Let be a compact Riemannian surface, be a positive function on . The standard mean field equation can be stated as follows
in , where are given distinct points, and denotes the Dirac measure with pole at . Here the area of is assumed to be constant and stands for the Laplace Beltrami operator with respect to .
Clearly, the above PDE is invariant under adding a constant. Hence, is normalized to satisfy
Let be the Green function with pole at , that is,
In terms of , one has
Thus, and vanishes exactly at . The mean field equation now reads as follows
We will discuss some techniques related to such an equation later.