On minimal surfaces with free boundaries in given homotopy classes
type: Item
instance of: scholarly article
title: On minimal surfaces with free boundaries in given homotopy classes
zbMATH Open document ID: 0576.53005
DOI: 10.1016/S0294-1449(16)30400-0
author: Peter Tolksdorf
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1985-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78094
review text: Let S be a compact smooth embedded surface in \({\mathbb{R}}^ 3\) and let B be the unit disc in \({\mathbb{R}}^ 2\). The author considers the problem of finding a surface that minimizes the area (Dirichlet integral) among all surfaces having a conformal parametrization \(u: B\to u(B)\subset {\mathbb{R}}^ 3\) with boundary values \(u| \partial B\) in a given nontrivial homotopy class of curves \(\gamma\) : \(\partial B\to S\). He shows that this problem is solved whenever a certain pinching condition is satisfied. For homology classes, the corresponding result is due to \textit{N. Davids} [Am. J. Math. 64, 348-362 (1942)] who used a method by R. Courant. The author obtains this result also for homotopy classes because he keeps the lengths of the boundary curves of the minimizing sequence as short as possible. Thus he obtains the uniform continuity of these boundary curves with the aid of the Courant-Lebesgue-Lemma.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 35R35
zbMATH DE Number: 3921391
zbMATH Keywords: minimal surfaces
zbMATH Keywords: free boundary problem
zbMATH Keywords: pinching condition
zbMATH Keywords: homotopy classes
MaRDI profile type: Publication
cites work: Regularity and finiteness of solutions to the free boundary problem for minimal surfaces
cites work: Q4125078
cites work: Q2548786
cites work: Behavior of minimal surfaces with free boundaries
cites work: Geometric properties of minimal surfaces with free boundaries
cites work: An optimal estimate of the free boundary of a minima surface.
cites work: Topology of three dimensional manifolds and the embedding problems in minimal surface theory
cites work: Q4081953
cites work: Q4097854
cites work: Stationary minimal surfaces with boundary on a simplex
cites work: On a free boundary problem for minimal surfaces
cites work: A finiteness result in the free boundary value problem for minimal surfaces
cites work: Minimal Surfaces Spanning Closed Manifolds and Having Prescribed Topological Position
OpenAlex ID: W2259555598
description: scientific article; zbMATH DE number 3921391