Spectre et géométrie conforme des variétés compactes à bord

Abstract

We prove that on any compact manifold $M^n$ with boundary, there exist a conformal class $C$ such that for any riemannian metric $g\in C$, $λ_1(M^n,g)Vol(M^n,g)^{2/n}< n.Vol(S^n,g_{\textrm{can}})^{2/n}$ and $σ_1(M,g,ρ)\mathcal M(\partial M)Vol(M)^{\frac{2-n}n}<n.Vol(S^n,g_{\textrm{can}})^{2/n}$, where $λ_1(M^n,g)$ denotes the first positive eigenvalue of the Neumann laplacian on $(M,g)$, $σ_1(M,g,ρ)$ the first positive Steklov eigenvalue for the density $ρ$ on $\partial M$, and $\mathcal M(\partial M)=\int_{\partial M}ρdv_g$. 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 Möbius 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.