Results on coupled Ricci and harmonic map flows

TL;DR

This paper studies the harmonic-Ricci flow, a coupling of Ricci and harmonic map flows, exploring its geometric context, solitons, compactness properties, and explicit solutions on the Lie group Nil^3.

Contribution

It introduces the harmonic-Ricci flow as a natural geometric flow, relates it to locally R^N-invariant Ricci flow, and extends Hamilton's compactness theorem to groupoids.

Findings

Identification of harmonic-Ricci flow within principal bundle constructions

Existence of gradient solitons for the flow

Generalization of compactness theorem to e1tale Riemannian groupoids

Abstract

We explore the harmonic-Ricci flow---that is, Ricci flow coupled with harmonic map flow---both as it arises naturally in certain principal bundle constructions related to Ricci flow and as a geometric flow in its own right. We demonstrate that one natural geometric context for the flow is a special case of the locally $\mathbb{R}^N$-invariant Ricci flow of Lott, and provide examples of gradient solitons for the flow. We prove a version of Hamilton's compactness theorem for the flow, and then generalize it to the category of \'{e}tale Riemannian groupoids. Finally, we provide a detailed example of solutions to the flow on the Lie group $\Nil^3$.