Harmonic morphisms, Hermitian structures and symmetric spaces

Abstract: A harmonic morphism is a map between two Riemannian manifolds with the property that its composition with a local harmonic function on the codomain is a local harmonic function on the domain. Such a map is automatically a harmonic map, satisfying an additional partial conformality condition called horizontal (weak) conformality. In local coordinates, this amounts to a non-linear, over-determined system of partial differential equations. Therefore, the question of the existence of such maps, even in the local case, is very hard to answer in general. However, when the manifolds involved carry Hermitian structures, it is natural to search among the holomorphic maps for candidates for harmonic morphisms. In this thesis, we study in a series of articles the geometry of harmonic morphisms and conformal foliations, a closely related topic, in the context of Hermitian geometry. Several results are proved for holomorphic harmonic morphisms between Kähler manifolds, in particular when the domain is of non-negative sectional curvature or constant holomorphic sectional curvature. Many of these results also extend to holomorphic conformal foliations. A principal tool we use is a formula by Walczak which, in this context, takes a particularly simple form. We make a brief digression to apply this formula to harmonic morphisms of warped product type, which results in a decomposition theorem for such maps. We also study the four-dimensional, conformally flat case in detail. By a construction which originates in the ideas of twistor theory, we produce the first known examples of harmonic morphisms without totally geodesic fibres from real hyperbolic spaces of even dimension greater than four. These ideas are then generalized to obtain harmonic morphisms from several other Riemannian symmetric spaces of type I and III, such as the Grassmannians, where the question of local existence was an open problem. By further exploiting the concept of duality for Riemannian symmetric spaces, we also obtain harmonic morphisms from several Riemannian symmetric spaces of type II and IV.