Three-dimensional Ricci solitons which project to surfaces

TL;DR

This paper investigates 3D Ricci solitons that project onto surfaces via semi-conformal maps, providing a new framework for constructing and classifying such solitons, including non-gradient examples on Nil and Sol geometries.

Contribution

It introduces a reformulation of Ricci soliton equations using parameters of semi-conformal maps, enabling explicit construction and classification of solitons on all 3D geometries.

Findings

Complete classification of Ricci solitons on 3D geometries.

Existence of non-gradient solitons on Nil and Sol.

New ansatz for constructing solitons from surface data.

Abstract

We study $3$-dimensional Ricci solitons which project via a semi-conformal mapping to a surface. We reformulate the equations in terms of parameters of the map; this enables us to give an ansatz for constructing solitons in terms of data on the surface. A complete description of the soliton structures on all the $3$-dimensional geometries is given, in particular, non-gradient solitons are found on Nil and Sol.