One cannot hear the shape of a drum
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- by Carolyn Gordon, David L. Webb and Scott Wolpert PDF
- Bull. Amer. Math. Soc. 27 (1992), 134-138 Request permission
Abstract:
We use an extension of Sunada’s theorem to construct a nonisometric pair of isospectral simply connected domains in the Euclidean plane, thus answering negatively Kac’s question, "can one hear the shape of a drum?" In order to construct simply connected examples, we exploit the observation that an orbifold whose underlying space is a simply connected manifold with boundary need not be simply connected as an orbifold.References
- Pierre Bérard, Transplantation et isospectralité. I, Math. Ann. 292 (1992), no. 3, 547–559 (French). MR 1152950, DOI 10.1007/BF01444635 —, Variétés Riemanniennes isospectrales non isometriques, Sem. Bourbaki, vol. 705, 1988/89.
- Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313, DOI 10.1007/BFb0064643
- Robert Brooks, Constructing isospectral manifolds, Amer. Math. Monthly 95 (1988), no. 9, 823–839. MR 967343, DOI 10.2307/2322897
- Robert Brooks, On manifolds of negative curvature with isospectral potentials, Topology 26 (1987), no. 1, 63–66. MR 880508, DOI 10.1016/0040-9383(87)90021-8
- Robert Brooks and Richard Tse, Isospectral surfaces of small genus, Nagoya Math. J. 107 (1987), 13–24. MR 909246, DOI 10.1017/S0027763000002518
- Peter Buser, Isospectral Riemann surfaces, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 2, 167–192 (English, with French summary). MR 850750, DOI 10.5802/aif.1054
- Peter Buser, Cayley graphs and planar isospectral domains, Geometry and analysis on manifolds (Katata/Kyoto, 1987) Lecture Notes in Math., vol. 1339, Springer, Berlin, 1988, pp. 64–77. MR 961473, DOI 10.1007/BFb0083047
- Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
- Dennis M. DeTurck, Audible and inaudible geometric properties, Rend. Sem. Fac. Sci. Univ. Cagliari 58 (1988), no. suppl., 1–26. Conference on Differential Geometry and Topology (Sardinia, 1988). MR 1122855
- Dennis M. DeTurck and Carolyn S. Gordon, Isospectral deformations. I. Riemannian structures on two-step nilspaces, Comm. Pure Appl. Math. 40 (1987), no. 3, 367–387. MR 882070, DOI 10.1002/cpa.3160400306
- Dennis M. DeTurck and Carolyn S. Gordon, Isospectral deformations. II. Trace formulas, metrics, and potentials, Comm. Pure Appl. Math. 42 (1989), no. 8, 1067–1095. With an appendix by Kyung Bai Lee. MR 1029118, DOI 10.1002/cpa.3160420803
- Carolyn S. Gordon, When you can’t hear the shape of a manifold, Math. Intelligencer 11 (1989), no. 3, 39–47. With an appendix by Dennis DeTurck. MR 1007037, DOI 10.1007/BF03025190
- Carolyn S. Gordon and Edward N. Wilson, Isospectral deformations of compact solvmanifolds, J. Differential Geom. 19 (1984), no. 1, 241–256. MR 739790
- Akira Ikeda, On lens spaces which are isospectral but not isometric, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 3, 303–315. MR 597742, DOI 10.24033/asens.1384
- Mark Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), no. 4, 1–23. MR 201237, DOI 10.2307/2313748
- J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 542. MR 162204, DOI 10.1073/pnas.51.4.542
- Peter Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, DOI 10.1112/blms/15.5.401
- Toshikazu Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. (2) 121 (1985), no. 1, 169–186. MR 782558, DOI 10.2307/1971195 W. P. Thurston, The geometry and topology of 3-manifolds, mimeographed lecture notes, Princeton Univ., 1976-79.
- Hajime Urakawa, Bounded domains which are isospectral but not congruent, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 3, 441–456. MR 690649, DOI 10.24033/asens.1433
- Marie-France Vignéras, Variétés riemanniennes isospectrales et non isométriques, Ann. of Math. (2) 112 (1980), no. 1, 21–32 (French). MR 584073, DOI 10.2307/1971319 H. Weyl, Über die Asymptotische Verteilung der Eigenwerte, Nachr. Konigl. Ges. Wiss. Göttingen (1911), 110-117.
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 27 (1992), 134-138
- MSC (2000): Primary 58G25; Secondary 35R30
- DOI: https://doi.org/10.1090/S0273-0979-1992-00289-6
- MathSciNet review: 1136137