Equivariant mean field flow

We consider a gradient flow associated to the mean field equation on (M,g)(M,g), a compact Riemannian surface without boundary. We prove that this flow exists for all time. Moreover, letting GG be a group of isometry acting on (M,g)(M,g), we obtain the convergence of the flow to a solution of the mean field equation under suitable hypothesis on the orbits of points of MM under the action of GG.