Harmonic Morphisms with 1-dim Fibres on 4-dim Einstein Manifolds

Harmonic morphisms are smooth maps between Riemannian manifolds which preserve Laplace's equation. They are characterised as harmonic maps which are horizontally weakly conformal 14, 20]. R.L. Bryant 7] proved that there are precisely two types of harmonic morphisms with one-dimensional bres which can be deened on a constant curvature space of dimension at least four. Here we prove that, on an Einstein four-manifold, there are precisely three types of harmonic morphisms with one-dimensional bres, the third type being new.