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

TL;DR

This paper introduces a family of coupled geometric flows depending on a parameter, analyzing their convergence and estimates, especially for cases where the parameter differs from one, with applications to Riemann surfaces.

Contribution

It defines a new one-parameter family of coupled flows generalizing previous work and explores their convergence properties and estimates, highlighting the case when the parameter is not equal to one.

Findings

Estimates for derivatives follow from $C^0$ bounds when $ eq 1$.

Monotonicity of energy functionals aids in establishing flow convergence.

Flow convergence demonstrated on Riemann surfaces.

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.