Stability of 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 Ricciflow called the Ricci Yang–Mills flow, due to theseequations being a coupling of Ricci flow and theYang–Mills heat flow. We use maximal regularity theory andideas of Simonett concerning the asymptotic behavior ofabstract quasilinear parabolic partial differentialequations to study the stability of the volume-normalizedRicci Yang–Mills flow at Einstein Yang–Mills metrics indimension two. In certain cases, we show the presence of acenter manifold of fixed points, whereas in others, we showthe existence of an asymptotically stable fixed point.

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Published 1 January 2010