zbMATH — the first resource for mathematics

Isospectral surfaces of genus two and three. (English) Zbl 1270.58020
Two Riemannian manifolds are said to be isospectral if they have the same eigenvalues of the Laplace operator. The literature contains a diverse collection of pairs of isospectral non-isometric manifolds satisfying various geometric constraints; the paper under review focuses on surfaces of low genus. In genus two, the authors construct the following examples: isospectral non-isometric surfaces with variable curvature; isospectral potentials on a surface of constant curvature \(-1\); and isospectral but non-isometric Riemannian orbifolds. In genus three, they obtain examples of isospectral non-isometric surfaces with variable curvature. All the examples consist of infinite families, and all are constructed by gluing copies of a basic building block. Isospectrality is proved via Sunada’s theorem. The authors give concrete descriptions of the groups used in Sunada’s theorem, which allows the reader to check their claims directly.
58J53 Isospectrality
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
Full Text: DOI
[1] Buser, Geometry and Spectra of Compact Riemann Surfaces (1992)
[2] DOI: 10.1016/S0040-9383(97)00086-4 · Zbl 0936.58013 · doi:10.1016/S0040-9383(97)00086-4
[3] DOI: 10.5802/aif.1054 · Zbl 0579.53036 · doi:10.5802/aif.1054
[4] DOI: 10.2307/1971319 · Zbl 0445.53026 · doi:10.2307/1971319
[5] Brooks, Nagoya Math. J. 107 pp 13– (1987) · Zbl 0605.58041 · doi:10.1017/S0027763000002518
[6] Thurston, The geometry and topology of three manifolds (1997)
[7] Bosma, Algebraic number theory pp 67– (2002)
[8] DOI: 10.2307/1971195 · Zbl 0585.58047 · doi:10.2307/1971195
[9] Brooks, Holomorphic functions and moduli pp 203– (1986)
[10] DOI: 10.1016/0040-9383(87)90021-8 · Zbl 0617.53048 · doi:10.1016/0040-9383(87)90021-8
[11] DOI: 10.2307/2313748 · Zbl 0139.05603 · doi:10.2307/2313748
[12] DOI: 10.1112/jlms/s2-48.3.565 · Zbl 0767.58042 · doi:10.1112/jlms/s2-48.3.565
[13] DOI: 10.1016/0040-9383(80)90015-4 · Zbl 0465.58027 · doi:10.1016/0040-9383(80)90015-4
[14] DOI: 10.1007/BF01444635 · Zbl 0735.58008 · doi:10.1007/BF01444635
[15] DOI: 10.1016/S1874-5741(00)80009-6 · doi:10.1016/S1874-5741(00)80009-6
[16] DOI: 10.1006/jabr.1999.8260 · Zbl 0958.20011 · doi:10.1006/jabr.1999.8260
[17] Buser, Geometry and analysis on manifolds pp 64– (1987)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.