Titlebook: Introduction to Isospectrality; Alberto Arabia Textbook 2022 Springer Nature Switzerland AG 2022 hear the shape of a drum.Laplacian.spectr

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Alberto ArabiaA self-contained account of the key contributions of Sunada, Buser, Bérard, Gordon, Webb and Wolpert.Provides a detailed construction of contractible, non-isometric isospectral surfaces.Includes 190 f

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Chapter 1 Introduction,This book is about ., a technique developed in the 1990s which led to the discovery of the first known non-isometric isospectral drumhead membranes. But before that, this introductory chapter gives an overview of the whole subject to help appreciate the value of this technique.

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Chapter 2 The Wave Equation on Flat Manifolds,While Sunada’s method applies to general Riemannian manifolds, the constructions of the Buser’s surfaces 1.6.5 and the GWW domains 1.6.6, to which this book is devoted, involve only flat manifolds of dimension 2 with piecewise linear boundaries (PLB-. in short) and local isometries.

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,Chapter 3 The Sunada–Bérard–Buser Method,In Chapter ., we introduced flat manifolds and surfaces with PL-boundaries, which are spaces with enough structure to give meaning to the Wave Equation. We also introduced local isometries to relate these spaces to each other, thereby defining a rich and flexible category of spaces where we can test isospectrality.

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,Chapter 4 The Gordon–Webb–Wolpert Isospectral Domains,In this chapter we explain the approach of Gordon–Webb–Wolpert, Buser–Semmler and Conway–Doyle to progressively transform the Buser surfaces . , for . obtaining the plane domains . and . and proving isospectrality at each transformation step.

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Appendix C The Path-Distance on a Hausdorf Connected Flat Manifold,Once the notion of . on a flat manifold . is clarified, we can consider defining the distance between two points . and . in a connected component of ., as the lower bound of the lengths of paths joining . to ..