The index theorem for \(k\)-fold connected minimal surfaces
type: Item
instance of: scholarly article
title: The index theorem for \(k\)-fold connected minimal surfaces
zbMATH Open document ID: 0564.58008
DOI: 10.1007/BF01455300
author: Ursula Thiel
published in: Mathematische Annalen
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/182975
review text: Section 0 is an introduction to Plateau's problem. Section 1 describes the analytical properties of conformality operators. Section 2 shows that: (1) the set of minimal \(k\)-fold connected immersions in \(R^ n\) of branching type \(\lambda\) whose boundary curve belongs to \({\mathcal A}=\{\alpha: \partial \Omega_{{\bar \rho}}\to R^ n|\) \(\alpha\) immersion of class \(H^ r\}\) can be viewed as a differentiable submanifold of \({\mathcal A}\times {\mathcal D}\), where \({\mathcal D}=\cup_{\rho \in P}{\mathcal D}_{\rho}\), \({\mathcal D}_{\rho}=\{u: \partial \Omega_{\rho}\to \partial \Omega_{{\bar \rho}}|\) u diffeomorphism of class \(H^ s\) with deg u\(=1\) on each component of \(\partial \Omega_{\rho}\}\); (2) at a minimal immersion, the conformality operator restricted to the manifold of surfaces of branching type \(\lambda\) is a submersion. Section 3 gives geometric applications.
reviewed by: Constantin Udrişte
Mathematics Subject Classification ID: 58D05
Mathematics Subject Classification ID: 49Q05
zbMATH DE Number: 3899791
zbMATH Keywords: index theorem
zbMATH Keywords: minimal immersion
zbMATH Keywords: Plateau's problem
zbMATH Keywords: conformality operators
zbMATH Keywords: immersions
MaRDI profile type: Publication
cites work: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II
cites work: The index theorem for classical minimal surfaces
cites work: Q4125078
cites work: On embedded minimal surfaces
cites work: Q5628582
cites work: The existence of embedded minimal surfaces and the problem of uniqueness
cites work: Q5518064
cites work: Q4081953
cites work: Q5667036
cites work: Stabilität mehrfach zusammenhängender Minimalflächen
cites work: An Infinite Dimensional Version of Sard's Theorem
cites work: Plateau's problem for embedded minimal surfaces of the type of the disc
cites work: On the number of simply connected minimal surfaces spanning a curve
cites work: Q3259806
cites work: Q5332327
description: scientific article; zbMATH DE number 3899791