Communications in Analysis and Geometry
Volume 19 (2011)
Stability of hyperbolic space under Ricci flow
Pages: 1023 – 1047
We study the Ricci flow of initial metrics which are$C^0$-perturbations of the hyperbolic metric on $\H^n$. Ifthe perturbation is bounded in the $L^2$-sense, and small enough inthe $C^0$-sense, then we show the following: In dimensions four andhigher, the scaled Ricci harmonic map heat flow of such a metricconverges smoothly, uniformly and exponentially fast in all$C^k$-norms and in the $L^2$-norm to the hyperbolic metric as timeapproaches infinity. We also prove a related result for the Ricciflow and for the two-dimensional conformal Ricci flow.