Critical sinh-Gordon flow with non-negative weight functions

The aim of this article is twofold: on one side we introduce and study the properties of a critical sinh-Gordon type flow ∂∂teu=Δgu+8π(h1eu∫Σh1eudVg−1)−ρ2(h2e−u∫Σh2e−udVg−1), 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.