Skip to main content
SpringerLink
Log in

Spectral Theory and Geometry

  • Chapter
First European Congress of Mathematics

Part of the book series: Progress in Mathematics ((PM,volume 3))

  • 947 Accesses

Abstract

Let M be a smooth differentiable manifold of dimension n and pick a Riemannian metric g on M. To the Riemannian manifold (M,g) one can associate a number of natural elliptic differential operators which arise from the geometric structure of (M,g). Usually these operators act in the space C ∞(E) of smooth sections of some vector bundle E over M. If (M,g) is a complete Riemannian manifold, then many of these operators give rise to self—adjoint operators in the Hilbert space L 2(E) of L 2—sections of E.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M.F. Atiyah, V.K. Patodi, I.M. Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43–69.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Barbasch, H. Moscoviei, L2-index and the Selberg trace formula, J. Fund. Anal. 53 (1983), 151–201.

    Article  MATH  Google Scholar 

  3. P. Bérard, Variétés riemanniennes isospectrales non isométriques, Asterisque 177–178 (1989), 127–154.

    Google Scholar 

  4. P. Bérard, Transplantation et isospectralité I, Math. Ann. 292 (1992), 547–559.

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Bérard, M. Berger, Le spectre d’une variété riemannienne en 1982, in: Spectra of riemannian manifolds, Kaigai Publications, 1983, 139–194.

    Google Scholar 

  6. M. Berger, P. Gauduchon, E. Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Math. 194, Springer-Verlag, Berlin- Heidelberg-New York (1971).

    Google Scholar 

  7. J-M. Bismut, J. Cheeger, Transgressed Euler classes of SL(2n, Z) vector bundles, adiabatic limits of eta invariants and special values of L-functions, Ann. Scient. Éc. Norm. Sup., 4e série, 25 (1992), 335–391.

    MathSciNet  MATH  Google Scholar 

  8. J.-M. Bismut, W. Zhang, Reidemeister, Milnor and Ray-Singer metrics: an extension of a theorem of Cheeger and Müller, preprint Université de Paris-Sud Nr. 92–08, Orsay, 1992.

    Google Scholar 

  9. A. Borei, J.-P. Serre, Corners and arithmetic groups, Commun. Math. Helv. 48 (1973), 436–491.

    Article  Google Scholar 

  10. N.V. Borisov, W. Müller, R. Schrader, Relative index theory and supersymmetric scattering theory, Comm. Math. Phys. 114 (1988), 475–513.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Brüning, L2-index theorems for certain complete manifolds, J. Diff. Geom. 32 (1990), 491–532.

    MATH  Google Scholar 

  12. J. Brüning, L2-index theorems for complete manifolds with ends of rank one type, Duke Math. J. 66 (1992), 257–309.

    Article  MathSciNet  MATH  Google Scholar 

  13. U. Bunke, Relative index theory, J. Funct. Analysis 105 (1992), 63–76.

    Article  MathSciNet  MATH  Google Scholar 

  14. P. Cartier, A.Voros, Une nouvelle interprétation de la formule des traces de Selberg, in: The Grothendieck Festschrift, Vol II. The Grothendieck Festschrift, Progress in Math. 87, Birkhäuser, Boston, 1990, 1–67.

    MathSciNet  Google Scholar 

  15. J. Chazarain, Formule de Poisson pour les variétés riemanniennes, Invent. Math. 24 (1974), 65–82.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Cheeger, Analytic torsion and the heat equation, Ann. of Math. 109 (1979), 259–322.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Cheeger, On the spectral geometry of spaces with cone-like singularities, Proc. Nat Acad. Sci. U.S.A. 76: 5 (1979), 2103–2106.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Cheeger, Spectral geometry of singular Riemannian spaces, J. Diff. Geom. 18 (1983), 575–657.

    MathSciNet  MATH  Google Scholar 

  19. Y. Colin de Verdière, Pseudo-Laplacians II, Ann. Inst. Fourier 33 (1983), 87–113.

    Article  MATH  Google Scholar 

  20. Y. Colin de Verdière, Une nouvelle démonstration du prolongement méromorphe de séries d’Eisenstein, C. R. Acad. Sc. Paris, séries I, 293 (1981), 361–363.

    MATH  Google Scholar 

  21. Y. Colin de Verdière, Spectre du laplacien et longueurs des géodésiques périodiques, Comp. Math. 27 (1973), I 83–106; II 159– 184.

