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Spectre et géométrie conforme des variétés compactes à bord

Part of: Spectral theory and eigenvalue problems Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  28 October 2014

Pierre Jammes
Pierre Jammes*
Affiliation:
Laboratoire J.-A. Dieudonné, Université Nice Sophia Antipolis — CNRS (UMR 7351), Parc Valrose, 06108 Nice Cedex 02, France email [email protected]
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Abstract

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.

Keywords

first Neumann and Steklov eigenvalueconformal volume

MSC classification

Primary: 35P15: Estimation of eigenvalues, upper and lower bounds 58J50: Spectral problems; spectral geometry; scattering theory
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 12 , December 2014 , pp. 2112 - 2126
Copyright
© The Author 2014 

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