Stability of the Ricci Yang-mills Flow at Einstein Yang-mills Metrics

Let P be a principal U(1)-bundle over a closed manifold M . On P , one can define a modified version of the Ricci flow called the Ricci Yang-Mills flow, due to these equations being a coupling of Ricci flow and the Yang-Mills heat flow. We use maximal regularity theory and ideas of Simonett concerning the asymptotic behavior of abstract quasilinear parabolic partial differential equations to study the stability of the volume-normalized Ricci Yang-Mills flow at Einstein Yang-Mills metrics in dimension two. In certain cases, we show the presence of a center manifold of fixed points, while in others, we show the existence of an asymptotically stable fixed point. By writing the Ricci flow equations for a metric on a U(1)-bundle in the KaluzaKlein ansatz with fixed fiber size, one obtains a coupled system of equations–an equation that resembles the Ricci flow on the base metric and an equation that resembles the Yang-Mills heat flow on the connection 1-form. We call this coupled system of equations the Ricci Yang-Mills flow. Recall that the Yang-Mills heat flow is well-behaved in low dimensions, while the Ricci flow can become singular even in dimension two. Thus, one hopes to exploit the nice behavior of the Yang-Mills heat flow part of the system to obtain convergence results of the Ricci Yang-Mills flow. In 1982, Richard Hamilton [7] proposed the Ricci flow as a means to study 3manifolds with positive Ricci curvature. Specifically, let (M, g) be an n-dimensional Riemannian manifold with metric g. The Ricci flow equations are defined to be