Ricci Yang-Mills flow on surfaces

TL;DR

This paper investigates the Ricci Yang-Mills flow on surfaces, establishing conditions for long-term existence and convergence to constant scalar curvature metrics, and classifying gradient solitons.

Contribution

It provides new results on the existence, long-term behavior, and classification of solutions to the Ricci Yang-Mills flow on surfaces.

Findings

Flow existence depends on isoperimetric constant bounds.

Flow exists for all time on higher genus surfaces.

Normalized flow converges to constant scalar curvature metrics.

Abstract

We study the behaviour of the Ricci Yang-Mills flow for U(1) bundles on surfaces. We show that existence for the flow reduces to a bound on the isoperimetric constant. In the presence of such a bound, we show that on $S^2$, if the bundle is nontrivial, the flow exists for all time. For higher genus surfaces the flow always exists for all time. The volume normalized flow always exists for all time and converges to a constant scalar curvature metric with the bundle curvature $F$ parallel. Finally, in an appendix we classify all gradient solitons of this flow on surfaces.