Conformal Mappings onto Simply and Multiply Connected Circular Arc Polygon Domains

204 Pages·2016·3.75 MB·English

Conformal Mappings onto Simply and Multiply
Connected Circular Arc Polygon Domains
Dissertationsschrift zur Erlangung des naturwissenschaftlichen
Doktorgrades der Julius-Maximilians-Universita¨t Wu¨rzburg
vorgelegt von
Ulrich Josef Bauer
aus
Dietfurt/To¨ging, Deutschland
Wu¨rzburg 2015
Eingereicht am 23.04.2015
bei der Fakult¨at Mathematik und Informatik
der Julius-Maximilians-Universit¨at Wu¨rzburg
1. Gutachter: Prof. Dr. Stephan Ruscheweyh
2. Gutachter: Prof. Dr. Wolfgang Lauf
Tag der Disputation: 20.11.2015
Contents
Contents III
Acknowledgments VII
1 Introduction 1
1.1 The Mapping Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Historical Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Main Results of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Structure of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Basic Information 7
2.1 M¨obius Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The Schwarzian Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Solving the Schwarzian Derivative . . . . . . . . . . . . . . . . . 10
2.2.2 Further Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Kleinian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Schottky Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Poincar´e Theta Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Simply Connected Circular Arc Polygon Domains 31
3.1 The Schwarzian Derivative of the Mappings . . . . . . . . . . . . . . . . 33
3.1.1 The Parameter Problem . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Moving the Conformal Center . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Comparison to the Schwarz-Christoffel Transformation . . . . . . . . . . 40
3.4 Domains Symmetric with Respect to Rotations . . . . . . . . . . . . . . 43
3.4.1 One Prevertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.2 Two Prevertices . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Multiply Connected Circular Arc Polygon Domains 49
4.1 Properties of the Schwarzian Derivative . . . . . . . . . . . . . . . . . . 50
4.1.1 Construction: Transformation . . . . . . . . . . . . . . . . . . . . 55
4.1.2 Construction: Reflection . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.3 Behavior of the Parameters γ . . . . . . . . . . . . . . . . . . . . 61
4.1.4 Idea Regarding the Convergence . . . . . . . . . . . . . . . . . . 65
4.1.5 Note about the Boundary Values . . . . . . . . . . . . . . . . . . 66
4.2 Mappings onto Doubly Connected Domains . . . . . . . . . . . . . . . . 67
4.2.1 Construction of the Schwarzian Derivative . . . . . . . . . . . . . 70
4.2.2 Mappings from Generic Doubly Connected Circular Domains . . 77
4.2.3 Validity of the Mappings . . . . . . . . . . . . . . . . . . . . . . 80
4.2.4 Univalence of the Mappings . . . . . . . . . . . . . . . . . . . . . 82
4.3 Known Mappings onto Doubly Connected Domains . . . . . . . . . . . . 84
4.3.1 Mapping Formula of Crowdy and Fokas . . . . . . . . . . . . . . 84
4.3.2 Schwarz-Christoffel Mapping of the Annulus . . . . . . . . . . . . 85
III
4.3.3 Nehari’s Mapping onto a Slit Domain . . . . . . . . . . . . . . . 94
4.4 Mappings onto Domains of Connectivity Three or Greater . . . . . . . . 98
4.4.1 Construction Revisited: Poincar´e Theta Series . . . . . . . . . . 99
4.4.2 Correction Terms for the Poincar´e Theta Series . . . . . . . . . . 101
4.4.3 Analytic Remainder . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4.4 Analytic Remainder Construction I: Laurent Series . . . . . . . . 115
4.4.5 Analytic Remainder Construction II: First Order Poles. . . . . . 119
4.4.6 Analytic Remainder Construction III: Second Order Poles . . . . 120
4.5 Known Mappings onto Domains of Connectivity Three or Greater . . . 130
4.5.1 Schwarz-Christoffel Mapping of Multiply Connected Domains . . 130
4.5.2 Convergence of the Schwarz-Christoffel Transformation . . . . . 132
4.5.3 Recent Results Regarding the Convergence of the SC Mapping . 136
5 Numerical Approaches to the CAPD Mapping 137
5.1 Solving the Schwarzian Derivative . . . . . . . . . . . . . . . . . . . . . 137
5.2 Choosing Suitable Paths for an Evaluation . . . . . . . . . . . . . . . . . 137
5.2.1 Reaching the Boundary Components . . . . . . . . . . . . . . . . 138
5.2.2 Reaching a Single Point . . . . . . . . . . . . . . . . . . . . . . . 142
5.3 Mappings onto Doubly Connected Domains . . . . . . . . . . . . . . . . 144
5.3.1 Calculating the Schwarzian Derivative . . . . . . . . . . . . . . . 146
5.3.2 Analytic Remainder . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.4 Mappings onto Domains of Connectivity Three or Greater . . . . . . . . 155
5.4.1 Calculating the Schwarzian Derivative . . . . . . . . . . . . . . . 156
5.4.2 Basis Functions for the Analytic Remainder . . . . . . . . . . . . 161
5.4.3 Validity of the Mappings . . . . . . . . . . . . . . . . . . . . . . 163
5.4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.5 Approach to the Parameter Problem . . . . . . . . . . . . . . . . . . . . 169
5.5.1 Numerical Solving the Parameter Problem . . . . . . . . . . . . . 169
5.5.2 Geometry of Circular Arc Polygon Domains . . . . . . . . . . . . 171
5.6 Current State of the Schwarz-Christoffel Transformation . . . . . . . . . 173
6 Conclusion 175
6.1 Simply Connected CAP Domains . . . . . . . . . . . . . . . . . . . . . . 175
6.2 Multiply Connected CAP Domains . . . . . . . . . . . . . . . . . . . . . 176
6.3 Numeric for CAPD Mappings . . . . . . . . . . . . . . . . . . . . . . . . 177
A Additional Notes 178
A.1 Restricting Equations for the CAPD Mapping onto Polygonal Domains 178
A.2 Proof of the Convergence Lemma 4.15 . . . . . . . . . . . . . . . . . . . 179
A.3 Calculation of the Pre-Schwarzian of Nehari’s Slit Mapping . . . . . . . 183
A.4 Validity of the Mappings by Methods of Differential Equations . . . . . 185
IV
B Basic Calculations for the Evaluation of the Schwarzian Derivative 187
B.1 System of Differential Equations . . . . . . . . . . . . . . . . . . . . . . 187
B.1.1 Initial Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
B.2 Boundary Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
B.2.1 Calculate the Circles Providing the Boundary . . . . . . . . . . . 188
B.2.2 Calculation of the Vertices. . . . . . . . . . . . . . . . . . . . . . 189
References 192
Index 196
V

