Isospectral Riemann surfaces

Buser, Peter

Annales de l'Institut Fourier, Tome 36 (1986) no. 2, pp. 167-192.

L’article donne de nouveaux exemples de surfaces de Riemann compactes qui sont non isométriques et ont le même spectre du laplacien. Ces exemples sont donnés pour le genre $g=5$ et pour tous les $g\ge 7$.

Dans une seconde partie nous construisons des surfaces isospectrales plongées dans ${\mathbf{R}}^3$ qui se réalisent par des modèles en papier.

We construct new examples of compact Riemann surfaces which are non isometric but have the same spectrum of the Laplacian. Examples are given for genus $g=5$ and for all $g\ge 7$. In a second part we give examples of isospectral non isometric surfaces in ${\mathbf{R}}^3$ which are realizable by paper models.

[1] P. Buser, Riemannsche Flächen mit Eigenwerten in (0, 1/4), Comment. Math. Helvetici, 52 (1977), 25-34. | MR | Zbl

[2] J. Chavel, Eigenvalues in Riemannian Geometry, Academic Press. Orlando etc., 1984. | Zbl

[3] F. Gassman, Bemerkungen zur vorstehenden Arbeit von Hurwitz, Math. Z., 25 (1926), 665-675.

[4] I. M. Gel'Fand, Automorphic functions and the theory of representations, Proc. Internat. Congress Math., (Stockholm, 1962), 74-85. | Zbl

[5] I. Gerst, On the theory of n-th power residues and a conjecture of Kronecker, Acta Arithmetica, 17 (1970), 121-139. | MR | Zbl

[6] H. Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen, Math., Ann., 138 (1959), 1-26. | MR | Zbl

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

[8] J. Milnor, Eigenvalues of the Laplace operators on certain manifolds, Proc. Nat. Sci. USA, 51 (1964), 542. | MR | Zbl

[9] R. Perlis, On the equation ξk(s) = ξk'(s), Journal of Number Theory, 9 (1977), 342-360. | Zbl

[10] E. Rees, Notes on geometry, Springer, Berlin, 1983. | MR | Zbl

[11] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc., 20 (1956), 47-87. | MR | Zbl

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

[13] T. Sunada, Gel'fand's problem on unitary representations associated with discrete subgroups of PSL2 (R), Bull. Amer. Meth. Soc., 12 (1985), 237-238. | MR | Zbl

[14] S. Tanaka, Selberg's Trace Formula and Spectrum, Osaka J. Math., (1966), 205-206. | MR | Zbl

[15] W. Thurston, The geometry and topology of 3-manifolds, Princeton Lecture Notes.

[16] H. Urakawa, Bounded domains which are isospectral but not congruent, Ann. Sci. Ec. Norm. Sup., 4e série, t. 15 (1982), 441-456. | EuDML | Numdam | MR | Zbl

[17] M. F. Vignéras, Exemples de sous-groupes discrets non conjugués de PSL (2, R) qui ont même fonction zêta de Selberg, C.R.A.S., Paris, 287 (1978). | MR | Zbl

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

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

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