In the first part of the talk, I will mention some
convergence, stability results for the Ricci flow equation on compact
manifolds and the uniqueness of a limiting metric to the Ricci flow
equation. In the second part of the talk, I will show a relation between
the Ricci flow equation on R^2 and the logarithmic fast diffusion
equation of a conformal factor. I will discuss the asymptotics of the
Ricci flow on R^2 in the case of a finite initial mass.