manuscripta mathematica

, Volume 121, Issue 2, pp 191–200 | Cite as

Collapsing to Riemannian manifolds with boundary and the convergence of the eigenvalues of the Laplacian

  • Junya TakahashiEmail author


We prove that the eigenvalues of the Laplacian acting on functions converge to those of the limit manifold for a special collapsing family of closed Riemannian manifolds without curvature bounds. The proof uses L 2-analysis.

Mathematics Subject Classification (2000)

Primary 58J50 Secondary 35P15 Secondary 53C23 Secondary 58J32 


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  1. 1.
    Anné, C. Spectre du Laplacien et écracement d’anses. Ann. Sci. Éc. Norm. Sup. (4), 20, 271–280 (1987)Google Scholar
  2. 2.
    Anné C., Colbois B. (1995) Spectre du Laplacien agissant sur les p-formes différentielles et écrasement d’anses. Math. Ann. 303, 545–573CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Chavel, I. Eigenvalues in Riemannian geometry. Academic, New York, No. 115 (1984)Google Scholar
  4. 4.
    Chavel I., Feldman E. (1981) Spectra of manifolds with small handles. Comment. Math. Helv. 56, 83–102CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Cheeger J., Colding T.H. (2000) On the structure of spaces with Ricci curvature bounded below. III. J. Diff. Geom. 54, 37–74MathSciNetzbMATHGoogle Scholar
  6. 6.
    Fukaya K. (1987) Collapsing of Riemannian manifolds and eigenvalues of Laplace operator. Invent. Math. 87, 517–547CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Kasue A. (2002) Convergence of Riemannian manifolds and Laplace operators. I. Ann. Inst. Fourier (Grenoble) 52, 1219–1257MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kasue A., Kumura H., Ogura Y. (1997) Convergence of heat kernels on a compact manifold. Kyushu J. Math. 51, 453–524CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Kuwae K., Shioya T. (2003) Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry. Commun. Anal. Geom. 11, 599–673MathSciNetzbMATHGoogle Scholar
  10. 10.
    Takahashi J. (2002) Collapsing of connected sums and the eigenvalues of the Laplacian. J. Geom. Phys. 40, 201–208CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Takahashi J. (2005) Vanishing of cohomology groups and large eigenvalues of the Laplacian on p-forms. Math. Zeit. 250, 43–57CrossRefzbMATHGoogle Scholar
  12. 12.
    Taylor, M.E. Partial differential equations I. basic theory. Appl. Math. Sci. 115, (1996)Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Division of Mathematics, Graduate School of Information SciencesTôhoku UniversitySendaiJapan

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