Harmonic morphisms onto Riemann surfaces and generalized analytic functions (Q1085853)

We study harmonic morphisms from domains in \({\mathbb{R}}^ 3\) and \(S^ 3\) to a Riemann surface N, obtaining the classification of such in terms of holomorphic mappings from a covering space of N into certain Grassmannians. We show that the only non-constant submersive harmonic morphism defined on the whole of \(S^ 3\) to a Riemann surface is essentially the Hopf map. Comparison is made with the theory of analytic functions. In particular we consider multiple-valued harmonic morphisms defined on domains in \({\mathbb{R}}^ 3\) and show how a cutting and glueing procedure may be applied to obtain a single-valued harmonic morphism from a certain 3-manifold. This is similar to the way in which the Riemann surface of a multiple-valued analytic function is constructed.