Small eigenvalues and thick-thin decomposition in negative curvature

Hamenstädt, Ursula

doi:10.5802/aif.3345

Small eigenvalues and thick-thin decomposition in negative curvature
[Petites valeurs propres et décomposition en parties épaisse et mince en courbure négative]

Hamenstädt, Ursula ¹

¹ Universität Bonn Endenicher Allee 60 53115 Bonn (Germany)

Annales de l'Institut Fourier, Tome 69 (2019) no. 7, pp. 3065-3093.

Résumé
Abstract

Soit $M$ une variété Riemannienne complète orientée, de dimension $n \geq 3$ et de volume finie. Supposons que la courbure de $M$ soit contenue dans $[- b^{2}, - 1]$ , et soit $M = M_{thick} \cup M_{thin}$ la décomposition en sa partie épaisse et sa partie mince. Soit $λ_{k} (M)$ la $k$ -ième valeur propre de l’opérateur Laplacien, avec conditions au bord de Neumann. Nous démontrons que $λ_{k} (M_{thick}) / 3 \leq λ_{k} (M)$ pour tout $k$ tel que $λ_{k} (M) < {(n - 2)}^{2} / 12$ . Si $M$ est hyperbolique et de dimension 3, alors $λ_{k} (M) \leq C log (vol (M_{thin}) + 2) λ_{k} (M_{thick})$ pour un nombre $C > 0$ fixé pourvu que $λ_{k} (M_{thick}) < 1 / 96$ .

Let $M$ be a finite volume oriented complete Riemannian manifold of dimension $n \geq 3$ and curvature in $[- b^{2}, - 1]$ , with thick-thin decomposition $M = M_{thick} \cup M_{thin}$ . Denote by $λ_{k} (M_{thick})$ the $k$ -th eigenvalue for the Laplacian on $M_{thick}$ , with Neumann boundary conditions. We show that $λ_{k} (M_{thick}) / 3 \leq λ_{k} (M)$ for all $k$ for which $λ_{k} (M) < {(n - 2)}^{2} / 12$ . If $M$ is hyperbolic and of dimension 3 then $λ_{k} (M) \leq C log (vol (M_{thin}) + 2) λ_{k} (M_{thick})$ for a fixed number $C > 0$ provided that $λ_{k} (M_{thick}) < 1 / 96$ .

Publié le : 2020-06-26

DOI : 10.5802/aif.3345

Classification : 58J50, 53C20
Keywords: Spectrum of the Laplacian, Neumann boundary conditions, manifolds of pinched negative curvature, thick-thin decomposition
Mot clés : Spectre du Laplacien, conditions au bord Neumann, variétés de courbure négative pincée, décomposition en parties épaisse et mince

Affiliations des auteurs :

Hamenstädt, Ursula ¹

¹ Universität Bonn Endenicher Allee 60 53115 Bonn (Germany)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Small eigenvalues and thick-thin decomposition in negative curvature},
     journal = {Annales de l'Institut Fourier},
     pages = {3065--3093},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {7},
     year = {2019},
     doi = {10.5802/aif.3345},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3345/}
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Hamenstädt, Ursula. Small eigenvalues and thick-thin decomposition in negative curvature. Annales de l'Institut Fourier, Tome 69 (2019) no. 7, pp. 3065-3093. doi : 10.5802/aif.3345. https://aif.centre-mersenne.org/articles/10.5802/aif.3345/

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