A Course in Complex Analysis and Riemann Surfaces Wilhelm Schlag Graduate Studies in Mathematics Volume 154 American Mathematical Society https://doi.org/10.1090//gsm/154 A Course in Complex Analysis and Riemann Surfaces Wilhelm Schlag Graduate Studies in Mathematics Volume 154 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed Rafe Mazzeo (Chair) Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 30-01, 30F10, 30F15, 30F20, 30F30, 30F35. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-154 Library of Congress Cataloging-in-Publication Data Schlag, Wilhelm, 1969– A course in complex analysis and Riemann surfaces / Wilhelm Schlag. pages cm. – (Graduate studies in mathematics ; volume 154) Includes bibliographical references and index. ISBN 978-0-8218-9847-5 (alk. paper) 1. Riemann surfaces–Textbooks. 2. Functions of complex variables–Textbooks. I. Title. QA333.S37 2014 515.93–dc23 2014009993 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2014 by Wilhelm Schlag. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 191817161514 Contents Preface vii Acknowledgments xv Chapter 1. From i to z: the basics of complex analysis 1 §1.1. The field of complex numbers 1 §1.2. Holomorphic, analytic, and conformal 4 §1.3. The Riemann sphere 9 §1.4. M¨obius transformations 11 §1.5. The hyperbolic plane and the Poincar´edisk 15 §1.6. Complex integration, Cauchy theorems 18 §1.7. Applications of Cauchy’s theorems 23 §1.8. Harmonic functions 33 §1.9. Problems 36 Chapter 2. From z to the Riemann mapping theorem: some finer points of basic complex analysis 41 §2.1. The winding number 41 §2.2. The global form of Cauchy’s theorem 45 §2.3. Isolated singularities and residues 47 §2.4. Analytic continuation 56 §2.5. Convergence and normal families 60 §2.6. The Mittag-Leffler and Weierstrass theorems 63 §2.7. The Riemann mapping theorem 69 §2.8. Runge’s theorem and simple connectivity 74 iii iv Contents §2.9. Problems 79 Chapter 3. Harmonic functions 85 §3.1. The Poisson kernel 85 §3.2. The Poisson kernel from the probabilistic point of view 91 §3.3. Hardy classes of harmonic functions 95 §3.4. Almost everywhere convergence to the boundary data 100 §3.5. Hardy spaces of analytic functions 105 §3.6. Riesz theorems 109 §3.7. Entire functions of finite order 111 §3.8. A gallery of conformal plots 117 §3.9. Problems 122 Chapter 4. Riemann surfaces: definitions, examples, basic properties 129 §4.1. The basic definitions 129 §4.2. Examples and constructions of Riemann surfaces 131 §4.3. Functions on Riemann surfaces 143 §4.4. Degree and genus 146 §4.5. Riemann surfaces as quotients 148 §4.6. Elliptic functions 151 §4.7. Covering the plane with two or more points removed 160 §4.8. Groups of M¨obius transforms 164 §4.9. Problems 174 Chapter 5. Analytic continuation, covering surfaces, and algebraic functions 179 §5.1. Analytic continuation 179 §5.2. The unramified Riemann surface of an analytic germ 185 §5.3. The ramified Riemann surface of an analytic germ 189 §5.4. Algebraic germs and functions 192 §5.5. Algebraic equations generated by compact surfaces 199 §5.6. Some compact surfaces and their associated polynomials 206 §5.7. ODEs with meromorphic coefficients 211 §5.8. Problems 221 Chapter 6. Differential forms on Riemann surfaces 225 §6.1. Holomorphic and meromorphic differentials 225 §6.2. Integrating differentials and residues 227 Contents v §6.3. The Hodge-∗ operator and harmonic differentials 230 §6.4. Statement and examples of the Hodge decomposition 236 §6.5. Weyl’s lemma and the Hodge decomposition 244 §6.6. Existence of nonconstant meromorphic functions 250 §6.7. Examples of meromorphic functions and differentials 258 §6.8. Problems 266 Chapter 7. The Theorems of Riemann-Roch, Abel, and Jacobi 269 §7.1. Homology bases and holomorphic differentials 269 §7.2. Periods and bilinear relations 273 §7.3. Divisors 280 §7.4. The Riemann-Roch theorem 285 §7.5. Applications and general divisors 289 §7.6. Applications to algebraic curves 292 §7.7. The theorems of Abel and Jacobi 295 §7.8. Problems 303 Chapter 8. Uniformization 305 §8.1. Green functions and Riemann mapping 306 §8.2. Perron families 310 §8.3. Solution of Dirichlet’s problem 314 §8.4. Green’s functions on Riemann surfaces 317 §8.5. Uniformization for simply-connected surfaces 326 §8.6. Uniformization of non-simply-connected surfaces 335 §8.7. Fuchsian groups 338 §8.8. Problems 349 Appendix A. Review of some basic background material 353 §A.1. Geometry and topology 353 §A.2. Algebra 363 §A.3. Analysis 365 Bibliography 371 Index 377 Preface During their first year at the University of Chicago, graduate students in mathematics take classes in algebra, analysis, and geometry, one of each every quarter. The analysis courses typically cover real analysis and measure theory, functional analysis and applications, and complex analysis. This book grew out of the author’s notes for the complex analysis classes which he taught during the Spring quarters of 2007 and 2008. These courses covered elementary aspects of complex analysis such as the Cauchy integral theorem, the residue theorem, Laurent series, and the Riemann mapping theorem, but also more advanced material selected from Riemann surface theory. Needless to say, all of these topics have been covered in excellent text- books as well as classic treatises. This book does not try to compete with the works of the old masters such as Ahlfors [1, 2], Hurwitz–Courant [44], Titchmarsh [80], Ahlfors–Sario [3], Nevanlinna [67], and Weyl [88]. Rather, it is intended as a fairly detailed, yet fast-paced introduction to those parts of the theory of one complex variable that seem most useful in other ar- eas of mathematics (geometric group theory, dynamics, algebraic geometry, number theory, functional analysis). There is no question that complex analysis is a cornerstone of a math- ematics specialization at every university and each area of mathematics re- quires at least some knowledge of it. However, many mathematicians never take more than an introductory class in complex variables which ends up being awkward and slightly outmoded. Often this is due to the omission of Riemann surfaces and the assumption of a computational, rather than a geometric point of view. The author has therefore tried to emphasize the intuitive geometric un- derpinnings of elementary complex analysis that naturally lead to Riemann vii viii Preface surface theory. Today this is either not taught at all, given an algebraic slant, or is presented from a sophisticated analytical perspective, leaving the students without any foundation, intuition, historical understanding, let alone a working knowledge of the subject. This book intends to develop the subject of Riemann surfaces as a nat- ural continuation of the elementary theory without which basic complex analysis would indeed seem artificial and antiquated. At the same time, we do not overly emphasize the algebraic aspects such as applications to elliptic curves. The author feels that those students who wish to pursue this direction will be able to do so quite easily after mastering the material in this book. Because of this, as well as numerous other omissions (e.g., zeta, theta, and automorphic functions, Serre duality, Dolbeault cohomol- ogy) and the reasonably short length of the book, it is to be considered as an “intermediate introduction”. Partly due to the fact that the Chicago first year curriculum covers a fair amount of topology and geometry before complex analysis, this book assumes knowledge of basic notions such as homotopy, the fundamental group, differential forms, cohomology and homology, and from algebra we require knowledge of the notions of groups and fields, and some familiarity with the resultant of two polynomials (but the latter is needed only for the definition of the Riemann surfaces of an algebraic germ). However, for the most part merely the most elementary familiarity of these concepts is assumed and we collect the few facts that we do need in Appendix A. As far as analytical prerequisites are concerned, they are fairly low, not extending far beyond multi-variable calculus and basic Hilbert space theory (in Chapter 6 we use orthogonal projections). One exception to this occurs in Sections 3.3, 3.4, and 3.5, which use the weak and weak-∗ topologies in Lp and the space of measures (Riesz representation theorem). Again, what we need is recalled in the appendix. Let us now describe the contents of the individual chapters in more detail. Chapter 1 introduces the concept of differentiability over C,the calculus of ∂z,∂z¯, the Cauchy-Riemann equations, power series, the M¨obius (or fractional linear) transformations and the Riemann sphere. Applications of these transformations to hyperbolic geometry (the Poincar´e disk and the upper half-plane models) are also discussed.