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. 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.

Table of Contents

1. Hyperbolic Structures

2. Trigonometry

3. Y-Pieces and Twist Parameters

4. The Collar Theorem

5. Bers' Constant and the Hairy Torus

6. The Teichmuller Space

7. The Spectrum of the Laplacian

8. Small Eigenvalues

9. Closed Geodesies and Huber's Theorem

10. Wolpert's Theorem

11. Sunada's Theorem

12. Examples of lsospectral Riemann Surfaces

13. The Size of lsospectral Families

14. Perturbations of the Laplacian in Teichmuller Space

Review

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)