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Covering Spaces for Domains in the Complex Plane Covering Spaces for Domains in the Complex Plane Hjörtur Björnsson FacultyFaculty of of Physical Physical Sciences Sciences UniversityUniversity of of Iceland Iceland 20202020 COVERING SPACES FOR DOMAINS IN THE COMPLEX PLANE Hjörtur Björnsson 60 ECTS thesis submitted in partial fulfillment of a Magister Scientiarum degree in Mathematics Advisor Reynir Axelsson Faculty Representative Ragnar Sigurðsson M.Sc. committee Reynir Axelsson Jón Ingólfur Magnússon Faculty of Physical Sciences School of Engineering and Natural Sciences University of Iceland Reykjavík, May 2020 Covering Spaces for Domains in the Complex Plane 60 ECTS thesis submitted in partial fulfillment of a M.Sc. degree in Mathematics Copyright c 2020 Hjörtur Björnsson All rights reserved Faculty of Physical Sciences School of Engineering and Natural Sciences University of Iceland Dunhagi 5 IS-107, Reykjavík, Reykjavík Iceland Telephone: 525 4000 Bibliographic information: Hjörtur Björnsson, 2020, Covering Spaces for Domains in the Complex Plane, M.Sc. thesis, Faculty of Physical Sciences, University of Iceland. Printing: Háskólaprent, Fálkagata 2, 107 Reykjavík Reykjavík, Iceland, May 2020 For Stefania Abstract In this thesis we present a modern version of a proof published by Tibor Radó in 1922, which shows that the universal covering space for a domain G in Cnf0; 1g is bi- holomorphically equivalent to the unit disk. Radó uses multivalued functions which we replace by mappings on appropriate covering spaces. We use Riemann domains for this purpose, which are essentially Riemann surfaces with global coordinates. Útdráttur Í þessari ritgerð fjöllum við um nútímalega útgáfu af sönnun sem Tibor Radó gaf út árið 1922. Hún sýnir að allsherjarþekjurúm sérhvers svæðis G í C n f0; 1g er fágað einsmóta einingarhringskífunni. Radó notaðist við marggild föll í sinni grein, en við skoðum í staðinn varpanir á viðeigandi þekjurúmum. Við notumst við Riemann-svæði, en þau eru sértilfelli af almennum Riemann-flötum, þar sem við höfum víðfeðm hnitaföll. v Contents Þakkarorð ix 1. Introduction1 1.1. Background . .2 1.1.1. The Uniformization Theorem . .2 1.1.2. Parametrization of Algebraic Curves . .3 1.1.3. Classification of Riemann Surfaces . .4 1.2. Proof of the Riemann Mapping Theorem . .6 1.3. Picard’s Little Theorem . .8 2. Preliminaries9 2.1. Topological Preliminaries . .9 2.1.1. Local Homeomorphisms and Sections . .9 2.1.2. Chains and Step Polygons . 11 2.2. Riemann Domains . 19 3. Uniformization Theorem for Domains in C 25 3.1. The Set Σ(G; y0) ............................. 26 3.2. Constructing a Surjective Function . 29 A. Appendix 33 A.1. Path Homotopy and the Fundamental Group . 33 A.2. Covering Spaces . 35 Bibliography 37 vii Þakkarorð Ég vil þakka Reyni Axelssyni og Jón Ingólfi Magnússyni, fyrir alla þá hjálp sem þeir hafa gefið mér við vinnslu þessarar ritgerðar. Ég vil einnig sérstaklega þakka Sigurði Frey Hafstein fyrir þann mikla stuðning og þolinmæði sem hann hefur sýnt mér í gegnum árin. Síðast en ekki síst vil ég þakka Stefaniu Crotti, grazie di cuore. ix 1. Introduction In 1922 Radó published a proof of the famous Riemann mapping theorem [10], by L.Fejér and F.Riesz. Theorem 1.0.1 (Riemann mapping theorem). Let G ⊂ C be a simply connected domain which is not all of C. Then there exists a biholomorphic map f : G ! E, where E is the open unit disk fz 2 C j jzj < 1g. Furthermore we can, for any p 2 G, choose f in such a way that f(p) = 0, and the map f is then uniquely determined up to a rotation. Their approach was a new one: they showed that the Riemann mapping theorem could be obtained as a solution of an extremal problem. The argument was very short, only about one full page. Many modern textbooks that contain a proof of the Riemann mapping theorem use similar methods, although modernized. The proof is presented in section 1.2. In his paper, Radó goes on to prove the following theorem: Theorem 1.0.2. Let G be a domain in C, such that the complement of G contains at least two points. Then there exists a holomorphic covering map p: E ! G. Radó uses a similar idea as Fejér and Riesz used to prove Theorem 1.0.1. He starts by considering a certain class G of multivalued holomorphic functions from G to E which is defined in such a way the inverse of any function f 2 G is a well defined single valued holomorphic function from the image of f in E to G. By picking a specific solution to an extremal problem Radó manages to construct a multivalued holomorphic surjection f : G ! E, whose inverse is a well defined holomorphic covering map f −1 : E ! G. In this thesis we present a more modern version of Radó’s proof. Instead of