Solving Beltrami Equations by Circle Packing
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The Circle Packing Theorem
Alma Mater Studiorum · Università di Bologna SCUOLA DI SCIENZE Corso di Laurea in Matematica THE CIRCLE PACKING THEOREM Tesi di Laurea in Analisi Relatore: Pesentata da: Chiar.mo Prof. Georgian Sarghi Nicola Arcozzi Sessione Unica Anno Accademico 2018/2019 Introduction The study of tangent circles has a rich history that dates back to antiquity. Already in the third century BC, Apollonius of Perga, in his exstensive study of conics, introduced problems concerning tangency. A famous result attributed to Apollonius is the following. Theorem 0.1 (Apollonius - 250 BC). Given three mutually tangent circles C1, C2, 1 C3 with disjoint interiors , there are precisely two circles tangent to all the three initial circles (see Figure1). A simple proof of this fact can be found here [Sar11] and employs the use of Möbius transformations. The topic of circle packings as presented here, is sur- prisingly recent and originates from William Thurston's famous lecture notes on 3-manifolds [Thu78] in which he proves the theorem now known as the Koebe-Andreev- Thurston Theorem or Circle Packing Theorem. He proves it as a consequence of previous work of E. M. Figure 1 Andreev and establishes uniqueness from Mostov's rigid- ity theorem, an imporant result in Hyperbolic Geometry. A few years later Reiner Kuhnau pointed out a 1936 proof by german mathematician Paul Koebe. 1We dene the interior of a circle to be one of the connected components of its complement (see the colored regions in Figure1 as an example). i ii A circle packing is a nite set of circles in the plane, or equivalently in the Riemann sphere, with disjoint interiors and whose union is connected. -
Quasiconformal Mappings, Complex Dynamics and Moduli Spaces
Quasiconformal mappings, complex dynamics and moduli spaces Alexey Glutsyuk April 4, 2017 Lecture notes of a course at the HSE, Spring semester, 2017 Contents 1 Almost complex structures, integrability, quasiconformal mappings 2 1.1 Almost complex structures and quasiconformal mappings. Main theorems . 2 1.2 The Beltrami Equation. Dependence of the quasiconformal homeomorphism on parameter . 4 2 Complex dynamics 5 2.1 Normal families. Montel Theorem . 6 2.2 Rational dynamics: basic theory . 6 2.3 Local dynamics at neutral periodic points and periodic components of the Fatou set . 9 2.4 Critical orbits. Upper bound of the number of non-repelling periodic orbits . 12 2.5 Density of repelling periodic points in the Julia set . 15 2.6 Sullivan No Wandering Domain Theorem . 15 2.7 Hyperbolicity. Fatou Conjecture. The Mandelbrot set . 18 2.8 J-stability and structural stability: main theorems and conjecture . 21 2.9 Holomorphic motions . 22 2.10 Quasiconformality: different definitions. Proof of Lemma 2.78 . 24 2.11 Characterization and density of J-stability . 25 2.12 Characterization and density of structural stability . 27 2.13 Proof of Theorem 2.90 . 32 2.14 On structural stability in the class of quadratic polynomials and Quadratic Fatou Conjecture . 32 2.15 Structural stability, invariant line fields and Teichm¨ullerspaces: general case 34 3 Kleinian groups 37 The classical Poincar´e{Koebe Uniformization theorem states that each simply connected Riemann surface is conformally equivalent to either the Riemann sphere, or complex line C, or unit disk D1. The quasiconformal mapping theory created by M.A.Lavrentiev and C. -
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 Dynamical Gaskets Generated by Rational Maps, Kleinian Groups, and Schwarz Reflections
ON DYNAMICAL GASKETS GENERATED BY RATIONAL MAPS, KLEINIAN GROUPS, AND SCHWARZ REFLECTIONS RUSSELL LODGE, MIKHAIL LYUBICH, SERGEI MERENKOV, AND SABYASACHI MUKHERJEE Abstract. According to the Circle Packing Theorem, any triangulation of the Riemann sphere can be realized as a nerve of a circle packing. Reflections in the dual circles generate a Kleinian group H whose limit set is an Apollonian- like gasket ΛH . We design a surgery that relates H to a rational map g whose Julia set Jg is (non-quasiconformally) homeomorphic to ΛH . We show for a large class of triangulations, however, the groups of quasisymmetries of ΛH and Jg are isomorphic and coincide with the corresponding groups of self- homeomorphisms. Moreover, in the case of H, this group is equal to the group of M¨obiussymmetries of ΛH , which is the semi-direct product of H itself and the group of M¨obiussymmetries of the underlying circle packing. In the case of the tetrahedral triangulation (when ΛH is the classical Apollonian gasket), we give a piecewise affine model for the above actions which is quasiconformally equivalent to g and produces H by a David surgery. We also construct a mating between the group and the map coexisting in the same dynamical plane and show that it can be generated by Schwarz reflections in the deltoid and the inscribed circle. Contents 1. Introduction 2 2. Round Gaskets from Triangulations 4 3. Round Gasket Symmetries 6 4. Nielsen Maps Induced by Reflection Groups 12 5. Topological Surgery: From Nielsen Map to a Branched Covering 16 6. Gasket Julia Sets 18 arXiv:1912.13438v1 [math.DS] 31 Dec 2019 7. -
