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Spectre et géométrie conforme des variétés compactes à bord
- Part of
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- Journal:
- Compositio Mathematica / Volume 150 / Issue 12 / December 2014
- Published online by Cambridge University Press:
- 28 October 2014, pp. 2112-2126
- Print publication:
- December 2014
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- Article
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- You have access
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We prove that on any compact manifold
$M^{n}$ with boundary, there exists a conformal class 
$C$ such that for any Riemannian metric 
$g\in C$ of unit volume, the first positive eigenvalue of the Neumann Laplacian satisfies 
${\it\lambda}_{1}(M^{n},g)<n\,\text{Vol}(S^{n},g_{\text{can}})^{2/n}$. We also prove a similar inequality for the first positive Steklov eigenvalue. The proof relies on a handle decomposition of the manifold. We also prove that the conformal volume of 
$(M,C)$ is 
$\text{Vol}(S^{n},g_{\text{can}})$, and that the Friedlander–Nadirashvili invariant 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.