multival- ued functions on G, we consider well defined holomorphic functions on appropriate covering spaces of G, satisfying the condition that they are injective with respect to the covering map (see Definition 2.2.4). This condition guarantees that a certain 1 1. Introduction ‘inverse’ of these functions is a well defined holomorphic map (see Lemma 2.2.5). Using the same idea as Radó, we prove Theorem 3.0.1, which says that for any domain G in C, there exists a holomorphic covering map p: Y ! G, where Y is biholomorphically equivalent to either C or E. The end result is an argument pleas- antly similar to a method used to prove the Riemann mapping theorem in many modern textbooks, albeit much longer. In section 1.1 we give a brief exposition on the Uniformization theorem and its consequences, and how it relates to our thesis. In section 1.2 we present Radó’s original argument to prove the Riemann mapping theorem, and we end this chapter by showing in section 1.3 how Picard’s little theorem is a simple corollary of the main result of this thesis. Chapter 2 presents some topological preliminaries and our main tool, the Riemann domains, and finally, in chapter 3 we have the main proof of our thesis. We have also included appendix A which contains some basic definitions and lemmas on homotopy and covering spaces. 1.1. Background The main theorem of this thesis, Theorem 3:0:1, mentioned in the introduction, can be viewed as a special case of the famous Uniformization theorem. The theorem, stated in the next section deals with general Riemann surfaces, but in this thesis we are working with domains of C. We give an overview of the theorem and its consequences with simplified steps to reduce unnecessary complexities that would be outside the scope of this thesis. 1.1.1. The Uniformization Theorem The Uniformization theorem is simple to state, but it took over a century for math- ematicians to formulate it and give a convincing proof. It wasn’t until the year 1907 that Henri Poincaré and Paul Koebe gave what is considered the first proofs of the theorem. The book "Uniformization of Riemann Surfaces: Revisiting a hundred- year-old theorem"[3] gives a detailed account of the very interesting history of the Uniformization theorem and presents proofs of various versions of it, up to the final form stated here below. Theorem 1.1.1 (Uniformization theorem). Every simply connected Riemann sur- face is biholomorphically equivalent to the Riemann sphere, the complex plane, or the unit disc. 2 1.1. Background The modern version of the Uniformization theorem is a classification result, but historically uniformization is about parametrization. The theorem arises from the study of algebraic curves and differential equations. In the next two subsections we give a brief description of parametrization of algebraic curves as a motivation, and classification of Riemann surfaces by the Uniformization theorem. 1.1.2. Parametrization of Algebraic Curves To explain the connection between the Uniformization theorem and parametrization, we give a simplified description of a certain class of algebraic curves called elliptic curves. An elliptic curve is a cubic polynomial of the form y2 = x3 + ax + b where a,b are complex numbers such that 4a3 + 27b2 6= 0. We want to parametrize this elliptic curve, and for the purpose let us assume that there exists a function z 7! (x(z); y(z)) = (f(z); f 0(z)). Then f must satisfy the differential equation (f 0)2 = f 3 + af + b: We can ‘solve’ this differential equation formally with the integrals Z df Z dx z = = p : pf 3 + af + b x3 + ax + b This type of integral is know as an elliptic integral, since it arises when evaluating the arc length of ellipses. The integral depends not only on which branch of the square root we use, but also on the path of integration. In this sense we can regard z as multivalued function of x. The key idea to obtain the parametrization is that we look at the inverse of this multivalued function, which, as turns out, gives us x as a single valued periodic function of z, and therefore we have parametrized the elliptic curve. The above calculations can be made precise [6, Ch. 10.6]. This idea brings us to the topic of doubly periodic meromorphic functions. One example is the Weierstrass } function which is meromorphic on C with periods 1 and i, that is to say }(z + 1) = }(z) and }(z + i) = }(z) for all z, and has double poles at every point on the lattice Λ = fm + in j m; n 2 Zg, see [6, Ch. 10.6]. This function is given by the series 1 X 1 1 }(z) = + − ; z2 (z − m − ni)2 (m + ni)2 m;n2Z (m;n)6=(0;0) and there are complex valued constants C1;C2 2 C such that 0 2 3 (} (z)) = 4 (}(z)) − C1}(z) − C2 3 1.
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