Study on the Problem of the Number Ring Transformation
The Research on Circle Family and Sphere Family The Original Topic: The Research and Generalizations on Several Kinds of Partition Problems in Combinatorics The Affiliated High School of South China Normal University Yan Chen Zhuang Ziquan Faculty Adviser:Li Xinghuai 153 The Research on Circle Family and Sphere Family 【Abstract】 A circle family is a group of separate or tangent circles in the plane. In this paper, we study how many parts at most a plane can be divided by several circle families if the circles in a same family must be separate (resp. if the circles can be tangent). We also study the necessary conditions for the intersection of two circle families. Then we primarily discuss the similar problems in higher dimensional space and in the end, raise some conjectures. 【Key words】 Circle Family; Structure Graph; Sphere Family; Generalized Inversion 【Changes】 1. Part 5 ‘Some Conjectures and Unsolved Problems’ has been rewritten. 2. Lemma 4.2 has been restated. 3. Some small mistakes have been corrected. 154 1 The Definitions and Preliminaries To begin with, we introduce some newly definitions and related preliminaries. Definition 1.1 Circle Family A circle family of the first kind is a group of separate circles;A circle family of the second kind is a group of separate or tangent circles. The capacity of a circle family is the number of circles in a circle family,the intersection of circle families means there are several circle families and any two circles in different circle families intersect. Definition 1.2 Compaction If the capacity of a circle family is no less than 3,and it intersects with another circle family with capacity 2,we call such a circle family compact. -
15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm
15 BASIC PROPERTIES OF CONVEX POLYTOPES Martin Henk, J¨urgenRichter-Gebert, and G¨unterM. Ziegler INTRODUCTION Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their im- portance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimiza- tion. In this chapter we try to give a short introduction, provide a sketch of \what polytopes look like" and \how they behave," with many explicit examples, and briefly state some main results (where further details are given in subsequent chap- ters of this Handbook). We concentrate on two main topics: • Combinatorial properties: faces (vertices, edges, . , facets) of polytopes and their relations, with special treatments of the classes of low-dimensional poly- topes and of polytopes \with few vertices;" • Geometric properties: volume and surface area, mixed volumes, and quer- massintegrals, including explicit formulas for the cases of the regular simplices, cubes, and cross-polytopes. We refer to Gr¨unbaum [Gr¨u67]for a comprehensive view of polytope theory, and to Ziegler [Zie95] respectively to Gruber [Gru07] and Schneider [Sch14] for detailed treatments of the combinatorial and of the convex geometric aspects of polytope theory. 15.1 COMBINATORIAL STRUCTURE GLOSSARY d V-polytope: The convex hull of a finite set X = fx1; : : : ; xng of points in R , n n X i X P = conv(X) := λix λ1; : : : ; λn ≥ 0; λi = 1 : i=1 i=1 H-polytope: The solution set of a finite system of linear inequalities, d T P = P (A; b) := x 2 R j ai x ≤ bi for 1 ≤ i ≤ m ; with the extra condition that the set of solutions is bounded, that is, such that m×d there is a constant N such that jjxjj ≤ N holds for all x 2 P . -
Planar Separator Theorem a Geometric Approach
Planar Separator Theorem A Geometric Approach based on A Framework for ETH-Tight Algorithms and Lower Bounds in Geometric Intersection Graphs by Mark de Berg, Hans L. Bodlaender, S´andor Kisfaludi-Bak, D´anielMarx and Tom C. van der Zanden Unit Disk Graphs • represent vertices as unit disks, i.e., disks with diameter 1 • disks intersect iff corresponding vertices are adjacent a d a b d b f $ e f e g g h h c c Unit Disk Graphs • represent vertices as unit disks, i.e., disks with diameter 1 • disks intersect iff corresponding vertices are adjacent • some unit disk graphs are planar a d a b d b f $ e f e g g h h c c Unit Disk Graphs • represent vertices as unit disks, i.e., disks with diameter 1 • disks intersect iff corresponding vertices are adjacent • some unit disk graphs are planar • some planar graphs are not unit disk graphs c c d b a d a 6$ b e f e f Geometric Separators Geometric Separators Geometric Separators Geometric Separators H Observation: Disks intersecting the boundary of H separate disks strictly inside H from disks strictly outside H . Small Balanced Separators Let G = (V ; E) be a graph. H ⊆ V is • a separator if there is a partition H; V1; V2 of V so that no edge in E has onep endpoint in V1 and one endpoint in V2, • small if jHj 2 O( n), • balanced if jV1j; jV2j ≤ βn for some constant β Small Balanced Geometric Separators H p Claim: There exists an H intersecting O( n) disks with ≤ 36=37n disks strictly inside H and ≤ 36=37n disks strictly outside H , i.e., H is a small balanced separator. -
