The Calabi flow on Kähler Surfaces with bounded Sobolev constant (I)

Abstract

We consider the formation of singularities along the Calabi flow by assuming the uniformly bounded Sobolev constants. On Kahler surfaces we prove that if curvature tensor is not uniformly bounded, then one can form a singular model called deepest bubble; such deepest bubble has to be a scalar flat ALE Kahler metric. In certain Kahler classes on toric Fano surfaces, the Sobolev constants are a priori bounded along the Calabi flow with small Calabi energy. We can also show in certain cases no deepest 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 \({\mathbb{CP}^2}\) blown up three points at generic position. Our result also implies existence of constant scalar curvature metrics on \({\mathbb{CP}^2}\) blown up three points at generic position in the Kahler classes where the exceptional divisors have the same area.