On the extension and smoothing of the Calabi flow on complex tori

TL;DR

This paper investigates the Calabi flow on complex tori, introducing a new method to bound curvature and demonstrating immediate smoothing of weak Kähler metrics in two dimensions, linking to Mabuchi energy minimizers.

Contribution

It develops an explicit curvature bound for the Calabi flow and proves immediate smoothing for weak Kähler metrics when dimension is two.

Findings

Curvature of Calabi flow can be explicitly bounded.

Weak Kähler metrics become smooth immediately in 2D.

Weak minimizers of Mabuchi energy are smooth in this setting.

Abstract

In this paper, we continue to study the Calabi flow on complex tori. We develop a new method to obtain an explicit bound of the curvature of the Calabi flow. As an application, we show that when $n=2$, the Calabi flow starting from a weak K\"ahler metric will become smooth immediately. It implies that in our settings, the weak minimizer of the Mabuchi energy is a smooth one.