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 {equation*} {∂/∂ t}e^u=Δ_gu+8π({h_1e^u/∫_Σh_1e^udV_g}-1)-ρ_2({{h_2e⁻ᵘ}{∫_Σh_2e⁻ᵘdV_g}}-1), {equation*} where ρ₂<8π, h₁,h₂ are non-negative weight functions and $Σ$ 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.