Function theory and index for minimal surfaces of genus 1. I: Fredholm bundles of holomorphic functions on punched tori
type: Item
instance of: scholarly article
title: Function theory and index for minimal surfaces of genus 1. I: Fredholm bundles of holomorphic functions on punched tori
zbMATH Open document ID: 0601.58057
DOI: 10.1007/BF01195359
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: Part I is devoted to the function theoretical aspects of the minimal surface theory. We get manifold structures for the fiber bundle \({\mathbb{A}}^{mj}\) of all double periodic holomorphic \(({\mathbb{C}}^ j\)- valued) functions of Sobolev class \(H^{m,2}\) defined on s-fold punched tori; the base space of the bundle is a 3s-dimensional manifold \({\mathbb{P}}\) (the Teichmüller space) defining the conformal types of the tori. The model space of the fibers is a \(2j(s+1)\)-codimensional subspace of the Hilbert space \(A^ m(B,{\mathbb{C}}^ s)^ j\) of holomorphic functions in the unit ball. In the manifold \(A^{mj}\) we consider for \(\mu:=(\mu _ 1,...,\mu _ n)\in {\mathbb{N}}^ n\) the subset \({\mathbb{A}}^{mj}_{\mu ^ *}\) of all functions having 1) precisely n zeros of the orders \(\mu _ 1,...,\mu _ n\), 2) well-defined real primitives, that is Re\(\oint f=0\) for closed paths. \({\mathbb{A}}^{mj}_{\mu ^ *}\) is a submanifold of real codimension \(2j(| \mu | -n)+j(s+1)\). There are several results concerning the ''internal'' topological and conformal structure of \({\mathbb{A}}^{m,j}\).
Mathematics Subject Classification ID: 58J22
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 32L05
zbMATH DE Number: 3969737
zbMATH Keywords: function theoretical aspects of the minimal surface theory
zbMATH Keywords: Teichmüller space
zbMATH Keywords: conformal types of the tori
zbMATH Keywords: topological and conformal structure
zbMATH Keywords: surfaces of higher genus
zbMATH Keywords: double periodic functions
zbMATH Keywords: indextheory
zbMATH Keywords: plateau problem
zbMATH Keywords: Riemann-Hilbert problem
zbMATH Keywords: variation of conformal structures
Wikidata QID: Q122803057
author: Karl-Heinz Schüffler
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01195359
OpenAlex ID: W231294788
description: scientific article; zbMATH DE number 3969737