    MATH  Google Scholar 

  22. P. Deift, E. Trubowitz, Inverse scattering on the line. Comm. Pure Appl. Math. 32 (1979), 121–251.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. Derezinski, Asymptotic completeness of long range N- body quantum systems, preprint Ecole Polytechnique Nr. 1023, Dec. 1991.

    Google Scholar 

  24. H. Donnelly, On the cuspidal spectrum of finite volume symmetric spaces, J. Diff. Geom. 17 (1982), 239–253.

    MathSciNet  MATH  Google Scholar 

  25. J.J. Duistermaat, V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), 39–79.

    Article  MathSciNet  MATH  Google Scholar 

  26. L.D. Faddejev, Expansion in eigenfunctions of the Laplace operator in the fundamental domain of a discrete group on the Lobacevskii plane, Trudy Mose. Mat. Obsc. 17 (1967), 323–350.

    Google Scholar 

  27. D. Fried, Lefschetz formulas for flows. Contemp. Math. 58, Part III, (1987), 19–69.

    MathSciNet  Google Scholar 

  28. D. Fried, Analytic torsion and closed geodesies on hyperbolic manifolds, Invent. Math. 84 (1986), 523–540.

    Article  MathSciNet  MATH  Google Scholar 

  29. D.S. Freed, R.E. Gompf, Computer calculations of Witten’s 3- manifold invariant, Comm. Math. Phys. 141 (1991). 79–117.

    Article  MathSciNet  MATH  Google Scholar 

  30. I.M. Gelfand, Automorphic functions and the theory of representations, in: Proc. Int. Cong. Math., Stockholm (1962), 74–85.

    Google Scholar 

  31. P. Gilkey, Invariance Theory, The Heat Equation and the Atiyah - Singer Index Theorem, Mathematics Lecture Series 11, Publish or Perish, Wilmington, 1984.

    Google Scholar 

  32. P. Gilkey, Leading terms in the asymptotics of the heat equation, Contemp. Math. 73 (1988), 79–85.

    MathSciNet  Google Scholar 

  33. C. Gordon, E. Wilson, Isospectral deformations of compact solv- manifolds, J. Diff. Geom. 19 (1984), 241–256.

    MathSciNet  MATH  Google Scholar 

  34. C. Gordon, D. Webb, S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math. 110 (1992), 1–22.

    Article  MathSciNet  MATH  Google Scholar 

  35. G.M. Graf, Asymptotic completeness for N-body short-range quantum systems: A new proof, Comm. Math. Phys. 132 (1990), 73–101.

    Article  MathSciNet  MATH  Google Scholar 

  36. M. Gromov, H.B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publicationes Math. IHES 58 (1983), 83–196.

    MathSciNet  MATH  Google Scholar 

  37. M. Gromov, M.A. Shubin, Von Neumann spectra near zero, Geom. Anal, and Funct. Anal. 1 (1991), 375–404.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. L. Guillopé, Théorie spectrale de quelques variétés à bouts, Ann. Scient. Éc. Norm. Sup. 22 (1989), 137–160.

    MATH  Google Scholar 

  40. D.A. Hejhal, The Selberg trace formula for PSL(2,R), Lecture Notes in Math 1001 (1983), Vol. II, Springer-Verlag, Berlin- Heidelberg-New York.

    Google Scholar 

  41. A. Ikeda, On lens spaces which are isospectral but not isometric, Ann. Scient. Éc. Norm. Sup. 4e série, 13 (1980), 303–315.

    MATH  Google Scholar 

  42. D. Johnson, A geometric form of Casson’s invariant, and its connection to Reidemeister torsion, unpublished lecture notes.

    Google Scholar 

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

    Article  MATH  Google Scholar 

  44. T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin-Heidelberg-New York, 1966.

    MATH  Google Scholar 

  45. S.-Y. Koyama, Determinant expression of Selberg zeta functions I, Trans. Amer. Math. Soc. 324 (1991), 149–168.

    Article  MathSciNet  MATH  Google Scholar 

  46. R.P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Math. 544 (1976), Springer-Verlag, Berlin- Heidelberg-New York.

