Anomaly Flow: Shi-Type Estimates and Long-time Existence

We consider the long-time existence of the anomaly flow on a compact complex $3$-fold with general slope parameter ${\alpha }^{\mathrm{\prime }}$. In particular, we obtain integral Shi-type estimates for the flow by adapting a integration-by-parts type argument instead of the usual maximum principle techniques. Following this, we prescribe a sufficient smallness condition on ${\alpha }^{\mathrm{\prime }}$ in order to extend the flow on $[0,\tau )$ to $[0,\tau +\epsilon )$.