Introduction to isospectrality
type: Item
instance of: scholarly article
title: Introduction to isospectrality
zbMATH Open document ID: 1505.58001
DOI: 10.1007/978-3-031-17123-9
author: Alberto Arabia
published in: Universitext
publication date: 2022-10-01T00:00:00Z
review text: This book gives a comprehensive presentation of a celebrated counterexample in spectral geometry which is related to a famous question: Can we hear the shape of a drum? This is an often quoted title of Mark Kac's celebrated article. In mathematical language, this question is aimed to characterize the geometry of the Riemannian manifold from the spectrum of the Beltrami-Laplacian, which opens the way to the vast field of research of spectral geometry. The main evidence for this question is the Weyl's law that the spectrum determines the volume. However, in [Proc. Natl. Acad. Sci. USA 51, 542 (1964; Zbl 0124.31202)], \textit{J. W. Milnor} constructed a counterexample for this question by exhibiting two non-isomorphic isospectral tori of dimension 16. In the 1980s, Sunada developped a method for systematically building isospectral spaces as follows. Let \(G\) be a finite group acting by orientation-preserving isometries on an orientable compact Riemannian manifold \(M\). If \(\Gamma_1\), \(\Gamma_2\) are two almost-conjugate subgroups of \(G\) acting freely on \(M\), then the quotient spaces \(M_i:=\Gamma_i\backslash M\) are isospectral compact Riemannian orientable manifolds. We say \(\Gamma_1\), \(\Gamma_2\) are almost-conjugate if for every conjugacy class \(C\) of \(G\), \(|C\cap \Gamma_1|=|C\cap \Gamma_2|\). Note that for every finite group \(G\), there exists an oriented compact connected manifold of dimension \(\geq 4\) whose fundamental group is isomorphic to \(G\). So we can find many counterexamples by constructing the almost-conjugate subgroups \((\Gamma_1, \Gamma_2)\). But the method had two important flaws concerning Kac's question. First, we need the dimension \(\geq 4\). Second, the manifolds \(\Gamma_i\backslash M\) cannot be contractible. The limitation of Sunada's method were progressively overcome by the work of Buser and Berard. In [Ann. Inst. Fourier 36, No. 2, 167--192 (1986; Zbl 0579.53036)], \textit{P. Buser} obtained a procedure for systematica…
reviewed by: Bo Liu
Mathematics Subject Classification ID: 58-01
Mathematics Subject Classification ID: 35-01
Mathematics Subject Classification ID: 58J53
Mathematics Subject Classification ID: 35R01
zbMATH DE Number: 7595710
zbMATH Keywords: spectral geometry
zbMATH Keywords: isospectrality
zbMATH Keywords: flat surfaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-031-17123-9
OpenAlex ID: W4295338269
description: scientific article; zbMATH DE number 7595710