Advertisement

Mathematische Zeitschrift

, Volume 250, Issue 1, pp 43–57 | Cite as

Vanishing of cohomology groups and large eigenvalues of the Laplacian on p-forms

  • Junya TakahashiEmail author
Article

Abstract.

For collapsing of closed Riemannian manifolds, the first positive eigenvalue of the Laplacian on p-forms may or may not tend to infinity. In special cases, we show that the existence of the first positive eigenvalue tending to infinity is related to vanishing of cohomology groups of generic fibers.

Keywords

Riemannian Manifold Large Eigenvalue Cohomology Group Positive Eigenvalue Generic Fiber 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anné, C.: Principle de Dirichlet pour les formes différentielles. Bull. Soc. math. France 117, 445–450 (1989)Google Scholar
  2. 2.
    Aubry, E., Colbois, B. Ghanaat, P., Ruh, E.A.: Curvature, Harnack’s inequality, and a spectral characterization of nilmanifolds. Ann. Global Anal. Geom. 23, 227–246 (2003)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bär, C.: Metrics with harmonic spinors. Geom. Func. Anal. 6, 899–942 (1996)Google Scholar
  4. 4.
    Cheeger, J., Gromov, M.: Collapsing Riemannian manifolds while keeping their curvature bounded I. J. Diff. Geom. 23, 309–346 (1986)zbMATHGoogle Scholar
  5. 5.
    Cheng, S.Y.: Eigenvalue comparison theorems and its geometric applications. Math. Zeit. 143, 289–297 (1975)zbMATHGoogle Scholar
  6. 6.
    Dai, X.: Adiabatic limits, non-multiplicity of signature and the Leray spectral sequence. J. Am. Math. Soc. 4, 265–321 (1991)zbMATHGoogle Scholar
  7. 7.
    Forman, R.: Spectral sequences and adiabatic limits. Commun. Math. Phys. 168, 57–116 (1995)zbMATHGoogle Scholar
  8. 8.
    Gentile, G., Pagliara, V.: Riemannian metrics with large first eigenvalue on forms of degree p. Proc. Am. Math. Soc. 123, 3855–3858 (1995)Google Scholar
  9. 9.
    Gilkey, P.B., Leahy, J.V., Park, J.H.: Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture. Studies in Advanced Mathematics, Chapman & Hall/CRC, 1999Google Scholar
  10. 10.
    Li, P., Yau, S.T.: Estimates of eigenvalues of a compact Riemannian manifold. Proc. Symp. Pure Math. 36, 205–239 (1980)zbMATHGoogle Scholar
  11. 11.
    Lott, J.: Collapsing and the differential form Laplacian : the case of a smooth limit space. Duke Math. J. 114, 267–306 (2002)zbMATHGoogle Scholar
  12. 12.
    Lott, J.: Collapsing and the differential form Laplacian : the case of a singular limit space. Preprint, math.DG/0201289, 2002Google Scholar
  13. 13.
    Mazzeo, R., Melrose, R.: The adiabatic limit, Hodge cohomology and Leray’s spectral sequence for a fibration. J. Diff. Geom. 31, 185–213 (1990)zbMATHGoogle Scholar
  14. 14.
    McGowan, J.: The p-spectrum of the Laplacian on compact hyperbolic three manifolds. Math. Ann. 279, 725–745 (1993)Google Scholar
  15. 15.
    Prokhorenkov, I.: Morse-Bott functions and the Witten Laplacian. Commun. Anal. Geom. 7, 841–918 (1999)zbMATHGoogle Scholar
  16. 16.
    Shioya, T., Yamaguchi, T.: Collapsing three-manifolds under a lower curvature bound. J. Diff. Geom. 56, 1–66 (2000)Google Scholar
  17. 17.
    Yamaguchi, T.: Collapsing 4-manifolds under a lower curvature bound. Preprint, 2002Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

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

Personalised recommendations