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Comptes Rendus Mathématique
Volume 355, n° 2
pages 216-221 (février 2017)
Doi : 10.1016/j.crma.2016.11.019
Received : 31 December 2015 ;  accepted : 9 December 2016
Variation of Laplace spectra of compact “nearly” hyperbolic surfaces
Variation du spectre de Laplace des surfaces compactes « presque » hyperboliques
 

Mayukh Mukherjee
 Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany 

Abstract

We use the real analyticity of the Ricci flow with respect to time proved by B. Kotschwar to extend a result of P. Buser, namely, we prove that the Laplace spectra of negatively curved compact orientable surfaces having the same genus  , the same area and the same curvature bounds vary in a “controlled way”, of which we give a quantitative estimate in our main theorem. The basic technical tool is a variational formula that provides the derivative of an eigenvalue branch under the normalized Ricci flow. In a related manner, we also observe how the above-mentioned real analyticity result can lead to unexpected conclusions concerning the spectral properties of generic metrics on a compact surface of genus  .

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Résumé

Nous utilisons l'analyticité réelle du flot de Ricci par rapport au temps, démontrée par B. Kotschwar, pour étendre un résultat de P. Buser. Précisément, nous montrons que le spectre de Laplace des surfaces compactes, orientables, de courbure négative, de même genre  , même aire et mêmes bornes pour la courbure, varie de « façon contrôlée ». Nous donnons une estimation quantitative de cette variation dans notre théorème principal. Notre outil technique de base est une formule variationnelle donnant la dérivée d'une branche de valeur propre sous l'action du flot de Ricci normalisé. Par analogie, nous indiquons comment le résultat d'analyticité réelle ci-dessus peut conduire à des conclusions inattendues sur les propriétés du spectre des métriques génériques sur une surface compacte, de genre  .

The full text of this article is available in PDF format.
1  We use   to denote the area element.
2  We note that the real analyticity of the  's is not absolutely essential for the ensuing arguments, but it definitely makes said arguments simpler.


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