Geometry and Spectra of Compact Riemann Surfaces (Progress in Mathematics) [Hardcover]

Peter Buser (Author)

Book Description

Publication Date: October 16, 1993 | ISBN-10: 3764334061 | ISBN-13: 978-3764334062

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.

--This text refers to an alternate Hardcover edition.

Editorial Reviews

Review

"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 --This text refers to an alternate Hardcover edition.

From the Back Cover

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 in textbook form at the graduate level, with only minimal requisites in either differential geometry or complex Riemann surface theory. The second part of the book is a self-contained introduction to the spectrum of the Laplacian based on the heat equation. Later chapters deal with recent developments on isospectrality, Sunada’s construction, a simplified proof of Wolpert’s theorem, and an estimate of the number of pairwise isospectral non-isometric examples which depends only on 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. — Mathematical Reviews This 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 --This text refers to the Paperback edition.

Product Details

• Hardcover: 456 pages
• Publisher: Birkhauser Verlag AG (October 16, 1993)
• Language: English
• ISBN-10: 3764334061
• ISBN-13: 978-3764334062
• Average Customer Review: 5.0 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #10,448,497 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

5 of 5 people found the following review helpful:

5.0 out of 5 stars Great book., December 30, 2007

By 

Theodore A. Fails - See all my reviews
(REAL NAME)   

This review is from: Geometry and Spectra of Compact Riemann Surfaces (Progress in Mathematics) (Hardcover)

I am no expert in this field, but I do not understand why this great book has not gotten wider exposure. The author gives a wonderful intro to the "geometry of compact Riemann surfaces based on hyperbolic geometry and on cutting and pasting." (Quoting from the preface.)

Perhaps Amazon has not helped his cause by miss-listing the author as J. Buser instead of Peter Buser, and compounding the problem by not returning this book on a search by author for Peter Buser.

All in all, this (fairly) advanced amateur gives Peter Buser's book on Riemann Surfaces five stars.

5.0 out of 5 stars Experts use this book, December 26, 2010

By 

Chris Judge (Bloomington) - See all my reviews
(REAL NAME)   

This review is from: Geometry and Spectra of Compact Riemann Surfaces (Progress in Mathematics) (Kindle Edition)

Just want to reply to the other review. I am an expert and I use this book quite often. I give it to my graduate students as a first source on the hyperbolic geometry of Riemann surfaces, especially Fenchel-Nielsen coordinates, length spectrum etc. It is a very readable introduction.