×

Spectre du Laplacien et écrasement d’anses. (Spectrum of the Laplacian and down breaking handles). (French) Zbl 0634.58035

Author’s abstract. We show, by the study of the convergence of the spectrum of the Laplacian in the case of a manifold \(X_ 1\) with handles breaking down to a manifold of lower dimension \(X_ 2\) with boundary in \(X_ 1\), that the suitable limiting operator is the Laplacian with Dirichlet boundary conditions on \(X_ 2\).
Reviewer: D.Perrone

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] C. ANNÉ , Spectre du laplacien et limites de variétés avec perte de dimension I , Prépublication de l’Institut Fourier, 1985 .
[2] C. ANNÉ , Perturbation du spectre X-TUB\epsilon Y (conditions de Neumann) (Séminaire de Théorie Spectrale et Géométrie de l’Institut Fourier, vol. 4, 1986 , p. 17-23). Numdam | MR 1046060 | Zbl 1002.58508 · Zbl 1002.58508
[3] C. ANNÉ , Écrasement d’anses et spectre du laplacien , Prépublications de l’Institut Fourier, n^\circ 67, 1986 .
[4] L. BÉRARD-BERGERY et J. P. BOURGUIGNON , Laplacians and Riemannian Submersions with Totally Geodesic Fibres (Illinois J. of Math., vol. 26, 1982 , p. 181-200). Article | MR 84m:58153 | Zbl 0483.58021 · Zbl 0483.58021
[5] G. BESSON , A Kato Type Inequality for Riemannian submersions with Totally Geodesic Fibers , (Annals of Global Analysis and Geometry, 1986 ). MR 89b:58215 | Zbl 0631.53035 · Zbl 0631.53035
[6] M. BERGER , P. GAUDUCHON et E. MAZET , Le spectre d’une variété riemannienne (Lecture Notes in Math., vol. 194, Springer-Verlag, 1971 ). MR 43 #8025 | Zbl 0223.53034 · Zbl 0223.53034
[7] G. COURTOIS , Spectre des variété privées d’un \epsilon -tube , Prépublications de l’Institut Fourier, 1986 .
[8] I. CHAVEL , Eigenvalues in Riemannian Geometry , Academic Press, 1984 . MR 86g:58140 | Zbl 0551.53001 · Zbl 0551.53001
[9] I. CHAVEL et E. A. FELDMAN , Spectra of Domains in Compact Manifolds (J. Fcn’l Anal., vol. 30, 1978 , p. 198-222). MR 80c:58027 | Zbl 0392.58016 · Zbl 0392.58016
[10] I. CHAVEL et E. A. FELDMAN , Isoperimetric Constants of Manifolds with Small handles (Math. Zeit., vol. 184, 1983 , p. 435-448). Article | MR 85e:58149 | Zbl 0525.53054 · Zbl 0525.53054
[11] Y. COLIN DE VERDIÈRE , Résonances (Séminaire de Théorie Spectrale et Géométrie de l’Institut Fourier, 1984 , p. 85. Numdam | Zbl 1002.58511 · Zbl 1002.58511
[12] J. J. DUISTERMATT , Fourier Integral Operators (Lecture Notes of the Courant Institute of Mathematical Sciences, New York, 1973 ).
[13] J. DODZIUK et B. RANDOL , Lower Bounds for \lambda 1 on a Finite-volume Hyperbolique Manifold (à paraître), 1986 . Zbl 0594.53036 · Zbl 0594.53036
[14] K. FUKAYA , Collapsing Riemannian manifolds to Lower Dimentional on [J. Diff. Geom., 1986 (à paraître)].
[15] K. FUKAYA , Collapsing of Riemannian Manifolds and Eigenvalues of the Laplace Operator [Invent. Math., 1986 (à paraître)]. Zbl 0589.58034 · Zbl 0589.58034
[16] T. KATO , Perturbation Theory for Linear Operators (Lecture Notes in Math., vol. 132, Springer-Verlag, 1976 ). MR 53 #11389 | Zbl 0342.47009 · Zbl 0342.47009
[17] S. OZAWA , Spectra of Domains with Small spherical Neumann Boundary (J. Fac. Sc. Univ. of Tokyo, vol. 30, n^\circ 2, 1983 , p. 259-277). MR 85k:35174 | Zbl 0541.35061 · Zbl 0541.35061
[18] J. RAUCH et M. TAYLOR , Potential and Scattering theory on Wildly Perturbed Domains (J. Fcn’l Anal., 1975 , p. 27-59). MR 51 #13476 | Zbl 0293.35056 · Zbl 0293.35056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.