Uniqueness and Symmetry for the Mean Field Equation on Arbitrary Flat Tori

We study the following mean field equation on a flat torus T := C/(Z + Z tau): Delta u + rho(e(u)/integral(T)e(u) - 1/vertical bar T vertical bar) = 0, where tau is an element of C, Im tau > 0, and vertical bar T vertical bar denotes the total area of the torus. We first prove that the solutions are evenly symmetric about any critical point of u provided that rho <= 8 pi. Based on this crucial symmetry result, we are able to establish further the uniqueness of the solution if rho <= min {8 pi, lambda(1) (T)vertical bar T vertical bar}. Furthermore, we also classify all one-dimensional solutions by showing that the level sets must be closed geodesics.