Title:The Calabi flow on Kähler surface with bounded Sobolev constant

Abstract: We consider the formation of singularities along the Calabi flow with the assumption of the uniform Sobolev constant. In particular, on Kähler surface we show that any "maximal bubble" has to be a scalar flat ALE Kähler metric. In some certain classes on toric Fano surface, the Sobolev constant is a priori bounded along the Calabi flow with small Calabi energy. Also we can show in certain case no maximal bubble can form along the flow, it follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently. To illustrate our results more clearly, we focus on an example on CP^2 blown up three points at generic position. Our result also implies existence of constant scalar curvature metrics on CP^2 blown up three points at generic position in the Kähler classes where the exceptional divisors have the same area.