On deformations of the spectrum of a Finsler-Laplacian that preserve the length spectrum

Barthelmé, Thomas

doi:10.24033/bsmf.2743

Barthelmé, Thomas ^1,²

¹Pennsylvania State University, State College, PA, USA
²Queen’s University, Kingston, ON, Canada

Bulletin de la Société Mathématique de France, Volume 145 (2017) no. 3, pp. 421-448

Abstract (Original language)
Résumé

The main result of this article is the construction of non-reversible Finsler metrics in negative curvature such that $4 λ_{1} > h^{2}$ , where $λ_{1}$ is the bottom of the $L^{2}$ -spectrum of a previously defined Finsler-Laplacian and $h$ the topological entropy of the flow. This gives a counter-example to a classical inequality in Riemannian geometry. We also show that the spectrum of that Finsler-Laplacian can detect changes in the Finsler metric that the marked length spectrum cannot.

Le résultat principal de cet article est la construction d’une famille de métriques de Finsler, non-réversible, en courbure négative satisfaisant $4 λ_{1} > h^{2}$ , où $λ_{1}$ est le bas du spectre $L^{2}$ d’un laplacien en géométrie de Finsler et $h$ est l’entropie topologique du flot géodésique. Ce résultat fournit un contre-exemple, pour les métriques de Finsler, à une inégalité classique de géométrie riemannienne. Nous montrons également que le spectre de ce laplacien détecte certains changements de la métrique qui sont invisible pour le spectre des longueurs.

Received: 2015-06-25
Accepted: 2016-05-23
Published online: 2017-01-28

MR Zbl

DOI: 10.24033/bsmf.2743

Classification: 58J60, 53C60
Keywords: Finsler-Laplacian, topological entropy, length spectrum
Mots-clés : Laplacien Finsler, entropy topologique, spectre des longueurs

Author's affiliations:

Barthelmé, Thomas ^{1
,

2}

¹ Pennsylvania State University, State College, PA, USA
² Queen’s University, Kingston, ON, Canada

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Barthelmé, Thomas. On deformations of the spectrum of a Finsler-Laplacian that preserve the length spectrum. Bulletin de la Société Mathématique de France, Volume 145 (2017) no. 3, pp. 421-448. doi: 10.24033/bsmf.2743

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