Sunada's method and the covering spectrum

TL;DR

This paper investigates whether the covering spectrum is a spectral invariant by analyzing Sunada's method, showing that under certain conditions it is, but generally it is not, with implications for geometric analysis.

Contribution

The paper identifies a group theoretic condition under which Sunada's method produces manifolds with identical covering spectra and constructs counterexamples where the spectra differ.

Findings

Covering spectrum is not a spectral invariant in general.

A specific group theoretic condition ensures identical covering spectra.

Counterexamples exist in dimension 3 and higher.

Abstract

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\`{e}re's classical result on constructing metrics where the first $k$ eigenvalues of the Laplace spectrum have been prescribed.