Quasiconformal Mappings, Complex Dynamics and Moduli Spaces
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The Riemann Mapping Theorem Christopher J. Bishop
The Riemann Mapping Theorem Christopher J. Bishop C.J. Bishop, Mathematics Department, SUNY at Stony Brook, Stony Brook, NY 11794-3651 E-mail address: [email protected] 1991 Mathematics Subject Classification. Primary: 30C35, Secondary: 30C85, 30C62 Key words and phrases. numerical conformal mappings, Schwarz-Christoffel formula, hyperbolic 3-manifolds, Sullivan’s theorem, convex hulls, quasiconformal mappings, quasisymmetric mappings, medial axis, CRDT algorithm The author is partially supported by NSF Grant DMS 04-05578. Abstract. These are informal notes based on lectures I am giving in MAT 626 (Topics in Complex Analysis: the Riemann mapping theorem) during Fall 2008 at Stony Brook. We will start with brief introduction to conformal mapping focusing on the Schwarz-Christoffel formula and how to compute the unknown parameters. In later chapters we will fill in some of the details of results and proofs in geometric function theory and survey various numerical methods for computing conformal maps, including a method of my own using ideas from hyperbolic and computational geometry. Contents Chapter 1. Introduction to conformal mapping 1 1. Conformal and holomorphic maps 1 2. M¨obius transformations 16 3. The Schwarz-Christoffel Formula 20 4. Crowding 27 5. Power series of Schwarz-Christoffel maps 29 6. Harmonic measure and Brownian motion 39 7. The quasiconformal distance between polygons 48 8. Schwarz-Christoffel iterations and Davis’s method 56 Chapter 2. The Riemann mapping theorem 67 1. The hyperbolic metric 67 2. Schwarz’s lemma 69 3. The Poisson integral formula 71 4. A proof of Riemann’s theorem 73 5. Koebe’s method 74 6. -
On the Combinatorics of External Rays in the Dynamics of the Complex Henon
ON THE COMBINATORICS OF EXTERNAL RAYS IN THE DYNAMICS OF THE COMPLEX HENON MAP. A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Ricardo Antonio Oliva May 1998 c Ricardo Antonio Oliva 1998 ALL RIGHTS RESERVED Addendum. This is a slightly revised version of my doctoral thesis: some typing and spelling mistakes have been corrected and a few sentences have been re-worded for better legibility (particularly in section 4.3). Also, to create a nicer pdf document with hyperref, the title of section 3.3.2 has been made shorter. The original title was A model for a map with an attracting fixed point as well as a period-3 sink: the (3-1)-graph. ON THE COMBINATORICS OF EXTERNAL RAYS IN THE DYNAMICS OF THE COMPLEX HENON MAP. Ricardo Antonio Oliva , Ph.D. Cornell University 1998 We present combinatorial models that describe quotients of the solenoid arising from the dynamics of the complex H´enon map 2 2 2 fa,c : C → C , (x, y) → (x + c − ay, x). These models encode identifications of external rays for specific mappings in the H´enon family. We investigate the structure of a region of parameter space in R2 empirically, using computational tools we developed for this study. We give a combi- natorial description of bifurcations arising from changes in the set of identifications of external rays. Our techniques enable us to detect, predict, and locate bifurca- tion curves in parameter space. We describe a specific family of bifurcations in a region of real parameter space for which the mappings were expected to have sim- ple dynamics. -
ONE HUNDRED YEARS of COMPLEX DYNAMICS the Subject of Complex Dynamics, That Is, the Behaviour of Orbits of Holomorphic Functions
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Liverpool Repository ONE HUNDRED YEARS OF COMPLEX DYNAMICS MARY REES The subject of Complex Dynamics, that is, the behaviour of orbits of holomorphic functions, emerged in the papers produced, independently, by Fatou and Julia, almost 100 years ago. Although the subject of Dynami- cal Systems did not then have a name, the dynamical properties found for holomorphic systems, even in these early researches, were so striking, so unusually comprehensive, and yet so varied, that these systems still attract widespread fascination, 100 years later. The first distinctive feature of iter- ation of a single holomorphic map f is the partition of either the complex plane or the Riemann sphere into two sets which are totally invariant under f: the Julia set | closed, nonempty, perfect, with dynamics which might loosely be called chaotic | and its complement | open, possibly empty, but, if non-empty, then with dynamics which were completely classified by the two pioneering researchers, modulo a few simply stated open questions. Before the subject re-emerged into prominence in the 1980's, the Julia set was alternately called the Fatou set, but Paul Blanchard introduced the idea of calling its complement the Fatou set, and this was immediately universally accepted. Probably the main reason for the remarkable rise in interest in complex dynamics, about thirty-five years ago, was the parallel with the subject of Kleinian groups, and hence with the whole subject of hyperbolic geome- try. A Kleinian group acting on the Riemann sphere is a dynamical system, with the sphere splitting into two disjoint invariant subsets, with the limit set and its complement, the domain of discontinuity, having exactly similar properties to the Julia and Fatou sets. -
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. -
Early Days in Complex Dynamics: a History of Complex Dynamics in One Variable During 1906–1942, by Daniel S
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 50, Number 3, July 2013, Pages 503–511 S 0273-0979(2013)01408-3 Article electronically published on April 17, 2013 Early days in complex dynamics: a history of complex dynamics in one variable during 1906–1942, by Daniel S. Alexander, Felice Iavernaro, and Alessandro Rosa, History of Mathematics, 38, American Mathematical Society, Providence, RI, London Mathematical Society, London, 2012, xviii+454 pp., ISBN 978-0- 8218-4464-9, US $99.00 Some time ago, on my way to a special lecture at the Fields Institute, I overheard a couple of graduate students whispering in the corridor: - Are you coming to Shishikura’s talk? -Who’sthat? - The guy who proved that the boundary of the Mandelbrot set has Hausdorff dimension two. - Sweet! That was not the topic of the talk (in fact, HD(∂M) = 2 had been proven 15 years earlier), so it was telling that they even knew what the notions were in the statement. To me, this conversation is symbolic of how much the language of complex dynamics has permeated mathematical culture since the early 1980s, when it was preposterous to think of doing serious research on the properties of quadratic polynomials.1 Back then much of the subject’s popularity had to do with the pretty pictures. A few lines of computer code were enough to start exploring an endless variety of fractal shapes, and ever since then, Julia sets have decorated calculus books, art galleries, and cereal boxes. But there was serious research to be done, and complex dynamics is today a vital branch of dynamical systems that has developed deep connections to differential equations, geometric group theory, harmonic analysis, algebraic number theory, and many other subjects. -
Generalized Solutions of a System of Differential Equations of the First Order and Elliptic Type with Discontinuous Coefficients
UNIVERSITY OF JYVASKYL¨ A¨ UNIVERSITAT¨ JYVASKYL¨ A¨ DEPARTMENT OF MATHEMATICS INSTITUT FUR¨ MATHEMATIK AND STATISTICS UND STATISTIK REPORT 118 BERICHT 118 GENERALIZED SOLUTIONS OF A SYSTEM OF DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND ELLIPTIC TYPE WITH DISCONTINUOUS COEFFICIENTS B. V. BOJARSKI JYVASKYL¨ A¨ 2009 UNIVERSITY OF JYVASKYL¨ A¨ UNIVERSITAT¨ JYVASKYL¨ A¨ DEPARTMENT OF MATHEMATICS INSTITUT FUR¨ MATHEMATIK AND STATISTICS UND STATISTIK REPORT 118 BERICHT 118 GENERALIZED SOLUTIONS OF A SYSTEM OF DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND ELLIPTIC TYPE WITH DISCONTINUOUS COEFFICIENTS B. V. BOJARSKI JYVASKYL¨ A¨ 2009 Editor: Pekka Koskela Department of Mathematics and Statistics P.O. Box 35 (MaD) FI{40014 University of Jyv¨askyl¨a Finland ISBN 978-951-39-3486-6 ISSN 1457-8905 Copyright c 2009, by B. V. Bojarski and University of Jyv¨askyl¨a University Printing House Jyv¨askyl¨a2009 Foreword to the translation of Generalized solutions of a sys- tem of dierential equations of rst order and of elliptic type with discontinuous coecients A remarkable feature of quasiconformal mappings in the plane is the in- terplay between analytic and geometric arguments in the theory. The utility of the analytic approach is principally due to the explicit representation formula f = z + Cµ(z) + C(µT µ)(z) + C(µT (µT µ))(z) + ..., (1) valid for a suitably normalized quasiconformal homeomorphism f of the plane, with compactly supported dilatation µ. Here T stands for the Beurling-Ahlfors singular integral operator and C is the Cauchy transform. It was previously known that T is an isometry on L2, and the above formula easily yields that in this case belongs to 1,2 2 However, this knowledge alone does not f Wloc (R ). -
Radnell-Schippers-Staubach, 2017
