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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

Chapter:
(p.206) 7 Twistor methods
Source:
Harmonic Morphisms Between Riemannian Manifolds
Author(s):

Paul Baird

John C. Wood

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198503620.003.0007

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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