Variétés riemanniennes isospectrales non isométriques

Pierre Bérard

Séminaire Bourbaki (1988-1989)

  • Volume: 31, page 127-154
  • ISSN: 0303-1179

How to cite

top

Bérard, Pierre. "Variétés riemanniennes isospectrales non isométriques." Séminaire Bourbaki 31 (1988-1989): 127-154. <http://eudml.org/doc/110105>.

@article{Bérard1988-1989,
author = {Bérard, Pierre},
journal = {Séminaire Bourbaki},
keywords = {eigenvalue; spectrum; isospectrality},
language = {fre},
pages = {127-154},
publisher = {Société Mathématique de France},
title = {Variétés riemanniennes isospectrales non isométriques},
url = {http://eudml.org/doc/110105},
volume = {31},
year = {1988-1989},
}

TY - JOUR
AU - Bérard, Pierre
TI - Variétés riemanniennes isospectrales non isométriques
JO - Séminaire Bourbaki
PY - 1988-1989
PB - Société Mathématique de France
VL - 31
SP - 127
EP - 154
LA - fre
KW - eigenvalue; spectrum; isospectrality
UR - http://eudml.org/doc/110105
ER -

References

top
  1. [1] Bérard P. — Transplantation des fonctions et isospectralité, en préparation. 
  2. [1] Bérard P., Berger M.— Le spectre d'une variété riemannienne en 1982, in Spectra of Riemannian manifolds, Kaigai Publications (1983), 139-194, [Proc. France-Japan 1984 Seminar Kyoto ed. Berger, Murakami, Ochiai], ou dans Lecture Notes in Math. Springer, 1207 (1986). 
  3. [1] Berger M. — Geometry of the spectrum, Proc. Symp. Pure Math.27 Part 2, Amer. Math. Soc.1975, 129-152. Zbl0311.53055MR383459
  4. [1] Berger M., Gauduchon P., Mazet E. — Le spectre d'une variété riemannienne, Lecture Notes in Math. Springer194, 1971. Zbl0223.53034MR282313
  5. [1] Berry J.P. — Tores isospectraux en dimension 3, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 163-166. Zbl0464.58018MR610309
  6. [1] Bleecker D. — Determination of a Riemannian metric form the first variation of its spectrum, Amer. J. Math., 107 (1985), 815-831. Zbl0577.58032MR796904
  7. [1] Brooks R. — On manifolds of negative curvature with isospectral potentials, Topology, 26 (1987), 63-66. Zbl0617.53048MR880508
  8. [2] Brooks R. — Constructing isospectral manifolds, Amer. Math. Monthly, 95 (1988), 823-839. Zbl0673.58046MR967343
  9. [3] Brooks R. — Isospectral potentials on a surface of genus 3, Math. Sciences Reach. Institute series, Ed. D. Drasin, "Holomorphic functions and moduli", Springer, 1988, 203-208. Zbl0657.58047MR955822
  10. [1] Brooks R., Perry P., Yang P. — Isospectral sets of conformally equivalent metrics, Duke Math. J., à paraître. Zbl0667.53037
  11. [1] Brooks R., Tse R. — Isospectral surfaces of small genus, Nagoya Math. J., 107 (1987), 13-24. Zbl0605.58041MR909246
  12. [1] Brossard J., Carmona R. — Can one hear the dimension of a fractal, Comm. Math. Phys., 104 (1986), 103-122. Zbl0607.58043MR834484
  13. [1] Brüning J. — On the compactness of isospectral potentials, Comm. Partial Diff. Eq., 9 (1984), 687-698. Zbl0547.58039MR745021
  14. [1] Brüning J., Heintze E. — Spektrale Starrheit gewisser Drehflächen, Math. Ann., 269 (1984), 95-101. Zbl0553.53028MR756778
  15. [1] Buser P. — Isospectral Riemann surfaces, Ann. Inst. Fourier, 36 (1986), 167- 192. Zbl0579.53036MR850750
  16. [2] Buser P. — Riemannsche Flächen und Längenspektrum vom Trigonometrischen Standpunkte aus, Habilitationsschrift, Bonn, 1980. 
  17. [3] Buser P. — Sur le spectre de longueurs des surfaces de Riemann, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 487-498. Zbl0466.32012MR612543
  18. [4] Buser P. — Cayley graphs and planar isospectral domains, Proc. Taniguchi Symp. "Geometry and Analysis on manifolds"1987, Lecture Notes in Math. Springer , 1339 (1988), 64-77. Zbl0647.53034MR961473
  19. [5] Buser P. — Geometry and spectrum of compact Riemann surfaces, livre en préparation. Zbl0770.53001
  20. [1] Buser P., Courtois G. — Finite parts of the spectrum of a Riemann surface, Prépublication de l'Institut Fourier n° 121, Grenoble , 1989. Zbl0711.58033MR1060691
  21. [1] Buser P., Semmler K.D. — The geometry and spectrum of the one holed torus, Comment. Math. Helv., 63 (1988), 259-274. Zbl0649.53028MR948781
  22. [1] Chang P.K. — PhD Thesis, University of Pennsylvania, 1988. 
  23. [1] Chang P., Deturck D. — On hearing the shape of a triangle, Proc. Amer. Math. Soc., à paraître. Zbl0721.58053
  24. [1] Chang S.Y.A., Yang P. — Compactness of isospectral conformal metrics on S3, Comment. Math. Helv., à paraître. Zbl0679.53038
  25. [2] Chang S.Y.A., Yang P. — The conformal deformation equation and isospectral sets of conformal metrics, Preprint, U. Southem Cal., 1988. MR925123
  26. [3] Chang S.Y.A., Yang P. — A compactness theorem for conformal metrics on 3-manifolds and application to isospectral metrics, Preprint, U. Southern Cal., 1988. 
  27. [1] Chavel I. — Eigenvalues in Riemannian geometry, Acad. Press, 1984. Zbl0551.53001MR768584
  28. [1] Chen Sheng. — On the construction of isospectral but non isometric Riemannian manifolds, Preprint, 1988Louisiana State University Baton Rouge. 
  29. [1] Craioveanu M., Puta M. — Introducere în Geometria Spectrală, Editions République Socialiste Roumaine, Bucarest, 1988. MR1011669
  30. [1] Deturck D.M. — Audible and inaudible geometric properties, Preprint U. Pennsylvania, 1988. MR1122855
  31. [1] Deturck D.M., Gordon C.S. — Isospectral deformations I : Riemannian structures on two-step nilspaces, Comm. Pure Appl. Math., 40 (1987), 367-387. Zbl0649.53025MR882070
  32. [2] Deturck D.M., Gordon C.S. — Isospectral metrics and finite Riemannian coverings, Contemporary Mathematics, Amer. Math. Soc., 64 (1987), 79-92. Zbl0646.58030MR881457
  33. [3] Deturck D.M., Gordon C.S. — Isospectral Riemannian metrics and potentials, Proc. Amer. Math. Soc., 17 (1987), 137-139. Zbl0629.58008MR888890
  34. [4] Deturck D.M., Gordon C.S. — Isospectral Riemannian metrics and potentials, Preprint Univ. Pennsylvania, 1987. MR888890
  35. [1] Deturck D., Gluck H., Gordon C., Webb D. — You cannot hear the size of a homology class, Comment. Math. Helv., à paraître. Zbl0694.53037
  36. [1] Durso C. — PhD Thesis, M.I.T., 1988. 
  37. [1] Ejiri N. — A construction of nonflat, compact irreducible Riemannian manifolds which are isospectral but not isometric, Math. Z., 168 (1979), 207-212. Zbl0396.53022MR544590
  38. [1] Fay J. — Analytic torsion and Prym differentials, Ann. of Math. Stud., 97 (1981), 107-122. Zbl0458.30025MR624809
  39. [1] Fleckinger-Pellé J., Lapidus M.L. — Tambour fractal : vers une résolution de la conjecture de Weyl-Berry pour les valeurs propes du laplacien, C. R. Acad. Sci. Paris Sér. I Math., 306 (1988), 171. Zbl0654.35079MR930556
  40. [1] Gilkey P.B. — Invariance theory for the heat equation and the Atiyah-Singer index theorem, Publish or Perish Inc., 1984. Zbl0565.58035MR783634
  41. [2] Gilkey P.B. — On spherical space-forms with meta-cyclic fundamental group which are isospectral but not equivariant cobordant, Compositio Math., 56 (1985), 171-200. Zbl0593.58041MR809865
  42. [3] Gilkey P.B. — Leading terms in the asymptotics of the heat equation, Contemporary Math., 73 (1988), 79-85. Zbl0661.58034MR954631
  43. [1] Gordon C.S. — Riemannian manifolds isospectral on functions but not on 1- forms, J. Diff. Geom., 24 (1986), 79-96. Zbl0585.53036MR857377
  44. [2] Gordon C.S. — The Laplace spectrum versus the length spectra of Riemannian manifolds, in Nonlinear Problems in Geometry (D.M. De Turck, ed.) Contemporary Math., 51 (1986), 63-80. Zbl0591.53042MR848934
  45. [1] Gordon C.S., Wilson E.N. — Isospectral deformations of compact solvmanifolds, J. Diff. Geom., 19 (1984), 241-256. Zbl0523.58043MR739790
  46. [2] Gordon C.S., Wilson E.N. — The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J., 33 (1986), 253-271. Zbl0599.53038MR837583
  47. [3] Gordon C.S., Wilson E.N. — Isometry groups of Riemannian solvmanifolds, Trans. Amer. Math. Soc., 307 (1988), 245-269. Zbl0664.53022MR936815
  48. [1] Guillemin V.W. — Lectures on spectral theory of elliptic operators, Duke Math. J., 44 (1977), 485-517. Zbl0463.58024MR448452
  49. [2] Guillemin V.W. — Inverse spectral problems in geometry, Preprint M.I.T., 1988. 
  50. [1] Guillemin V., Kazhdan D. — Some inverse spectral results for negatively curved 2-manifolds, Topology, 19 (1980), 153-180. Zbl0456.58031MR579579
  51. [2] Guillemin V., Kazhdan D. — Some inverse spectral results for negatively curved n-manifolds, Proc. Symp. Pure Math., Geometry of the Laplace Operator, Amer. Math. Soc., 36 (1980), 153-180. Zbl0456.58031MR573432
  52. [1] Guralnick R.M. — Subgroups inducing the same permutation representation, J. of Algebra, 81 (1983), 312-319. Zbl0527.20005MR700287
  53. [1] Haas A. — Length spectra as moduli for hyperbolic surfaces, Duke Math. J., 5 (1985), 922-935. Zbl0595.30052MR816393
  54. [1] Hejhal D. — The Selberg trace formula and the Riemann zeta function, Duke Math. J., 43 (1976), 441-482. Zbl0346.10010MR414490
  55. [2] Hejhal D. — The Selberg trace formula for PSL(2, R), Vol. I : Lecture Notes in Math. Springer548, 1976 Vol. II : Lecture Notes in Math. Springer1001, 1983. Zbl0347.10018MR439755
  56. [1] Ikeda A. — Isospectral problem for spherical space forms, in spectra of Riemannian Manifolds, ed. by M. Berger, S. Murakami and T. Ochiai, Kaigai Publications (1983), 57-63. 
  57. [2] Ikeda A. — Riemannian manifolds p-isospectral but not (p + 1)-isospectral, Preprint Hiroshima University, 1989. Zbl0704.53037MR1040537
  58. [1] Kac M. — Can one hear the shape of a drum ?, Amer. Math. Monthly, 73 (1966), 1-23. Zbl0139.05603MR201237
  59. [1] Keen L. — Collars on Riemann surfaces, Princeton U. Press, Ann. of Math. Stud., 79 (1974), 263-268. Zbl0304.30014MR379833
  60. [1] Kitaoka Y. — Positive definite quadratic forms with the same representation numbers, Arch. Math., 28 (1977), 495-497. Zbl0361.10019MR441864
  61. [1] Kneser M. — Lineare Relationen zwischen Darstellungszahlen quadratischer Formen, Math. Ann., 168 (1967), 31-39. Zbl0146.05901MR205943
  62. [1] Kuwabara R. — On isospectral deformations of Riemannian metrics, Compositio Math., 40 (1980), 319-324. Zbl0406.53038MR571054
  63. [2] Kuwabara R. — On the characterization of flat metrics by the spectrum, Comment. Math. Helv., 55 (1980), 427-444. Zbl0448.58027MR593057
  64. [3] Kuwabara R. — On isospectral deformations of Riemannian metrics II, Compositio Math., 47 (1982), 195-205. Zbl0505.53019MR677020
  65. [1] Mc Kean H.P. — Selberg's trace formula as applied to a compact Riemann surface, Comm. Pure Appl. Math., 25 (1972), 225-246. MR473166
  66. [2] Mc Kean H.P., Singer I.M. — Curvature and the eigenvalues of the Laplacian, J. Differential Geom., 1 (1967), 43-69. Zbl0198.44301MR217739
  67. [1] Melrose R. — Isospectral sets of drumheads are compact in C∞, Preprint M.S.R.I. Berkeley, 1984. 
  68. [1] Milnor J. — Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci., 51 (1964), 542. Zbl0124.31202MR162204
  69. [1] Min-oo M. — Spectral rigidity for manifolds with negative curvature operator, Contemp. Math. Nonlinear Problems in Geometry, 51 (1986), 99-103. Zbl0591.53041MR848937
  70. [1] Mumford D. — A remark on Mahler's compactness theorem, Proc. Amer. Math. Soc., 28 (1971), 289-294. Zbl0215.23202MR276410
  71. [1] Onofri E. — On the positivity of the effective action in a theory of random surfaces, Comm. Math. Phys., 86 (1982), 321-326. Zbl0506.47031MR677001
