Isospectral plane domains and surfaces via Riemannian orbifolds
One cannot hear the shape of a drum: this is the answer the authors give in the present article to a famous question attributed to S. Bochner and put forward by \textit{Mark Kac} in Am. Math. Mon. 73, No. 4, Part II, 1-23 (1966; Zbl 0139.056). If \(M\) is a domain in the Euclidean plane, then the Dirichlet eigenvalues of the Laplacian are essentially the frequencies produced by a drum of shape \(M\). Mathematically speaking and put into slightly more general form, the question of Kac was therefore to establish if two compact isometric Riemannian manifolds, perhaps with boundary, are isometric (when the boundary is nonempty one may consider the Dirichlet or the Neumann spectrum of the Laplace operator on the manifold). Put in this generality it has been known for quite some time that the answer is no: examples of isospectral but not isometric manifolds have been produced by many mathematicians, starting with J. Milnor in 1964. None of these examples referred to plane domains (the example of Milnor was in dimension 16), so the problem in its original form remained open until the present very beautiful article in which the authors exhibit explicit isospetral nonisometric plane domains. It is perhaps worth mentioning that quite some effort has been devoted to the relation between the spectrum of the Laplace operator and the geometry of the manifold. (Results of this type have a long tradition, starting perhaps with work of P. Debye and continuing to the present day through work of H. Weyl, R. Courant, L. Hörmander, V. Ivrij, R. Melrose, D. G. Vasil'ev, to mention just a few.) In particular it is known that one can recover the volume of the manifold and -- often -- information on the boundary, from the spectral asymptotics.
scientific article; zbMATH DE number 167382
scientific article