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  • © 2010

Geometry and Spectra of Compact Riemann Surfaces

Birkhäuser

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  • ISBN: 978-0-8176-4992-0
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Table of contents (14 chapters)

  1. Front Matter

    Pages i-xvi
  2. Hyperbolic Structures

    • Peter Buser
    Pages 1-30
  3. Trigonometry

    • Peter Buser
    Pages 31-62
  4. Y-Pieces and Twist Parameters

    • Peter Buser
    Pages 63-93
  5. The Collar Theorem

    • Peter Buser
    Pages 94-121
  6. Bers’ Constant and the Hairy Torus

    • Peter Buser
    Pages 122-137
  7. The Teichmüller Space

    • Peter Buser
    Pages 138-181
  8. The Spectrum of the Laplacian

    • Peter Buser
    Pages 182-209
  9. Small Eigenvalues

    • Peter Buser
    Pages 210-223
  10. Closed Geodesics and Huber’s Theorem

    • Peter Buser
    Pages 224-267
  11. Wolpert’s Theorem

    • Peter Buser
    Pages 268-282
  12. Sunada’s Theorem

    • Peter Buser
    Pages 283-310
  13. Examples of Isospectral Riemann Surfaces

    • Peter Buser
    Pages 311-339
  14. The Size of Isospectral Families

    • Peter Buser
    Pages 340-361
  15. Back Matter

    Pages 409-456

About this book

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

Keywords

  • Laplace operator
  • Riemann surfaces
  • Sunada’s construction
  • Wolpert’s theorem
  • complex Riemann surface theory
  • differential geometry
  • equation
  • geometry
  • proof
  • theorem

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)

Authors and Affiliations

  • , Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Lausanne-Ecublens, Switzerland

    Peter Buser

Bibliographic Information

Buying options

eBook USD 109.00
Price excludes VAT (USA)
  • ISBN: 978-0-8176-4992-0
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 149.99
Price excludes VAT (USA)