Skip to main content
SpringerLink
Log in
Book cover

Schrödinger Operators pp 401–426Cite as

Inverse spectral problems on compact Riemannian manifolds

  • Conference paper
  • First Online:

  • 172 Accesses

Part of the book series: Lecture Notes in Physics ((LNP,volume 345))

This is a preview of subscription content, log in via an institution.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. T. Aubin. Équations différentielles non linéares et problème de Yamabe concernant la courbure scalaire. J. Math. Pures. Appl. 55 (1976), 533–598.

    Google Scholar 

  2. T. Aubin. Nonlinear analysis on manifolds. Monge-Ampére equations. Berlin, Heidelberg, New York: Springer-Verlag, 1982.

    Google Scholar 

  3. P. Bérard. Spectral Geometry: Direct and Inverse Problems. Berlin, Heidelberg, New York: Springer-Verlag, 1986 (Lecture Notes in Mathematics, No. 1207).

    Google Scholar 

  4. M. Berger, P. Gauduchon, E. Mazet. Le Spectre d'une Variété Riemannienne (Lecture Notes in Mathematics, No. 194). Berlin, Heidelberg, New York: Springer-Verlag, 1982.

    Google Scholar 

  5. R. Brooks. On manifolds of negative curvature with isospectral potentials. Topology 26 (1987), 63–66.

    Google Scholar 

  6. R. Brooks, P. Perry, P. Yang. Isospectral sets of conformally equivalent metrics. Preprint, University of Southern California, 1987.

    Google Scholar 

  7. R. Brooks, R. Tse. Isospectml surfaces of small genus. Nagoya Math. J. 107 (1987), 13–24.

    Google Scholar 

  8. J. Brüning. On the compactness of isospectral potentials. Comm. P. D. E. 9 (1984), 687–698.

    Google Scholar 

  9. S. Y. A. Chang, P. Yang. Compactness of isospectral conformal metrics on S 3. To appear in Comm. Math. Helv.

    Google Scholar 

  10. S. Y. A. Chang, P. Yang. Isospectral conformal metrics on 3-Manifolds. Preprint, University of Southern California, 1988.

    Google Scholar 

  11. S. Y. A. Chang, P. Yang. The conformal deformation equation and Isospectral sets of conformal metrics. To appear in Contemporary Math.

    Google Scholar 

  12. I. Chavel. Eigenvalues in Riemannian Geometry. (Pure and Applied Mathematics, vol. 115). New York: Academic Press, 1984.

    Google Scholar 

  13. R. Courant, D. Hilbert. Methods of Mathematical Physics, vol. 1. New York: Interscience Publishers, 1953.

    Google Scholar 

  14. H. Cycon, R. Froese, W. Kirsch, B. Simon. Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry. (Texts and monographs in Physics). New York: Springer-Verlag, 1987.

    Google Scholar 

  15. D. DeTurck, C. Gordon. Isospectral Deformations, I. Riemannian Structures on Two-Step Nilspaces. C. P. A. M. 40 (1987), 367–387.

    Google Scholar 

  16. D. DeTurck, C. Gordon. Isospectral Deformations, II. Metrics and potentials on general manifolds. Preprint, 1987.

    Google Scholar 

  17. D. DeTurck, C. Gordon. Isospectral Riemannian metrics and potentials. Bull A. M. S. 17 (1987), 137–140.

    Google Scholar 

  18. J. Duistermaat, V. Guillemin. The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 24 (1975), 39–80.

    Google Scholar 

  19. C. J. Earle. Teichmüller Theory. In Discrete Groups and Automorphic Functions, ed. W. J. Harvey (London, New York, San cisco: Academic Press, 1977), pp. 143–162.

    Google Scholar 

  20. P. B. Gilkey. Recursion relations and the asymptotic behavior of the eigenvalues of the Laplacian. Compositio. Math. 38 (1979), 201–240.

    Google Scholar 

  21. P. B. Gilkey. Invariance theory, the heat equation, and the Atiyah-Singer Index Theorem. Wilmington: Publish or Perish, 1984.

    Google Scholar 

  22. P. B. Gilkey. Leading terms in the asymptotics of the heat equation. Preprint, 1987.

    Google Scholar 

  23. C. Gordon, E. Wilson. Isospectral deformations of compact solvmanifolds. J. Diff. Geom. 19 (1984), 241–256.

    Google Scholar 

  24. V. Guillemin, D. Kazhdan. Some inverse spectral results for negatively curved 2-manifolds. Topology 19 (1980), 301–312.

    Google Scholar 

  25. M. Kac. Can one hear the shape of a drum? Amer. Math. Monthly 73 (1966), 1–23.

    Google Scholar 

  26. J. Lee, M. Parker. The Yamabe Problem. Bull. Amer. Math. Soc. 17 (1987), 37–81.

    Google Scholar 

  27. J. Lelong-Ferrand. Transformations conformes et quasiconformes des variétés Riemanniennes: Applications à la démonstration d'une conjecture de A. Lichnerozvicz. C. R. Acad. Sci. Paris 269 (1969), 583–586.

