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

Let (Σ , g) be a closed Riemannian surface, G = { σ 1 , ... , σ N } be an isometric group acting on it. Denote a positive integer ℓ = inf x ∈ Σ I (x) , where I (x) is the number of all distinct points of the set { σ 1 (x) , ... , σ N (x) }. A sufficient condition for existence of solutions to the mean field equation Δ g u = 8 π ℓ h e u ∫ Σ h e u d v g − 1 Vol g (Σ) is given. This recovers results of Ding–Jost–Li–Wang, Asian J. Math. (1997) 230–248 when ℓ = 1 or equivalently G = { Id } , where Id is the identity map. [ABSTRACT FROM AUTHOR]