QUASICONFORMAL TEICHMULLER¨ THEORY AS AN ANALYTICAL FOUNDATION FOR TWO-DIMENSIONAL CONFORMAL FIELD THEORY DAVID RADNELL, ERIC SCHIPPERS, AND WOLFGANG STAUBACH Abstract. The functorial mathematical definition of conformal field theory was first formulated approximately 30 years ago. The underlying geometric category is based on the moduli space of Riemann surfaces with parametrized boundary components and the sewing operation. We survey the recent and careful study of these objects, which has led to significant connections with quasiconformal Teichm¨uller theory and geometric function theory. In particular we propose that the natural analytic setting for conformal field theory is the moduli space of Riemann surfaces with so-called Weil-Petersson class parametrizations. A collection of rig- orous analytic results is advanced here as evidence. This class of parametrizations has the required regularity for CFT on one hand, and on the other hand are natural and of interest in their own right in geometric function theory. 1. Introduction 1.1. Introduction. Two-dimensional conformal field theory (CFT) models a wide range of physical phenomena. Its mathematical structures connect to many branches of mathematics, including complex geometry and analysis, representation theory, algebraic geometry, topology, and stochastic analysis. There are several mathematical notions of CFT, each of which is relevant to probing the particular mathematical structures one is interested in. Without attempting any overview of this vast subject, we first highlight some early literature, and then explain the purpose of this review. The conformal symmetry group in two-dimensional quantum field theory goes back to at least the Thirring model from 1958, although this was perhaps not fully recognized immediately. -
Solving Beltrami Equations by Circle Packing
transactions of the american mathematical society Volume 322, Number 2, December 1990 SOLVING BELTRAMI EQUATIONS BY CIRCLE PACKING ZHENG-XU HE Abstract. We use Andreev-Thurston's theorem on the existence of circle pack- ings to construct approximating solutions to the Beltrami equations on Riemann surfaces. The convergence of the approximating solutions on compact subsets will be shown. This gives a constructive proof of the existence theorem for Beltrami equations. 1. Introduction A circle packing in the plane or in a closed Riemannian 2-manifold of con- stant curvature is a collection of closed round disks with disjoint interiors. The nerve of such circle packing is the embedded graph whose vertices are the centers of the disks and whose edges are the shortest geodesic arcs joining the centers of each pair of intersecting disks. By Andreev-Thurston's theorem, for any tri- 2 2 angulation of the Riemann sphere S , there is a circle packing on S , unique up to a Möbius transformation, whose nerve is isotopic to the 1-skeleton of the triangulation (see [T, Corollary 13.6.2, or MR]). Based on this fact, William P. Thurston suggested a geometric approach to the Riemann mapping theorem. Recently, Burt Rodin and Dennis Sullivan [RS] carried out this ap- proach and verified the convergence of Thurston's scheme. In this paper, we will use this circle packing method to construct approximations to solutions /:fi-»C of the Beltrami equation: (1.1) d,fi(z) = k(z)dzfi(z) a.e. z = x + iyeSi, where Q. is some open Jordan domain in C, and k: Q —>C is some measurable function with (1.2) ||A||00= esssup|A(z)|<l. -
On Conformal Welding Homeomorphisms Associated to Jordan Curves
Annales Academire Scientiarum Fennicre Series A. I. Mathematica Volumen 15, 1990, 293-306 ON CONFORMAL WELDING HOMEOMORPHISMS ASSOCIATED TO JORDAN CURVES Y. Katznelson, Subhashis Nag, and Dennis P. Sullivan Abstract tr'or any Jordan curve C on the Riemann sphere, the conformal welding home- omorphism, Weld(C), of the circle or real line onto itself is obtained by comparing the bLundary values of the Riemann mappings of the unit disk or upper half-plane onto the two complementa,ry Jordan regions separated by C. In Part I we show that a family of infinitely spiralling curves lie intermediate between smooth curves and curves with corners, in the sense that the welding for the spirals has a non-differentiable but Lipschitz singularity. We also show how the behaviour of Riemann mapping germs under analytic continuation implies that spirals were the appropriate curves to produce such weldings. A formula relating explicitly the rate of spiralling to the extent of non-differentiablity is proved by two methods, and related to known theorems for chord-arc curves. Part II studies families of Jordan curves possessing the same welding. We utilise curves with positive area and the Ahlfors-Bers generalised Riemann map- ping theorem to build an infinite dimensional "Teichmiiller space" of curves all having the same welding homeomorphism. A point of view on welding as a prob- Iem of extension of conformal structure is also derived in this part. Introduction In this paper we address two questions regarding the conformal welding home- omorphism, Wetd(C), associated to any Jordan curve C on the Riemann sphere Ö. -
