One cannot hear the shape of a drum

Gordon, Carolyn; Webb, David L.; Wolpert, Scott

doi:10.1090/S0273-0979-1992-00289-6

One cannot hear the shape of a drum

Authors: Carolyn Gordon, David L. Webb and Scott Wolpert
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

Pierre Bérard, Transplantation et isospectralité. I, Math. Ann. 292 (1992), no. 3, 547–559 (French). MR 1152950, DOI https://doi.org/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
Robert Brooks, Constructing isospectral manifolds, Amer. Math. Monthly 95 (1988), no. 9, 823–839. MR 967343, DOI https://doi.org/10.2307/2322897
Robert Brooks, On manifolds of negative curvature with isospectral potentials, Topology 26 (1987), no. 1, 63–66. MR 880508, DOI https://doi.org/10.1016/0040-9383%2887%2990021-8
Robert Brooks and Richard Tse, Isospectral surfaces of small genus, Nagoya Math. J. 107 (1987), 13–24. MR 909246, DOI https://doi.org/10.1017/S0027763000002518
Peter Buser, Isospectral Riemann surfaces, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 2, 167–192 (English, with French summary). MR 850750
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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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
Mark Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), no. 4, 1–23. MR 201237, DOI https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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
Marie-France Vignéras, Variétés riemanniennes isospectrales et non isométriques, Ann. of Math. (2) 112 (1980), no. 1, 21–32 (French). MR 584073, DOI https://doi.org/10.2307/1971319

H. Weyl, Über die Asymptotische Verteilung der Eigenwerte, Nachr. Konigl. Ges. Wiss. Göttingen (1911), 110-117.