Skip to main content

Stability of harmonic maps and eigenvalues of laplacian

Part of the Lecture Notes in Mathematics book series (LNM,volume 1201)

Keywords

  • Riemannian Manifold
  • Compact Riemannian Manifold
  • Jacobi Operator
  • Riemannian Submersion
  • Compact Domain

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. Bérard Bergery & J.P. Bourguignon, Laplacians and Riemannian submersions with totally geodesic fibers, Ill. J. Math., 26(1982),181–200.

    MATH  Google Scholar 

  2. P.Bérard & S.Gallot, Inégalités isopérimetriques pour l'equation de la chaleur et application a l'estimation de quelques invariants, Seminaire Goulaouic-Meyer-Schwartz, n o 15, 1983.

    Google Scholar 

  3. M. Berger, P. Gauduchon & E. Mazet, Le spectre d'une variété riemannienne, Lecture Notes in Math., no 194, Springer, Berlin, 1971.

    CrossRef  MATH  Google Scholar 

  4. H. Donnely, Spectral invariants of the second variation operator, Ill. J. Math., 21(1977),185–189.

    MathSciNet  Google Scholar 

  5. J. Eells & L. Lemaire, A report on harmonic maps, Bull. London Math. Soc., 10(1978),1–68.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. J.Eells & L.Lemaire, Selected topics in harmonic maps, Region. Conf. series Math., Amer. Math. Soc., no 50, 1982.

    Google Scholar 

  7. P. Gilkey, The spectral geometry of real and complex manifolds, Proc. Sympos. Pure Math., 27(1975),265–280.

    CrossRef  MathSciNet  Google Scholar 

  8. P. Gilkey, The spectral geometry of a Riemannian manifold, J. Diff. Geom., 10(1975),601–618.

    MathSciNet  MATH  Google Scholar 

  9. D. Gromoll, W. Klingenberg & W. Meyer, Riemansche Geometrie im Grossen, Lecture Notes in Math., no 55, Springer, Berlin,1968.

    CrossRef  MATH  Google Scholar 

  10. T. Hasegawa, Spectral geometry of closed minimal submanifolds in a space form, real and complex, Kodai Math. J., 3(1980),224–252.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. P.F. Leung, On the stability of harmonic maps, Lecture Notes in Math., n o 949, Springer, Berlin, (1982),122–129.

    CrossRef  Google Scholar 

  12. A. Lichnérowicz, Géométrie des groupe de transformations, Travaux Recherches Math., III, Dunod, Paris, 1958.

    MATH  Google Scholar 

  13. E. Mazet, La formule de la variation seconde de l'energie au voisinage d'une application harmonique, J. Diff. Geom., 8(1973),279–296.

    MathSciNet  MATH  Google Scholar 

  14. M.Obata, Riemannian manifolds admitting a solution of a certain system of differential equations, Proc. U.S.-Japan Semin. Diff. Geom., Kyoto, Japan (1965),101–114.

    Google Scholar 

  15. Y.Ohnita, Stability of harmonic maps and standard minimal immersions, a preprint.

    Google Scholar 

  16. T. Sakai, On eigenvalues of Laplacian and curvature of Riemannian manifold, Tohoku Math. J., 23(1971),589–603.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. J. Simons, Minimal varieties in Riemannian manifolds, Ann. Math., 88(1968),62–105.

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. R.T. Smith, The second variation formula for harmonic mappings, Proc. Amer. Math. Soc.,, 47(1975),229–236.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. T. Sunada, Holomorphic mappings into compact quotient of symmetric bounded domains, Nagoya Math. J., 64(1976),159–175.

    MathSciNet  MATH  Google Scholar 

  20. S. Tanno, Eigenvalues of the Laplacian of Riemannian manifolds, Tohoku Math. J., 25(1973),391–403.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. H.Urakawa, Stability of harmonic maps and eigenvalues of Laplacian, a preprint.

    Google Scholar 

  22. H.Urakawa, The first eigenvalue of the Laplacian for a positively curved homogeneous Riemannian manifold, a preprint.

    Google Scholar 

  23. H.Urakawa, Nullities and indicies of Yang-Mills fields over Einstein manifolds with positive Ricci tensor, a preprint.

    Google Scholar 

  24. H.Urakawa, Spectral geometry of the second variation operator of harmonic maps, a prerint.

    Google Scholar 

  25. Y.L. Xin, Some results on stable harmonic maps, Duke Math. J., 47(1980)609–613.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. S.T.Yau, Survey on partial differential equations in differential geometry, Seminar on Diff. Geom., Ann. Math. Studies, n o 102, Princeton, (1982),3–71.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1986 Springer-Verlag

About this paper

Cite this paper

Urakawa, H. (1986). Stability of harmonic maps and eigenvalues of laplacian. In: Shiohama, K., Sakai, T., Sunada, T. (eds) Curvature and Topology of Riemannian Manifolds. Lecture Notes in Mathematics, vol 1201. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0075663

Download citation

  • DOI: https://doi.org/10.1007/BFb0075663

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16770-9

  • Online ISBN: 978-3-540-38827-2

  • eBook Packages: Springer Book Archive