On a rigidity result for the first conformal eigenvalue of the Laplacian

Given (M, g) a smooth compact Riemannian manifold without boundary of dimension n >= 3, we consider the first conformal eigenvalue which is by definition the supremum of the first eigenvalue of the Laplacian among all metrics conformal to g of 2 volume 1. We prove that it is always greater than n omega(n/2)(n), the value it takes in the conformal class of the round sphere, except if (M, g) is conformally diffeomorphic to the standard sphere.