Critical sinh-Gordon flow with non-negative weight functions

TL;DR

This paper introduces and analyzes a critical sinh-Gordon flow on Riemannian surfaces with non-negative weights, proving convergence under certain conditions and extending previous results.

Contribution

It extends the analysis of sinh-Gordon flows to include non-negative weights and proves convergence to solutions, broadening the scope of prior work.

Findings

Flow converges to solutions under geometric conditions

Extension of Zhou's 2008 results to non-negative weights

Careful blow-up analysis supports convergence proof

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=\Delta_gu+8\pi\left({\frac{h_1e^u}{\int_{\Sigma}h_1e^udV_g}}-1\right)-\rho_2\left({\frac{h_2e^{-u}}{\int_{\Sigma}h_2e^{-u}dV_g}}-1\right), \end{equation*} where $\rho_2<8\pi$, $h_1,h_2$ are non-negative weight functions and $\Sigma$ 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.