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 g 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 Kahler-Einstein metric on a KS manifold.