On convergence criteria for the coupled flow of Li-Yuan-Zhang

Abstract: A one-parameter family of coupled flows depending on a parameter $\kappa>0$ is introduced which reduces when $\kappa=1$ to the coupled flow of a metric $\omega$ with a $(1,1)$-form $\alpha$ due recently to Y. Li, Y. Yuan, and Y. Zhang. It is shown in particular that, for $\kappa\not=1$, estimates for derivatives of all orders would follow from $C^0$ estimates for $\omega$ and $\alpha$. Together with the monotonicity of suitably adapted energy functionals, this can be applied to establish the convergence of the flow in some situations, including on Riemann surfaces. Very little is known as yet about the monotonicity and convergence of flows in presence of couplings, and conditions such as $\kappa\not=1$ seem new and may be useful in the future.