HARMONIC MORPHISMS AND HERMITIAN STRUCTURES ON EINSTEIN 4-MANIFOLDS

Source abstract

We show that a submersive harmonic morphism from an orientable Einstein 4-manifold M 4 to a Riemann surface, or a conformal foliation of M 4 by minimal surfaces, determines an (integrable) Hermitian structure with respect to which it is holomorphic. Conversely, any nowhere-Kähler Hermitian structure of an orientable anti-self-dual Einstein 4-manifold arises locally in this way. In the case M 4 =ℝ 4 we show that a Hermitian structure, viewed as a map into S 2 , is a harmonic morphism; in this case and for S 4 , [Formula: see text] we determine all (submersive) harmonic morphisms to surfaces locally, and, assuming a non-degeneracy condition on the critical points, globally.