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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.
MSC:
 58J53 Isospectrality 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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