Results on coupled Ricci and harmonic map flows

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 étale Riemannian groupoids. Finally, we provide a detailed example of solutions to the flow on the Lie group $\Nil^3$.

25 pages. Results on stability have been moved to a different paper. Exposition has been updated and typos corrected