A concise course in complex analysis and Riemann surfaces Wilhelm Schlag Contents Preface v Chapter 1. From i to z: the basics of complex analysis 1 1. The field of complex numbers 1 2. Differentiability and conformality 3 3. M¨obius transforms 7 4. Integration 12 5. Harmonic functions 19 6. The winding number 21 7. Problems 24 Chapter 2. From z to the Riemann mapping theorem: some finer points of basic complex analysis 27 1. The winding number version of Cauchy’s theorem 27 2. Isolated singularities and residues 29 3. Analytic continuation 33 4. Convergence and normal families 36 5. The Mittag-Leffler and Weierstrass theorems 37 6. The Riemann mapping theorem 41 7. Runge’s theorem 44 8. Problems 46 Chapter 3. Harmonic functions on D 51 1. The Poisson kernel 51 2. Hardy classes of harmonic functions 53 3. Almost everywhere convergence to the boundary data 55 4. Problems 58 Chapter 4. Riemann surfaces: definitions, examples, basic properties 63 1. The basic definitions 63 2. Examples 64 3. Functions on Riemann surfaces 67 4. Degree and genus 69 5. Riemann surfaces as quotients 70 6. Elliptic functions 73 7. Problems 77 Chapter 5. Analytic continuation, covering surfaces, and algebraic functions 79 1. Analytic continuation 79 2. The unramified Riemann surface of an analytic germ 83 iii iv CONTENTS 3. The ramified Riemann surface of an analytic germ 85 4. Algebraic germs and functions 88 5. Problems 100 Chapter 6. Differential forms on Riemann surfaces 103 1. Holomorphic and meromorphic differentials 103 2. Integrating differentials and residues 105 3. The Hodge operator and harmonic differentials 106 4. Statement and∗ examples of the Hodge decomposition 110 5. Problems 115 Chapter 7. Hodge’s theorem and the L2 existence theory 119 1. Weyl’s lemma and the Hodge decomposition 119 2. Existence of nonconstant meromorphic functions 123 3. Problems 128 Chapter 8. The Theorems of Riemann-Roch, Abel, and Jacobi 129 1. Homology bases, periods, and Riemann’s bilinear relations 129 2. Divisors 136 3. The proof of the Riemann-Roch theorem 137 4. Applications and general divisors 139 5. The theorems of Abel and Jacobi 142 6. Problems 142 Chapter 9. The Dirichlet problem and Green functions 145 1. Green functions 145 2. The potential theory proof of the Riemann mapping theorem 147 3. Existence of Green functions via Perron’s method 148 4. Behavior at the boundary 151 Chapter 10. Green functions and the classification problem 155 1. Green functions on Riemann surfaces 155 2. Hyperbolic Riemann surfaces admit Green functions 156 3. Problems 160 Chapter 11. The uniformization theorem 161 1. The statement for simply connected surfaces 161 2. Hyperbolic, simply connected, surfaces 161 3. Parabolic, simply connected, surfaces 162 Chapter 12. Hints and Solutions 165 Chapter 13. Review of some facts from algebra and geometry 191 1. Geometry and topology 191 2. Algebra 194 Bibliography 197 Preface During their first year at the University of Chicago, graduate students in mathe- matics take classes in algebra, analysis, and geometry, one of each every quarter. The analysis classes typically cover real analysis and measure theory, functional analysis, and complex analysis. This book grew out of the author’s notes for the complex analysis class which he taught during the Spring quarter of 2007 and 2008. The course covered elementary aspects of complex analysis such as the Cauchy integral theorem, the residue theorem, Laurent series, and the Riemann mapping theorem with Riemann surface the- ory. Needless to say, all of these topics have been covered in excellent textbooks as well as classic treatise. This book does not try to compete with the works of the old masters such as Ahlfors [1], Hurwitz–Courant [20], Titchmarsh [39], Ahlfors–Sario [2], Nevanlinna [34], Weyl [41]. Rather, it is intended as a fairly detailed yet fast paced guide through those parts of the theory of one complex variable that seem most useful in other parts of mathematics. There is no question that complex analysis is a corner stone of the analysis education at every university and each area of mathematics requires at least some knowledge of it. However, many mathematicians never take more than an introductory class in complex variables that often appears awkward and slightly out- moded. Often, this is due to the omission of Riemann surfaces and the assumption of a computational, rather than geometric point of view. Therefore, the authors has tried to emphasize the very intuitive geometric underpinnings of elementary complex analysis which naturally lead to Riemann surface theory. As for the latter, today it is either not taught at all or sometimes given a very algebraic slant which does not appeal to more analytically minded students. This book intends to develop the subject of Riemann sur- faces as a natural continuation of the elementary theory without which the latter would indeed seem artificial and antiquated. At the same time, we do not overly emphasize the algebraic aspect such as 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 such omissions as well as the reasonably short length of the book it is to be considered “intermediate”. Partly because of the fact that the Chicago first year curriculum covers topology and geometry this book assumes knowledge of basic notions such as homotopy, the fundamen- tal group, differential forms, co-homology 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, only the most basic knowledge of these concepts is assumed and we collect the few facts that we do need in Chapter 13. Let us now describe the contents of the individual chapters in more detail. Chap- ter 1 introduces the concept of differentiability over C, the calculus of ∂z, ∂z¯, M¨obius (or fractional linear) transformations and some applications of these transformations to v vi PREFACE hyperbolic geometry. In particular, we prove the Gauss-Bonnet theorem in that case. Next, we develop integration and Cauchy’s theorem in various guises, then apply this to the study of analyticity, and harmonicity, the logarithm and the winding number. We conclude the chapter with some brief comments about co-homology and the fundamental group. Chapter 2 refines the Cauchy formula by extending it to zero homologous cycles, i.e., those cycles which do not wind around any point outside of the domain of holomorphy. We then classify isolated singularities, prove the Laurent expansion and the residue theorems with applications. After that, Chapter 2 studies analytic continuation and presents the monodromy theorem. Then, we turn to convergence of analytic functions and normal families with application to the Mittag-Leffler and Weierstrass theorems in the entire plane, as well as the Riemann mapping theorem. The chapter concludes with Runge’s theorem. In Chapter 3 we study the Dirichlet problem on the unit disk. This means that we solve the boundary value problem for the Laplacian on the disk via the Poisson kernel. We present the usual Lp based Hardy classes of harmonic functions on the disk, and discuss the question of representing them via their boundary data both in the Lp and the almost every sense. We then sketch the more subtle theory of homolomorphic functions in the Hardy class, or equivalently of the boundedness properties of the conjugate harmonic functions (with the F.& M. Riesz theorem and the notion of inner and outer functions being the most relevant here). The theory of Riemann surfaces begins with Chapter 4. This chapter covers the basic definitions of such surfaces and the analytic functions on them. Elementary results such as the Riemann-Hurwitz formula for the branch points are discussed and several examples of surfaces and analytic functions defined on them are presented. In particular, we show how to define Riemann surfaces via discontinuous group actions and give examples of this procedure. The chapter closes with a discussion of tori and some aspects of the classical theory of meromorphic functions on these tori (doubly periodic or elliptic functions). Chapter 5 presents another way in which Riemann surfaces arise naturally, namely via analytic continuation. Historically, the desire to resolve unnatural issues related to “multi-valued functions” (most importantly for algebraic functions) lead Riemann to introduce his surfaces. Even though the underlying ideas leading from a so-called analytic germ to its Riemann surface are very geometric and intuitive (and closely related to covering spaces in topology), their rigorous rendition requires some patience as ideas such as “analytic germ”, “branch point”, “(un)ramified Riemann surface of an analytic germ” etc., need to be defined precisely. This typically proceeds via some factorization procedure of a larger object (i.e., equivalence classes of sets which are indistinguishable from the point of view of the particular object we wish to construct). The chapter also develops some basic aspects of algebraic functions and their Riemann surfaces. At this point the reader will need to be familiar with the resultant of two polynomials. In particular, we will see that every (!) compact Riemann surface is obtained through analytic continuation of some algebraic germ. This uses the machinery of Chapter 5 together with a potential theoretic result that guarantees the existence of a non-constant meromorphic function on every Riemann surface, which we prove in Chapter 7. Chapter 6 introduces differential forms on Riemann surfaces and their integrals. Needless to say, the only really important class are the 1-forms and we define harmonic, holomorphic and meromorphic forms and the residues in the latter case.