Stability of the Ricci flow at Ricci-flat metrics

Abstract

If g is a metric whose Ricci flow g(t) converges, one may ask if the same is true for metrics g̃ that are small perturbations of g. We use maximal regularity theory and center manifold analysis to study flat and Ricci-flat metrics. We show that if g is flat, there is a unique exponentially-attractive center manifold at g consisting entirely of equilibria for the flow. Adding a continuity argument, we prove stability for any metric whose Ricci flow converges to a flat metric. We obtain a slightly weaker stability result for a Kähler-Einstein metric on a K3 manifold. 1 Partially supported by NSF grant PHY9800732.