Minoration conforme du spectre du laplacien de Hodge-de Rham

Abstract

Let $M^n$ be a n-dimensional compact manifold, with $n\ge 3$. For any conformal class C of riemannian metrics on M, we set $μ_k^c(M,C)=\inf_{g\in C}μ_{[\frac n2],k}(M,g)\Vol(M,g)^{\frac2n}$, where ${\mu }_{p,k}(M,g)$ is the k-th eigenvalue of the Hodge laplacian acting on coexact p-forms. We prove that $0<{\mu }_k^c(M,C)\le {\mu }_k^c(S^n,[g_{can}])\le k^{\frac{2}{n}}{\mu }_1^c(S^n,[g_{can}])$.

11 pages, 2 figures, in french