Constructing Complex Structures on Surfaces Via the Beltrami Equation
Constructing complex structures on surfaces via the Beltrami equation Sayantan Khan University of Michigan Tuesday 30th October, 2018 Abstract Coming up with concrete examples of Riemann surfaces of genus 2 or higher is not very easy. One way to do so is to start with an orientable Riemannian manifold of dimension 2, and look at the conformal class of that metric, and that canonically corresponds to a complex structure on the surface. In this talk, we’ll see how to get the complex structure from the conformal structure by solving the Beltrami equation. 1 The problem with holomorphic maps Much like how a smooth manifold is defined in a first course on the subject, one can define Riemann surfaces in an analogous manner. Definition 1.1 (Riemann surface). A Riemann surface X is a topological space which is Hausdorff and second countable such that every point x 2 X has an open set U around it, and a homeomorphism φ to an open subset 0 0 -1 of C such that for any other choice of homeomorphism φ to an open subset of C, φ ◦ φ is a holomorphic function. For all purposes, it seems like a fairly innocuous definition, and just a generalization of the definition of a smooth surface. But anyone who has taken a complex analysis course knows well that holomorphic functions are often annoyingly rigid. Unlike smooth functions, you can’t just construct a holomorphic function to satisfy any property you want: for instance, the only holomorphic function that vanishes in an open set is the constant function 0. -
Uniformization of Surface Laminations Annales Scientifiques De L’É.N.S
ANNALES SCIENTIFIQUES DE L’É.N.S. ALBERTO CANDEL Uniformization of surface laminations Annales scientifiques de l’É.N.S. 4e série, tome 26, no 4 (1993), p. 489-516 <http://www.numdam.org/item?id=ASENS_1993_4_26_4_489_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1993, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. scient. EC. Norm. Sup., 46 serie, t. 26, 1993, p. 489 a 516. UNIFORMIZATION OF SURFACE LAMINATIONS BY ALBERTO CANDEL ABSTRACT. — A surface lamination is a metric space that carries a foliation with leaves of dimension two. Given a riemannian metric along the leaves we study the problem of finding another such metric, in the same conformal class, for which all leaves have the same constant curvature. As for surfaces, the existence of such metric is determined by the Euler characteristics of the lamination. These numbers are obtained by evaluating the invariant transverse measures on the curvature form of the given metric. We prove that there is a metric of curvature — 1 (resp. 1) if and only if all Euler characteristics are negative (resp. positive). Using harmonic measures we prove a similar statement holds for flat metrics. -
An Introduction to Riemann Surfaces
Treball de final de grau Grau de Matem´atiques Facultat de Matem´atiques Universitat de Barcelona AN INTRODUCTION TO RIEMANN SURFACES Sergio Ruiz Bonilla Director: Ignasi Mundet i Riera Departament d'Algebra´ i Geometria. UB Barcelona, June 29, 2015 Introduction In this document we give a first view to Riemann surface theory. Starting from definition and examples in chapter I, in the next chapter one sees the relation be- tween the well known oriented smooth surfaces and this new object, with the result that any oriented smooth surfaces is equivalent to a Riemann surface. In order to prove this result, almost-complex structures and isothermal coordinates (be- tween others) are explained, and the key point is the existence of these isothermal coordinates for a smooth surface as we will see. Finally in Chapter III we stablish the relation between Riemann surfaces and algebraic curves. First we construct a Riemann surface from a polynomial, which is relatively easy, and then we give and prove the Main Theorem for Riemann sur- faces using Hilbert space techniques and some tools like the Riesz Representation Theorem. This Main Theorem is the key to prove the existence of meromorphic functions on a Riemann surface and the fact that any compact Riemann surface arises from a polynomial. Acknowledgments I would like to thank my advisor Ignasi Mundet, for all the time spent on me, and my parents, for supporting as much as parents can support their son. i ii Contents 1 Basic definitions and examples 1 1.1 Riemann surfaces . 1 1.2 The Riemann sphere .