Acknowledgments
I would like to express my deepest gratitude to all those who supported me on my way
to finish this thesis.
I wish to especially thank my supervisors Professor Dr. Stephan Ruscheweyh and Pro-
fessor Dr. Wolfgang Lauf for providing me this great opportunity and for all their
support.
I want to thank all current and former members of chair IV of the Department of
Mathematics in Wu¨rzburg for their backing. I also owe thanks to those of the OTH
Regensburg who supported me in any way.
Last but not least, I am grateful to my parents, my sisters and my girlfriend for their
patience and support.
VII

1 Introduction
1.1 The Mapping Problem
Since the publication of the famous Riemann mapping theorem, it is known that every
simply connected domain can be conformally mapped onto the unit disk. It was ex-
tended later to also ensure the conformal mapping of any multiply connected domain
onto a conformally equivalent canonical region like a circular domain (Definition 4.1).1
While the existence of such mappings for all finitely connected domains is guaranteed,
the search for the mapping formulas has since remained an open topic.
The Schwarz-Christoffel transformation and its variations represent a class of mappings
onto polygonal domains and other similar domains. A historical summary of the evolu-
tion of the Schwarz-Christoffel mapping can be found in Section 1.2.
The two common features of these mappings are that each of them conformally maps
a canonical domain onto a domain whose boundary contains several vertices and that
the mapping formulas contain at least two kinds of parameters. The issue with these
parameters is that they define the shape of the image domain, but the actual shape can
only be seen by evaluating the mapping. The search for the correct parameters for a
given image domain is known as the “parameter problem”.
The goal of this thesis is to investigate conformal mappings onto circular arc polygon
domains, i.e. domains that are bounded by polygons consisting of circular arcs instead
ofline segments. Thisincludesthe investigationof thethirdkindof parametersofthese
mappings in the case of simply connected image domains. A new version for conformal
mappings onto multiply connected domains will also be constructed. We will moreover
discuss the necessary concepts of the numerics to give an approach to the parameter
problem. A summary of the results of our thesis can be found in Section 1.3 and an
outline of the different chapters can be found in Section 1.4.
1.2 Historical Overview
The first known reference to conformal mappings from the upper half plane onto polyg-
onaldomains(Figure1)canbefoundinanarticlebyElwinBrunoChristoffelpublished
in the year 1867. Some years later Hermann Amandus Schwarz discovered the same
mappings independently of Christoffel. Referring to both mathematicians, the mapping
onto a polygonal domain is known as the “Schwarz-Christoffel transformation”.
While all domains can be approximated by polygonal domains, the common mapping
formula (Lemma 3.5) has two major drawbacks. First, the mappings are from standard
domains like the upper half plane onto polygonal domains. However, in most scenarios,
the mappings are needed the other way around, i.e. from a polygonal domain onto a
1The corresponding theorems and more canonical regions can, for example, be found in the book of
Conway [Con95, Chapter 15].
1
1 Introduction
f
Figure1: Schwarz-Christoffeltransformation: Theupperhalfplaneismapped
onto a domain bounded by a polygon.
standard domain.
The second problem is in choosing the correct parameters. The mapping formula con-
tains two kinds of parameters (the preimages of the corners of the polygon and the
interior angle at these corners) and the shape of the image domain depends on these
choices, but the result of an actual mapping can only be attained by evaluating the
mapping. This problem is normally simply referred to as the “parameter problem” in
the context of the Schwarz-Christoffel transformation.
Christoffel and Schwarz also expanded this topic to different forms of the mapping
formula by changing some of the properties of the image domains. This includes map-
pings onto the exterior of a polygon, domains with curved boundary parts and domains
bounded by circular arc polygons (Figure 2). The circular arc polygon domains, e.g.
f
Figure 2: Circular arc polygon domain mapping: The unit disk is mapped
onto a domain bounded by a polygon, where the sides are circular arcs.
domains, where the boundary is a polygon while the straight sides are replaced by cir-
cular arcs, will be the main topic of this thesis.
While they have discovered many mapping formulas, the parameter problem restricted
them to simple examples.
This topic became of interest again in the second half of the twentieth century. With
2