A Mean Field Type Flow on a Closed Riemannian Surface with the Action of an Isometric Group

Abstract

Let $(\Sigma,g)$ be a closed Riemannian surface, $\mathbf{G} = \{ \sigma_1,\ldots,\sigma_N \}$ be an isometric group acting on it. Denote a positive integer $\ell = \min_{x \in \Sigma} I(x)$, where $I(x)$ is the number of all distinct points of the set $\{ \sigma_1(x),\ldots,\sigma_N(x) \}$. By a method of flow due to Castéras (Pacific J. Math. 2015), we prove that the solution to the mean field equation \[ -\Delta_g u = 8\pi \ell \left( \frac{he^u}{\int_{\Sigma} he^u \, dv_g} - \frac{1}{\operatorname{Vol}_g(\Sigma)} \right) \] exists under given conditions. This gives a new proof of Yang and Zhu's result in (Internat. J. Math. 2020). The case $\ell = 1$ was studied by Li and Zhu (Calc. Var. Partial Differential Equations 2019).