sigma) \cap S^{2}(1)\) is empty. The basic tool in the proof is the maximum principle.
scientific article; zbMATH DE number 1117472