Harmonic morphisms from three-dimensional Euclidean and spherical space forms

TL;DR

This paper classifies all harmonic morphisms from three-dimensional Euclidean and spherical space forms to surfaces, showing they are compositions of standard morphisms and weakly conformal maps.

Contribution

It extends previous classifications by describing all harmonic morphisms from any three-dimensional Euclidean or spherical space form to surfaces.

Findings

All such harmonic morphisms are compositions of standard morphisms and weakly conformal maps.

The paper lists all standard harmonic morphisms for these space forms.

Provides a complete description of harmonic morphisms from these space forms to surfaces.

Abstract

P. Baird and the second author studied harmonic morphisms from a three-dimensional simply-connected space form to a surface and obtained a complete local and global classification of them. In this paper, we obtain a description of all harmonic morphisms from any three-dimensional Euclidean and spherical space form to a surface, namely that any such harmonic morphism is the composition of a standard harmonic morphism and a weakly conformal map. We list the standard harmonic morphisms.