The global existence and convergence of the Calabi flow on ${\mathbb{C}}^n/{\mathbb{Z}}^n+i{\mathbb{Z}}^n$

Abstract

In this note, we study the long time existence of the Calabi flow on $X={\mathbb{C}}^n/{\mathbb{Z}}^n+i{\mathbb{Z}}^n$. Assuming the uniform bound of the total energy, we establish the non-collapsing property of the Calabi flow by using Donaldson's estimates and Streets' regularity theorem. Next we show that the curvature is uniformly bounded along the Calabi flow on $X$ when the dimension is 2, partially confirming Chen's conjecture. Moreover, we show that the Calabi flow exponentially converges to the flat Kähler metric for arbitrary dimension if the curvature is uniformly bounded, partially confirming Donaldson's conjecture.

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• Ricci Flow and the Poincare Conjecture
• Interior estimates for solutions of Abreu's equation
• The Calabi flow on Kähler surface with bounded Sobolev constant
• The Calabi flow with small initial energy
• Stability of Calabi flow near an extremal metric
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Cited by in corpus (3)

• Convexity of the extended K-energy and the large time behaviour of the weak Calabi flow
• Kähler-Ricci flow, Kähler-Einstein metric, and K-stability
• Kähler non-collapsing, eigenvalues and the Calabi flow