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Isospectral surfaces of small genus

Published online by Cambridge University Press:  22 January 2016

Roberts Brooks
Affiliation:
Department of Mathematics, University of Southern California Los Angeles, California 90089-1113, U. S. A.
Richard Tse
Affiliation:
Department of Mathematics, University of Southern California Los Angeles, California 90089-1113, U. S. A.
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In this note, we will construct simple examples of isospectral surfaces. In what follows, we will use the term “surface” to mean a surface endowed with a Riemannian metric, while the term “Riemann surface” will be reserved for a surface endowed with a metric of constant curvature. We will show:

THEOREM 1. There exist pairs of surfaces S1 and S2 of genus 3, such that S1 and S2 are isospectral but not isometric.

THEOREM 2. There exist pairs of Riemann surfaces S1 and S2 of genus 4 and 6, which are isospectral but not isometric.

THEOREM 3. There exist unoriented surfaces S1 and S2 of Euler characteristic X(S1) = X(S2) = — 6 which are isospectral but not isometric.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1987

References

[1] Beardon, A., The Geometry of Discontinuous Groups, Springer Verlag 1983.Google Scholar
[2] Brooks, R., On manifolds of negative curvature with isospectral potentials, Topology, 26 (1987), 6366.CrossRefGoogle Scholar
[3] Buser, P., Isospectral Riemann surfaces, Ann. Inst. Fourier, XXXVI (1986), 167192.CrossRefGoogle Scholar
[4] Guralnick, R., Subgroups inducing the same permutation representation, J. Algebra, 81 (1983), 312319.CrossRefGoogle Scholar
[5] Sunada, T., Riemannian coverings and isospectral manifolds, Ann. of Math., 121 (1985), 169186.CrossRefGoogle Scholar
[6] Vigneras, M. F., Variétés riemannienes isospectrales et non isometriques, Ann. of Math., 112 (1980), 2132.CrossRefGoogle Scholar
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