Degree theory on oriented infinite-dimensional varieties and the Morse number of minimal surfaces spanning a curve in \({\mathbb{R}}^ n\)
type: Item
instance of: scholarly article
title: Degree theory on oriented infinite-dimensional varieties and the Morse number of minimal surfaces spanning a curve in \({\mathbb{R}}^ n\)
zbMATH Open document ID: 0622.58006
DOI: 10.1007/BF01169005
published in: Manuscripta Mathematica
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155023
review text: The author introduces for almost all closed curves \(\Gamma\) in \({\mathbb{R}}^ n\) (n\(\geq 3)\) an invariant, called the ''Morse number''. In a certain sense, this invariant counts the number of minimal surfaces spanning \(\Gamma\). The author shows that this invariant is equal to 1 and as a consequence, he proves a strong version of the mountain-pass lemma of Morse-Tompkins. The construction of the Morse number and the proof rely on degree theory for Fredholm vector fields on Banach manifolds. For dimension \(n=3\), an important difficulty has to be overcome: due to the index theorem of Böhme and the author, almost all wires \(\Gamma\) in \({\mathbb{R}}^ n\) bound finitely many nondegenerate minimal surfaces for \(n\geq 4\), while for \(n=3\) this is not true.
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 58E15
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 4009364
zbMATH Keywords: ''Morse number
zbMATH Keywords: minimal surfaces
zbMATH Keywords: mountain-pass lemma
author: Anthony J. Tromba
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4009364