Eine globalanalytische Behandlung des Douglas'schen Problems. (A global analysis treatment of the Douglas problem)
type: Item
instance of: scholarly article
title: Eine globalanalytische Behandlung des Douglas'schen Problems. (A global analysis treatment of the Douglas problem)
zbMATH Open document ID: 0606.58047
DOI: 10.1007/BF01169007
published in: Manuscripta Mathematica
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155025
review text: We consider the Douglas problem of genus 0. Starting point is the global analysis for minimal surfaces of the type of the disc which was developed by A. J. Tromba. The set of all k-connected minimal surfaces of genus 0 has a product structure, which is a consequence of the variation of the conformal type. The base space is the space of domain parameters and the fibres are the manifolds of k-connected minimal surfaces of constant conformal type. It is possible to develop a global analysis also for the more general situation considered here with the following results: The map which assigns to every minimal surface its boundary curve (in the sense of the Douglas problem) is a Fredholm operator. The index depends on the number and the total order of the branch points. The analysis allows to prove isolatedness and stability results in a relatively simple way. In the meantime the author has given a global analytic description for surfaces of genus 1, double periodic minimal surfaces, and the index theorem is proved in this case.
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 58E12
Mathematics Subject Classification ID: 30F15
Mathematics Subject Classification ID: 35G30
Mathematics Subject Classification ID: 32G05
zbMATH DE Number: 3980096
zbMATH Keywords: Teichmüller space
zbMATH Keywords: Douglas problem
zbMATH Keywords: minimal surfaces
zbMATH Keywords: variation of the conformal type
zbMATH Keywords: Fredholm operator
zbMATH Keywords: isolatedness and stability
zbMATH Keywords: index theorem
author: Karl-Heinz Schüffler
MaRDI profile type: Publication
Wikidata QID: Q122803063
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OpenAlex ID: W1252912079
description: scientific article; zbMATH DE number 3980096