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VISUALIZING THE RIEMANN MAPPING A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree Master of Science In Mathematics By Vrej Mikaelian 2014 SIGNATURE PAGE THESIS: VISUALIZING THE RIEMANN MAPPING AUTHOR: Vrej Mikaelian DATE SUBMITTED: Spring 2014 Department of Mathematics and Statistics Dr. John Arlo Caine Thesis Committee Chair Mathematics & Statistics Dr. Mitsuo Kobayashi Mathematics & Statistics Dr. John Rock Mathematics & Statistics ii ACKNOWLEDGMENTS I would like to express my sincere gratitude to my advisor Dr. John Arlo Caine for his enthusiasm, immense knowledge, and support in completing this work. I could not have imagined having a better advisor and mentor for my thesis. I would also like to thank the rest of my thesis committee: Dr. Kobayahsi and Dr. Rock, for giving me the knowledge and skills to complete my graduate coursework. I am also grateful to my professors: Dr. Michael Green, Dr. Patricia Hale, Dr. Amber Rosin, and Dr. Robin Todd Wilson for their help to improve while working as a Graduate Teaching Associate and support in finding a full time job a year before completing my graduate coursework. I also want to thank my beautiful wife, Katya. She supported me, helped me to generate nice pictures, loved me, and brought up to the world a beautiful little baby daughter for me during my graduate school years. Lastly, and most importantly, I wish to thank my grandmother who raised me, and my grandfather who taught me and planted in me the idea to pursue graduate studies in mathematics during my childhood years. To their memory I dedicate this thesis. - Vrej Mikaelian iii ABSTRACT The Riemann Mapping Theorem states that any proper simply connected open subset of the complex plane is conformally equivalent to the unit disk. This is an amazing result from complex analysis first proved by Riemann in his 1851 PhD thesis. Since then, several other mathematicians have introduced different, or more compact, versions of the proof of the theorem. This project concerns a visual approach to the proof of the Riemann Mapping Theorem via transformations of circle packings. In chapter 1 we explain the connection between transformations of circle packings and conformal maps. In chapter 2 we illustrate the proof of the theorem due to Riesz. In 1985 Thurston proposed a scheme for constructing the Riemann map as a limit of transformations of circle packings. We explain Thurston’s conjecture and the proof by Rodin-Sullivan in chapter 3. iv List of Figures 1.1 A Riemann mapping. .......................... 2 1.2 A sequence of finite Riemann mappings under fpzq “ z3 : ...... 8 z´i 1.3 A sequence of finite Riemann mappings under fpzq “ z`i : ..... 9 2.1 A classical conformal mapping. .................... 12 3.1 Circle packings of Ω and D. ...................... 21 3.2 Two circle packings and their carriers. 22 3.3 Angle distortions under f: ....................... 28 3.4 n circles surround the unit disk. .................... 29 3.5 Eight unequal vs. equal size circles surround the unit disk. 30 3.6 An n-cycle surrounding the unit disk. 31 v Contents Signature Page ................................. ii Acknowledgements .............................. iii Abstract ..................................... iv List of Figures ................................. v 1 Introduction ................................. 1 1.1 The Riemann Mapping Theorem ................... 1 1.1.1 The Geometry of Conformal Maps. 2 1.2 Thurston’s idea and the method of circle packings . 7 1.3 The Rodin-Sullivan Theorem ...................... 10 2 The Riemann Mapping Theorem .................... 11 2.1 The Proof of Uniqueness ........................ 12 2.2 The Proof of Existence ......................... 13 2.2.1 Reduction to the case Ω Ă D .................. 13 2.2.2 Existence of f :Ω Ď D Ñ D .................. 15 2.2.3 f :Ω Ñ D is surjective ..................... 17 3 Thurston’s Conjecture and Rodin-Sullivan’s Theorem ...... 20 vi 3.1 Convergence of Domain and Range Carriers. 24 3.2 The Convergence of the Functions fE as e Ñ 0: ............ 28 Bibiliography .................................. 