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Spectral geometry: direct and inverse problems. With an appendix by G. Besson. (English) Zbl 0608.58001
Lecture Notes in Mathematics, 1207. Berlin etc.: Springer-Verlag XIII, 272 p. DM 38.50 (1986).
This book deals with some aspects of spectral geometry. Let M be a compact Riemannian manifold and let spec(M) be the spectrum of the Laplace-Beltrami operator. In a direct spectral problem, one wants information on the spectrum in terms of the underlying geometry, in indirect problems one seeks geometric and topological information from the spectrum. Both types of problems are relevant to questions arising in physics.
The first three chapters are introductory in nature. The first chapter gives some motivations from mathematical physics and seeks to show how some mathematical objects arise from physical principles. The second chapter reviews the basic notions (geodesics, curvature, etc.) from Riemannian geometry. The third chapter introduces the variational characterizations of the eigenvalues.
Chapters four through six are the heart of the book. Chapter four deals with isoperimetric methods. A new proof of Cheeger’s lower bound for the first non-zero eigenvalue is given. Chapter five takes up the heat equation and discusses the isoperimetric inequality for the heat equation. Chapter six gives some applications of isoperimetric inequalities to the heat equation which is closely related to the compactness theorems of M. Gromov. Chapter seven gives a brief summary of some results in spectral geometry.
The book also contains several appendixes. Appendix A by G. Besson discusses the symmetrization process and gives relationships between Riemannian geometry, spectral geometry, and operator theory on a Hilbert space. Appendix B (A guide to the literature) is joint with M. Berger and was first published in ”Spectra of Riemannian manifolds” Kaigai Publ., Tokyo (1983). It is a comprehensive annotated bibliography of the literature up to 1982 organized by both subject and author. Appendix C contains some recent material updating this bibliography.
The book is well organized and nicely written with many excellent examples. It is a major addition to the subject containing both standard material as well as new results.
Reviewer: P.Gilkey

58-02 Research exposition (monographs, survey articles) pertaining to global analysis
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J35 Heat and other parabolic equation methods for PDEs on manifolds
53C20 Global Riemannian geometry, including pinching
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