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**Eigenvalues in Riemannian geometry. With a chapter by Burton Randol. With an appendix by Jozef Dodziuk.**
*(English)*
Zbl 0551.53001

So far there was only one monograph on the Laplace operator from the point of view of Riemannian geometry [M. Berger, P. Gauduchon and E. Mazet, Le spectre d’une variété riemannienne (1971; Zbl 0223.53034)]. The bibliography of Berard and Berger on this topic, circulating as preprint, contains about 600 new titles since then, which proves the explosion of the subject in the last years. Chavel calls his book an introduction, but it is more than that. Except for an appendix (on the Laplacian on forms, by J. Dodziuk) the subject matter is restricted to the Laplacian acting on functions.

The first chapter summarizes the basic facts, the second gives examples. In chapter III the lowest Dirichlet eigenvalue of a geodesic disk is compared to the corresponding eigenvalue for space forms. The theorems of Obata and Toponogov are consequences. Chapters IV and V study isoperimetric inequalities (the solution of the Rayleigh conjecture by Faber-Krahn, isoperimetric constants and estimates of eigenvalues). Chapters VI-IX are devoted to the heat equation, and chapter IX to topological perturbations with negligible spectral effects. In chapters X and XI the author treats compact surfaces of constant negative curvature and the relation between low eigenvalues and the geometry and topology of the surface. Chapter XI (by B. Randol) is devoted to the Selberg trace formula, emphasizing the differential geometric point of view, and restricting the discussion to hyperbolic spaces. The final chapter collects miscellanea. - This book deserves widespread distribution!

The first chapter summarizes the basic facts, the second gives examples. In chapter III the lowest Dirichlet eigenvalue of a geodesic disk is compared to the corresponding eigenvalue for space forms. The theorems of Obata and Toponogov are consequences. Chapters IV and V study isoperimetric inequalities (the solution of the Rayleigh conjecture by Faber-Krahn, isoperimetric constants and estimates of eigenvalues). Chapters VI-IX are devoted to the heat equation, and chapter IX to topological perturbations with negligible spectral effects. In chapters X and XI the author treats compact surfaces of constant negative curvature and the relation between low eigenvalues and the geometry and topology of the surface. Chapter XI (by B. Randol) is devoted to the Selberg trace formula, emphasizing the differential geometric point of view, and restricting the discussion to hyperbolic spaces. The final chapter collects miscellanea. - This book deserves widespread distribution!

Reviewer: U.Simon

### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

53C20 | Global Riemannian geometry, including pinching |

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |

58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |