The global existence and convergence of the Calabi flow on Cn/Zn+iZn

In this note, we study the long time existence of the Calabi flow on X=Cn/Zn+iZn. Assuming the uniform bound of the total energy, we establish the non-collapsing property of the Calabi flow by using Donaldsonʼs estimates and Streetsʼ regularity theorem. Next we show that the curvature is uniformly bounded along the Calabi flow on X when the dimension is 2, partially confirming Chenʼs 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 Donaldsonʼs conjecture.