This chapter discusses how twistor methods can be used to construct nonconstant harmonic morphisms from (orientable) Einstein 4-manifolds to Riemann surfaces. It is shown that any such map induces an (integrable) Hermitian structure J on the 4-manifold with respect to which the map is holomorphic. The fibres of the map are ‘superminimal’, i.e., J is parallel along them. Conversely, a Hermitian structure induces (local) harmonic morphisms with these properties. Thus, the problem of finding harmonic morphisms is converted into that of finding Hermitian structures and superminimal surfaces in an Einstein 4-manifold; a problem that can be solved by twistor theory. This gives explicit constructions of all harmonic morphisms from domains of Euclidean 4-space, the 4-sphere, and complex projective 2-space to surfaces. The last section discusses harmonic morphisms from other Einstein manifolds, including the join of two complex projective 2-spaces endowed with the Page metric.
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