Variation of Laplace spectra of compact “nearly” hyperbolic surfaces

Mayukh Mukherjee

doi:10.1016/j.crma.2016.11.019

Differential geometry

Variation of Laplace spectra of compact “nearly” hyperbolic surfaces

Presented by: the Editorial Board

Mayukh Mukherjee ¹

¹ Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany

Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 216-221.

Abstract
Résumé

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 $γ \geq 2$ , 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 $γ \geq 2$ .

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 $γ \geq 2$ , 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 $γ \geq 2$ .

Received: 2015-12-31
Accepted: 2016-12-09
Published online: 2016-12-27

DOI: 10.1016/j.crma.2016.11.019

Author's affiliations:

Mayukh Mukherjee ¹

¹ Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany

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     author = {Mayukh Mukherjee},
     title = {Variation of {Laplace} spectra of compact {\textquotedblleft}nearly{\textquotedblright} hyperbolic surfaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {216--221},
     publisher = {Elsevier},
     volume = {355},
     number = {2},
     year = {2017},
     doi = {10.1016/j.crma.2016.11.019},
     language = {en},
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Mayukh Mukherjee. Variation of Laplace spectra of compact “nearly” hyperbolic surfaces. Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 216-221. doi : 10.1016/j.crma.2016.11.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.11.019/

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