Duke Mathematical Journal

Pour toute surface hyperbolique de genre g, λ2g2>1/4

Jean-Pierre Otal and Eulalio Rosas

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Résumé

Nous étudions l'influence de la topologie d'une surface hyperbolique sur le nombre des valeurs propres de son Laplacien qui sont inférieures ou égales à 1/4. Le premier résultat de l'article est un énoncé du type “trou spectral”, son titre, dans lequel λj est la j-ième des valeurs propres λ0=0<λ1⋅⋅⋅λj⋅⋅⋅ du Laplacien et g est le genre de la surface. Une construction classique dûe à Buser montre que ce résultat est optimal. Nous donnons aussi un énoncé du même type pour les surfaces de volume fini. Les méthodes prolongent celles de [O], qui utilisaient de manière essentielle l'approche topologique par Sévennec de la question de la majoration de la multiplicité de la deuxième valeur propre des opérateurs de Schrödinger [Se].

Abstract

We study the influence of the topology of a hyperbolic surface on the number of its Laplace eigenvalues that are at most 1/4. The first result of the article is a “spectral gap” statement, namely, its title where λj is the jth of the eigenvalues λ0=0<λ1⋅⋅⋅λj⋅⋅⋅ of the Laplace operator and where g is the genus of the surface. A classical construction due to Buser shows that this result is sharp. We give a similar statement for finite volume surfaces. The methods develop those found in [O], which used in an essential way the topological approach of Sévennec to the question of bounding the multiplicity of the second eigenvalue for Schrödinger operators [Se]

Article information

Source
Duke Math. J., Volume 150, Number 1 (2009), 101-115.

Dates
First available in Project Euclid: 15 September 2009

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1253020546

Digital Object Identifier
doi:10.1215/00127094-2009-048

Mathematical Reviews number (MathSciNet)
MR2560109

Zentralblatt MATH identifier
1179.30041

Subjects
Primary: 30F
Secondary: 35P05: General topics in linear spectral theory 35P15: Estimation of eigenvalues, upper and lower bounds

Citation

Otal, Jean-Pierre; Rosas, Eulalio. Pour toute surface hyperbolique de genre $g$ , $\lambda_{2g-2}&gt; 1/4$. Duke Math. J. 150 (2009), no. 1, 101--115. doi:10.1215/00127094-2009-048. https://projecteuclid.org/euclid.dmj/1253020546


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