Harmonic morphisms between Riemannian manifolds

Paul Baird and John C. Wood

London Mathematical Society Monographs, No. 29,
Oxford University Press

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Harmonic morphisms are maps which preserve Laplace's equation. More explicitly, a map between Riemannian manifolds is called a harmonic morphism if its composition with any locally defined harmonic function on the codomain is a harmonic function on the domain; it thus `pulls back' germs of harmonic functions to germs of harmonic functions. Examples include harmonic functions, conformal mappings in the plane, holomorphic mappings with values in a Riemann surface and certain submersions arising from Killing fields and geodesic fields. Their study involves many different branches of mathematics: we shall discuss aspects of the theory of foliations, polynomials induced by Clifford systems and orthogonal multiplications, twistor and mini-twistor spaces and Hermitian structures. We shall explore relations with topology, including Seifert fibre spaces and circle actions, and with isoparametric functions and the Beltrami fields equation of hydrodynamics.

This monograph represents the first attempt to document the theory of harmonic morphisms between Riemannian manifolds in book form. Chapter 1 gives the flavour of the subject in a way which should be understandable to a competent final-year undergraduate student. For the rest of the book, we shall assume that the reader is familiar with the basic concepts of differential geometry as found, for example, in the first two volumes of M. Spivak's book: A comprehensive introduction to Riemannian geometry, though we shall remind readers of key concepts at appropriate points. Apart from some occasional technicalities of analysis or partial differential equations for which we refer the reader to the appropriate literature, this book is self-contained; in particular, it is not necessary to know anything about harmonic maps or partial differential equations to read it. We hope that it will be a useful book both for graduate students wishing to learn about harmonic morphisms and related topics on harmonic maps, and for researchers in the field or in related fields.

As shown by B. Fuglede and T. Ishihara in 1978/9, a harmonic morphism is a harmonic map with an additional property called horizontal weak conformality (or semiconformality). However, there are major differences with studies of the general theory of harmonic mappings. Firstly, Fuglede established that any non-constant harmonic morphism is an open mapping, in particular, it must preserve or decrease dimension. Secondly, the second order equation of harmonicity and the first order equation of horizontal weak conformality form an overdetermined system of equations: solutions are not guaranteed even locally; their existence depends in an essential way on the topology of the manifolds and the Riemannian structures on them, for example, there is no harmonic morphism from the (n+1)-sphere to the n-sphere whatever metrics they have, when n > 3. This contrasts with the theory of harmonic mappings, where a theorem of Eells and Ferreira (1991) shows that, provided the dimension of the domain manifold is at least 3, any homotopy class of maps between Riemannian manifolds contains a (smooth) harmonic representative with respect to a metric on the domain conformally equivalent to the given one.

On the other hand, we find a remarkable duality between the theory of harmonic morphisms to a surface and the theory of conformal minimal immersions from a surface giving Weierstrass representations for harmonic morphisms and Bernstein theorems, for example, the Hopf map is essentially the only harmonic morphism from the 3-sphere to the 2-sphere, both with their standard metrics.