Communications in Analysis and Geometry

Volume 19 (2011)

Number 5

Stability of hyperbolic space under Ricci flow

Pages: 1023 – 1047



Oliver C. Schnürer (Fachbereich Mathematik und Statistik, Universität Konstanz, Germany)

Felix Schulze (Freie Universität Berlin, Germany)

Miles Simon (Institut für Analysis und Numerik, Magdeburg, Germany)


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.

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