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

TL;DR

This paper studies the Calabi flow on K"ahler surfaces, showing under certain conditions that the flow exists globally and converges to extremal metrics, with detailed analysis of singularity formation and specific examples.

Contribution

It proves that on K"ahler surfaces with bounded Sobolev constant, the Calabi flow avoids singularities and converges to extremal metrics, especially on blow-ups of CP^2.

Findings

Maximal bubbles are scalar flat ALE K"ahler metrics.

Under certain conditions, the flow remains smooth and converges.

Existence of constant scalar curvature metrics on blow-ups of CP^2.

Abstract

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. 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 K\"ahler classes where the exceptional divisors have the same area.