Geometry and Spectra of Compact Riemann Surfaces [electronic resource]

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Annotation This classic monograph is a self-contained introduction to the geometry of Riemann surfaces of constant curvature -1 and their length and eigenvalue spectra. It focuses on two subjects: the geometric theory of compact Riemann surfaces of genus greater than one, and the relationship of the Laplace operator with the geometry of such surfaces.The first part of the book is written at the graduate level, with only minimal requisites in either differential geometry or complex Riemann surface theory. The second part of the book is an introduction to the spectrum of the Laplacian based on head equations. Later chapters deal with recent developments on isospectrality, Sunadaâs construction, a simplified proof of Wolpertâs theorem, and an estimate of the number of pairwise isospectral non-isometric examples which depend only one genus.Researchers and graduate students interested in compact Riemann surfaces will find this book a useful reference.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. âMathematical ReviewsThis is a thick and leisurely book which will repay repeated study with many pleasant hours â 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 âstate of the artâ in the theory of the Laplaceâ 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. âBulletin of the AMSThe book is very well written and quite accessible; there is an excellent bibliography at the end. âZentralblatt MATH

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