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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 $g\in C$ ofunit volume, the first positive eigenvalue of the Neumann Laplacian satisfies $\lambda_1(M^n,g)< n Vol(S^n,g_{\textrm{can}})^{2/n}$. 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 $Vol(S^n,g_{\textrm{can}})$, 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.
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https://hal.archives-ouvertes.fr/hal-00769258
Contributor : Pierre Jammes <>
Submitted on : Sunday, December 30, 2012 - 2:32:50 PM
Last modification on : Monday, October 12, 2020 - 2:28:05 PM

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Pierre Jammes. Spectre et géométrie conforme des variétés compactes à bord. Compositio Mathematica, Foundation Compositio Mathematica, 2014, 150 (12), pp.2112-2126. ⟨10.1112/S0010437X14007696⟩. ⟨hal-00769258⟩

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