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 $\phi : (M, g)\to (N, h)$

between two Riemannian manifolds 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

Riemannian 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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