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 H. M.

This text covers Riemann surface theory from elementary aspects to the fontiers of current research. Open and closed surfaces are treated with emphasis on the compact case. Basic tools are developed to describe the analytic, geometric, and algebraic properties of Riemann surfaces and the Abelian varities associated with these surfaces. For this new edition, the material has been brought up-to-date, and erros have been corrected. The book should be of interest no only to pure mathematicians, but also to physicists interested in string theory and related topics.

0 An Overview.- 0.1. Topological Aspects, Uniformization, and Fuchsian Groups.- 0.2. Algebraic Functions.- 0.3. Abelian Varieties.- 0.4. More Analytic Aspects.- I Riemann Surfaces.- I.1. Definitions and Examples.- I.2. Topology of Riemann Surfaces.- I.3. Differential Forms.- I.4. Integration Formulae.- II Existence Theorems.- II. 1. Hilbert Space Theory-A Quick Review.- II.2. Weyl's Lemma.- II.3. The Hilbert Space of Square Integrable Forms.- II.4. Harmonic Differentials.- II.5. Meromorphic Functions and Differentials.- III Compact Riemann Surfaces.- III. 1. Intersection Theory on Compact Surfaces.- III.2. Harmonic and Analytic Differentials on Compact Surfaces.- III.3. Bilinear Relations.- III.4. Divisors and the Riemann-Roch Theorem.- III.5. Applications of the Riemann-Roch Theorem.- III.6. Abel's Theorem and the Jacobi Inversion Problem.- III.7. Hyperelliptic Riemann Surfaces.- III.8. Special Divisors on Compact Surfaces.- III.9. Multivalued Functions.- III. 10. Projective Imbeddings.- III. 11. More on the Jacobian Variety.- III. 12. Torelli's Theorem.- IV Uniformization.- IV. 1. More on Harmonic Functions (A Quick Review).- IV.2. Subharmonic Functions and Perron's Method.- IV.3. A Classification of Riemann Surfaces.- IV.4. The Uniformization Theorem for Simply Connected Surfaces.- IV.5. Uniformization of Arbitrary Riemann Surfaces.- IV.6. The Exceptional Riemann Surfaces.- IV. 7. Two Problems on Moduli.- IV.8. Riemannian Metrics.- IV.9. Discontinuous Groups and Branched Coverings.- IV. 10. Riemann-Roch-An Alternate Approach.- IV. 11. Algebraic Function Fields in One Variable.- V Automorphisms of Compact Surfaces-Elementary Theory.- V.l. Hurwitz's Theorem.- V.2. Representations of the Automorphism Group on Spaces of Differentials.- V.3. Representationof Aut M on H1(M).- V.4. The Exceptional Riemann Surfaces.- VI Theta Functions.- VI. 1. The Riemann Theta Function.- VI.2. The Theta Functions Associated with a Riemann Surface.- VI.3. The Theta Divisor.- VII Examples.- VII. 1. Hyperelliptic Surfaces (Once Again).- VII.2. Relations Among Quadratic Differentials.- VII.3. Examples of Non-hyperelliptic Surfaces.- VII.4. Branch Points of Hyperelliptic Surfaces as Holomorphic Functions of the Periods.- VII.5. Examples of Prym Differentials.- VII.6. The Trisecant Formula.