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Transactions of the American Mathematical Society

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The eigenvalue spectrum as moduli for flat tori


Author: Scott Wolpert
Journal: Trans. Amer. Math. Soc. 244 (1978), 313-321
MSC: Primary 53C99; Secondary 58G99
DOI: https://doi.org/10.1090/S0002-9947-1978-0514879-9
MathSciNet review: 0514879
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Abstract: A flat torus T carries a natural Laplace Beltrami operator. It is a conjecture that the spectrum of the Laplace Beltrami operator determines T modulo isometries. We prove that, with the exception of a subvariety in the moduli space of flat tori, this conjecture is true. A description of the subvariety is given.


References [Enhancements On Off] (What's this?)

  • [1] M. Berger, P. Gauduchon and E. Mazet, Le spectre d'une variété Riemannienne, Springer-Verlag, Berlin, 1971. MR 0282313 (43:8025)
  • [2] M. Berger, Geometry of the spectrum, Proc. Sympos. Pure Math., vol. 27, Part 2, Amer. Math. Soc., Providence, R. I., 1975, pp. 129-152. MR 0383459 (52:4340)
  • [3] J. W. S. Cassels, An introduction to the geometry of numbers, Springer-Verlag, Berlin,1975, p. 135. MR 1434478 (97i:11074)
  • [4] I. M. Gel'fand, M. I. Graev and I. I. Pyaetskii-Shapiro, Representation theory and automorphic functions, Saunders, Philadelphia, 1969. MR 0233772 (38:2093)
  • [5] R. C. Gunning, Lectures on modular forms, Ann. of Math. Studies, no. 48, Princeton Univ. Press, Princeton, N. J., 1962. MR 0132828 (24:A2664)
  • [6] H. P. McKean, Selberg's trace formula as applied to a compact Riemann surface, Comm. Pure Appl. Math. 25 (1972), 225-246. MR 0473166 (57:12843a)
  • [7] J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 542. MR 0162204 (28:5403)
  • [8] S. Wolpert, The eigenvalue spectrum as moduli for compact Riemann surfaces, Bull. Amer. Math. Soc. 83 (1977), 1306-1308. MR 0499329 (58:17228)
  • [9] -, The length spectrum as moduli for compact Riemann surfaces (to appear).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0514879-9
Keywords: Flat torus, Laplace Beltrami operator, spectrum, moduli, lattice, quadratic forms
Article copyright: © Copyright 1978 American Mathematical Society

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