Function theory and index theory for minimal surfaces of genus 1. III: The index theorem for minimal surfaces of genus 1
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'').
scientific article; zbMATH DE number 3969739