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Isospectral Riemann surfaces

Peter Buser

Annales de l'institut Fourier (1986)

  • Volume: 36, Issue: 2, page 167-192
  • ISSN: 0373-0956

Abstract

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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 ≥ 7 . In a second part we give examples of isospectral non isometric surfaces in R 3 which are realizable by paper models.

How to cite

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Buser, Peter. "Isospectral Riemann surfaces." Annales de l'institut Fourier 36.2 (1986): 167-192. <http://eudml.org/doc/74711>.

@article{Buser1986,
abstract = {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 $\{\bf R\}^ 3$ which are realizable by paper models.},
author = {Buser, Peter},
journal = {Annales de l'institut Fourier},
keywords = {Riemann surfaces; spectrum of the Laplacian; isospectral},
language = {eng},
number = {2},
pages = {167-192},
publisher = {Association des Annales de l'Institut Fourier},
title = {Isospectral Riemann surfaces},
url = {http://eudml.org/doc/74711},
volume = {36},
year = {1986},
}

TY - JOUR
AU - Buser, Peter
TI - Isospectral Riemann surfaces
JO - Annales de l'institut Fourier
PY - 1986
PB - Association des Annales de l'Institut Fourier
VL - 36
IS - 2
SP - 167
EP - 192
AB - 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 ${\bf R}^ 3$ which are realizable by paper models.
LA - eng
KW - Riemann surfaces; spectrum of the Laplacian; isospectral
UR - http://eudml.org/doc/74711
ER -

References

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  1. [1] P. BUSER, Riemannsche Flächen mit Eigenwerten in (0, 1/4), Comment. Math. Helvetici, 52 (1977), 25-34. Zbl0348.53027MR55 #7924
  2. [2] J. CHAVEL, Eigenvalues in Riemannian Geometry, Academic Press. Orlando etc., 1984. Zbl0551.53001
  3. [3] F. GASSMAN, Bemerkungen zur vorstehenden Arbeit von Hurwitz, Math. Z., 25 (1926), 665-675. 
  4. [4] I. M. GEL'FAND, Automorphic functions and the theory of representations, Proc. Internat. Congress Math., (Stockholm, 1962), 74-85. Zbl0138.07102
  5. [5] I. GERST, On the theory of n-th power residues and a conjecture of Kronecker, Acta Arithmetica, 17 (1970), 121-139. Zbl0233.10002MR44 #1643
  6. [6] H. HUBER, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen, Math., Ann., 138 (1959), 1-26. Zbl0089.06101MR22 #99
  7. [7] H. P. MCKEAN, Selberg's trace formula as applied to a compact Riemann surface, Comm. Pure Appl. Math., 25 (1972), 225-246. 
  8. [8] J. MILNOR, Eigenvalues of the Laplace operators on certain manifolds, Proc. Nat. Sci. USA, 51 (1964), 542. Zbl0124.31202MR28 #5403
  9. [9] R. PERLIS, On the equation ξk(s) = ξk'(s), Journal of Number Theory, 9 (1977), 342-360. Zbl0389.12006
  10. [10] E. REES, Notes on geometry, Springer, Berlin, 1983. Zbl0498.51001MR84c:51001b
  11. [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. Zbl0072.08201MR19,531g
  12. [12] T. SUNADA, Riemannian coverings and isospectral manifolds, Annals of Math., 121 (1985), 169-186. Zbl0585.58047MR86h:58141
  13. [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. Zbl0581.22013MR86c:22018
  14. [14] S. TANAKA, Selberg's Trace Formula and Spectrum, Osaka J. Math., (1966), 205-206. Zbl0202.11601MR36 #312
  15. [15] W. THURSTON, The geometry and topology of 3-manifolds, Princeton Lecture Notes. Zbl0483.57007
  16. [16] H. URAKAWA, Bounded domains which are isospectral but not congruent, Ann. Sci. Ec. Norm. Sup., 4e série, t. 15 (1982), 441-456. Zbl0505.58036MR84g:58106
  17. [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). Zbl0387.10013MR58 #10826
  18. [18] M.F. VIGNÉRAS, Variétés riemanniennes isospectrales et non isométriques, Ann. of Math., 112 (1980), 21-32. Zbl0445.53026MR82b:58102
  19. [19] S. WOLPERT, The eigenvalue spectrum as moduli for compact Riemann surfaces, Bull. Amer. Math. Soc., 83 (1977), 1306-1308. Zbl0368.32009MR58 #17228
  20. [20] S. WOLPERT, The length spectra as moduli for compact Riemann surfaces, Ann. of Math., 109 (1979), 323-351. Zbl0441.30055MR80j:58067

Citations in EuDML Documents

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  1. Ori Parzanchevski, On G -sets and isospectrality
  2. Pierre Bérard, Transplantation et isospectralité I
  3. Carolyn Gordon, William Kirwin, Dorothee Schueth, David Webb, Quantum Equivalent Magnetic Fields that Are Not Classically Equivalent
  4. Carolyn S. Gordon, Ruth Gornet, Dorothee Schueth, David L. Webb, Edward N. Wilson, Isospectral deformations of closed riemannian manifolds with different scalar curvature
  5. Pierre Bérard, Variétés riemanniennes isospectrales non isométriques
  6. Laurent Charles, Álvaro Pelayo, San Vũ Ngoc, Isospectrality for quantum toric integrable systems

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