Holomorphic harmonic morphisms from four-dimensional non-Einstein manifolds

TL;DR

This paper constructs specific 4D Riemannian Lie groups with conformal foliations that are holomorphic but not Kähler, demonstrating they are not Einstein manifolds, thus addressing an open question in harmonic morphism theory.

Contribution

It provides explicit examples of non-Einstein 4D Riemannian Lie groups with holomorphic harmonic morphisms, expanding understanding of complex harmonic morphisms in non-Einstein settings.

Findings

Constructed 4D Riemannian Lie groups with conformal foliations

Foliations are holomorphic but not Kähler

These groups are proven not to be Einstein manifolds

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.