    Google Scholar 

  47. P.D. Lax, R.S. Phillips, Scattering theory for automorphic forms, Annals Math. Studies 87, Princeton, N.J. (1976).

    Google Scholar 

  48. J. Lott, Heat kernels on covering spaces and topological invariants, J. Diff. Geom. 35 (1992), 471–510.

    MathSciNet  MATH  Google Scholar 

  49. J. Lott, M. Rothenberg, Analytic torsion for group actions, J. Diff. Geom. 34 (1991), 431–481.

    MathSciNet  MATH  Google Scholar 

  50. J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  52. H. Maaß, Über eine neue Art von nichtanalytischen automorphen Formen, Math. Annalen 121 (1949), 141–183.

    Article  MATH  Google Scholar 

  53. D.H. Mayer, The thermodynamic formalism approach to Selberg’s zeta function for PSL(2,Z), Bull. Amer. Math. Soc. 25 (1991), 55–60.

    Article  MathSciNet  MATH  Google Scholar 

  54. H.P. McKean, Selberg’s trace formula as applied to a compact Riemann surface, Comm. Pure Appl Math. 25 (1972), 225–246.

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  56. C. Moeglin, J.-L. Waldspurger, Le spectre résiduel de GL(n), Ann. Scient. Éc. Norm. Sup., 4e série, 22 (1989), 605–674.

    MathSciNet  MATH  Google Scholar 

  57. C. Moeglin, Sur les formes automorphes de carré integrable, in: Proc. Int. Cong. Math., Kyoto, 1990, Vol. II, 815–819.

    Google Scholar 

  58. R. Melrose, Isospectral drumheads are compact in C∞, preprint.

    Google Scholar 

  59. R. Melrose, Weyl asymptotic for the phase in obstacle scattering Comm. Partial Diff. Equations 13 (1988), 1431–1439.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  61. H. Moscovici, L2 index of elliptic operators on locally symmetric spaces of finite volume, Contemp. Math. 10 (1982), 129–138.

    MathSciNet  MATH  Google Scholar 

  62. H. Moscovici, R.J. Stanton, R-torsion and zeta functions for locally symmetric manifolds, Invent. Math. 105 (1991), 185–216.

    Article  MathSciNet  MATH  Google Scholar 

  63. W. Müller, Analytic torsion and R-torsion of Riemannian manifolds, Advances in Math. 28 (1978), 233–305.

    Article  MATH  Google Scholar 

  64. W. Müller, Analytic torsion and R-torsion for unimodular repre-sentations, J. Amer. Math, Soc. 6 (1993), 721–753.

    Article  MathSciNet  MATH  Google Scholar 

  65. W. Müller, Manifolds with cusps of rank one, Lecture Notes in Math. 1244, Springer-Ver lag, Berlin-Heidelberg-New York (1987).

    Google Scholar 

  66. W. Müller, Spectral theory for Riemannian manifolds with cusps and a related trace formula, Math. Nachrichten 111 (1983), 197– 288.

    Article  Google Scholar 

  67. W. Müller, The trace class conjecture in the theory of automorphic forms, Ann. of Math. 130 (1989), 473–529.

    Article  MathSciNet  MATH  Google Scholar 

  68. W. Müller, Spectral geometry and scattering theory for certain complete surfaces of finite volume, Invent. Math. 109 (1992), 265–305.

    Article  MathSciNet  MATH  Google Scholar 

  69. W. Müller, L2-index theory, eta invariants and values of L- functions, Contemp. Math. 105 (1990), 145–189.

    Google Scholar 

  70. W. Miiller, L2-index and resonances, in: Geometry and Analysis on Manifolds, Lecture Notes in Math. 1339, Springer, Berlin (1988), 203–211.

    Google Scholar 

  71. S. Novikov, M. Shubin, Morse inequalities and von Neumann invariants of nonSimply connected manifolds, Uspekhi Matem. Nauk 41 (1986), 222.

    Google Scholar 

  72. B. Osgood, R. Phillips, P. Sarnak, Extremals of determinants of Laplacians, J. Funct Anal 80 (1988), 148–211.

    Article  MathSciNet  MATH  Google Scholar 

  73. B. Osgood, R. Phillips, P. Sarnak, Compact isospectral sets of surfaces, J. Funct Anal 80 (1988), 212–234.

    Article  MathSciNet  MATH  Google Scholar 

  74. B. Osgood, R. Phillips, P. Sarnak, Moduli spaces, heights and isospectral sets of plane domains, Ann. Math. 129 (1989), 293–362.

    Article  MathSciNet  MATH  Google Scholar 

  75. L.B. Parnovski, Spectral asymptotics of the Laplace operators on surfaces with hyperbolic ends, preprint, Univ. Augsburg, 1993.

