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

TL;DR

This paper establishes conditions for the existence of solutions to a mean field equation on a closed Riemannian surface with symmetry group actions, generalizing previous results for trivial group actions.

Contribution

It provides a new sufficient condition for the existence of solutions to the mean field equation considering isometric group actions, extending earlier work to more symmetric settings.

Findings

Derived a sufficient condition for solution existence.

Generalized previous results to nontrivial symmetry groups.

Connected solutions to geometric symmetry properties.

Abstract

Let $(\Sigma,g)$ be a closed Riemannian surface, $\textbf{G}=\{\sigma_1,\cdots,\sigma_N\}$ 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 $$\Delta_g u=8\pi\ell\left(\frac{he^u}{\int_\Sigma he^udv_g}-\frac{1}{{\rm Vol}_g(\Sigma)}\right)$$ is given. This recovers results of Ding-Jost-Li-Wang (Asian J Math 1997) when $\ell=1$ or equivalently $\textbf{G}=\{Id\}$, where $Id$ is the identity map.