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

TL;DR

This paper investigates the long-term behavior of the Calabi flow on a specific complex manifold, establishing conditions for its existence, curvature bounds, and convergence to flat metrics, thereby confirming parts of two conjectures.

Contribution

It proves the long-time existence, curvature bounds, and exponential convergence of the Calabi flow on certain complex manifolds, partially confirming two major conjectures in the field.

Findings

Calabi flow exists long-term under energy bounds

Curvature remains bounded in dimension 2

Flow converges exponentially to flat metrics under curvature bounds

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 K\"ahler metric for arbitrary dimension if the curvature is uniformly bounded, partially confirming Donaldson's conjecture.