Constant mean curvature surfaces with boundary in Euclidean three-space
type: Item
instance of: scholarly article
title: Constant mean curvature surfaces with boundary in Euclidean three-space
zbMATH Open document ID: 0980.53013
author: Rafael López
published in: Tsukuba Journal of Mathematics
publication date: 1999-10-04T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49Q10
zbMATH DE Number: 1347378
zbMATH Keywords: planar boundary
zbMATH Keywords: constant mean curvature surfaces
zbMATH Keywords: graph
zbMATH Keywords: maximum principle
zbMATH Keywords: balancing formula
reviewed by: Wolfgang Kühnel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496163774
OpenAlex ID: W4206043211
DOI: 10.21099/TKBJM/1496163774
description: scientific article; zbMATH DE number 1347378