Collapsing of Riemannian manifolds and eigenvalues of Laplace operator. (English) Zbl 0589.58034

In this paper, we discuss the perturbation of eigenvalues of Laplace operators under deformation of manifolds, which does not preserve topological types. The main result concerns the case when curvatures and diameters are uniformly bounded. The most interesting phenomenon occurring under this restriction is that manifolds collapse to a lower dimensional space. It is proved that, in that case, the eigenvalues converge to that of the differential operator on the limit space which, in general, has a regular singularity at the singular point of the space. There are also some discussions on the case in which the curvatures are not bounded. The discussion in that case is not systematic but gives several examples which seem to the author to be typical.


58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C20 Global Riemannian geometry, including pinching
Full Text: DOI EuDML


[1] Anné, C.: Lecture at Marseille Luminy (1984)
[2] Beale, J.T.: Scattering Frequency of Resonators. Comm. Pure Appl. Math.26, 549-565 (1973) · Zbl 0261.35065
[3] Bemelmans, J., Min-Oo, Ruh, E.: Smoothing Riemannian metrics. Math. Z.188, 69-74 (1984) · Zbl 0536.53044
[4] Cheng, S.Y.: Eigenvalue comparison theorems and its geometric applications. Math. Z.143, 289-297 (1975) · Zbl 0329.53035
[5] Fukaya, K.: Collapsing Riemannian manifolds to lower dimensional one (to appear in J. Differ. Geom.) · Zbl 0703.53042
[6] Fukaya, K.: A boundary of the set of Riemannian manifolds with bounded curvatures and diameters (Preprint) · Zbl 0652.53031
[7] Gallot, S.: Inéqualité isopérimétrique, courbure de Ricci et invariants géometriques II. C.R. Acad. Sc. Paris296, 365-368 (1982)
[8] Gromov, M., Lafontaine, J., Pansu, P.: Structures métriques pour les variétés riemanniennes, Paris: Cedic Fernand/Nathan, 1981
[9] Li, P.: On the Sovolev constant and thep-spectrum of a compact Riemannian manifold. Ann. Sci. Ec. Norm. Super.13, 451-469 (1980)
[10] Mitchel, J.: On Carnot-Carathéodory metrics. J. Differ. Geom.21, 25-45 (1985)
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