Sunada’s method and the covering spectrum

Bart de Smit Universiteit Leiden Ruth Gornet University of Texas at Arlington Craig J. Sutton Dartmouth College

Differential Geometry mathscidoc:1609.10203

Journal of Differential Geometry, 86, (3), 501-537, 2010
In 2004, Sormani and Wei introduced the covering spectrum: a geometric invariant that isolates part of the length spectrum of a Riemannian manifold. In their paper they observed that certain Sunada isospectral manifolds share the same covering spectrum, thus raising the question of whether the covering spectrum is a spectral invariant. In the present paper we describe a group theoretic condition under which Sunada’s method gives manifolds with identical covering spectra. When the group theoretic condition of our method is not met, we are able to construct Sunada isospectral manifolds with distinct covering spectra in dimension 3 and higher. Hence, the covering spectrum is not a spectral invariant. The main geometric ingredient of the proof has an interpretation as the minimum-marked-length-spectrum analogue of Colin de Verdi`ere’s classical result on constructing metrics where the first k eigenvalues of the Laplace spectrum have been prescribed.
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  author={Bart de Smit, Ruth Gornet, and Craig J. Sutton},
  booktitle={Journal of Differential Geometry},
Bart de Smit, Ruth Gornet, and Craig J. Sutton. SUNADA’S METHOD AND THE COVERING SPECTRUM. 2010. Vol. 86. In Journal of Differential Geometry. pp.501-537.
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