Introduction to isospectrality
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 systematically constructing pairs of isospectral flat surfaces. But Buser surfaces were not yet embedded in \(\mathbb{R}^2\). In [Astérisque 177--178, 127--154 (1989; Zbl 0703.53035)], \textit{P. Bérard} relaxed the freeness requirement on the group action. In [Invent. Math. 110, No. 1, 1--22 (1992; Zbl 0778.58068)], \textit{C. Gordon} et al. construced two contractible isospectral non-isometric surfaces in \(\mathbb{R}^2\) with piecewise linear boundaries, which do indeed give a negative answer to the Kac's question. This book is mainly about the Sunada's method and the materials are limited to a moderate level of mathematics to fully understand the isospectrality of GWW surfaces. This book is accessible to third-year university students, which describes a complete and elegant solution to a long-standing mathematical problem. It consists of four chapters and four appendices: 1. Introduction 2. The Wave Equation on Flat Surfaces 3. The Sunada-Bérard-Buser Method 4. The Gordon-Webb-Wolpert Isospectral Membranes Appendix A. Representation of Finite Groups and Almost-Conjugate Groups Appendix B. The Laplacian as Isometry-Invariant Differential Operator Appendix C. The Path-Distance on a Hausdorff Connected Flat Manifold Appendix D. Group Quotients of Flat Manifolds
scientific article; zbMATH DE number 7595710