The study of harmonic morphisms in the 3-dimensional Euclidean space goes back to a paper of Jacobi from 1848. This was then introduced into the setting of Riemannian geometry, in the late 1970s by Fuglede and Ishihara, independently. A harmonic morphism between two Riemannian manifolds (M, g) and (N, h) is a map that pulls back real-valued harmonic functions on (N,h) to harmonic functions on (M,g).

In 1983 Baird and Eells have shown that in the case when the codomain is a surface, the regular fibres of a harmonic morphism form a minimal conformal foliation on the domain. These are interesting geometric objects and our main motivation for studying harmonic morphisms in this particular case.

Harmonic morphisms are solutions to an over-determined non-linear system of partial differential equations. They do not have a general existence theory. There even exist rather simple 3-dimensional Lie groups for which one can show that local solutions do not exist.

In this talk we will explain the general theory and give a survey of what is known when (M,g) is a Lie group or a symmetric space and (N,h) is the flat complex plane.

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