Geometry and Spectra of Compact Riemann Surfaces / Edition 1

by Peter Buser

Overview

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.

Product Details

ISBN-13:	9780817649913
Publisher:	Birkhäuser Boston
Publication date:	11/04/2010
Series:	Modern Birkhäuser Classics
Edition description:	Softcover reprint of hardcover edition 2010
Pages:	456
Product dimensions:	6.14(w) x 9.21(h) x 0.24(d)

Table of Contents

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.

Editorial Reviews

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 Răileanu, 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)

From the Publisher

The focus is on two subjects: the 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 as a graduate textbook and contains the Fenchel-Nielsen approach to Teichmuller space, hyperbolic trigonometry and Bers' partition theorem. The second part introduces the spectrum of the Laplacian based on the heat equation, and other coverage includes: recent developments on isospectrality, Sunada's construction, and an estimate of the number of pairwise isospectral non-isometric examples which depend only on genus. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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