Anomaly Flow II: Estimates and Long-time Existence for the Coupled Flow

In this follow-up to \cite{Sua25}, we extend the results to the coupled anomaly flow which incorporates the evolution of a Hermitian metric on a holomorphic vector bundle over a compact complex $3$-fold with general slope parameter ${\alpha }^{\mathrm{\prime }}$. This yields a similar sufficient smallness condition on the slope parameter ${\alpha }^{\mathrm{\prime }}$ in order to extend the flow on $[0,\tau )$ to $[0,\tau +\epsilon )$. We also augment the argument of \cite{Sua25} slightly to fix the peculiarity of the units in the condition for ${\alpha }^{\mathrm{\prime }}$ for the $k=1$ case.