Paul Baird

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

Three-dimensional Ricci solitons which project to surfaces

Abstract We study 3-dimensional Ricci solitons which project via a semi-conformal mapping to a surface. We reformulate the equations in terms of parameters of the map; this enables us to give an ansatz for constructing solitons in terms of data on the surface. A complete description of the soliton structures on all the 3-dimensional geometries is given, in particular, non-gradient solitons are found on Nil and Sol.

On Constructing Biharmonic Maps and Metrics

We construct biharmonic nonharmonic maps between Riemannian manifoldsM and N by first making the ansatz that ϕM → N be aharmonic map and then deforming the metric conformally on M to renderϕ biharmonic. The deformation will, in general, destroy theharmonicity of ϕ. We call a metric which renders the identity mapbiharmonic, a biharmonic metric. On an Einstein manifold, theonly conformally equivalent biharmonic metrics are defined byisoparametric functions.

Liouville-type theorems for biharmonic maps between Riemannian manifolds

Abstract We prove Liouville type theorems for biharmonic maps from complete manifolds and from Euclidean balls.

Riemannian twistors and Hermitian structures on low‐dimensional space forms

A Riemannian Kerr Theorem is described, which associates to an integrable Hermitian structure (with singularities) on S4 a complex analytic surface in CP3. To such a Hermitian structure can be associated a harmonic morphism defined on each of the four‐dimensional space forms with values in a Riemann surface. This unifies the classifications for the three‐dimensional space forms of Baird and Wood and the recent classifications for R4 and S4 by Wood, into one simple equation. As a consequence we derive the classification for harmonic morphisms defined on four‐dimensional hyperbolic space with values in a Riemann surface and a new implicit form for harmonic morphisms defined on three‐dimensional hyperbolic space. Unification of the Kerr Theorems for S4 and for compactified Minkowski space M is achieved at the complex level after suitably embedding the spaces in compactified, complexified Minkowski space.

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.

Polynômes semi-conformes et morphismes harmoniques

Abstract. We prove a Liouville type theorem for harmonic morphisms from

Harmonic Maps and Morphisms from Multilinear Norm-Preserving Mappings

{\bf R}^m

Twistor Theory on a Finite Graph

to

A Bchner technique for harmonic mappings from a 3-manifold to a surface

{\bf R}^n\,\, (n\geq 3)