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Harmonic Morphisms Between Riemannian Manifolds$
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Paul Baird and John C. Wood

Print publication date: 2003

Print ISBN-13: 9780198503620

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198503620.001.0001

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Twistor methods

Twistor methods

(p.206) 7 Twistor methods
Harmonic Morphisms Between Riemannian Manifolds

Paul Baird

John C. Wood

Oxford University Press

This chapter discusses how twistor methods can be used to construct nonconstant harmonic morphisms from (orientable) Einstein 4-manifolds to Riemann surfaces. It is shown that any such map induces an (integrable) Hermitian structure J on the 4-manifold with respect to which the map is holomorphic. The fibres of the map are ‘superminimal’, i.e., J is parallel along them. Conversely, a Hermitian structure induces (local) harmonic morphisms with these properties. Thus, the problem of finding harmonic morphisms is converted into that of finding Hermitian structures and superminimal surfaces in an Einstein 4-manifold; a problem that can be solved by twistor theory. This gives explicit constructions of all harmonic morphisms from domains of Euclidean 4-space, the 4-sphere, and complex projective 2-space to surfaces. The last section discusses harmonic morphisms from other Einstein manifolds, including the join of two complex projective 2-spaces endowed with the Page metric.

Keywords:   Einstein manifold, 4-manifold, Hermitian structure, superminimal, Page metric

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