Geometry and Spectra of Compact Riemann Surfaces

  • Peter Buser

Part of the Modern Birkhäuser Classics book series (MBC)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Peter Buser
    Pages 1-30
  3. Peter Buser
    Pages 31-62
  4. Peter Buser
    Pages 63-93
  5. Peter Buser
    Pages 94-121
  6. Peter Buser
    Pages 122-137
  7. Peter Buser
    Pages 138-181
  8. Peter Buser
    Pages 182-209
  9. Peter Buser
    Pages 210-223
  10. Peter Buser
    Pages 224-267
  11. Peter Buser
    Pages 268-282
  12. Peter Buser
    Pages 283-310
  13. Peter Buser
    Pages 311-339
  14. 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


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

Authors and affiliations

  • Peter Buser
    • 1
  1. 1., Département de MathématiquesEcole Polytechnique Fédérale de LausanneLausanne-EcublensSwitzerland

Bibliographic information