Long time existence of Ricci flow on compact surfaces of negative curvature

Question

Is there a long time existence for the Ricci flow on compact negatively curved surfaces? I just read that the normalized Ricci flow has a long time solution converging to a metric of constant negative curvature. I was wondering what happens to the ordinary Ricci flow, or if there is any condition under which one has long time existence? Thanks!

Answer 1

The normalized Ricci flow $\partial_\tau \hat g = - 2 \text{Rc}_{\hat g}+\int \text{Scal}_\hat g d\mu_\hat g$ is related to the standard Ricci flow $\partial_t g = -2 \text{Rc}_g$ by $$\hat g(\tau) = \frac{g(t(\tau))}{\text{Vol}(t(\tau))}$$ where the rescaled time satisfies $t'(\tau) = \text{Vol}(t(\tau))$. Since we are working on a surface $\Sigma$, we have

$$ \frac{\partial}{\partial t} d\mu_g = -\text{Scal}_gd\mu_g = -2\kappa_g d\mu_g$$

and thus Gauss-Bonnet gives us

$$ \text{Vol}'(t) = -4 \pi \chi(\Sigma) \implies \text{Vol}(t) = \text{Vol}(0) - 4 \pi \chi t.$$

Thus if we fix $t(0) = 0$, the rescaled time is $$t(\tau) = \frac{\text{Vol}(0)}{4 \pi \chi}\left(1 - e^{-4\pi \chi \tau}\right).$$

Since $\Sigma$ has negative curvature, we know $\chi < 0$ and thus $t \to +\infty$ as $\tau \to +\infty$; so the long-time existence of the normalized Ricci flow implies that of the standard Ricci flow. Since the volume grows linearly, we don't get convergence of $g(t)$; but we get "convergence of shape", since $\hat g(\tau)$ and $g(t(\tau))$ differ only by a scale factor.