34 vii Chapter 1 Introduction 1.1 The Riemann Mapping Theorem The study of mapping properties of holomorphic functions has been of interest to many mathematicians interested in complex analytic function theory. Let D “ tz P C : |z ´ 0| ă 1u be the unit disk. Given an open proper subset Ω of the complex plane C; what conditions on Ω guarantee the existence of a holomorphic bijective function f :Ω Ñ D with f ´1 : D Ñ Ω being holomorphic as well? Finding a solution to this problem would allow mathematicians to transfer questions about analytic functions on Ω; which may have very little geometric structure, to D which has more useful properties. In his PhD thesis in 1851, Bernhard Riemann solved this problem. Theorem 1.1.1. The Riemann Mapping Theorem For any simply connected proper open subset Ω of C; there exists a holomorphic bijection F of Ω onto D the unit disk with a holomorphic inverse. Furthermore, if 1 z0 P Ω; then there exists a unique such F with F pz0q “ 0 and F pz0q ¡ 0: (Figure 1 1.1 illustrates the theorem). Figure 1.1: A Riemann mapping. Riemann proved his theorem using the Dirichlet principle with an assumption that the boundary of Ω is piecewise smooth. For many years there were contro­ versies about this proof until later when David Hilbert was able to prove that the Dirichlet principle is valid under the hypotheses that Riemann was working with. In 1912, Constantin Caratheodory introduced a new proof of the theorem using Riemann surfaces. Later in 1922, Leopold Fejer and Frigyes Riesz published an even shorter version of the proof using Montel’s theorem whereas in Riemann’s proof the desired mapping was obtained as the solution of an extremal problem. This proof is given in Chapter 2. 1.1.1 The Geometry of Conformal Maps. In this subsection we will define conformal mappings and visualize their geometric structure. Then we will show the beautiful connection between conformal bijections and bijective holomorphic functions with nonzero complex derivative. Lastly, we will restate the Riemann Mapping Theorem using the theory of conformal maps. We define a conformal map as a map that preserves angles and orientation at 2 each point of its domain. Here are two examples to get the idea. Example 1.1.2. Let Ω “ tz P C : Repzq ¡ 0u and define a mapping f :Ω Ñ C i π iargpzq i π to be fpzq “ e 4 z with z “ |z|e : Then |fpzq| “ |e 4 ||z| “ |z| and argpfpzqq “ π π π π i 4 i 4 iargp z q ip4 `argp z qq argp e zq “ arg p|z|e e q “ argp|z|e q “ 4 `argpzq : This shows that f π is a rigid rotation about the origin by 4 : The map f preserves angles and orientation which means that it is an example of a conformal map. Example 1.1.3. Let Ω “ tz P C : Repzq ¡ 0u: Define f :Ω Ñ C to be the complex conjugate function defined by fpzq “ z: Then f is the reflection across the real axis. Thus, f is not a conformal map since it preserves angles but reverses the orientation. Let us now take a look at what it means for a map to preserve orientation. Let f “ u ` vi where u :Ω Ñ R and v :Ω Ñ R; so that f :Ω Ñ C determines 2 fR :Ω Ñ R by fRpx; yq “ pupx; yq; vpx; yqq: Definition 1.1.4. A map f preserves orientation at px0; y0q if and only if DfRpx0; y0q has positive determinant. i π 1 x´y x`y Example 1.1.5. Let f z e 4 z ? 1 i x yi ? i ? u vi: p q “ “ 2 p ` qp ` q “ p 2 q ` p 2 q “ ` x´y x`y So f x; y ? ; ? which implies that, Rp q “ pp 2 q p 2 qq u u ?1 ? 1 x y 2´ 2 detpDfRpx0; y0qq “ det “ det “ 1 ¡ 0: v v ??1 1 x y 2 2 ˚ ‹ ˚ ‹ ˝ ‚ ˝ ‚ This shows that f preserves orientation at each point as our intuition suggests. Example 1.1.6. Let fpzq “ z “ x ´ yi: So fRpx; yq “ px; ´yq which implies that, 1 0 det Df x ; y det 1 0: p Rp 0 0qq “ ¨ ˛ “ ´ ă 0 ´1 ˚ ‹ ˝ ‚ 3 Example 1.1.7. Let fp zq “ z2 ` z “ x2 ´ y2 ` x ` p2xy ´ yqi “ upx; yq ` vpx; yqi: 2 2 So fRpx; yq “ px ´ y ` x; 2xy ´ yq which implies that, ux uy 2x ` 1 ´2y det Df x ; y det det 4y2 4x2 1: p Rp 0 0qq “ ¨ ˛ “ ¨ ˛ “ ´ ´ ` vx vy 2y 2x ´ 1 ˚ ‹ ˚ ‹ ˝ ‚ ˝ ‚ 2 2 1 2 So, detpDfp x0 ; y 0qq ¡ 0 if and only if x ` y ă p2 q : This shows that f preserves 1 orientation for all values inside the circle of radius 2 centered at 0 and does not preserve orientation on and outside that circle. Now, to show what it means to preserve angles, we will measure angles between curves by computing angles between the tangent vectors. Meanwhile, we will apply the derivative Dfpx0; y0q to see what fR does to the tangent vectors at px0; y0q: a11 a12 Definition 1.1.8. A two-by-two matrix A “ ¨ ˛ preserves angles if and a21 a22 ˚ ‹ only if there exists p ¡ 0 such that AT A “ pI: ˝ ‚ This is because A preserves angles (as a linear transformation on R2) if it pre­ ~x~¨y T serves dot products up to a scalar p (because we have cos θ “ ? ? ). AA “ pI ~x¨~x ~y¨~y x1 2 2 means that A~x ¨ A~y “ p~x ¨ ~y for all ~x; ~y P R with ~x “ ¨ ˛ P R : This is x2 ˚ ‹ true if and only if pA~xqT pA~yq “ p~xT ~y if and only if ~xT pAA˝T q~y‚ “ p~x T ~y if and only if AT A “ pI which means that if θ1 is the angle between A~x and A~y then A~x¨A~y p~x¨~y p~x¨~y ~x¨~y cospθ1 q “ ? ? “ ? ? “ ?? ““ cospθq: A~x¨A~x A~y¨A~y ~x¨~x p~y¨~y pk~xk pk~yk k~xkk~yk Definition 1.1.9.
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