Harmonic morphisms and circle actions on 3- and 4-manifolds

Abstract

Harmonic morphisms are considered as a natural generalization of the analytic functions of Riemann surface theory. It is shown that any closed analytic 3-manifold supporting a non-constant harmonic morphism into a Riemann surface must be a Seifert fibre space. Harmonic morphisms φ:M→N from a closed 4-manifold to a 3-manifold are studied. These determine a locally smooth circle action on M with possible fixed points. This restricts the topology of M. In all cases, a harmonic morphism φ:M→N from a closed (n+1)-dimensional manifold to an n-dimensional manifold (n≥2, with M, N analytic in the case n=2) determines a locally smooth circle action on M.