Embedded Plateau problem
type: Item
instance of: scholarly article
zbMATH Open document ID: 1242.53069
DOI: 10.1090/S0002-9947-2011-05486-3
author: Baris Coskunuzer
published in: Transactions of the American Mathematical Society
publication date: 2012-04-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1005.1723
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 49Q05
zbMATH DE Number: 6024126
zbMATH Keywords: Plateau problem
zbMATH Keywords: embedded Plateau problem
zbMATH Keywords: area minimizing surface
zbMATH Keywords: minimal disk
zbMATH Keywords: area minimizing disk
MaRDI profile type: Publication
arXiv ID: 1005.1723
Wikidata QID: Q125622755
cites work: Verzweigungspunkte von H-Flächen. II
cites work: Q3207672
cites work: Coincidence set of minimal surfaces for the thin obstacle
cites work: Shrinkwrapping and the taming of hyperbolic 3-manifolds
cites work: Shapes of embedded minimal surfaces
cites work: Solution of the Problem of Plateau
cites work: Embeddedness of minimal surfaces with total boundary curvature at most \(4\pi\)
cites work: Q5565773
cites work: Two dimensional variational problems with thin obstacles
cites work: On the geometric and topological rigidity of hyperbolic 3-manifolds
cites work: On embedded minimal surfaces
cites work: Regularity of minimizing surfaces of prescribed mean curvature
cites work: Optimal regularity for the Signorini problem
cites work: Area inequalities for embedded disks spanning unknotted curves
cites work: The Existence of Least Area Surfaces in 3-Manifolds
cites work: On the coincidence set in variational inequalities
cites work: The problem of Plateau on a Riemannian manifold
cites work: Topology of three dimensional manifolds and the embedding problems in minimal surface theory
cites work: The classical Plateau problem and the topology of three-dimensional manifolds. The embedding of the solution given by Douglas-Morrey and an analytic proof of Dehn's lemma
cites work: The existence of embedded minimal surfaces and the problem of uniqueness
cites work: Variational problems with inequalities as boundary conditions or how to fashion a cheap hat for Giacometti's brother
cites work: A proof of the regularity everywhere of the classical solution to Plateau's problem
cites work: Extreme curves bound embedded minimal surfaces of the type of the disc
title: Embedded Plateau problem
review text: The Plateau problem asks for the existence of an area minimizing disk for a given simple closed curve in a manifold. After this problem was solved for \(3\)-dimensional Euclidean space by Douglas and Rado, it was generalized to Riemannian manifolds including a regularity of solutions by many people. However, these area minimizing disks may not be embedded, even though the curves bound an embedded disk in the ambient manifold. In decades, this question of embeddedness of the area minimizing disk has been studied.NEWLINENEWLINEIn this paper, the author considers the same embeddedness question. But instead of considering the question ``for which curves must the area minimizing disks be embedded?'', he analyzes the structure of the surface which minimizes area among the embedded disks whose boundary is any given simple closed curve. He proves that if \(\Gamma\) is a simple closed curve bounding an embedded disk in a closed \(3\)-manifold \(M\), then there exists a disk \(\Sigma\) with \(\partial \Sigma = \Gamma\) such that \(\Sigma\) minimizes the area among all the embedded disks bounding \(\Gamma\). Moreover, \(\Sigma\) is minimal and smoothly embedded everywhere except where the boundary \(\Gamma\) meets the interior of \(\Sigma\).
reviewed by: Gabjin Yun
description: scientific article; zbMATH DE number 6024126