Fixed Points, Koebe Uniformization and Circle Packings
Annals of Mathematics, 137 (1993), 369-406 Fixed points, Koebe uniformization and circle packings By ZHENG-XU HE and ODED SCHRAMM* Contents Introduction 1. The space of boundary components 2. The fixed-point index 3. The Uniqueness Theorem 4. The Schwarz-Pick lemma 5. Extension to the boundary 6. Maximum modulus, normality and angles 7. Uniformization 8. Domains in Riemann surfaces 9. Uniformizations of circle packings Addendum Introduction A domain in the Riemann sphere C is called a circle domain if every connected component of its boundary is either a circle or a point. In 1908, P. Koebe [Kol] posed the following conjecture, known as Koebe's Kreisnormierungsproblem: A ny plane domain is conformally homeomorphic to a circle domain in C. When the domain is simply connected, this is the con tent of the Riemann mapping theorem. The conjecture was proved for finitely connected domains and certain symmetric domains by Koebe himself ([K02], [K03]); for domains with various conditions on the "limit boundary compo nents" by R. Denneberg [De], H. Grotzsch [Gr], L. Sario [Sa], H. Meschowski *The authors were supported by N.S.F. Grants DMS-9006954 and DMS-9112150, respectively. The authors express their thanks to Mike Freedman, Dennis Hejhal, Al Marden, Curt McMullen, Burt Rodin, Steffen Rohde and Bill Thurston for conversations relating to this work. Also thanks are due to the referee, and to Steffen Rohde, for their careful reading and subsequent corrections. The paper of Sibner [Si3l served as a very useful introduction to the subject. I. Benjamini, O. Häggström (eds.), Selected Works of Oded Schramm, Selected Works in Probability and Statistics, DOI 10.1007/978-1-4419-9675-6_6, C Springer Science+Business Media, LLC 2011 105 370 z.-x. -
Geometric Representations of Graphs
1 Geometric Representations of Graphs Laszl¶ o¶ Lovasz¶ Institute of Mathematics EÄotvÄosLor¶andUniversity, Budapest e-mail: [email protected] December 11, 2009 2 Contents 0 Introduction 9 1 Planar graphs and polytopes 11 1.1 Planar graphs .................................... 11 1.2 Planar separation .................................. 15 1.3 Straight line representation and 3-polytopes ................... 15 1.4 Crossing number .................................. 17 2 Graphs from point sets 19 2.1 Unit distance graphs ................................ 19 2.1.1 The number of edges ............................ 19 2.1.2 Chromatic number and independence number . 20 2.1.3 Unit distance representation ........................ 21 2.2 Bisector graphs ................................... 24 2.2.1 The number of edges ............................ 24 2.2.2 k-sets and j-edges ............................. 27 2.3 Rectilinear crossing number ............................ 28 2.4 Orthogonality graphs ................................ 31 3 Harmonic functions on graphs 33 3.1 De¯nition and uniqueness ............................. 33 3.2 Constructing harmonic functions ......................... 35 3.2.1 Linear algebra ............................... 35 3.2.2 Random walks ............................... 36 3.2.3 Electrical networks ............................. 36 3.2.4 Rubber bands ................................ 37 3.2.5 Connections ................................. 37 4 Rubber bands 41 4.1 Rubber band representation ............................ 41 3 4 CONTENTS -
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. -
Quasiconformal Mappings, from Ptolemy's Geography to the Work Of
Quasiconformal mappings, from Ptolemy’s geography to the work of Teichmüller Athanase Papadopoulos To cite this version: Athanase Papadopoulos. Quasiconformal mappings, from Ptolemy’s geography to the work of Teich- müller. 2016. hal-01465998 HAL Id: hal-01465998 https://hal.archives-ouvertes.fr/hal-01465998 Preprint submitted on 13 Feb 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. QUASICONFORMAL MAPPINGS, FROM PTOLEMY'S GEOGRAPHY TO THE WORK OF TEICHMULLER¨ ATHANASE PAPADOPOULOS Les hommes passent, mais les œuvres restent. (Augustin Cauchy, [204] p. 274) Abstract. The origin of quasiconformal mappings, like that of confor- mal mappings, can be traced back to old cartography where the basic problem was the search for mappings from the sphere onto the plane with minimal deviation from conformality, subject to certain conditions which were made precise. In this paper, we survey the development of cartography, highlighting the main ideas that are related to quasicon- formality. Some of these ideas were completely ignored in the previous historical surveys on quasiconformal mappings. We then survey early quasiconformal theory in the works of Gr¨otzsch, Lavrentieff, Ahlfors and Teichm¨uller,which are the 20th-century founders of the theory. -
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 ).