Hostname: page-component-7479d7b7d-68ccn Total loading time: 0 Render date: 2024-07-11T21:25:33.940Z Has data issue: false hasContentIssue false
  • English
  • Français

Isospectral surfaces of genus two and three

Published online by Cambridge University Press:  27 February 2012

DENNIS BARDEN  and
HYUNSUK KANG
DENNIS BARDEN
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Cambridge CB3 0WB. e-mail: db127@dpmms.cam.ac.uk
HYUNSUK KANG*
Affiliation:
Department of Mathematics, Korea Institute for Advanced Study, Hoegiro 87, Dong-daemun-gu, Seoul 130-722, Korea. e-mail: h.kang@kias.re.kr
*
Corresponding author
Get access Rights & Permissions [Opens in a new window]

Abstract

We give examples of isospectral non-isometric surfaces of genus two and three with variable curvature, as well as hyperbolic orbifolds of genus two. We apply the first result to construct isospectral potentials on a Riemann surface of genus two.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 1 , July 2012 , pp. 99 - 110
Copyright
Copyright © Cambridge Philosophical Society 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bérard, P.Transplantation et isospectralité. I. Math. Ann. 292 No. 3 (1992), 547–559.CrossRefGoogle Scholar
[2]Bérard, P.Transplantation et isospectralité. II. J. London Math. Soc. (2) 48 No. 3 (1993), 565–576.CrossRefGoogle Scholar
[3]Brooks, R.On manifolds of negative curvature with isospectral potentials. Topology 26 No. 1 (1987), 63–66.CrossRefGoogle Scholar
[4]Brooks, R. Isospectral potentials on a surface of genus three. Holomorphic functions and moduli Vol. 1 (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ. 10 (Springer, 1988), 203–208.Google Scholar
[5]Bosma, W. and de Smit, B. On arithmetically equivalent number fields of small degree. Algebraic number theory (Sydney, 2002) Lecture Notes in Comput. Sci. 2369 (Springer, Berlin, 2002), 67–79.Google Scholar
[6]Brooks, R. and Tse, R.Isospectral surfaces of small genus. Nagoya Math. J. 107 (1987), 13–24.CrossRefGoogle Scholar
[7]Buser, P.Isospectral Riemann surfaces. Ann. Inst. Fourier 36 (1986), 167–192.CrossRefGoogle Scholar
[8]Buser, P. Cayley graphs and planar isospectral domains. Geometry and analysis on manifolds (Katata/Kyoto, 1987) Lecture Notes in Math. 1339 (Springer, Berlin, 1988) 64–77.Google Scholar
[9]Buser, P.Geometry and Spectra of Compact Riemann Surfaces (Birkhauser, Boston 1992).Google Scholar
[10]Croke, C. B. and Sharafutdinov, V. A.Spectral rigidity of a compact negatively curved manifold. Topology 37 No. 6 (1998), 1265–1273.CrossRefGoogle Scholar
[11]de Smit, B. and Lenstra, H.W. Jr., Linearly equivalent actions of solvable groups. J. Algebra 228 (2000), 270–285.CrossRefGoogle Scholar
[12]Fechner, J. Die Konstruktion isospektraler Mannigfaltigkeiton von kleinem Geschlecht. Diploma thesis. Humboldt-Universität zu Berlin (2007).Google Scholar
[13]Gordon, C. S. Survey of isospectral manifolds. Handbook of Differential Geometry 1 (Elsevier Science B.V. 2000), 747–778.Google Scholar
[14]Guillemin, V. and Kazhdan, D.Some inverse spectral results for negatively curved two-manifolds. Topology 19 (1980), 301–312.CrossRefGoogle Scholar
[15]Kac, M.Can one hear the shape of a drum? Amer. Math. Monthly 73 (1966), 1–23.CrossRefGoogle Scholar
[16]Kang, H. D-isospectral metrics and isospectral properties. PhD Thesis. University of Cambridge (2006).Google Scholar
[17]Sunada, T.Riemannian coverings and isospectral manifolds. Ann. Math. 121 (1985), 169–186.CrossRefGoogle Scholar
[18]Thurston, W.The geometry and topology of three manifolds. (MSRI publications, 1997).Google Scholar
[19]Vignéras, M. F.Variétés riemannienes isospectrales et non isometriques. Ann. Math. 112 (1980), 21–32.CrossRefGoogle Scholar