Geometry and Spectra of Compact Riemann Surfaces

Abstract

This book consists of two parts. The first part is a very detailed presentation of the geometry of Riemann surfaces from a cutting-and-pasting approach. Starting with hyperbolic trigonometry, it develops Riemann surface theory through Teichmüller space, providing along the way a wealth of information that researchers in the field have long wanted to see in print. The second part is a self-contained exposition of the spectrum of the Laplacian of Riemann surfaces, with particular attention given to the problem of isospectral surfaces (particularly the theorems of Wolpert and Sunada). 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. The book is pitched at the level of graduate students. The reviewer is unable to imagine anyone who can understand the title not benefiting greatly from the book.