Function theory and index for minimal surfaces of genus 1. I: Fredholm bundles of holomorphic functions on punched tori (Q1081175)

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 \({\mathbb{A}}^{m,j}\).