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Harmonic Morphisms from Lie Groups and Symmetric Spaces - Some Existence Theory

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Sigmundur Gudmundsson
Lund University
Thu, 2019-07-18 16:30 - 17:30
MPIM Lecture Hall

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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