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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.

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