On stability of the hyperbolic space form under the normalized Ricci flow

TL;DR

This paper proves that small, fast-decaying perturbations of the hyperbolic metric in dimensions greater than five will exponentially converge back to the hyperbolic space under the normalized Ricci flow.

Contribution

It establishes exponential convergence of the normalized Ricci flow to hyperbolic space for small, rapidly decaying perturbations in high dimensions.

Findings

Flow converges exponentially to hyperbolic metric

Convergence holds for perturbations decaying fast at infinity

Results are valid for dimensions greater than five

Abstract

This paper studies the normalized Ricci flow from a slight perturbation of the hyperbolic metric on $\mathbb H^n$. It's proved that if the perturbation is small and decays sufficiently fast at the infinity, then the flow will converge exponentially fast to the hyperbolic metric when the dimension $n>5$.