Surfaces of constant mean curvature with boundary in a sphere
type: Item
instance of: scholarly article
title: Surfaces of constant mean curvature with boundary in a sphere
zbMATH Open document ID: 0899.53007
author: Rafael López
published in: Osaka Journal of Mathematics
publication date: 1998-10-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1117472
zbMATH Keywords: Euclidean 3-space
zbMATH Keywords: surface with prescribed boundary
zbMATH Keywords: constant mean curvature surface
reviewed by: Thomas Hasanis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1117472