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Consider a compact surface $M$ of genus $\gamma > 1$ (I am using the more usual letter "$g$" to denote metric), and the normalized Ricci flow on it. It is known that at time $t$, the scalar curvature $R$ satisfies $$ |R - r| < Ce^{rt}, $$ where $r = \frac{\int_M R d\mu}{\int_M d\mu}$ is the average scalar curvature of $M$, and $C$ is a constant depending only on the initial metric $g_0$.

I was wondering if certain special examples are known where the convergence takes place much more quickly under the normalized Ricci flow. For example, a nice answer could be: there is this special surface of genus $\gamma_1$ with special starting metric $g'_0$ such that at time $t$, we have $$ |R - r| < Ce^{-e^{-rt}}. $$ Any ideas, references, will be highly appreciated.

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