Regularity scales and convergence of the Calabi flow

TL;DR

This paper introduces regularity scales to analyze the Calabi flow, proving convergence on extremal Kähler surfaces and supporting Donaldson's conjecture, with extensions to higher dimensions under scalar curvature bounds.

Contribution

It defines regularity scales for the Calabi flow and establishes convergence results, partially confirming Donaldson's conjecture in complex dimension 2.

Findings

Convergence of the Calabi flow on extremal Kähler surfaces.

Partial confirmation of Donaldson's conjecture in complex dimension 2.

Extension of results to higher dimensions with scalar curvature bounds.

Abstract

We define regularity scales to study the behavior of the Calabi flow. Based on estimates of the regularity scales, we obtain convergence theorems of the Calabi flow on extremal Kahler surfaces, under the assumption of global existence of the Calabi flow solutions. Our results partially confirm Donaldson's conjectural picture for the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra assumption that the scalar curvature is uniformly bounded.