Uniqueness of solutions for a mean field equation on torus

We consider on a two-dimensional flat torus T the following equation When the fundamental domain of the torus is (0,a)×(0,b) (a b), we establish that the constants are the unique solutions whenever and this result is sharp if . A similar conclusion is obtained for general two-dimensional torus by considering the length of the shortest closed geodesic. These results are derived by comparing the isoperimetric profile of the torus T with the one of the two-dimensional canonical sphere which has same volume as T.