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

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