Minoration conforme du spectre du laplacien de Hodge-de Rham

Let M ⁿ be an n-dimensional compact manifold, with n ≥ 3. For any conformal class C of riemannian metrics on M, we set , where μ _p,k (M,g) is the kth eigenvalue of the Hodge laplacian acting on coexact p-forms. We prove that . We also prove that if g is a smooth metric such that , and n = 0,2,3 mod 4, then there is a non-zero corresponding eigenform of degree with constant length. As a corollary, on a four-manifold with non vanishing Euler characteristic, there is no such smooth extremal metric.