Harmonic morphisms with fibers of dimension one

TL;DR

This paper classifies harmonic morphisms with one-dimensional fibers in higher dimensions, providing local and global results using differential systems and moving frames, especially for space forms with constant curvature.

Contribution

It offers new classification theorems for harmonic morphisms with one-dimensional fibers, including local, finite, and global classifications in higher dimensions.

Findings

Local classification for harmonic morphisms with specified target metric

Finiteness theorem for such morphisms with specified domain metric

Complete classification when the domain is a space form of constant curvature

Abstract

I prove three classification results about harmonic morphisms whose fibers have dimension one. All are valid when the domain is at least of dimension 4. (The character of this overdetermined problem is very different when the dimension of the domain is 3 or less.) The first result is a local classification for such harmonic morphisms with specified target metric, the second is a finiteness theorem for such harmonic morphisms with specified domain metric, and the third is a complete classification of such harmonic morphisms when the domain is a space form of constant sectional curvature The methods used are exterior differential systems and the moving frame. The fundamental results are local, but, because of the rigidity of the solutions, they allow a complete global classification.