Consider a compact surface M of genus γ>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|<Cert,

where r=∫MRdμ∫Mdμ is the average scalar curvature of M, and C is a constant depending only on the initial metric g0.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 γ1 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.

**Answer**

**Attribution***Source : Link , Question Author : user81712 , Answer Author : Community*