A maximum principle at infinity for minimal surfaces and applications
type: Item
instance of: scholarly article
title: A maximum principle at infinity for minimal surfaces and applications
zbMATH Open document ID: 0667.49024
DOI: 10.1215/S0012-7094-88-05736-5
author: Harold Rosenberg
author: Rémi Langevin
published in: Duke Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: The main results: (1) Let \(f_ 1\), \(f_ 2\) be solution of the exterior Plateau problem on \(\Omega =\{X=(x_ 1,x_ 2)|\) \(x^ 2_ 1+x^ 2_ 2\geq 1\}\). Suppose \(f_ 1(X)\geq f_ 2(X)\) for \(X\in \Omega\) and there exists a sequence \(X_ i\in \Omega\) such that \(| X_ i| \to \infty\) and \(| f_ 1(X_ i)-f_ 2(X_ i)| \to 0\). Then \(f_ 1=f_ 2.\) (2) Minimal herissons with catenoid type ends are rigid. (3) Let f solve the exterior Plateau problem on \(\Omega\). Then \(0\leq a(f)\leq 1\), where a(f) is the coefficient of log R in the asymptotic expansion of f near infinity. If \(a(f)=1\), then \(f(R)=\cosh^{-1}(R)\), i.e., f defines a catenoid.
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 58E12
zbMATH DE Number: 4092216
zbMATH Keywords: minimal surfaces
zbMATH Keywords: exterior Plateau problem
zbMATH Keywords: catenoid
reviewed by: Constantin Udrişte
MaRDI profile type: Publication
cites work: Q3806286
cites work: A survey of the geometric results in the classical theory of minimal surfaces
cites work: Q5611582
cites work: Complete minimal surfaces and minimal herissons
description: scientific article; zbMATH DE number 4092216