Small eigenvalues and thick-thin decomposition in negative curvature
Annales de l'Institut Fourier, Volume 69 (2019) no. 7, pp. 3065-3093.

Let M be a finite volume oriented complete Riemannian manifold of dimension n3 and curvature in [-b 2 ,-1], with thick-thin decomposition M=M thick 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 )/3λ 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)Clog(vol(M thin )+2)λ k (M thick ) for a fixed number C>0 provided that λ k (M thick )<1/96.

Soit M une variété Riemannienne complète orientée, de dimension n3 et de volume finie. Supposons que la courbure de M soit contenue dans [-b 2 ,-1], et soit M=M thick 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 )/3λ 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)Clog(vol(M thin )+2)λ k (M thick ) pour un nombre C>0 fixé pourvu que λ k (M thick )<1/96.

Published online:
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
Hamenstädt, Ursula 1

1 Universität Bonn Endenicher Allee 60 53115 Bonn (Germany)
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Hamenstädt, Ursula. Small eigenvalues and thick-thin decomposition in negative curvature. Annales de l'Institut Fourier, Volume 69 (2019) no. 7, pp. 3065-3093. doi : 10.5802/aif.3345. http://www.numdam.org/articles/10.5802/aif.3345/

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