Stability of hyperbolic space under Ricci flow

Abstract

We study the Ricci flow of initial metrics which are C0-perturbations of the hyperbolic metric on Hn. If the perturbation is bounded in the L2-sense, and small enough in the C0-sense, then we show the following: In dimensions four and higher, the scaled Ricci harmonic map heat flow of such a metric converges smoothly, uniformly and exponentially fast in all Cknorms and in the L2-norm to the hyperbolic metric as time approaches infinity. We also prove a related result for the Ricci flow and for the two-dimensional conformal Ricci flow.