Riemann Surfaces Lecture Notes
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Arxiv:1003.6025V1 [Math.HO] 31 Mar 2010 Karl Stein (1913-2000)
Karl Stein (1913-2000) Alan Huckleberry Karl Stein was born on the first of January 1913 in Hamm in Westfalen, grew up there, received his Abitur in 1932 and immediately thereafter be- gan his studies in M¨unster. Just four years later, under the guidance of Heinrich Behnke, he passed his Staatsexam, received his promotion and became Behnke’s assistant. Throughout his life, complex analysis, primarily in higher dimensions (“mehrere Ver¨anderliche”), was the leitmotif of Stein’s mathematics. As a fresh Ph.D. in M¨unster in 1936, under the leadership of the master Behnke, he had already been exposed to the fascinating developments in this area. The brilliant young Peter Thullen was proving fundamental theorems, Henri Cartan had visited M¨unster, and Behnke and Thullen had just writ- ten the book on the subject. It must have been clear to Stein that this was the way to go. Indeed it was! The amazing phenomenon of analytic continuation in higher dimensions had already been exemplified more than 20 years be- fore in the works of Hartogs and E. E. Levi. Thullen’s recent work had gone much further. In the opposite direction, Cartan and Thullen had arXiv:1003.6025v1 [math.HO] 31 Mar 2010 proved their characterization of domains in Cn which admit a holomor- phic function which can not be continued any further. Behnke himself was also an active participant in mathematics research, always bringing new ideas to M¨unster. This was indeed an exciting time for the young researcher, Karl Stein. Even though the pest of the Third Reich was already invading academia, Behnke kept things going for as long as possible. -
The Universal Covering Space of the Twice Punctured Complex Plane
The Universal Covering Space of the Twice Punctured Complex Plane Gustav Jonzon U.U.D.M. Project Report 2007:12 Examensarbete i matematik, 20 poäng Handledare och examinator: Karl-Heinz Fieseler Mars 2007 Department of Mathematics Uppsala University Abstract In this paper we present the basics of holomorphic covering maps be- tween Riemann surfaces and we discuss and study the structure of the holo- morphic covering map given by the modular function λ onto the base space C r f0; 1g from its universal covering space H, where the plane regions C r f0; 1g and H are considered as Riemann surfaces. We develop the basics of Riemann surface theory, tie it together with complex analysis and topo- logical covering space theory as well as the study of transformation groups on H and elliptic functions. We then utilize these tools in our analysis of the universal covering map λ : H ! C r f0; 1g. Preface First and foremost, I would like to express my gratitude to the supervisor of this paper, Karl-Heinz Fieseler, for suggesting the project and for his kind help along the way. I would also like to thank Sandra for helping me out when I got stuck with TEX and Daja for mathematical feedback. Contents 1 Introduction and Background 1 2 Preliminaries 2 2.1 A Quick Review of Topological Covering Spaces . 2 2.1.1 Covering Maps and Their Basic Properties . 2 2.2 An Introduction to Riemann Surfaces . 6 2.2.1 The Basic Definitions . 6 2.2.2 Elementary Properties of Holomorphic Mappings Between Riemann Surfaces . -
Moduli Spaces of Riemann Surfaces
MODULI SPACES OF RIEMANN SURFACES ALEX WRIGHT Contents 1. Different ways to build and deform Riemann surfaces 2 2. Teichm¨uller space and moduli space 4 3. The Deligne-Mumford compactification of moduli space 11 4. Jacobians, Hodge structures, and the period mapping 13 5. Quasiconformal maps 16 6. The Teichm¨ullermetric 18 7. Extremal length 24 8. Nielsen-Thurston classification of mapping classes 26 9. Teichm¨uller space is a bounded domain 29 10. The Weil-Petersson K¨ahlerstructure 40 11. Kobayashi hyperbolicity 46 12. Geometric Shafarevich and Mordell 48 References 57 These are course notes for a one quarter long second year graduate class at Stanford University taught in Winter 2017. All of the material presented is classical. For the softer aspects of Teichm¨ullertheory, our reference was the book of Farb-Margalit [FM12], and for Teichm¨ullertheory as a whole our comprehensive ref- erence was the book of Hubbard [Hub06]. We also used Gardiner and Lakic's book [GL00] and McMullen's course notes [McM] as supple- mental references. For period mappings our reference was the book [CMSP03]. For Geometric Shafarevich our reference was McMullen's survey [McM00]. The author thanks Steve Kerckhoff, Scott Wolpert, and especially Jeremy Kahn for many helpful conversations. These are rough notes, hastily compiled, and may contain errors. Corrections are especially welcome, since these notes will be re-used in the future. 2 WRIGHT 1. Different ways to build and deform Riemann surfaces 1.1. Uniformization. A Riemann surface is a topological space with an atlas of charts to the complex plane C whose translation functions are biholomorphisms. -
The Measurable Riemann Mapping Theorem
CHAPTER 1 The Measurable Riemann Mapping Theorem 1.1. Conformal structures on Riemann surfaces Throughout, “smooth” will always mean C 1. All surfaces are assumed to be smooth, oriented and without boundary. All diffeomorphisms are assumed to be smooth and orientation-preserving. It will be convenient for our purposes to do local computations involving metrics in the complex-variable notation. Let X be a Riemann surface and z x iy be a holomorphic local coordinate on X. The pair .x; y/ can be thoughtD C of as local coordinates for the underlying smooth surface. In these coordinates, a smooth Riemannian metric has the local form E dx2 2F dx dy G dy2; C C where E; F; G are smooth functions of .x; y/ satisfying E > 0, G > 0 and EG F 2 > 0. The associated inner product on each tangent space is given by @ @ @ @ a b ; c d Eac F .ad bc/ Gbd @x C @y @x C @y D C C C Äc a b ; D d where the symmetric positive definite matrix ÄEF (1.1) D FG represents in the basis @ ; @ . In particular, the length of a tangent vector is f @x @y g given by @ @ p a b Ea2 2F ab Gb2: @x C @y D C C Define two local sections of the complexified cotangent bundle T X C by ˝ dz dx i dy WD C dz dx i dy: N WD 2 1 The Measurable Riemann Mapping Theorem These form a basis for each complexified cotangent space. The local sections @ 1 Â @ @ Ã i @z WD 2 @x @y @ 1 Â @ @ Ã i @z WD 2 @x C @y N of the complexified tangent bundle TX C will form the dual basis at each point. -
From Riemann Surfaces to Complex Spaces Reinhold Remmert∗
From Riemann Surfaces to Complex Spaces Reinhold Remmert∗ We must always have old memories and young hopes Abstract This paper analyzes the development of the theory of Riemann surfaces and complex spaces, with emphasis on the work of Rie- mann, Klein and Poincar´e in the nineteenth century and on the work of Behnke-Stein and Cartan-Serre in the middle of this cen- tury. R´esum´e Cet article analyse le d´eveloppement de la th´eorie des surfaces de Riemann et des espaces analytiques complexes, en ´etudiant notamment les travaux de Riemann, Klein et Poincar´eauXIXe si`ecle et ceux de Behnke-Stein et Cartan-Serre au milieu de ce si`ecle. Table of Contents 1. Riemann surfaces from 1851 to 1912 1.1. Georg Friedrich Bernhard Riemann and the covering principle 1.1∗. Riemann’s doctorate 1.2. Christian Felix Klein and the atlas principle 1.3. Karl Theodor Wilhelm Weierstrass and analytic configurations AMS 1991 Mathematics Subject Classification: 01A55, 01A60, 30-03, 32-03 ∗Westf¨alische Wilhelms–Universit¨at, Mathematisches Institut, D–48149 Munster,¨ Deutschland This expos´e is an enlarged version of my lecture given in Nice. Gratias ago to J.-P. Serre for critical comments. A detailed exposition of sections 1 and 2 will appear elsewhere. SOCIET´ EMATH´ EMATIQUE´ DE FRANCE 1998 204 R. REMMERT 1.4. The feud between G¨ottingen and Berlin 1.5. Jules Henri Poincar´e and automorphic functions 1.6. The competition between Klein and Poincar´e 1.7. Georg Ferdinand Ludwig Philipp Cantor and countability of the topology 1.8. -
The Cousin Problems and the Emergence of the Sheaf Concept Author(S): Renaud Chorlay Source: Archive for History of Exact Sciences, Vol
From Problems to Structures: the Cousin Problems and the Emergence of the Sheaf Concept Author(s): Renaud Chorlay Source: Archive for History of Exact Sciences, Vol. 64, No. 1 (January 2010), pp. 1-73 Published by: Springer Stable URL: http://www.jstor.org/stable/41342411 Accessed: 23-05-2017 00:50 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Springer is collaborating with JSTOR to digitize, preserve and extend access to Archive for History of Exact Sciences This content downloaded from 132.248.9.8 on Tue, 23 May 2017 00:50:12 UTC All use subject to http://about.jstor.org/terms Arch. Hist. Exact Sci. (2010) 64:1-73 DOI 10. 1007/s00407-009-0052-3 From Problems to Structures: the Cousin Problems and the Emergence of the Sheaf Concept Renaud Chorlay Received: 19 November 2008 / Published online: 30 July 2009 © Springer- Verlag 2009 Abstract Historical work on the emergence of sheaf theory has mainly concen- trated on the topological origins of sheaf cohomology in the period from 1945 to 1950 and on subsequent developments. However, a shift of emphasis both in time-scale and disciplinary context can help gain new insight into the emergence of the sheaf concept. -
BIHOLOMORPHIC TRANSFORMATIONS 1. Introduction This Is a Short Overview for the Beginners
BIHOLOMORPHIC TRANSFORMATIONS BUMA FRIDMAN AND DAOWEI MA 1. Introduction This is a short overview for the beginners (graduate students and advanced undergrad- uates) on some aspects of biholomorphic maps. A mathematical theory identifies objects that are \similar in every detail" from the point of view of the corresponding theory. For instance, in algebra we use the notion of isomorphism of algebraic objects (groups, rings, vector spaces, etc.), while in topology homeomorphic topological spaces are considered similar in every detail. The idea is to classify the objects under consideration. In complex analysis the corresponding notion is the biholomorphism of the main objects in the theory: complex manifolds. Two complex manifolds M1;M2 are biholomorphic if there is a bijective holomorphic map F : M1 ! M2. Like in those other theories one wants to find the biholomorphic classification of complex manifolds. This attempt is certainly an important motivation to study biholomorphic transformations. The classification problem appears to be a very difficult problem, and only partial results (though very important and highly interesting) are known. The Riemann Mapping Theorem states that any two proper simply connected domains in the complex plane are conformally equivalent. This is an exception. The general case is: for any n ≥ 1 in Cn any two \randomly" picked domains are not biholomorphic. We will mention three examples to support this viewpoint. We also are pointing out that in case n = 1 the classification problem is well researched and mostly understood. For n ≥ 2 it is way more complicated, and by now the pursuit of it has produced many very interesting and deep results and also created useful tools for SCV. -
Complex Analysis Course Notes
NOTES FOR MATH 520: COMPLEX ANALYSIS KO HONDA 1. Complex numbers 1.1. Definition of C. As a set, C = R2 = (x; y) x; y R . In other words, elements of C are pairs of real numbers. f j 2 g C as a field: C can be made into a field, by introducing addition and multiplication as follows: (1) (Addition) (a; b) + (c; d) = (a + c; b + d). (2) (Multiplication) (a; b) (c; d) = (ac bd; ad + bc). · − C is an Abelian (commutative) group under +: (1) (Associativity) ((a; b) + (c; d)) + (e; f) = (a; b) + ((c; d) + (e; f)). (2) (Identity) (0; 0) satisfies (0; 0) + (a; b) = (a; b) + (0; 0) = (a; b). (3) (Inverse) Given (a; b), ( a; b) satisfies (a; b) + ( a; b) = ( a; b) + (a; b). (4) (Commutativity) (a; b) +− (c;−d) = (c; d) + (a; b). − − − − C (0; 0) is also an Abelian group under multiplication. It is easy to verify the properties − f g 1 a b above. Note that (1; 0) is the identity and (a; b)− = ( a2+b2 ; a2−+b2 ). (Distributivity) If z ; z ; z C, then z (z + z ) = (z z ) + (z z ). 1 2 3 2 1 2 3 1 2 1 3 Also, we require that (1; 0) = (0; 0), i.e., the additive identity is not the same as the multi- plicative identity. 6 1.2. Basic properties of C. From now on, we will denote an element of C by z = x + iy (the standard notation) instead of (x; y). Hence (a + ib) + (c + id) = (a + c) + i(b + d) and (a + ib)(c + id) = (ac bd) + i(ad + bc). -
Introduction to Holomorphic Dynamics: Very Brief Notes
Introduction to Holomorphic Dynamics: Very Brief Notes Saeed Zakeri Last modified: 5-8-2003 Lecture 1. (1) A Riemann surface X is a connected complex manifold of dimension 1. This means that X is a connected Hausdorff space, locally homeomorphic to R2, which is equipped with a complex structure f(Ui; zi)g. Here each Ui is an open subset of X, S =» the union Ui is X, each local coordinate zi : Ui ¡! D is a homeomorphism, and ¡1 whenever Ui \ Uj 6= ; the change of coordinate zjzi : zi(Ui \ Uj) ! zj(Ui \ Uj) is holomorphic. In the above definition we have not assumed that X has a countable basis for its topology. But this is in fact true and follows from the existence of a complex structure (Rado’s Theorem). (2) A map f : X ! Y between Riemann surfaces is holomorphic if w ± f ± z¡1 is a holomorphic map for each pair of local coordinates z on X and w on Y for which this composition makes sense. We often denote this composition by w = f(z).A holomorphic map f is called a biholomorphism or conformal isomorphism if it is a homeomorphism, in which case f ¡1 is automatically holomorphic. (3) Examples of Riemann surfaces: The complex plane C, the unit disk D, the Riemann b sphere C, complex tori T¿ = C=(Z © ¿Z) with Im(¿) > 0, open connected subsets of Riemann surfaces such as the complement of a Cantor set in C. (4) The Uniformization Theorem: Every simply connected Riemann surface is confor- mally isomorphic to Cb, C, or D. -
Tasty Bits of Several Complex Variables
Tasty Bits of Several Complex Variables A whirlwind tour of the subject Jiríˇ Lebl October 11, 2018 (version 2.4) . 2 Typeset in LATEX. Copyright c 2014–2018 Jiríˇ Lebl License: This work is dual licensed under the Creative Commons Attribution-Noncommercial-Share Alike 4.0 International License and the Creative Commons Attribution-Share Alike 4.0 International License. To view a copy of these licenses, visit https://creativecommons.org/licenses/ by-nc-sa/4.0/ or https://creativecommons.org/licenses/by-sa/4.0/ or send a letter to Creative Commons PO Box 1866, Mountain View, CA 94042, USA. You can use, print, duplicate, share this book as much as you want. You can base your own notes on it and reuse parts if you keep the license the same. You can assume the license is either the CC-BY-NC-SA or CC-BY-SA, whichever is compatible with what you wish to do, your derivative works must use at least one of the licenses. Acknowledgements: I would like to thank Debraj Chakrabarti, Anirban Dawn, Alekzander Malcom, John Treuer, Jianou Zhang, Liz Vivas, Trevor Fancher, and students in my classes for pointing out typos/errors and helpful suggestions. During the writing of this book, the author was in part supported by NSF grant DMS-1362337. More information: See https://www.jirka.org/scv/ for more information (including contact information). The LATEX source for the book is available for possible modification and customization at github: https://github.com/jirilebl/scv . Contents Introduction 5 0.1 Motivation, single variable, and Cauchy’s formula ..................5 1 Holomorphic functions in several variables 11 1.1 Onto several variables ................................ -
Complex Manifolds and Holomorphic Vector Bundles by L
Lecture 1 { Complex Manifolds and Holomorphic Vector Bundles by L. Ni A Complex manifold M m is a smooth manifold such that it is covered by holomorphic atlas ! Cn · −1 ('i : Ui ) with local transition mappings 'i 'j being holomorphic. This clearly makes it possible to discuss holomorphic functions, mappings etc. Examples: Cm, Riemann surfaces, complex projective space Pm, etc. Implicit function theorem also leads to the concept of submanifolds (e.g. zero set of a non-critical holomorphic function). More examples via coverings/quotiens and surgeries. In general, a topological manifold can be endowed with extra structures. Hence there are analytic manifolds, algebraic manifolds, topological manifolds, PL-manifolds, etc. The fundamental problem is to classify the simplest ones in each category. One can also ask if a weaker equivalence implies a stronger equivalence, e.g. the Poincar´econjecture. Conjecture (uniqueness): A complex manifold diffeomorphic to Pm must be biholomorphic to it. There is also the famous question: is there a holomorphic structure on S6? The embedding problem, which unites the extrinsic and intrinsic geometry to some degree, is also universal. Whitney(1936): any compact differential manifold of dimension n can be smoothly embed- ded into 2n + 1-dimensional Euclidean space R2n+1. This has motivated several important developments including Nash(1956): any C1 Riemannian manifold (M n; g) can be isomet- rically embedded into some Euclidean space Rd with d = d(n). Unfortunately it fails completely for compact complex manifolds due to that any holomor- phic function on a compact complex manifold must be a constant. On the other hand in this direction there are two important results: Remmert(1956)-Narasimhan(1960)-Bishop(1961, joined UCSD afterwards): every Stein manifold M m can be holomorphically embedded into C2m+1. -
Riemann Surfaces
3 Riemann surfaces In this chapter we define and give the first properties of Riemann surfaces. These are the holomorphic counterpart of the (real) differential manifolds. We will see how the Fuchsian groups studied in the previous chapter are related to a classification of these surfaces. The main sources for this chapter will be Milnor’s "Dynamics in one complex variables" and Beardon’s "The geometry of discrete groups". 3.1 Definitions Definition 3.1.1. A holomorphic function on a subset V of C is a map f : V C such æ that the limit f(z + h) f(z) f Õ(z):=lim ≠ h 0 h æ exists for every z V . œ Definition 3.1.2. A Riemann surface is a connected complex manifold of dimension 1. This means that a Riemann Surface is a connected topological space X which is Hausdorff (i.e., for every pair of points x = y X there exist open neighbourhoods ” œ U x and U y such that U U = ) endowed with a family of charts x – y – x fl y ” ÿ (U ,Ï ): j J { j j œ } (called an atlas for X) such that 1. U : j J is an open cover of X; { j œ } 2. each Ïj is a homeomorphism of Uj onto an open subset of C; 23 24 Riemann surfaces 3. if U = U U = , then i fl j ” ÿ 1 Ï Ï≠ : Ï (U) Ï (U) ¶ j j æ i is a holomorphic map between the (open) subsets Ïj(U) and Ïi(U) of C.