Ricci flow negative curvature

Question

We denote by $\mathbb{H}^n$ the hyperbolic plane of dimension $n$.

I don't know much about the Ricci flow, so my first question is probably naïve : can one define a normalized Ricci flow such that the hyperbolic metric on $\mathbb{H}^n$ is a stationary solution (constant in time) of the flow ?

My second question, Let $g$ be a compact perturbation of the hyperbolic metric. By compact I mean a smooth change of the metric on a compact set. If the previously defined flow exists, does the solution of this Ricci flow with initial data $g$ converge as $t \to \infty$ toward the hyperbolic metric ?

Answer 1

The answer to your first question is Yes. The equation $$ \tag{*} \partial_t g = -2\textrm{Ric} - 2(n-1)g $$ is a Ricci flow type equation that admits hyperbolic space as a static solution. In fact, if you have solution to (*) you can rescale time/space to obtain a Ricci flow and vice versa. This is proven in Lemma A.3 here.

For your second question, you need to assume that the change of metric on a compact set is "small" in some sense. Otherwise you could have singularities, etc on the compact set (although in $n=3$ you might be able to construct a flow with surgery that eventually converges back to hyperbolic space plus some other pieces, I don't think this has been done although I think its doable with known technology). However, if you assume $C^0$ closeness, then what you ask is proven by Schnurer--Schulze--Simon (see also Bamler for a generalization and further references)