黎曼曲面 第2版（影印版）

　　 It is gratifying to learn that there is new life in an old field that has been at the center of one's existence for over a quarter of a century. It is particularly pleasing that the subject of Riemann surfaces has attracted the attention of a new generation of mathematicians from (newly) adjacent fields (for example, those interested in hyperbolic manifolds and iterations of rational maps) and young physicists who have been convinced (certainly not by mathematicians) that compact Riemann surfaces may play an important role in their (string) universe. We hope that non-mathematicians as well as mathematicians (working in nearby areas to the central topic of this book) will also learn part of this subject for the sheer beauty and elegance of the material (work of Weierstrass, Jacobi, Riemann, Hilbert, Weyl) and as healthy exposure to the way (some) mathematicians write about mathematics.
　　

preface to the second edition
preface to the first edition
commonly used symbols
chapter 0
an overview
0.1. topological aspects, uniformization, and fuchsian groups
0.2. algebraic functions
0.3. abelian varieties
0.4. more analytic aspects
chapter i
riemann surfaces
i.1. definitions and examples
i.2. topology of riemann surfaces
i.3. differential forms
i.4. integration formulae
chapter ii
existence theorems
ii.1. hilbert space theory--a quick review
ii.2. weyl's lemma
ii.3. the hilbert space of square lntegrable forms

　　Preface to the Second Edition
　　 It is gratifying to learn that there is new life in an old field that has been at the center of one's existence for over a quarter of a century. It is particularly pleasing that the subject of Riemann surfaces has attracted the attention of a new generation of mathematicians from (newly) adjacent fields (for example, those interested in hyperbolic manifolds and iterations of rational maps) and young physicists who have been convinced (certainly not by mathematicians) that compact Riemann surfaces may play an important role in their (string) universe. We hope that non-mathematicians as well as mathematicians (working in nearby areas to the central topic of this book) will also learn part of this subject for the sheer beauty and elegance of the material (work of Weierstrass, Jacobi, Riemann, Hilbert, Weyl) and as healthy exposure to the way (some) mathematicians write about mathematics.
　　 We had intended a more comprehensive revision, including a fuller treatment of moduli problems and theta functions. Pressure of other commitments would have substantially delayed (by years) the appearance of the book we wanted to produce. We have chosen instead to make a few
　　modest additions and to correct a number of errors. We are grateful to the readers who pointed out some of our mistakes in the first edition; the responsibility for the remaining mistakes carried over from the first edition and for any new ones introduced into the second edition remains with the authors.
　　June 1991
　　Jerusalem
　　and
　　Stony Brook
　　H.M. FARKAS
　　 and
　　 I. KRA
　　Preface to the First Edition
　　The present volume is the culmination of ten years' work separately and jointly. The idea of writing this book began with a set of notes for a course given by one of the authors in 1970-1971 at the Hebrew University. The notes were refined several times and used as the basic content of courses given subsequently by each of the authors at the State University of New York at
　　Stony Brook and the Hebrew University.
　　 In this book we present the theory of Riemann surfaces and its many different facets. We begin from the most elementary aspects and try to bring the reader up to the frontier of present-day research. We treat both open and closed surfaces in this book, but our main emphasis is on the compact case.
　　In fact, Chapters III, V, VI, and VII deal exclusively with compact surfaces.Chapters I and II are preparatory, and Chapter IV deals with uniformization.
　　 All works on Riemann surfaces go back to the fundamental results of Riemann, Jacobi, Abel, Weierstrass, etc. Our book is no exception. In addition to our debt to these mathematicians of a previous era, the present work has been influenced by many contemporary mathematicians.
　　 At the outset we record our indebtedness to our teachers Lipman Bers and Harry Ernest Rauch, who taught us a great deal of what we know about this subject, and who along with Lars V. Ahlfors are responsible for the modem rebirth of the theory of Riemann surfaces. Second, we record our gratitude to our colleagues whose theorems we have freely written down without attribution. In particular, some of the material in Chapter III is the work of Henrik H. Martens, and some of the material in Chapters V and VI ultimately goes back to Robert D. M. Accola and Joseph Lewittes.
　　 We thank several colleagues who have read and criticized earlier versions of the manuscript and made many helpful suggestions: Bernard Maskit,Henry Laufer, Uri Srebro, Albert Marden, and Frederick P. Gardiner. The errors in the final version are, however, due only to the authors. We also thank the secretaries who typed the various versions: Carole Alberghine and Estella Shivers.
　　August 1979 H.M. FARKAS I. KRA