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Anyone familiar with the author´s hands-on approach to Riemann surfaces will be gratified by both the breadth and the depth of the topics considered here. The exposition is also extremely clear and thorough. Anyone not familiar with the author´s approach is in for a real treat. Mathematical ReviewsThis is a thick and leisurely book which will repay repeated study with many pleasant hours both for the beginner and the expert. It is fortunately more or less self-contained, which makes it easy to read, and it leads one from essential mathematics to the state of the art in the theory of the Laplace Beltrami operator on compact Riemann surfaces. Although it is not encyclopedic, it is so rich in information and ideas the reader will be grateful for what has been included in this very satisfying book. Bulletin of the AMS The book is very well written and quite accessible; there is an excellent bibliography at the end. Zentralblatt MATH

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Hyperbolic Structures.- Trigonometry.- Y-Pieces and Twist Parameters.- The Collar Theorem.- Bers´ Constant and the Hairy Torus.- The Teichmüller Space.- The Spectrum of the Laplacian.- Small Eigenvalues.- Closed Geodesics and Huber´s Theorem.- Wolpert´s Theorem.- Sunada´s Theorem.- Examples of Isospectral Riemann Surfaces.- The Size of Isospectral Families.- Perturbations of the Laplacian in Teichmüller Space.

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From the reviews:

"Anyone familiar with the author´s hands-on approach to Riemann surfaces will be gratified by both the breadth and the depth of the topics considered here. The exposition is also extremely clear and thorough. Anyone not familiar with the author´s approach is in for a real treat." -Mathematical Reviews

"Originally published as Volume 106 in the series Progress in Mathematics, this version is a reprint of the classic monograph, 1992 edition, consisting of two parts. ... An appendix is devoted to curves and isotopies. The book is a very useful reference for researches and also for graduate students interested in the geometry of compact Riemann surfaces of constant curvature -- 1 and their length and eigenvalue spectra." (Liliana Raileanu, Zentralblatt MATH, Vol. 1239, 2012)

"Geometry and Spectra of Compact Riemann Surfaces is a pleasure to read. There is a lot of motivation given, examples proliferate, propositions and theorems come equipped with clear proofs, and excellent drawings ... . a fine piece of scholarship and a pedagogical treat." (Michael Berg, The Mathematical Association of America, May, 2011)

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