In mathematics, a harmonic morphism is a (smooth) map
between Riemannian manifolds that pulls back real-valued harmonic functions on the codomain to harmonic functions on the domain.
Harmonic morphisms form a special class of harmonic maps, namely those that are horizontally (weakly) conformal.
is expressed by the non-linear system where
is a continuous function called the dilation.
Harmonic morphisms are therefore solutions to non-linear over-determined systems of partial differential equations, determined by the geometric data of the manifolds involved.
For this reason, they are difficult to find and have no general existence theory, not even locally.
is a surface, the system of partial differential equations that we are dealing with, is invariant under conformal changes of the metric
This means that, at least for local studies, the codomain can be chosen to be the complex plane with its standard flat metric.
In this situation a complex-valued function
is a harmonic morphisms if and only if and This means that we look for two real-valued harmonic functions
that are orthogonal and of the same norm at each point.
This shows that complex-valued harmonic morphisms
from Riemannian manifolds generalise holomorphic functions
from Kähler manifolds and possess many of their highly interesting properties.
The theory of harmonic morphisms can therefore be seen as a generalisation of complex analysis.
[1] In differential geometry, one is interested in constructing minimal submanifolds of a given ambient space
Harmonic morphisms are useful tools for this purpose.
This is due to the fact that every regular fibre
with values in a surface is a minimal submanifold of the domain with codimension 2.
[1] This gives an attractive method for manufacturing whole families of minimal surfaces in 4-dimensional manifolds
, in particular, homogeneous spaces, such as Lie groups and symmetric spaces.