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This 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. Research workers and graduate students interested in compact Riemann surfaces will find here a number of useful tools and insights to apply to their investigations.
This book deals with two subjects. The first subject is the geometric theory of compact Riemann surfaces of genus greater than one, the second subject is the Laplace operator and its relationship with the geometry of compact Riemann surfaces. The book grew out of the idea, a long time ago, to publish a Habili- tionsschrift, a thesis, in which I studied Bers' pants decomposition theorem and its applications to the spectrum of a compact Riemann surface. A basic tool in the thesis was cutting and pasting in connection with the trigono metry of hyperbolic geodesic polygons. As this approach to the geometry of a compact Riemann surface did not exist in book form, I took this book as an occasion to carry out the geometry in detail, and so it grew by several chapters. Also, while I was writing things up there was much progress in the field, and some of the new results were too challenging to be left out of the book. For instance, Sunada's construction of isospectral manifolds was fascinating, and I got hooked on constructing examples for quite a while. So time went on and the book kept growing. Fortunately, the interest in exis tence proofs also kept growing. The editor, for instance, was interested, and so was my family. And so the book finally assumed its present form. Many of the proofs given here are new, and there are also results which appear for the first time in print.
This book deals with two subjects. The first subject is the geometric theory of compact Riemann surfaces of genus greater than one, the second subject is the Laplace operator and its relationship with the geometry of compact Riemann surfaces. The book grew out of the idea, a long time ago, to publish a Habili- tionsschrift, a thesis, in which I studied Bers' pants decomposition theorem and its applications to the spectrum of a compact Riemann surface. A basic tool in the thesis was cutting and pasting in connection with the trigono metry of hyperbolic geodesic polygons. As this approach to the geometry of a compact Riemann surface did not exist in book form, I took this book as an occasion to carry out the geometry in detail, and so it grew by several chapters. Also, while I was writing things up there was much progress in the field, and some of the new results were too challenging to be left out of the book. For instance, Sunada's construction of isospectral manifolds was fascinating, and I got hooked on constructing examples for quite a while. So time went on and the book kept growing. Fortunately, the interest in exis tence proofs also kept growing. The editor, for instance, was interested, and so was my family. And so the book finally assumed its present form. Many of the proofs given here are new, and there are also results which appear for the first time in print.
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)
"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)