A concise course in complex analysis and Riemann surfaces

1 A concise course in complexanalysis and Riemann surfacesWilhelm SchlagContentsPrefacevChapter 1. Fromitoz: the basics of complex analysis11. The field of complex numbers12. Differentiability and conformality33. M obius transforms74. Integration125. Harmonic functions196. The winding number217. Problems24 Chapter 2. Fromzto the Riemann mapping theorem: some finer points of basiccomplex analysis271. The winding number version of Cauchy s theorem272. Isolated singularities and residues293. Analytic continuation334. Convergence and normal families365. The Mittag-Leffler and Weierstrass theorems376. The Riemann mapping theorem417. Runge s theorem448. Problems46 Chapter 3. Harmonic functions onD511.

2 The Poisson kernel512. Hardy classes of harmonic functions533. Almost everywhere convergence to the boundary data554. Problems58 Chapter 4. Riemann surfaces : definitions, examples, basic properties631. The basic definitions632. Examples643. Functions on Riemann surfaces674. Degree and genus695. Riemann surfaces as quotients706. Elliptic functions737. Problems77 Chapter 5. Analytic continuation, covering surfaces , and algebraic functions791. Analytic continuation792. The unramified Riemann surface of an analytic germ83iiiivCONTENTS3. The ramified Riemann surface of an analytic germ854. Algebraic germs and functions885. Problems100 Chapter 6. Differential forms on Riemann surfaces1031.

3 Holomorphic and meromorphic differentials1032. Integrating differentials and residues1053. The Hodge operator and harmonic differentials1064. Statement and examples of the Hodge decomposition1105. Problems115 Chapter 7. Hodge s theorem and theL2existence theory1191. Weyl s lemma and the Hodge decomposition1192. Existence of nonconstant meromorphic functions1233. Problems128 Chapter 8. The Theorems of Riemann -Roch, Abel, and Jacobi1291. Homology bases, periods, and Riemann s bilinear relations1292. Divisors1363. The proof of the Riemann -Roch theorem1374. Applications and general divisors1395. The theorems of Abel and Jacobi1426. Problems142 Chapter 9. The Dirichlet problem and Green functions1451.

4 Green functions1452. The potential theory proof of the Riemann mapping theorem1473. Existence of Green functions via Perron s method1484. Behavior at the boundary151 Chapter 10. Green functions and the classification problem1551. Green functions on Riemann surfaces1552. Hyperbolic Riemann surfaces admit Green functions1563. Problems160 Chapter 11. The uniformization theorem1611. The statement for simply connected surfaces1612. Hyperbolic, simply connected, surfaces1613. Parabolic, simply connected, surfaces162 Chapter 12. Hints and Solutions165 Chapter 13. Review of some facts from algebra and geometry1911. Geometry and topology1912. Algebra194 Bibliography197 PrefaceDuring their first year at the University of Chicago, graduate students in mathe-matics take classes in algebra, analysis , and geometry, oneof each every quarter.

5 Theanalysis classes typically cover real analysis and measuretheory, functional analysis , andcomplex analysis . This book grew out of the author s notes for the complex analysisclass which he taught during the Spring quarter of 2007 and 2008. The course coveredelementary aspects of complex analysis such as the Cauchy integral theorem, the residuetheorem, Laurent series, and the Riemann mapping theorem with Riemann surface the-ory. Needless to say, all of these topics have been covered inexcellent textbooks aswell as classic treatise. This book does not try to compete with the works of the oldmasters such as Ahlfors [1], Hurwitz Courant [20], Titchmarsh [39], Ahlfors Sario [2],Nevanlinna [34], Weyl [41].

6 Rather, it is intended as a fairly detailed yet fast pacedguide through those parts of the theory of one complex variable that seem most usefulin other parts of mathematics. There is no question that complex analysis is a cornerstone of the analysis education at every university and eacharea of mathematics requiresat least some knowledge of it. However, many mathematiciansnever take more than anintroductory class in complex variables that often appearsawkward and slightly out-moded. Often, this is due to the omission of Riemann surfacesand the assumption ofa computational, rather than geometric point of view. Therefore, the authors has triedto emphasize the very intuitive geometric underpinnings ofelementary complex analysiswhich naturally lead to Riemann surface theory.

7 As for the latter, today it is either nottaught at all or sometimes given a very algebraic slant whichdoes not appeal to moreanalytically minded students. This book intends to developthe subject of Riemann sur-faces as a natural continuation of the elementary theory without which the latter wouldindeed seem artificial and antiquated. At the same time, we donot overly emphasize thealgebraic aspect such as elliptic curves. The author feels that those students who wishto pursue this direction will be able to do so quite easily after mastering the material inthis book. Because of such omissions as well as the reasonably short length of the bookit is to be considered intermediate .Partly because of the fact that the Chicago first year curriculum covers topology andgeometry this book assumes knowledge of basic notions such as homotopy, the fundamen-tal group, differential forms, co-homology and homology, and from algebra we requireknowledge of the notions of groups and fields, and some familiarity with the resultant oftwo polynomials (but the latter is needed only for the definition of the Riemann surfacesof an algebraic germ).

8 However, only the most basic knowledge of these concepts isassumed and we collect the few facts that we do need in us now describe the contents of the individual chapters in more detail. Chap-ter 1 introduces the concept of differentiability overC, the calculus of z, z, M obius(or fractional linear) transformations and some applications of these transformations tovviPREFACE hyperbolic geometry. In particular, we prove the Gauss-Bonnet theorem in that , we develop integration and Cauchy s theorem in various guises, then apply this tothe study of analyticity, and harmonicity, the logarithm and the winding number. Weconclude the chapter with some brief comments about co-homology and the 2 refines the Cauchy formula by extending it to zero homologous cycles, ,those cycles which do not wind around any point outside of thedomain of then classify isolated singularities, prove the Laurentexpansion and the residuetheorems with applications.

9 After that, Chapter 2 studies analytic continuation andpresents the monodromy theorem. Then, we turn to convergence of analytic functionsand normal families with application to the Mittag-Leffler and Weierstrass theorems inthe entire plane, as well as the Riemann mapping theorem. Thechapter concludes withRunge s Chapter 3 we study the Dirichlet problem on the unit disk. This means that wesolve the boundary value problem for the Laplacian on the disk via the Poisson kernel. Wepresent the usualLpbased Hardy classes of harmonic functions on the disk, and discussthe question of representing them via their boundary data both in theLpand the almostevery sense. We then sketch the more subtle theory of homolomorphic functions in theHardy class, or equivalently of the boundedness propertiesof the conjugate harmonicfunctions (with the M.)

10 Riesz theorem and the notion of inner and outer functionsbeing the most relevant here).The theory of Riemann surfaces begins with Chapter 4. This chapter covers the basicdefinitions of such surfaces and the analytic functions on them. Elementary results suchas the Riemann -Hurwitz formula for the branch points are discussed and several examplesof surfaces and analytic functions defined on them are presented. In particular, we showhow to define Riemann surfaces via discontinuous group actions and give examples of thisprocedure. The chapter closes with a discussion of tori and some aspects of the classicaltheory of meromorphic functions on these tori (doubly periodic or elliptic functions).Chapter 5 presents another way in which Riemann surfaces arise naturally, namelyvia analytic continuation.