Uniqueness of solutions for a mean field equation on torus

We consider on a two-dimensional flat torus T the following equation [GRAPHIC] When the fundamental domain of the torus is (0, a) x (0, b) (a >= b), we establish that the constants are the unique solutions whenever [GRAPHIC] and this result is sharp if b/a >=, pi/4. 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. (c) 2005 Elsevier Inc. All rights reserved.