Critical sinh-Gordon flow with non-negative weight functions

The aim of this article is twofold: one one side we introduce and study the properties of a critical sinh-Gordon type flow \begin{equation*} {\frac{\partial}{\partial t}}e^u=Δ_gu+8π\left({\frac{h_1e^u}{\int_Σh_1e^udV_g}}-1\right)-ρ_2\left({\frac{h_2e^{-u}}{\int_Σh_2e^{-u}dV_g}}-1\right), \end{equation*} where ${\rho }_2<8\pi$, $h_1,h_2$ are non-negative weight functions and $\mathrm{\Sigma }$ is a closed Riemannian surface. Secondly, under suitable geometric conditions, we prove the convergence of the flow to a solution of the critical sinh-Gordon equation, extending the result of Zhou (2008) to the case of non-negative weights. The argument is based on a careful blow-up analysis. Some remarks about a Toda flow are also given.