Critical sinh-Gordon flow with non-negative weight functions
Qiang Fei, Aleks Jevnikar, Sang-Hyuck Moon
TL;DR
This work analyzes a critical sinh-Gordon flow on a closed surface with nonnegative weights, proving global existence and uniqueness for any initial data. It develops a detailed blow-up analysis to understand the asymptotic behavior of time-slices and derives a sharp energy lower bound in blow-up regimes, expressed via a singular mean-field functional and Green function data. Under a geometric condition, the authors construct a sub-barrier to prevent blow-up and demonstrate convergence of the flow to a stationary critical sinh-Gordon solution using a barrier argument and the Łojasiewicz–Simon gradient inequality. The results extend previous critical-case results to nonnegative weights and illuminate connections to Toda-type systems through the flow framework.
Abstract
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 $ρ_2<8π$, $h_1,h_2$ 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.