Harmonic morphisms from compact 3-manifolds

Abstract

AbstractThis chapter shows that a non-constant harmonic morphism with compact fibres from a 3-manifold endows it with the structure of a Seifert fibre space — a 3-manifold with a certain type of one-dimensional foliation with leaf space an orbifold. Conversely, any Seifert fibre space can be obtained from a harmonic morphism by smoothing the orbifold leaf space. A global version of the factorization theorem is provided, then the metrics on a 3-manifold which support a harmonic morphism to a surface are described locally and globally. It is shown how fundamental invariants of a one-dimensional foliation propagate along its leaves. The necessary curvature conditions where a Riemannian 3-manifold supports a non-constant harmonic morphism to a surface are given. These show that there are never more than two such harmonic morphisms up to equivalence, even locally, unless it is a space form.