The Calabi flow on K\"ahler surface with bounded Sobolev constant

Understand

We consider the formation of singularities along the Calabi flow with the assumption of the uniform Sobolev constant.

• In particular, on K\"ahler surface we show that any "maximal bubble" has to be a scalar flat ALE K\"ahler 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.