Stability of Ricci Yang—Mills flow at EinsteinYang—Mills metrics

摘要

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, whereas in others, we show the existence of an asymptotically stable fixed point.