Uniqueness of solutions for a mean field equation on torus

We consider on a two-dimensional flat torus T the following equationΔu+ρ(eu∫Teu−1|T|)=0. When the fundamental domain of the torus is (0,a)×(0,b)(0,a)×(0,b)(a⩾b)(a⩾b), we establish that the constants are the unique solutions wheneverρ⩽{8πifba⩾π4,32baifba⩽π4, and this result is sharp if ba⩾π4. 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.