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 $g\in C$, ${\lambda }_1(M^n,g)Vol(M^n,g{)}^{2/n}<n.Vol(S^n,g_{\text{can}}{)}^{2/n}$ and ${\sigma }_1(M,g,\rho )\mathcal{M}(\partial M)Vol(M{)}^{\frac{2-n}{n}}<n.Vol(S^n,g_{\text{can}}{)}^{2/n}$, 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 $\mathcal{M}(\partial M)={\int }_{\partial M}\rho d{v}_g$. The proof relies on a handle decomposition of the manifold. We also prove that the conformal volume of $(M,C)$ is $Vol(S^n,g_{\text{can}})$, 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.

19 pages, in French, 5 figures

References in corpus (4)

• Extremal eigenvalues of the Laplacian in a conformal class of metrics : the "conformal spectrum"
• The supremum of conformally covariant eigenvalues in a conformal class
• How large can the first eigenvalue be on a surface of genus two?
• Première valeur propre du laplacien, volume conforme et chirurgies