Riemannian twistors and Hermitian structures on low-dimensional space forms

Abstract

A Riemannian Kerr Theorem is described, which associates to an integrable Hermitian structure (with singularities) on S4 a complex analytic surface in CP3. To such a Hermitian structure can be associated a harmonic morphism defined on each of the four-dimensional space forms with values in a Riemann surface. This unifies the classifications for the three-dimensional space forms of Baird and Wood and the recent classifications for R4 and S4 by Wood, into one simple equation. As a consequence we derive the classification for harmonic morphisms defined on four-dimensional hyperbolic space with values in a Riemann surface and a new implicit form for harmonic morphisms defined on three-dimensional hyperbolic space. Unification of the Kerr Theorems for S4 and for compactified Minkowski space M is achieved at the complex level after suitably embedding the spaces in compactified, complexified Minkowski space.