The Kähler Ricci flow on Fano surfaces (I)

Suppose {(M, g(t)), 0 ≤ t < ∞} is a Kähler Ricci flow solution on a Fano surface. If {pipre}Rm{pipre} is not uniformly bounded along this flow, we can blowup at the maximal curvature points to obtain a limit complete Riemannian manifold X. We show that X must have certain topological and geometric properties. Using these properties, we are able to prove that {pipre}Rm{pipre} is uniformly bounded along every Kähler Ricci flow on toric Fano surface, whose initial metric has toric symmetry. In particular, such a Kähler Ricci flow must converge to a Kähler Ricci soliton metric. Therefore we give a new Ricci flow proof of the existence of Kähler Ricci soliton metrics on toric Fano surfaces.