Function theory and index theory for minimal surfaces of genus 1. III: The index theorem for minimal surfaces of genus 1
type: Item
instance of: scholarly article
title: Function theory and index theory for minimal surfaces of genus 1. III: The index theorem for minimal surfaces of genus 1
zbMATH Open document ID: 0601.58059
DOI: 10.1007/BF01189638
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: Part III [for part II see the preceding review (Zbl 0601.58058)]: First we show that it is possible to regard (via an injection \(\Lambda)\) the set \({\mathbb{M}}_{\mu}\) as a subset \(({\mathbb{U}}_{\mu})\) of the bundle \(A\times D\times {\mathbb{P}}\) where \(A:=set\) of s-tuples of oriented Jordan curves in \({\mathbb{R}}^ k\), \(D:=set\) of diffeomorphisms of the boundaries (isotopic to the identity). Now, define the ''Tromba-operator'' \(\pi\) : \({\mathbb{U}}_{\mu}\to A\), \(\pi (\alpha,u,p)=\alpha\) which associates to a minimal surface its oriented boundary curve (and the inverse \((\pi \cdot \Lambda)^{-1}\) describes the solutions of the corresponding Plateau problem). We then prove that \(\pi\) is fredholm and compute index \(\pi =2(n-(k-2)| \mu |)\) (''Index theorem'').
Mathematics Subject Classification ID: 58J22
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 32L05
zbMATH DE Number: 3969739
zbMATH Keywords: minimal surfaces
zbMATH Keywords: surfaces of higher genus
zbMATH Keywords: double periodic functions
zbMATH Keywords: Teichmüller theory
zbMATH Keywords: indextheory
zbMATH Keywords: plateau problem
zbMATH Keywords: Riemann-Hilbert problem
zbMATH Keywords: variation of conformal structures
Wikidata QID: Q122803048
author: Karl-Heinz Schüffler
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01189638
OpenAlex ID: W312170399
description: scientific article; zbMATH DE number 3969739