Function theory and index theory for minimal surfaces of genus 1. II. Manifold structures for double periodic minimal surfaces (Q1081176)

Part II [for part I see the preceding review] is concerned with the properties of the ''conformality operator'' \(K: {\mathbb{A}}^{m,k}\to {\mathbb{A}}^{m,1}\), \(K(f)=\sum ^{k}_{1}(f^ i)^ 2\). The zeros in \({\mathbb{A}}^{m,k}_{\mu ^ *}\) can be regarded (via \(f\mapsto Re\int f)\) as the set \({\mathbb{M}}_{\mu}\) of all minimal surfaces of the topological type of s-fold punched oriented surfaces of genus 1 which have the given type \(\mu\) of (interior) branch points. We prove: 1) K is analytic and 2) \(K: {\mathbb{A}}_{\mu *}^{m,k}\to {\mathbb{A}}^{m,1}\) has corank \(4| \mu |\) near its zeros. Therefore, \({\mathbb{M}}_{\mu}\) is a submanifold and the conformal type can be freely perturbed.