A harmonic morphism between two manifolds is a map which pulls back local harmonic functions to local harmonic functions. Such maps form a special class of harmonic maps and are formally analogous to holomorphic maps. P. Baird, the author of the article under review and others have made a systematic study of harmonic morphisms in various contexts. The present paper concerns the situation of morphisms \(f\) from a 4-manifold \(M\) to a 2-manifold \(N\). Its main purpose is to show that if \(M\) is an Einstein manifold, then there is an integrable Hermitian structure on the complement of the critical set of \(f\) such that \(f\) is holomorphic.
The setting for this result is in the study of 2-planes in the tangent space of \(M\). Indeed, the fibres of \(f\) for a general 4-manifold are minimal surfaces. The Einstein condition shows that these are actually superminimal. It is clear that the role of twistor spaces as they have been used in the theory of harmonic maps in the past by Eells, Salamon and others is important here, and indeed special features hold in the self-dual Einstein situation. The paper goes on to study in detail the cases \(M=\mathbb{R}^ 4\), \(S^ 4\), \(\mathbb{C} P^ 2\) and \(\mathbb{C} P^ 1\times\mathbb{C} P^ 1\). One particular result is that there exist examples of harmonic morphisms from open sets of \(\mathbb{R}^ 4\) to \(S^ 2\) which are not holomorphic with respect to any orthogonal complex structure.