HARMONIC MORPHISMS FROM EINSTEIN 4-MANIFOLDS TO RIEMANN SURFACES
MARINA VILLE
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Source: Crossref
Published: May 1, 2003
DOI: 10.1142/s0129167x0300179x
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If M and N are Riemannian manifolds, a harmonic morphism f : M → N is a map which pulls back local harmonic functions on N to local harmonic functions on M. If M is an Einstein 4-manifold and N is a Riemann surface, John Wood showed that such an f is holomorphic w.r.t. some integrable complex Hermitian structure defined on M away from the singular points of f. In this paper we extend this complex structure to the entire manifold M. It follows that there are no non-constant harmonic morphisms from [Formula: see text] or [Formula: see text] to a Riemann surface. The proof relies heavily on the real analyticity of the whole situation. We conclude by an example of a non-constant harmonic morphism from [Formula: see text] to [Formula: see text].
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