Ricci flows on surfaces related to the Einstein Weyl and Abelian vortex equations

TL;DR

This paper explores specific geometric equations on surfaces related to Einstein Weyl and Abelian vortex equations, demonstrating their preservation under Ricci flow and providing explicit solutions including solitons and singular flows.

Contribution

It introduces a unified framework for equations involving metrics and Killing fields, showing their invariance under Ricci flow and explicitly solving these equations.

Findings

Explicit solutions including steady gradient Ricci solitons and sausage metrics

Discovery of eternal, ancient, and immortal Ricci flows

Examples of Ricci flows with conical singularities

Abstract

There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of D. Calderbank) and a real version of a special case of the Abelian vortex equations, and it is shown that the property that a metric solve these equations is preserved by the Ricci flow. The equations are solved explicitly, and among the metrics obtained are all steady gradient Ricci solitons (e.g. the the cigar soliton) and the sausage metric; there are found other examples of eternal, ancient, and immortal Ricci flows, as well as some Ricci flows with conical singularities.