On the existence of harmonic morphisms from three-dimensional Lie groups

TL;DR

This paper classifies three-dimensional Riemannian Lie groups that admit harmonic morphisms to surfaces, expanding understanding of geometric structures and harmonic map theory.

Contribution

It provides a complete classification of 3D Riemannian Lie groups capable of harmonic morphisms to surfaces, a novel result in differential geometry.

Findings

Identified all 3D Riemannian Lie groups with harmonic morphisms to surfaces

Characterized the geometric conditions enabling such morphisms

Extended the theory of harmonic maps in the context of Lie groups

Abstract

In this paper we classify those three-dimensional Riemannian Lie groups which admit harmonic morphisms to surfaces.