Surfaces of constant mean curvature bounded by convex curves
type: Item
instance of: scholarly article
title: Surfaces of constant mean curvature bounded by convex curves
zbMATH Open document ID: 0899.53006
DOI: 10.1023/A:1004979310168
author: Rafael López
published in: Geometriae Dedicata
publication date: 1998-10-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1053731
zbMATH Keywords: Euclidean 3-space
zbMATH Keywords: constant mean curvature surface
reviewed by: Thomas Hasanis
MaRDI profile type: Publication
full work available at URL: a:1004979310168
OpenAlex ID: W105082770
description: scientific article; zbMATH DE number 1053731