Holomorphic harmonic morphisms from four-dimensional non-Einstein manifolds

2014/05/20 by Gudmundsson, Sigmundur
#53C12 #53C43 #58E20 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1405.5053

Abstract

We construct 4-dimensional Riemannian Lie groups carrying left-invariant conformal foliations with minimal leaves of codimension 2. We show that these foliations are holomorphic with respect to an (integrable) Hermitian structure which is not K" ahler. We then prove that the Riemannian Lie groups constructed are \it not Einstein manifolds. This answers an important open question in the theory of complex-valued harmonic morphisms from Riemannian 4-manifolds.

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