On Ricci solitons and twistorial harmonic morphisms

TL;DR

This paper investigates Ricci solitons in the context of twistorial harmonic morphisms, establishing uniqueness and triviality conditions for soliton flows in specific geometric constructions.

Contribution

It provides new results on the characterization and uniqueness of soliton flows associated with twistorial harmonic morphisms in four and three dimensions.

Findings

Uniqueness of soliton flows in the Gibbons-Hawking construction

Identification of a contour integral characterizing trivial solitons in Beltrami fields

Results are valid under real-analyticity assumptions

Abstract

We study the soliton flow on the domain of a twistorial harmonic morphism between Riemannian manifolds of dimensions four and three. Assuming real-analyticity, we prove that, for the Gibbons-Hawking construction, any soliton flow is uniquely determined by its restriction to any local section of the corresponding harmonic morphism. For the Beltrami fields construction, we identify a contour integral whose vanishing characterises the trivial soliton flows.