Spectre et g\'eom\'etrie conforme des vari\'et\'es compactes \`a bord

We prove that on any compact manifold $M^n$ with boundary, there exist a conformal class $C$ such that for any riemannian metric $g\in C$, $\lambda_1(M^n,g)Vol(M^n,g)^{2/n}< n.Vol(S^n,g_{\textrm{can}})^{2/n}$ and $\sigma_1(M,g,\rho)\mathcal M(\partial M)Vol(M)^{\frac{2-n}n}<n.Vol(S^n,g_{\textrm{can}})^{2/n}$, where $\lambda_1(M^n,g)$ denotes the first positive eigenvalue of the Neumann laplacian on $(M,g)$, $\sigma_1(M,g,\rho)$ the first positive Steklov eigenvalue for the density $\rho$ on $\partial M$, and $\mathcal M(\partial M)=\int_{\partial M}\rho dv_g$. The proof relies on a handle decomposition of the manifold. We also prove that the conformal volume of $(M,C)$ is $Vol(S^n,g_{\textrm{can}})$, and that the Friedlander-Nadirashvili and the M\"obius volume of $M$ are equal to those of the sphere. If $M$ is a domain in a space form, $C$ is the conformal class of the canonical metric.