Minoration conforme du spectre du laplacien de Hodge-de Rham

Abstract

Let Mn be an n-dimensional compact manifold, with n ≥ 3. For any conformal class C of riemannian metrics on M, we set \({\mu_k^c(M,C)=\inf_{g\in C}\mu_{\left[\frac n2\right],k}(M,g)\,{\rm Vol}(M,g)^{\frac2n}}\) , where μp,k(M,g) is the kth eigenvalue of the Hodge laplacian acting on coexact p-forms. We prove that \({0 < \mu_k^c(M,C)\leq \mu_k^c(S^n,[g_{\rm can}])\leq k^{\frac2n}\mu_1^c(S^n,[g_{\rm can}])}\) . We also prove that if g is a smooth metric such that \({\mu_{\left[\frac n2\right],1}(M,g)\,{\rm Vol}(M,g)^{\frac2n}=\mu_1^c(M,[g])}\) , and n = 0,2,3 mod 4, then there is a non-zero corresponding eigenform of degree \({\left[\frac{n-1}2\right]}\) with constant length. As a corollary, on a four-manifold with non vanishing Euler characteristic, there is no such smooth extremal metric.