    Google Scholar 

  76. R. Phillips, P. Sarnak, Perturbation theory for the Laplacian on automorphic functions, J. Amer. Math. Soc. 5 (1992), 1–32.

    Article  MathSciNet  MATH  Google Scholar 

  77. R. Phillips, P. Sarnak, On cusp forms for cofinite subgroups of PSL(2,R), Invent. Math. 80 (1985), 339–364.

    Article  MathSciNet  MATH  Google Scholar 

  78. J. Pöschel, E. Trubowitz, Inverse spectral theory, Academic Press, New York, 1987.

    MATH  Google Scholar 

  79. M. Pollicott, Some applications of the thermodynamic formalism to manifolds of constant negative curvature, Advances in Math. 85 (1991), 161–192.

    Article  MathSciNet  MATH  Google Scholar 

  80. D.B. Ray, I.M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145–210.

    Article  MathSciNet  MATH  Google Scholar 

  81. J. Roe, An index theorem for open manifolds I, II, J. Diff. Geom. 27 (1988), 87–113;

    MathSciNet  MATH  Google Scholar 

  82. J. Roe, Exotic cohomology and index theory for complete Riemannian manifolds, preprint, Oxford, 1990.

    Google Scholar 

  83. P. Sarnak, On cusp forms, Contemp. Math. 53 (1986), 393–407.

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  85. A. Selberg, Harmonic Analysis and discrete groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. B 20 (1956), 47–87.

    MathSciNet  MATH  Google Scholar 

  86. A. Selberg, Harmonic Analysis, in: Collected papers, Vol. Springer-Verlag, Berlin-Heidelberg-New York, 1989, 626–674.

    Google Scholar 

  87. I.M. Sigal, A. Soffer, N-particle scattering problem: asymptotic completeness for short-range systems, Ann. Math 126 (1987), 35–108.

    Article  MathSciNet  MATH  Google Scholar 

  88. M. Stern, M., L2-index theorems on locally symmetric spaces, Invent Math. 96 (1989), 231–282.

    Article  MathSciNet  MATH  Google Scholar 

  89. M.F. Vigneras, Variétés riemanniennes isospectrales et non isométriques, Ann. of Math. 112 (1980), 21–32.

    Article  MathSciNet  MATH  Google Scholar 

  90. H. Weyl, Ramifications, old and new, of the eigenvalue problem, Bull. Amer. Math. Soc. 56 (1950), 115–139.

    Article  MathSciNet  MATH  Google Scholar 

  91. E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1988), 351–399.

    Article  MathSciNet  Google Scholar 

  92. E. Witten, On quantum gauge theories in two dimensions, Commun. Math. Phys. 141 (1991), 153–209.

    Article  MathSciNet  MATH  Google Scholar 

  93. S. Wolpert, The length spectra as moduli for compact Riemann surfaces, Ann. of Math. 109 (1979), 323–351.

    Article  MathSciNet  MATH  Google Scholar 

  94. S. Wolpert, Disappearance of cusp forms in families, preprint 1992.

    Google Scholar 

  95. S. Zelditch, Kuznecov sum formulae and Szegö limit formulae on manifolds, Comm. Partial Diff. Equations 17 (1992), 221–260.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. MPI für Mathematik, Gottfried-Claren Str. 26, D-53225, Bonn, Germany

    Werner Müller

Authors
  1. Werner Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Laboratoire de Mathématiques Fondamentales, Université Pierre et Marie Curie, 4, place Jussieu, 75252, Paris Cedex 05, France

    Anthony Joseph  & Rudolf Rentschler  & 

  2. The Weizmann Institute of Science, 76100, Rehovot, Israel

    Anthony Joseph

  3. Mathématiques Bâtiment 425, Université de Paris-Sud, 91405, Orsay Cedex, France

    Fulbert Mignot

  4. Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, 4, place Jussieu, 75252, Paris Cedex, France

    François Murat

  5. Laboratoire de Statistique Médicale, Université Paris V, 45, rue des Saints Pères, 75270, Paris Cedex, France

    Bernard Prum

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Birkhäuser Verlag

About this chapter

Cite this chapter

Müller, W. (1994). Spectral Theory and Geometry. In: Joseph, A., Mignot, F., Murat, F., Prum, B., Rentschler, R. (eds) First European Congress of Mathematics . Progress in Mathematics, vol 3. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9110-3_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-9110-3_5

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9911-6

  • Online ISBN: 978-3-0348-9110-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation