DOCSLIB.ORG

Covering Spaces, Uniformization and Picard Theorems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Covering Spaces, Uniformization and Picard Theorems Covering Spaces, Uniformization and Picard Theorems Complex Analysis - Spring 2017 - Dupuy Abstract The purpose of this note to develop enough covering space theory to allow us to talk about the covering space proof of the Little Picard Theorem. ♠♠♠ Taylor: [add references] 1 Fundamental Groups Definition 1.1. Let (X; x0) be a pointed topological space. Exercise 1. 1. Check that π1(X; x0) is a group. 2. Assume that X is a path connected topological space. Show that π1(X; x0) ∼= π1(X; x1) as groups. Given f :(X; x ) (Y; y ) we get an induced map of fundamental groups 0 ! 0 f∗ : π (X; x ) π (Y; y ) 1 0 ! 1 0 f∗([γ]) = [f γ] ◦ where [γ] denotes the homotopy class of the loop γ : [0; 1] X. ! Example 1.2. 1. Let D be the unit disc. We have π (D 0 ; 1=2) = Z. 1 n f g ∼ 2. Let S1 = z C : z = 1 = @D. We have π (S1; 1) = Z. f 2 j j 1 The above example shows homotopy invariance of π1. We have that D 0 S1. nf g ∼ Example 1.3. Let X be the figure eight. We have π1(X; x0) = Z Z = α; β = 1 ∼ ∗ h g F2 the free group on two generators. The figure eight is an example of a \smash product". The figure eight is the smash product of S1 with itself. A smash product is where you take two topological spaces and identify two points between them. 1 G1 ∗ G2 is the free product of two groups. It is just the word build out of elements from G1 and G2 subject to the obvious relations. 1 Exercise 2. Show that the fundamental group of a smash product is the free product of the fundamental groups. 1 Example 1.4. 1. π1(P ; x0) = 1 (it doesn't matter what the base point is). 2. π (P1 0 ; x ). Note that is doesn't matter which point we remove and 1 n f g 0 we have P1 a = P1 = C. n f g ∼ n f1g ∼ 3. π (P1 0; 1 ; x ) = π (P1 ; 0 ; x ) = π (C 0 ; x ) = Z. 1 n f g 0 1 n f1 g 0 1 n f g 0 ∼ 4. π (P1 0; 1; ; x ) = Z Z. 1 n f 1g 0 ∼ ∗ What about an elliptic curve? Example 1.5. Let Λ = !1Z !2Z. Recall the fundamental domain for C=Λ. Consider our basepoint C=Λ ⊕z = 0 ! ! ! + ! . Let α be the path 3 0 ≡ 1 ≡ 2 ≡ 1 2 from 0 to !1 and β be the path from 0 to !2 Any loop can be pushed to the boundary we see that any element is generated by α and β. We can also see from the fundamental domain that αβα−1β−1 = 1 and hence that αβ = βα. This show that the group is commutative. It remains to show that α and β are not homotopic. We can try to work this out but we won't. Instead we use that C=Λ = S1 S1 as topological spaces and then use ∼ × that π (S1 S1; (x ; x )) = π (S1; x ) π (S1; x ) = Z Z. 2 1 × 1 2 ∼ 1 1 × 1 2 ∼ × To completely justify this we need the following exercise. Exercise 3. 1. Let γ; γ0 : J X Y be a morphisms of topological spaces. Write γ(t) = (α(t); β(t)) and! γ×0(t) = (α0(t); β0(t)) where α; α0 : J X and β; β0 : J Y are the maps obtained by projecting γ; γ0 to X and! Y respectively.! Show that γ γ0 α α0 and β β0: ∼ () ∼ ∼ 2. Show that the map π (X Y; (x ; y )) π (X; x ) π (Y; y ) given by 1 × 0 0 ! 1 0 × 1 0 [(α; β)] ([α0; β0]) 7! is a well-defined, bijective group homomorphism. 2Warning: Although it is true that for every lattice Λ ⊂ C we have C=Λ =∼ S1 × S1 as topological spaces we do not have C=Λ1 =∼ C=Λ2 as Riemann Surfaces for Λ1 6= Λ2 as general 3 3 3 topological spaces. The j-invariant, j = 1728g2(Λ) =∆(Λ) where ∆(Λ) = g2(Λ) − 27g2(Λ) determines whether we have isomorphic Riemann surfaces. 2 2 Covering spaces In this section we work with Manifolds. Definition 2.1. A covering map of X is a surjective p : Y X such that for every x X there exist some U x open with ! 2 3 −1 a p (U) = Vi i2I with Vi Y open such that p V : Vi U is a homeomorphism. ⊂ j i ! Morphism of covering spaces: Let p1 : Y1 X and p1 : Y2 X be covering maps. A morphisms of coverings of X is a! morphisms of topological! spaces f : Y Y such that the following diagram commutes: 1 ! 2 f Y1 / Y2 : p1 p2 ~ X The collection of objects and morphisms allow us to define a category Cov(X) of coverings of X. We denote the category of coverings of X by Cov(X): We denote the category of connected coverings by Cov(X)0: Definition 2.2. For a pointed topological space we define Cov(X; x0) and 0 Cov(X; x0) in the same way only using morphisms of pointed topological spaces. Remark 2.3. The space Y in the definition above is a covering space and we will use the term covering space and covering map interchangably understanding that covering spaces come with covering maps. Definition 2.4. An automorphism of a covering is called a deck transforma- tion. For a cover X Y the group of deck transformations are denoted by G(Y=X). ! Definition 2.5. A universal covering space is an initial object of Cov(X)0 [unique up to isomorphism]. Spelled-out version of universal covering space: suppose that Xe is a universal cover of X. This means for every cover Y X there exist some map Xe Y such that ! ! Xe / Y : X 3 Definition 2.6. Let p : Y X be a covering of topological spaces (or pointed topological spaces). Consider! a map γ : Z X of topological spaces (I can be anything here). A lift of γ is a map γ such! that p γ = γ. e ◦ e Lemma 2.7 (Path Lifting). Let p :(Y; y ) (X; x ) be a covering space. 0 ! 0 For any γ : [0; 1] (X; x0) such that γ(0) = x0) there exists a unique lifting γ : [0; 1] Y such! that e ! 1. γe is a lift. 2. γe(0) = y0 Sn Proof. Since [0; 1] is compact we can assume that γ([0; 1]) i=1 Ui X with −1 ` ⊂ ⊂ p (Ui) = j2Ji Vi;j local trivializations. We perform induction on n. If n = 1 then let V1;j1 be the unique open set containing y0. The lifting • of the path follows from the isomorphism V U . 1;j1 ∼= 1 Suppose now the theorem holds for covers containing m < n open sets. • Break γ into parts γ = γ1 γ2 such that γ1([0; 1]) and γ2([0; 1]) are con- · 3 tained in a union of fewer than n elements of = Ui : 1 i n . By U f ≤ ≤ g inductive hypothesis, γ1 has a unique lift γe1 with γe1(0) = y0. By induc- tive hypothesis, γ2 has a unique lift γe2 with γe2(0) = γe1(1). The conjuction γ := γ γ works. e e1 · e2 Lemma 2.8. Let Z = [0; 1]. Let p : Y X be a covering. Let γ : Z X be a map with a lift γ : Z Y . Any homotopy! H : [0; 1] Z X lifts to! a unique e ! × ! homotopy He : [0; 1] Z Y . × ! Proof. First we prove the uniqueness statement in the theorem. Suppose • He1; He2 : [0; 1] Z Y are two lifts of H : [0; 1] Z X with He1(0; z) = × ! × ! He2(0; z) = γ(z). Then for each z0 Z, and each i 1; 2 , the function e 2 2 f g Hei(t; z0) is a path lifting of H(t; z0) with Hei(0; z0) = γe(z0) the lifted point. By uniqueness of path lifting we have He1(t; z0) = He2(t; z0). Since I Z is compact there exists some n and some = Ui : 1 i n • × U f Sn ≤ ≤ g a collection of trivializing open sets such that H(I Z) i=1 Ui X; −1 ` × ⊂ ⊂ where the Ui are trivializing: p (Ui) = Vi;j where Ji is some index ∼ j2Ji set. Let's suppose in addition that such a collection is minimal. We perform induction on n. Base case of induction: Suppose n = 1. Then U = V ;j for some unique • 1 ∼ 1 1 V1;j1 y0 (we are just using something here to pin down the sheet in the cover3 that we want to lift to). We use this isomorphism to transport the homotopy. 3 If no such break point exists then for every subinterval [a; b] we have γ([a; b]) touching every one of these Ui's. Letting a ! b we get a discontinuity which is a contradiction. 4 y0 Y y1 p x0 X Figure 1: Here is a cover of a surface with three holes to a surface with holes. The map here is degree two. Note that loops on the bottom do not necessarily lift to loops on the top. To perform the inductive step we must suppose there exists a partition • [ [0; 1] Z = [0; 1] [0; 1] = [ti; ti ] [zj; zj ] (2.9) × × +1 × +1 1≤i≤r;1≤j≤s such that H([ti; ti+1] [zj; zj+1])intersects a proper subset of the Ui : 1 i n .
Load more

Recommended publications
  • 1.6 Covering Spaces 1300Y Geometry and Topology
    1.6 Covering spaces 1300Y Geometry and Topology 1.6 Covering spaces Consider the fundamental group π1(X; x0) of a pointed space. It is natural to expect that the group theory of π1(X; x0) might be understood geometrically. For example, subgroups may correspond to images of induced maps ι∗π1(Y; y0) −! π1(X; x0) from continuous maps of pointed spaces (Y; y0) −! (X; x0). For this induced map to be an injection we would need to be able to lift homotopies in X to homotopies in Y . Rather than consider a huge category of possible spaces mapping to X, we restrict ourselves to a category of covering spaces, and we show that under some mild conditions on X, this category completely encodes the group theory of the fundamental group. Definition 9. A covering map of topological spaces p : X~ −! X is a continuous map such that there S −1 exists an open cover X = α Uα such that p (Uα) is a disjoint union of open sets (called sheets), each homeomorphic via p with Uα. We then refer to (X;~ p) (or simply X~, abusing notation) as a covering space of X. Let (X~i; pi); i = 1; 2 be covering spaces of X. A morphism of covering spaces is a covering map φ : X~1 −! X~2 such that the diagram commutes: φ X~1 / X~2 AA } AA }} p1 AA }}p2 A ~}} X We will be considering covering maps of pointed spaces p :(X;~ x~0) −! (X; x0), and pointed morphisms between them, which are defined in the obvious fashion.
    [Show full text]
  • Riemann Surfaces
    RIEMANN SURFACES AARON LANDESMAN CONTENTS 1. Introduction 2 2. Maps of Riemann Surfaces 4 2.1. Defining the maps 4 2.2. The multiplicity of a map 4 2.3. Ramification Loci of maps 6 2.4. Applications 6 3. Properness 9 3.1. Definition of properness 9 3.2. Basic properties of proper morphisms 9 3.3. Constancy of degree of a map 10 4. Examples of Proper Maps of Riemann Surfaces 13 5. Riemann-Hurwitz 15 5.1. Statement of Riemann-Hurwitz 15 5.2. Applications 15 6. Automorphisms of Riemann Surfaces of genus ≥ 2 18 6.1. Statement of the bound 18 6.2. Proving the bound 18 6.3. We rule out g(Y) > 1 20 6.4. We rule out g(Y) = 1 20 6.5. We rule out g(Y) = 0, n ≥ 5 20 6.6. We rule out g(Y) = 0, n = 4 20 6.7. We rule out g(C0) = 0, n = 3 20 6.8. 21 7. Automorphisms in low genus 0 and 1 22 7.1. Genus 0 22 7.2. Genus 1 22 7.3. Example in Genus 3 23 Appendix A. Proof of Riemann Hurwitz 25 Appendix B. Quotients of Riemann surfaces by automorphisms 29 References 31 1 2 AARON LANDESMAN 1. INTRODUCTION In this course, we’ll discuss the theory of Riemann surfaces. Rie- mann surfaces are a beautiful breeding ground for ideas from many areas of math. In this way they connect seemingly disjoint fields, and also allow one to use tools from different areas of math to study them.
    [Show full text]
  • Real Proofs of Complex Theorems (And Vice Versa)
    REAL PROOFS OF COMPLEX THEOREMS (AND VICE VERSA) LAWRENCE ZALCMAN Introduction. It has become fashionable recently to argue that real and complex variables should be taught together as a unified curriculum in analysis. Now this is hardly a novel idea, as a quick perusal of Whittaker and Watson's Course of Modern Analysis or either Littlewood's or Titchmarsh's Theory of Functions (not to mention any number of cours d'analyse of the nineteenth or twentieth century) will indicate. And, while some persuasive arguments can be advanced in favor of this approach, it is by no means obvious that the advantages outweigh the disadvantages or, for that matter, that a unified treatment offers any substantial benefit to the student. What is obvious is that the two subjects do interact, and interact substantially, often in a surprising fashion. These points of tangency present an instructor the opportunity to pose (and answer) natural and important questions on basic material by applying real analysis to complex function theory, and vice versa. This article is devoted to several such applications. My own experience in teaching suggests that the subject matter discussed below is particularly well-suited for presentation in a year-long first graduate course in complex analysis. While most of this material is (perhaps by definition) well known to the experts, it is not, unfortunately, a part of the common culture of professional mathematicians. In fact, several of the examples arose in response to questions from friends and colleagues. The mathematics involved is too pretty to be the private preserve of specialists.
    [Show full text]
  • Uniformization of Riemann Surfaces Revisited
    UNIFORMIZATION OF RIEMANN SURFACES REVISITED CIPRIANA ANGHEL AND RARES¸STAN ABSTRACT. We give an elementary and self-contained proof of the uniformization theorem for non-compact simply-connected Riemann surfaces. 1. INTRODUCTION Paul Koebe and shortly thereafter Henri Poincare´ are credited with having proved in 1907 the famous uniformization theorem for Riemann surfaces, arguably the single most important result in the whole theory of analytic functions of one complex variable. This theorem generated con- nections between different areas and lead to the development of new fields of mathematics. After Koebe, many proofs of the uniformization theorem were proposed, all of them relying on a large body of topological and analytical prerequisites. Modern authors [6], [7] use sheaf cohomol- ogy, the Runge approximation theorem, elliptic regularity for the Laplacian, and rather strong results about the vanishing of the first cohomology group of noncompact surfaces. A more re- cent proof with analytic flavour appears in Donaldson [5], again relying on many strong results, including the Riemann-Roch theorem, the topological classification of compact surfaces, Dol- beault cohomology and the Hodge decomposition. In fact, one can hardly find in the literature a self-contained proof of the uniformization theorem of reasonable length and complexity. Our goal here is to give such a minimalistic proof. Recall that a Riemann surface is a connected complex manifold of dimension 1, i.e., a connected Hausdorff topological space locally homeomorphic to C, endowed with a holomorphic atlas. Uniformization theorem (Koebe [9], Poincare´ [15]). Any simply-connected Riemann surface is biholomorphic to either the complex plane C, the open unit disk D, or the Riemann sphere C^.
    [Show full text]
  • Uniformization of Riemann Surfaces
    CHAPTER 3 Uniformization of Riemann surfaces 3.1 The Dirichlet Problem on Riemann surfaces 128 3.2 Uniformization of simply connected Riemann surfaces 141 3.3 Uniformization of Riemann surfaces and Kleinian groups 148 3.4 Hyperbolic Geometry, Fuchsian Groups and Hurwitz’s Theorem 162 3.5 Moduli of Riemann surfaces 178 127 128 3. UNIFORMIZATION OF RIEMANN SURFACES One of the most important results in the area of Riemann surfaces is the Uni- formization theorem, which classifies all simply connected surfaces up to biholomor- phisms. In this chapter, after a technical section on the Dirichlet problem (solutions of equations involving the Laplacian operator), we prove that theorem. It turns out that there are very few simply connected surfaces: the Riemann sphere, the complex plane and the unit disc. We use this result in 3.2 to give a general formulation of the Uniformization theorem and obtain some consequences, like the classification of all surfaces with abelian fundamental group. We will see that most surfaces have the unit disc as their universal covering space, these surfaces are the object of our study in 3.3 and 3.5; we cover some basic properties of the Riemaniann geometry, §§ automorphisms, Kleinian groups and the problem of moduli. 3.1. The Dirichlet Problem on Riemann surfaces In this section we recall some result from Complex Analysis that some readers might not be familiar with. More precisely, we solve the Dirichlet problem; that is, to find a harmonic function on a domain with given boundary values. This will be used in the next section when we classify all simply connected Riemann surfaces.
    [Show full text]
  • Lecture 34: Hadamaard's Theorem and Covering Spaces
    Lecture 34: Hadamaard’s Theorem and Covering spaces In the next few lectures, we’ll prove the following statement: Theorem 34.1. Let (M,g) be a complete Riemannian manifold. Assume that for every p M and for every x, y T M, we have ∈ ∈ p K(x, y) 0. ≤ Then for any p M,theexponentialmap ∈ exp : T M M p p → is a local diffeomorphism, and a covering map. 1. Complete Riemannian manifolds Recall that a connected Riemannian manifold (M,g)iscomplete if the geodesic equation has a solution for all time t.Thisprecludesexamplessuch as R2 0 with the standard Euclidean metric. −{ } 2. Covering spaces This, not everybody is familiar with, so let me give a quick introduction to covering spaces. Definition 34.2. Let p : M˜ M be a continuous map between topological spaces. p is said to be a covering→ map if (1) p is a local homeomorphism, and (2) p is evenly covered—that is, for every x M,thereexistsaneighbor- 1 ∈ hood U M so that p− (U)isadisjointunionofspacesU˜ for which ⊂ α p ˜ is a homeomorphism onto U. |Uα Example 34.3. Here are some standard examples: (1) The identity map M M is a covering map. (2) The obvious map M →... M M is a covering map. → (3) The exponential map t exp(it)fromR to S1 is a covering map. First, it is clearly a local → homeomorphism—a small enough open in- terval in R is mapped homeomorphically onto an open interval in S1. 113 As for the evenly covered property—if you choose a proper open in- terval inside S1, then its preimage under the exponential map is a disjoint union of open intervals in R,allhomeomorphictotheopen interval in S1 via the exponential map.
    [Show full text]
  • COVERING SPACES, GRAPHS, and GROUPS Contents 1. Covering
    COVERING SPACES, GRAPHS, AND GROUPS CARSON COLLINS Abstract. We introduce the theory of covering spaces, with emphasis on explaining the Galois correspondence of covering spaces and the deck trans- formation group. We focus especially on the topological properties of Cayley graphs and the information these can give us about their corresponding groups. At the end of the paper, we apply our results in topology to prove a difficult theorem on free groups. Contents 1. Covering Spaces 1 2. The Fundamental Group 5 3. Lifts 6 4. The Universal Covering Space 9 5. The Deck Transformation Group 12 6. The Fundamental Group of a Cayley Graph 16 7. The Nielsen-Schreier Theorem 19 Acknowledgments 20 References 20 1. Covering Spaces The aim of this paper is to introduce the theory of covering spaces in algebraic topology and demonstrate a few of its applications to group theory using graphs. The exposition assumes that the reader is already familiar with basic topological terms, and roughly follows Chapter 1 of Allen Hatcher's Algebraic Topology. We begin by introducing the covering space, which will be the main focus of this paper. Definition 1.1. A covering space of X is a space Xe (also called the covering space) equipped with a continuous, surjective map p : Xe ! X (called the covering map) which is a local homeomorphism. Specifically, for every point x 2 X there is some open neighborhood U of x such that p−1(U) is the union of disjoint open subsets Vλ of Xe, such that the restriction pjVλ for each Vλ is a homeomorphism onto U.
    [Show full text]
  • Generalized Covering Spaces and the Galois Fundamental Group
    GENERALIZED COVERING SPACES AND THE GALOIS FUNDAMENTAL GROUP CHRISTIAN KLEVDAL Contents 1. Introduction 1 2. A Category of Covers 2 3. Uniform Spaces and Topological Groups 6 4. Galois Theory of Semicovers 9 5. Functoriality of the Galois Fundamental Group 15 6. Universal Covers 16 7. The Topologized Fundamental Group 17 8. Covers of the Earring 21 9. The Galois Fundamental Group of the Harmonic Archipelago 22 10. The Galois Fundamental Group of the Earring 23 References 24 Gal Abstract. We introduce a functor π1 from the category of based, connected, locally path connected spaces to the category of complete topological groups. We then compare this groups to the fundamental group. In particular, we show that there is a topological σ Gal group π1 (X; x) whose underlying group is π1(X; x) so that π1 (X; x) is the completion σ of π1 (X; x). 1. Introduction Gal In this paper, we give define a topological group π1 (X; x) called the Galois funda- mental group, which is associated to any based, connected, locally path connected space X. As the notation suggests, this group is intimately related to the fundamental group π1(X; x). The motivation of the Galois fundamental group is to look at automorphisms of certain covers of a space instead of homotopy classes of loops in the space. It is well known that if X has a universal cover, then π1(X; x) is isomorphic to the group of deck transformations of the universal cover. However, this approach breaks down if X is not semi locally simply connected.
    [Show full text]
  • Chapter 9 Partitions of Unity, Covering Maps ~
    Chapter 9 Partitions of Unity, Covering Maps ~ 9.1 Partitions of Unity To study manifolds, it is often necessary to construct var- ious objects such as functions, vector fields, Riemannian metrics, volume forms, etc., by gluing together items con- structed on the domains of charts. Partitions of unity are a crucial technical tool in this glu- ing process. 505 506 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ The first step is to define “bump functions”(alsocalled plateau functions). For any r>0, we denote by B(r) the open ball n 2 2 B(r)= (x1,...,xn) R x + + x <r , { 2 | 1 ··· n } n 2 2 and by B(r)= (x1,...,xn) R x1 + + xn r , its closure. { 2 | ··· } Given a topological space, X,foranyfunction, f : X R,thesupport of f,denotedsuppf,isthe closed set! supp f = x X f(x) =0 . { 2 | 6 } 9.1. PARTITIONS OF UNITY 507 Proposition 9.1. There is a smooth function, b: Rn R, so that ! 1 if x B(1) b(x)= 2 0 if x Rn B(2). ⇢ 2 − See Figures 9.1 and 9.2. 1 0.8 0.6 0.4 0.2 K3 K2 K1 0 1 2 3 Figure 9.1: The graph of b: R R used in Proposition 9.1. ! 508 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ > Figure 9.2: The graph of b: R2 R used in Proposition 9.1. ! Proposition 9.1 yields the following useful technical result: 9.1. PARTITIONS OF UNITY 509 Proposition 9.2. Let M be a smooth manifold.
    [Show full text]
  • Surfaces and Complex Analysis (Lecture 34)
    Surfaces and Complex Analysis (Lecture 34) May 3, 2009 Let Σ be a smooth surface. We have seen that Σ admits a conformal structure (which is unique up to a contractible space of choices). If Σ is oriented, then a conformal structure on Σ allows us to view Σ as a Riemann surface: that is, as a 1-dimensional complex manifold. In this lecture, we will exploit this fact together with the following important fact from complex analysis: Theorem 1 (Riemann uniformization). Let Σ be a simply connected Riemann surface. Then Σ is biholo- morphic to one of the following: (i) The Riemann sphere CP1. (ii) The complex plane C. (iii) The open unit disk D = fz 2 C : z < 1g. If Σ is an arbitrary surface, then we can choose a conformal structure on Σ. The universal cover Σb then inherits the structure of a simply connected Riemann surface, which falls into the classification of Theorem 1. We can then recover Σ as the quotient Σb=Γ, where Γ ' π1Σ is a group which acts freely on Σb by holmorphic maps (if Σ is orientable) or holomorphic and antiholomorphic maps (if Σ is nonorientable). For simplicity, we will consider only the orientable case. 1 If Σb ' CP , then the group Γ must be trivial: every orientation preserving automorphism of S2 has a fixed point (by the Lefschetz trace formula). Because Γ acts freely, we must have Γ ' 0, so that Σ ' S2. To see what happens in the other two cases, we need to understand the holomorphic automorphisms of C and D.
    [Show full text]
  • TEICHM¨ULLER SPACE Contents 1. Introduction 1 2. Quasiconformal
    TEICHMULLER¨ SPACE MATTHEW WOOLF Abstract. It is a well-known fact that every Riemann surface with negative Euler characteristic admits a hyperbolic metric. But this metric is by no means unique { indeed, there are uncountably many such metrics. In this paper, we study the space of all such hyperbolic structures on a Riemann surface, called the Teichm¨ullerspace of the surface. We will show that it is a complete metric space, and that it is homeomorphic to Euclidean space. Contents 1. Introduction 1 2. Quasiconformal maps 2 3. Beltrami Forms 3 4. Teichm¨ullerSpace 4 5. Fenchel-Nielsen Coordinates 6 References 9 1. Introduction Much of the theory of Riemann surfaces boils down to the following theorem, the two-dimensional equivalent of Thurston's geometrization conjecture: Proposition 1.1 (Uniformization theorem). Every simply connected Riemann sur- face is isomorphic either to the complex plane C, the Riemann sphere P1, or the unit disk, D. By considering universal covers of Riemann surfaces, we can see that every sur- face admits a spherical, Euclidean, or (this is the case for all but a few surfaces) hyperbolic metric. Since almost all surfaces are hyperbolic, we will restrict our attention in the following material to them. The natural question to ask next is whether this metric is unique. We see almost immediately that the answer is no (almost any change of the fundamental region of a surface will give rise to a new metric), but this answer gives rise to a new question. Does the set of such hyper- bolic structures of a given surface have any structure itself? It turns out that the answer is yes { in fact, in some ways, the structure of this set (called the Teichm¨uller space of the surface) is more interesting than that of the Riemann surface itself.
    [Show full text]
  • The Uniformization Theorem Donald E. Marshall the Koebe Uniformization Theorem Is a Generalization of the Riemann Mapping The- O
    The Uniformization Theorem Donald E. Marshall The Koebe uniformization theorem is a generalization of the Riemann mapping The- orem. It says that a simply connected Riemann surface is conformally equivalent to either the unit disk D, the plane C, or the sphere C∗. We will give a proof that illustrates the power of the Perron method. As is standard, the hyperbolic case (D) is proved by constructing Green’s function. The novel part here is that the non-hyperbolic cases are treated in a very similar manner by constructing the Dipole Green’s function. A Riemann surface is a connected Hausdorff space W , together with a collection of open subsets Uα ⊂ W and functions zα : Uα → C such that (i) W = ∪Uα (ii) zα is a homeomorphism of Uα onto the unit disk D, and −1 (iii) if Uα ∩ Uβ = ∅ then zβ ◦ zα is analytic on zα(Uα ∩ Uβ). The functions zα are called coordinate functions, and the sets Uα are called coordinate disks. We can extend our collection of coordinate functions and disks to form a base for −1 D 1 the topology on W by setting Uα,r = zα (r ) and zα,r = r zα|Uα,r , for r < 1. We can also compose each zα with a M¨obius transformation of the disk onto itself so that we can assume that for each coordinate disk Uα and each p0 ∈ Uα there is a coordinate function zα with zα(p0) = 0. For the purposes of the discussion below, we will assume that our collection {zα,Uα} includes all such maps, except that we will further assume that each Uα has compact closure in W , by restricting our collection.
    [Show full text]