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: 2020-06-26
DOI: https://doi.org/10.5802/aif.3345
Classification: 58J50,  53C20
Keywords: Spectrum of the Laplacian, Neumann boundary conditions, manifolds of pinched negative curvature, thick-thin decomposition
@article{AIF_2019__69_7_3065_0,
     author = {Hamenst\"adt, Ursula},
     title = {Small eigenvalues and thick-thin decomposition in negative curvature},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {7},
     year = {2019},
     pages = {3065-3093},
     doi = {10.5802/aif.3345},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2019__69_7_3065_0/}
}
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. https://aif.centre-mersenne.org/item/AIF_2019__69_7_3065_0/

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