Open Access
November 2010 Sunada's method and the covering spectrum
Bart de Smit, Ruth Gornet, Craig J. Sutton
J. Differential Geom. 86(3): 501-538 (November 2010). DOI: 10.4310/jdg/1303219428


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 Verdière’s classical result on constructing metrics where the first $k$ eigenvalues of the Laplace spectrum have been prescribed.


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Bart de Smit. Ruth Gornet. Craig J. Sutton. "Sunada's method and the covering spectrum." J. Differential Geom. 86 (3) 501 - 538, November 2010.


Published: November 2010
First available in Project Euclid: 19 April 2011

zbMATH: 1219.53040
MathSciNet: MR2785841
Digital Object Identifier: 10.4310/jdg/1303219428

Rights: Copyright © 2010 Lehigh University

Vol.86 • No. 3 • November 2010
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