Minoration conforme du spectre du laplacien de Hodge-de Rham

Abstract

Let $M^n$ be a n-dimensional compact manifold, with $n\geq3$. For any conformal class C of riemannian metrics on M, we set $\mu_k^c(M,C)=\inf_{g\in C}\mu_{[\frac n2],k}(M,g)\Vol(M,g)^{\frac2n}$, where $\mu_{p,k}(M,g)$ is the k-th eigenvalue of the Hodge laplacian acting on coexact p-forms. We prove that $0<\mu_k^c(M,C)\leq\mu_k^c(S^n,[g_{can}])\leq k^{\frac2n}\mu_1^c(S^n,[g_{can}])$.