Harmonic morphisms from solvable Lie groups

TL;DR

This paper introduces two novel methods for constructing harmonic morphisms from solvable Lie groups, expanding the classes of spaces where such mappings can be globally defined, and identifies specific groups lacking these morphisms.

Contribution

The paper presents two new methods for constructing harmonic morphisms from solvable Lie groups, applicable to various classes including nilpotent, Damek-Ricci, and symmetric spaces.

Findings

Global solutions from nilpotent Lie groups and symmetric spaces of rank ≥ 3

Global solutions from Damek-Ricci spaces and many symmetric spaces

Existence of 3D solvable Lie groups without complex harmonic morphisms

Abstract

In this paper we introduce two new methods for constructing harmonic morphisms from solvable Lie groups. The first method yields global solutions from any simply connected nilpotent Lie group and from any Riemannian symmetric space of non-compact type and rank $r\ge 3$. The second method provides us with global solutions from any Damek-Ricci space and many non-compact Riemannian symmetric spaces. We then give a continuous family of 3-dimensional solvable Lie groups not admitting any complex valued harmonic morphisms, not even locally.