    Google Scholar 

  28. A. N. Livčic. Some homological properties of U-systems. Mat. Zametki 10 (1971) 555–564.

    Google Scholar 

  29. H. P. McKean, I. M. Singer. Curvature and the eigenvalues of the Laplacian. J. Diff. Geom. 1 (1967), 43–69.

    Google Scholar 

  30. R. B. Melrose. Isospectral drumheads are C ∞. M. S. R. I. preprint, 1983.

    Google Scholar 

  31. R. B. Melrose. Private communication.

    Google Scholar 

  32. J. Milnor. Eigenvalues of the Laplacian on certain manifolds. Proc. Nat. Acad. Sci. U. S. A. 51 (1964), 542.

    Google Scholar 

  33. S. Minakshisundaram, A. Pleijel. Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds. Can. J. Math. 1, 242–256 (1949).

    Google Scholar 

  34. D. Mumford. A remark on Mahler's compactness theorem. Proc. Amer. Math. Soc. 28 (1971), 289–294.

    Google Scholar 

  35. M. Obata. The conjectures on conformal transformations of Riemannian manifolds. J. Diff. Geom. 6 (1971), 247–258.

    Google Scholar 

  36. E. Onofri. On the positivity of the effective action in a theory of random surfaces. Comm. Math. Phys. 86 (1982), 321–326.

    Google Scholar 

  37. B. Osgood, R. Phillips, P. Sarnak. Extremals of determinants of Laplacians. J. Func. Anal., to appear.

    Google Scholar 

  38. B. Osgood, R. Phillips, P. Sarnak. Compact isospectral sets of surfaces. J. Func. Anal., to appear.

    Google Scholar 

  39. B. Osgood, R. Phillips, P. Sarnak. Compact isospectral sets of plane domains. Proc. Nat. Acad. Sci., to appear.

    Google Scholar 

  40. B. Osgood, R. Phillips, P. Sarnak. Moduli space, heights, and isospectral sets of plane domains. Stanford University preprint, 1988.

    Google Scholar 

  41. J. P. Otal. Le spectra marqué des longueurs des surfaces à courbure négative. Orsay preprint, 1988.

    Google Scholar 

  42. A. Polyakov. Quantum geometry of bosonic strings. Physics Letters 103B (1981) 207–210.

    Google Scholar 

  43. A. Polyakov. Quantum geometry of fermionic strings. Physics Letters 103B (1981) 211–213.

    Google Scholar 

  44. M. Reed, B. Simon. Methods of Mathematical Physics, II. Fourier Analysis, Self-Adjointness. New York, San Francisco, London: Academic Press, 1975.

    Google Scholar 

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

    Google Scholar 

  46. R. Schoen. Conformal deformation of a Riemannian metric to constant scalar curvature. J. Diff. Geom. 20 (1984), 479–496.

    Google Scholar 

  47. R. Schoen. Recent progress in geometric partial differential equations. In Proceedings of the International Congress of Mathematicians, Berkeley, California, U. S. A., 1986, ed. Andrew M. Gleason (Providence, R. I.: American Mathematical Society, 1987), pp. 121–130.

    Google Scholar 

  48. R. Schoen. Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. Stanford University preprint, 1988.

    Google Scholar 

  49. T. Sunada. Riemannian coverings and isospectral manifolds. Ann. Math. 121 (1985), 169–186.

    Google Scholar 

  50. N. Trudinger. Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa (4) 3 (1985), 265–274.

    Google Scholar 

  51. H. Urakawa. Bounded domains which are isospectral but not isometric. Ann. Ec. Norm. Sup. 15 (1982), 441–456.

    Google Scholar 

  52. M. F. Vignéras. Variétés Riemanniennes isospectrales et non isométrique. Ann. Math. 112 (1980), 21–32.

    Google Scholar 

  53. J. Wolf. Spaces of Constant Curvature. New York: McGraw-Hill, 1967.

    Google Scholar 

  54. S. Wolpert. Asymptotics of the spectrum and the zeta function on the space of Riemann surfaces. Preprint.

    Google Scholar 

  55. H. Yamabe. On the deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12 (1960), 21–37.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kentucky, 40506, Lexington, Kentucky, USA

    Peter A. Perry

Authors
  1. Peter A. Perry
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Helge Holden Arne Jensen

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag

About this paper

Cite this paper

Perry, P.A. (1989). Inverse spectral problems on compact Riemannian manifolds. In: Holden, H., Jensen, A. (eds) Schrödinger Operators. Lecture Notes in Physics, vol 345. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51783-9_26

Download citation

  • DOI: https://doi.org/10.1007/3-540-51783-9_26

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51783-2

  • Online ISBN: 978-3-540-46807-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation