Variétés riemanniennes isospectrales non isométriques

Bérard, Pierre

Séminaire Bourbaki : volume 1988/89, exposés 700-714, Astérisque, no. 177-178 (1989), Exposé no. 705, 28 p.

[1] Bérard P. - Transplantation des fonctions et isospectralité, en préparation.

[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).

[1] Berger M. - Geometry of the spectrum, Proc. Symp. Pure Math. 27 Part 2, Amer. Math. Soc. 1975, 129-152. | Zbl | MR

[1] Berger M., Gauduchon P., Mazet E. - Le spectre d'une variété riemannienne, Lecture Notes in Math. Springer 194, 1971. | Zbl | MR

[1] Berry J. P. - Tores isospectraux en dimension 3, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 163-166. | Zbl | MR

[1] Bleecker D. - Determination of a Riemannian metric form the first variation of its spectrum, Amer. J. Math., 107 (1985), 815-831. | Zbl | MR

[1] Brooks R. - On manifolds of negative curvature with isospectral potentials, Topology, 26 (1987), 63-66. | Zbl | MR

[2] Brooks R. - Constructing isospectral manifolds, Amer. Math. Monthly, 95 (1988), 823-839. | Zbl | MR

[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. | Zbl | MR

[1] Brooks R., Perry P., Yang P. - Isospectral sets of conformally equivalent metrics, Duke Math. J., à paraître. | Zbl

[1] Brooks R., Tse R. - Isospectral surfaces of small genus, Nagoya Math. J., 107 (1987), 13-24. | Zbl | MR

[1] Brossard J., Carmona R. - Can one hear the dimension of a fractal, Comm. Math. Phys., 104 (1986), 103-122. | Zbl | MR

[1] Brüning J. - On the compactness of isospectral potentials, Comm. Partial Diff. Eq., 9 (1984), 687-698. | Zbl | MR

[1] Brüning J., Heintze E. - Spektrale Starrheit gewisser Drehflächen, Math. Ann., 269 (1984), 95-101. | Zbl | MR

[1] Buser P. - Isospectral Riemann surfaces, Ann. Inst. Fourier, 36 (1986), 167- 192. | Zbl | MR | Numdam

[2] Buser P. - Riemannsche Flächen und Längenspektrum vom Trigonometrischen Standpunkte aus, Habilitationsschrift, Bonn, 1980.

[3] Buser P. - Sur le spectre de longueurs des surfaces de Riemann, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 487-498. | Zbl | MR

[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. | Zbl | MR

[5] Buser P. - Geometry and spectrum of compact Riemann surfaces, livre en préparation. | Zbl

[1] Buser P., Courtois G. - Finite parts of the spectrum of a Riemann surface, Prépublication de l'Institut Fourier n° 121, Grenoble , 1989. | MR

[1] Buser P., Semmler K. D. - The geometry and spectrum of the one holed torus, Comment. Math. Helv., 63 (1988), 259-274. | Zbl | MR

[1] Chang P.K. - PhD Thesis, University of Pennsylvania, 1988.

[1] Chang P., Deturck D. - On hearing the shape of a triangle, Proc. Amer. Math. Soc., à paraître. | Zbl

[1] Chang S. Y. A., Yang P. - Compactness of isospectral conformal metrics on S3, Comment. Math. Helv., à paraître. | Zbl

[2] Chang S. Y. A., Yang P. - The conformal deformation equation and isospectral sets of conformal metrics, Preprint, U. Southem Cal., 1988. | MR

[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.

[1] Chavel I. - Eigenvalues in Riemannian geometry, Acad. Press, 1984. | Zbl | MR

[1] Chen Sheng. - On the construction of isospectral but non isometric Riemannian manifolds, Preprint, 1988 Louisiana State University Baton Rouge.

[1] Craioveanu M., Puta M. - Introducere în Geometria Spectrală, Editions République Socialiste Roumaine, Bucarest, 1988. | MR

[1] Deturck D. M. - Audible and inaudible geometric properties, Preprint U. Pennsylvania, 1988. | MR

[1] Deturck D. M., Gordon C. S. - Isospectral deformations I : Riemannian structures on two-step nilspaces, Comm. Pure Appl. Math., 40 (1987), 367-387. | Zbl | MR

[2] Deturck D. M., Gordon C. S. - Isospectral metrics and finite Riemannian coverings, Contemporary Mathematics, Amer. Math. Soc., 64 (1987), 79-92. | Zbl | MR

[3] Deturck D. M., Gordon C. S. - Isospectral Riemannian metrics and potentials, Proc. Amer. Math. Soc., 17 (1987), 137-139. | Zbl | MR

[4] Deturck D. M., Gordon C. S. - Isospectral Riemannian metrics and potentials, Preprint Univ. Pennsylvania, 1987. | MR

[1] Deturck D., Gluck H., Gordon C., Webb D. - You cannot hear the size of a homology class, Comment. Math. Helv., à paraître.

[1] Durso C. - PhD Thesis, M.I.T., 1988.

[1] Ejiri N. - A construction of nonflat, compact irreducible Riemannian manifolds which are isospectral but not isometric, Math. Z., 168 (1979), 207-212. | Zbl | MR

[1] Fay J. - Analytic torsion and Prym differentials, Ann. of Math. Stud., 97 (1981), 107-122. | Zbl | MR

[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. | Zbl | MR

[1] Gilkey P. B. - Invariance theory for the heat equation and the Atiyah-Singer index theorem, Publish or Perish Inc., 1984. | Zbl | MR

[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. | Zbl | MR | Numdam

[3] Gilkey P. B. - Leading terms in the asymptotics of the heat equation, Contemporary Math., 73 (1988), 79-85. | Zbl | MR

[1] Gordon C. S. - Riemannian manifolds isospectral on functions but not on 1- forms, J. Diff. Geom., 24 (1986), 79-96. | Zbl | MR

[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. | Zbl | MR

[1] Gordon C. S., Wilson E. N. - Isospectral deformations of compact solvmanifolds, J. Diff. Geom., 19 (1984), 241-256. | Zbl | MR

[2] Gordon C. S., Wilson E. N. - The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J., 33 (1986), 253-271. | Zbl | MR

[3] Gordon C. S., Wilson E. N. - Isometry groups of Riemannian solvmanifolds, Trans. Amer. Math. Soc., 307 (1988), 245-269. | Zbl | MR

[1] Guillemin V. W. - Lectures on spectral theory of elliptic operators, Duke Math. J., 44 (1977), 485-517. | Zbl | MR

[2] Guillemin V. W. - Inverse spectral problems in geometry, Preprint M.I.T., 1988.

[1] Guillemin V., Kazhdan D. - Some inverse spectral results for negatively curved 2-manifolds, Topology, 19 (1980), 153-180. | Zbl | MR

[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. | Zbl | MR

[1] Guralnick R. M. - Subgroups inducing the same permutation representation, J. of Algebra, 81 (1983), 312-319. | Zbl | MR

[1] Haas A. - Length spectra as moduli for hyperbolic surfaces, Duke Math. J., 5 (1985), 922-935. | Zbl | MR

[1] Hejhal D. - The Selberg trace formula and the Riemann zeta function, Duke Math. J., 43 (1976), 441-482. | Zbl | MR

[2] Hejhal D. - The Selberg trace formula for PSL(2, R), Vol. I : Lecture Notes in Math. Springer 548, 1976 Vol. II : Lecture Notes in Math. Springer 1001, 1983. | Zbl | MR

[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.

[2] Ikeda A. - Riemannian manifolds p-isospectral but not (p + 1)-isospectral, Preprint Hiroshima University, 1989. | MR

[1] Kac M. - Can one hear the shape of a drum ?, Amer. Math. Monthly, 73 (1966), 1-23. | Zbl | MR

[1] Keen L. - Collars on Riemann surfaces, Princeton U. Press, Ann. of Math. Stud., 79 (1974), 263-268. | Zbl | MR

[1] Kitaoka Y. - Positive definite quadratic forms with the same representation numbers, Arch. Math., 28 (1977), 495-497. | Zbl | MR

[1] Kneser M. - Lineare Relationen zwischen Darstellungszahlen quadratischer Formen, Math. Ann., 168 (1967), 31-39. | Zbl | MR

[1] Kuwabara R. - On isospectral deformations of Riemannian metrics, Compositio Math., 40 (1980), 319-324. | Zbl | MR | Numdam

[2] Kuwabara R. - On the characterization of flat metrics by the spectrum, Comment. Math. Helv., 55 (1980), 427-444. | Zbl | MR

[3] Kuwabara R. - On isospectral deformations of Riemannian metrics II, Compositio Math., 47 (1982), 195-205. | Zbl | MR | Numdam

[1] Mckean H. P. - Selberg's trace formula as applied to a compact Riemann surface, Comm. Pure Appl. Math., 25 (1972), 225-246. | MR

[2] Mckean H. P., Singer I. M. - Curvature and the eigenvalues of the Laplacian, J. Differential Geom., 1 (1967), 43-69. | Zbl | MR

[1] Melrose R. - Isospectral sets of drumheads are compact in C∞, Preprint M.S.R.I. Berkeley, 1984.

[1] Milnor J. - Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci., 51 (1964), 542. | Zbl | MR

[1] Min-Oo M. - Spectral rigidity for manifolds with negative curvature operator, Contemp. Math. Nonlinear Problems in Geometry, 51 (1986), 99-103. | Zbl | MR

[1] Mumford D. - A remark on Mahler's compactness theorem, Proc. Amer. Math. Soc., 28 (1971), 289-294. | Zbl | MR

[1] Onofri E. - On the positivity of the effective action in a theory of random surfaces, Comm. Math. Phys., 86 (1982), 321-326. | Zbl | MR

[1] Osgood B., Phillips R., Sarnak P. - Extremals of determinants of Laplacians, J. Funct. Anal., 80 (1988), 148-211. | Zbl | MR

[2] Osgood B., Phillips R., Sarnak P. - Compact isospectral sets of Riemann surfaces, J. Funct. Anal., 80 (1988), 212-234. | Zbl | MR

[3] Osgood B., Phillips R., Sarnak P. - Compact isospectral sets of plane domains, Proc. Nat. Acad. Sci. U.S.A., 85 (1988), 5359-5361. | Zbl | MR

[4] Osgood B., Phillips R., Sarnak P. - Moduli space, heights and isospectral sets of plane domains, Ann. of Math., à paraître. | Zbl

[1] Otal J. P. - Le spectre marqué des longueurs des surfaces à courbure négative, Preprint Orsay, 1988. | MR

[1] Patodi V. K. - Curvature and the fundamental solution of the heat equation, J. Indian Math. Soc., 34 (1970), 269-285. | Zbl | MR

[1] Perlis R. - On the equation ζK(s) = ζK'(s), J. Number Theory, 9 (1977), 342-360. | Zbl

[1] Sarnak P. - Determinants of Laplacians; heights and finiteness, Preprint Stanford U., 1988. | MR

[1] Stolz S. - Exotic structures on 4-manifolds detected by spectral invariants, Invent. Math., 94 (1988), 147-162. | Zbl | MR

[1] Sunada T. - Spectrum of a compact flat manifold, Comment. Math. Helv., 53 (1978), 613-621. | Zbl | MR

[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. | Zbl | MR

[3] Sunada T. - Riemannian coverings and isospectral manifolds, Ann. of Math., 121 (1985), 169-186. | Zbl | MR

[1] Tse R. - A lower bound for the number of isospectral surfaces of arbitrarily large genus, PhD Thesis U. Southern Cal., 1988. | MR

[1] Urakawa H. - Bounded domains which are isospectral but not congruent, Ann. Sci. Ecole Norm. Sup., 15 (1982), 441-456. | Zbl | MR | Numdam

[1] Uribe A. - Some remarks on the problem of isospectrality, Preprint Princeton, 1988.

[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. | Zbl | MR

[2] Vignéras M. F. - Variétés riemanniennes isospectrales et non isométriques, Ann. of Math., 112 (1980), 21-32. | Zbl | MR

[1] Witt E. - Eine Identität zwischen Modulformen zweiten Grades, Abh. Sem. Univ. Hamburg, 14 (1941), 289-322. | Zbl | MR | JFM

[1] Wolpert S. - The eigenvalue spectrum as moduli for compact Riemann surfaces, Bull. Amer. Math. Soc., 83 (1977), 1306-1308. | Zbl | MR

[2] Wolpert S. - The Eigenvalue Spectrum as Moduli for Flat Tori, Trans. Amer. Math. Soc., 244 (1978), 313-321. | Zbl | MR

[3] Wolpert S. - The length spectrum as moduli for compact Riemann surfaces, Ann. Math., 109 (1979), 323-351. | Zbl | MR

[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. | Zbl | MR