  72. [1] Osgood B., Phillips R., Sarnak P. — Extremals of determinants of Laplacians, J. Funct. Anal., 80 (1988), 148-211. Zbl0653.53022MR960228
  73. [2] Osgood B., Phillips R., Sarnak P. — Compact isospectral sets of Riemann surfaces, J. Funct. Anal., 80 (1988), 212-234. Zbl0653.53021MR960229
  74. [3] Osgood B., Phillips R., Sarnak P. — Compact isospectral sets of plane domains, Proc. Nat. Acad. Sci. U.S.A., 85 (1988), 5359-5361. Zbl0674.30021MR952815
  75. [4] Osgood B., Phillips R., Sarnak P. — Moduli space, heights and isospectral sets of plane domains, Ann. of Math., à paraître. Zbl0677.58045
  76. [1] Otal J.P. — Le spectre marqué des longueurs des surfaces à courbure négative, Preprint Orsay, 1988. MR1038361
  77. [1] Patodi V.K. — Curvature and the fundamental solution of the heat equation, J. Indian Math. Soc., 34 (1970), 269-285. Zbl0237.53039MR488181
  78. [1] Perlis R. — On the equation ζK(s) = ζK'(s), J. Number Theory, 9 (1977), 342-360. Zbl0389.12006
  79. [1] Sarnak P. — Determinants of Laplacians; heights and finiteness, Preprint Stanford U., 1988. MR1039364
  80. [1] Stolz S. — Exotic structures on 4-manifolds detected by spectral invariants, Invent. Math., 94 (1988), 147-162. Zbl0656.58032MR958593
  81. [1] Sunada T. — Spectrum of a compact flat manifold, Comment. Math. Helv., 53 (1978), 613-621. Zbl0422.53023MR511851
  82. [2] Sunada T. — Fundamental groups and Laplacians, Proc. Taniguchi Symp. "Geometry and Analysis on Manifolds"1987, Lecture Notes in Math. Springer , 1339 (1988), 248-277. Zbl0646.58027MR961485
  83. [3] Sunada T. — Riemannian coverings and isospectral manifolds, Ann. of Math., 121 (1985), 169-186. Zbl0585.58047MR782558
  84. [1] Tse R. — A lower bound for the number of isospectral surfaces of arbitrarily large genus, PhD ThesisU. Southern Cal., 1988. MR1034979
  85. [1] Urakawa H. — Bounded domains which are isospectral but not congruent, Ann. Sci. Ecole Norm. Sup., 15 (1982), 441-456. Zbl0505.58036MR690649
  86. [1] Uribe A. — Some remarks on the problem of isospectrality, Preprint Princeton, 1988. 
  87. [1] Vignéras M.F. — Exemples de sous-groupes discrets non conjugués de PSL(2, R) qui ont même fonction zêta de Selberg, C.R. Acad. Sci. Paris, 287 (1978), 47-49. Zbl0387.10013MR491604
  88. [2] Vignéras M.F. — Variétés riemanniennes isospectrales et non isométriques, Ann. of Math., 112 (1980), 21-32. Zbl0445.53026MR584073
  89. [1] Witt E. — Eine Identität zwischen Modulformen zweiten Grades, Abh. Sem. Univ. Hamburg, 14 (1941), 289-322. Zbl0025.01701MR5508JFM67.0296.01
  90. [1] Wolpert S. — The eigenvalue spectrum as moduli for compact Riemann surfaces, Bull. Amer. Math. Soc., 83 (1977), 1306-1308. Zbl0368.32009MR499329
  91. [2] Wolpert S. — The Eigenvalue Spectrum as Moduli for Flat Tori, Trans. Amer. Math. Soc., 244 (1978), 313-321. Zbl0405.58051MR514879
  92. [3] Wolpert S. — The length spectrum as moduli for compact Riemann surfaces, Ann. Math., 109 (1979), 323-351. Zbl0441.30055MR528966
  93. [4] Wolpert S. — Asymptotics of the spectrum and the Selberg zêta function on the space of Riemann surfaces, Comm. Math. Phys., 112 (1987), 283-315. Zbl0629.58029MR905169

Citations in EuDML Documents

top
  1. Bruno Colbois, Introduction au laplacien
  2. Pierre Bérard, Transplantation et isospectralité II
  3. Pierre Bérard, Transplantation et isospectralité I
  4. Carolyn S. Gordon, Ruth Gornet, Dorothee Schueth, David L. Webb, Edward N. Wilson, Isospectral deformations of closed riemannian manifolds with different scalar curvature
  5. Emmanuel Philippe, Le spectre des longueurs des surfaces hyperboliques : un exemple de rigidité.
  6. Pierre Pansu, Le flot géodésique des variétés riemanniennes à courbure négative

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.