John C. Wood

Harmonic morphisms between Riemannian manifolds

Introduction BASIC FACTS ON HARMONIC MORPHISMS 1. Complex-valued harmonic morphisms on three-dimensional Euclidean space 2. Riemannian manifolds and conformality 3. Harmonic mappings between Riemannian manifolds 4. Fundamental properties of harmonic morphisms 5. Harmonic morphisms defined by polynomials TWISTOR METHODS 6. Mini-twistor theory on three-dimensional space-forms 7. Twistor methods 8. Holomorphic harmonic morphisms 9. Multivalued harmonic morphisms TOPOLOGICAL AND CURVATURE CONSIDERATIONS 10. Harmonic morphisms from compact 3-manifolds 11. Curvature considerations 12. Harmonic morphisms with one-dimensional fibres 13. Reduction techniques FURTHER DEVELOPMENTS 14. Harmonic morphisms between semi-Riemannian manifolds Appendix Glossary of Notation Bibliography Index

Harmonic maps and integrable systems

Introduction and background material the geometry of surfaces sigma and chiral models the algebraic approach the twistor approach.

HARMONIC MORPHISMS AND HERMITIAN STRUCTURES ON EINSTEIN 4-MANIFOLDS

We show that a submersive harmonic morphism from an orientable Einstein 4-manifold M4 to a Riemann surface, or a conformal foliation of M4 by minimal surfaces, determines an (integrable) Hermitian structure with respect to which it is holomorphic. Conversely, any nowhere-Kahler Hermitian structure of an orientable anti-self-dual Einstein 4-manifold arises locally in this way. In the case M4=ℝ4 we show that a Hermitian structure, viewed as a map into S2, is a harmonic morphism; in this case and for S4, we determine all (submersive) harmonic morphisms to surfaces locally, and, assuming a non-degeneracy condition on the critical points, globally.

Harmonic maps into symmetric spaces and integrable systems

Let M = (M m , g), N = (N n , h) be C ∞ Riemannian manifolds of dimensions m,n respectively and let o: M m → N n be a C ∞ mapping between them.

Harmonic morphisms with one-dimensional fibres on Einstein manifolds

We prove that, from an Einstein manifold of dimension greater than or equal to five, there are just two types of harmonic morphism with one-dimensional fibres. This generalizes a result of R.L. Bryant who obtained the same conclusion under the assumption that the domain has constant curvature.

On the space of harmonic 2-spheres in CP2

Carrying further the work of T.A. Crawford, we show that each component of the space of harmonic maps from the 2-sphere to complex projective 2-space of degree d and energy 4πE is a smooth closed submanifold of the space of all Cj maps (j≥2). We achieve this by showing that the Gauss transform which relates them to spaces of holomorphic maps of given degree and ramification index is smooth and has injective differential.

HERMITIAN STRUCTURES AND HARMONIC MORPHISMS IN HIGHER DIMENSIONAL EUCLIDEAN SPACES

We construct new complex-valued harmonic morphisms from Euclidean spaces from functions which are holomorphic with respect to Hermitian structures. In particular, we give the first global examples of complex-valued harmonic morphisms from ℝn for each n>4 which do not arise from a Kahler structure; it is known that such examples do not exist for n≤4.

Harmonic maps and morphisms in 4 dimensions

We discuss (i) harmonic maps from surfaces to 4-manifolds, especially, twistor constructions and a recent application to the study of the space of harmonic maps from S 2 to CP 2 , (ii) harmonic morphisms from 4-manifolds to surfaces, especially relationships with Hermitian structures, shear-free ray congruences and a twistorial construction of foliations by superminimal surfaces of hyperbolic 4-space with given boundary values at innnity.

A note on the fundamental group of a manifold of negative curvature

Let Y be a compact connected C ∞ Riemannian manifold with negative sectional curvatures. Let G be a non-trivial subgroup of the fundamental group π 1 ( Y ). G is known to be cyclic if it is abelian (Preissmann (6)) or contains a subnormal abelian (hence cyclic) subgroup (Yau(9)). These results may be generalized as follows: Say that a group G is of type (α) if ∃a ∈ G , a ≠ e, such that for all b belonging to a set of generators for G we have a m b n = b q a p for some integers m, n, p, q with either m = p or n = q .

Some Aspects of Harmonic Maps from a Surface to Complex Projective Space

Let M and N be smooth compact Riemannian surfaces without boundary. If N has negative curvature, or more generally, if π2 (N) = 0, we have general existence theorems for harmonic maps (8, 18, 22, 23). The first interesting case where we have no 2 general existence theorem is when N is the two-sphere S2. In §1 we discuss some results in this case and see how they generalize when S2 = ℂ ℙ1 is replaced by the n-sphere Sn or complex projective n-space ℂ ℙn. In §2 we outline a development of the method introduced by Din-Zakrewski (4), Buns and Glaser-Stora (13) for manufacturing certain harmonic maps into ℂ ℙn from holomorphic ones. This method classifies all harmonic maps S2 → ℂ ℙn and all harmonic maps from the torus to ℂ ℙn of non-zero degree. Full details will appear in a paper by J. Eells and the author (11).