Stability of hyperbolic space under Ricci flow

TL;DR

This paper demonstrates that small C^0-perturbations of hyperbolic metrics on H^n in dimensions four and higher evolve under Ricci flow to converge smoothly and exponentially fast back to the hyperbolic metric, showing stability.

Contribution

It establishes the exponential convergence of Ricci flow for bounded L^2-perturbations of hyperbolic metrics in higher dimensions, extending stability results.

Findings

Convergence of Ricci harmonic map heat flow to hyperbolic metric

Exponential decay in all C^k-norms and L^2-norms

Results for Ricci flow and 2D conformal Ricci flow

Abstract

We study the Ricci flow of initial metrics which are C^0-perturbations of the hyperbolic metric on H^n. If the perturbation is bounded in the L^2-sense, and small enough in the C^0-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 C^k-norms and in the L^2-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.