Mean field equations on a closed Riemannian surface with the action of an isometric group

Abstract

Let $(\Sigma ,g)$ be a closed Riemannian surface, $\text{G}=\{{\sigma }_1,\cdots ,{\sigma }_N\}$ be an isometric group acting on it. Denote a positive integer $\ell ={\inf }_{x\in \Sigma }I(x)$, where $I(x)$ is the number of all distinct points of the set $\{{\sigma }_1(x),\cdots ,{\sigma }_N(x)\}$. A sufficient condition for existence of solutions to the mean field equation $${\Delta }_{g}u=8\pi \ell \left(\frac{h{e}^u}{{\int }_{\Sigma }h{e}^{u}d{v}_g}-\frac{1}{{\mathrm{Vol}}_g(\Sigma )}\right)$$ is given. This recovers results of Ding-Jost-Li-Wang (Asian J Math 1997) when $\ell =1$ or equivalently $\text{G}=\{Id\}$, where $Id$ is the identity map.

23 pages