Small eigenvalues and thick-thin decomposition in negative curvature

Hamenstädt, Ursula

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

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

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

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

[1] Baik, Hyungrul; Gekhtman, Ilya; Hamenstädt, Ursula The smallest positive eigenvalue of fibered hyperbolic 3-manifolds, Proc. Lond. Math. Soc., Volume 120 (2020) no. 5, pp. 704-741 | DOI | MR | Zbl

[2] Burger, Marc; Gelander, Tsachik; Lubotzky, Alexander; Mozes, Shahar Counting hyperbolic manifolds, Geom. Funct. Anal., Volume 12 (2002) no. 6, pp. 1161-1173 | DOI | MR | Zbl

[3] Burger, Marc; Schroeder, Viktor Volume, diameter and the first eigenvalue of locally symmetric spaces of rank one, J. Differ. Geom., Volume 26 (1987), pp. 273-284 | DOI | MR | Zbl

[4] Buser, Peter Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, 106, Birkhäuser, 1992 | MR | Zbl

[5] Buser, Peter; Colbois, Bruno; Dodziuk, Jozef Tubes and eigenvalues for negatively curved manifolds, J. Geom. Anal., Volume 3 (1993) no. 1, pp. 1-26 | DOI | MR | Zbl

[6] Chavel, Isaac Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, 115, Academic Press Inc., 1984 | MR | Zbl

[7] Chung, Fan R. K. Laplacians on graphs and Cheeger inequality, Combinatorics, Paul Erdős is eighty. Vol. 2 (Bolyai Society Mathematical Studies), Volume 2, János Bolyai Mathematical Society, 1996, pp. 157-172 | Zbl

[8] Dodziuk, Jozef A lower bound for the first eigenvalue of a finite-volume negatively curved manifold, Bol. Soc. Bras. Mat., Volume 18 (1987) no. 2, pp. 23-34 | DOI | MR | Zbl

[9] Dodziuk, Jozef; Randol, Burton Lower bounds for ${\lambda }_1$ on a finite-volume hyperbolic manifold, J. Differ. Geom., Volume 24 (1986), pp. 133-139 | DOI | MR | Zbl

[10] Eschenburg, Jost Comparison theorems and applications, Manuscr. Math., Volume 59 (1987), pp. 295-323 | DOI | Zbl

[11] Eschenburg, Jost Comparison theorems in Riemannian geometry, Lecture Notes Series, 3, Università degli Studi Trento, 1994

[12] Gelander, Tsachik Homotopy type and volume of locally symmetric manifolds, Duke Math. J., Volume 124 (2004) no. 3, pp. 459-515 | DOI | MR | Zbl

[13] Hamenstädt, Ursula Small eigenvalues of geometrically finite manifolds, J. Geom. Anal., Volume 14 (2004) no. 2, pp. 281-290 | DOI | MR | Zbl

[14] Hamenstädt, Ursula; Viaggi, Gabriele Small eigenvalues of random 3-manifolds (2019) (https://arxiv.org/abs/1903.08031)

[15] Lenzhen, A.; Souto, Juan (in preparation)

[16] Louder, Larsen; Souto, Juan Diameter and spectral gap for planar graphs (2012) (https://arxiv.org/abs/1204.4435v2)

[17] Mantuano, Tatiana Discretization of Compact Riemannian Manifolds Applied to the Spectrum of Laplacian, Ann. Global Anal. Geom., Volume 27 (2005) no. 1, pp. 33-46 | DOI | MR | Zbl

[18] Otal, Jean-Pierre; Rosas, Eulalio Pour toute surface hyperbolique de genre $g$, ${\lambda }_{2g-2}>1/4$, Duke Math. J., Volume 150 (2009) no. 1, pp. 101-155 | DOI | MR | Zbl

[19] Schoen, Richard A lower bound for the first eigenvalue of a negatively curved manifold, J. Differ. Geom., Volume 17 (1982) no. 2, pp. 233-238 | DOI | MR | Zbl

[20] White, Nina Spectral bounds on closed hyperbolic 3-manifolds, J. Lond. Math. Soc., Volume 87 (2013) no. 3, pp. 837-853 | DOI | MR | Zbl