Geometry and Spectra of Compact Riemann Surfaces: 106 - Rilegato

Buser, Peter

Rilegato

ISBN 10: 0817634061 ISBN 13: 9780817634063

Casa editrice: Birkhauser, 1992

Riassunto

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.The first part of the book is written in textbook form at the graduate level, with few requisites other than background in either differential geometry or complex Riemann surface theory. It begins with an account of the Fenchel-Nielsen approach to Teichmüller Space. Hyperbolic trigonometry and Bers' partition theorem (with a new proof which yields explicit bounds) are shown to be simple but powerful tools in this context. The second part of the book is a self-contained introduction to the spectrum of the Laplacian based on head equations. The approach chosen yields a simple proof that compact Riemann surfaces have the same eigenvalues if and only if they have the same length spectrum. Later chapters deal with recent developments on isospectrality, Sunada's construction, a simplified proof of Wolpert's theorem, and an estimate fo the number of pairwise isospectral non-isometric examples which depends only on genus.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.

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Recensione

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)

Contenuti

Preface.-Chapter 1: Hyperbolic Structures.-Chapter 2: Trigonometry.- Chapter 3: Y-Pieces and Twist Parameters.- Chapter 4:The Collar Theorem.- Chapter 5: Bers’ Constant and the Hairy Torus.- Chapter 6: The Teichmüller Space.- Chapter 7: The Spectrum of the Laplacian.- Chapter 8: Small Eigenvalues.- Chapter 9: Closed Geodesics and Huber’s Theorem.- Chapter 10: Wolpert’s Theorem.- Chapter 11: Sunada’s Theorem.- Chapter 12: Examples of Isospectral Riemann surfaces.- Chapter 13: The Size of Isospectral Families.- Chapter 14: Perturbations of the Laplacian in Hilbert Space.-Appendix: Curves and Isotopies.-Bibliography.-Index.-Glossary.

Dettagli dell?edizione

• Editore: Birkhauser
• Data di pubblicazione: 1992
• ISBN 10: 0817634061
• ISBN 13: 9780817634063
• Rilegatura: Copertina rigida
• Lingua: Inglese
• Numero edizione: 1
• Numero di pagine: 454