Conformally Invariant Metrics and Quasiconformal Mappings (Springer Monographs in Mathematics)

Preface
Notation and Terminology
Contents
List of Figures
Part I Introduction and Review
	1 Introduction
	2 A Survey of Quasiregular Mappings
Part II Conformal Geometry
	3 Möbius Transformations
		3.1 Poincaré Extension
			3.1.1 Stereographic Projection
			3.1.2 Balls in the Chordal Metric
			3.1.3 Absolute Ratio
		3.2 Automorphisms of the Unit Ball
			3.2.1 The Lipschitz Constant of Ta|Bn
			3.2.2 The Ahlfors Bracket
		3.3 Chordal Isometries
	4 Hyperbolic Geometry
		4.1 The Poincaré Half Space
		4.2 The Poincaré Unit Ball
		4.3 Bounds for Hyperbolic Distance
	5 Generalized Hyperbolic Geometries
		5.1 The Quasihyperbolic Metric
		5.2 Möbius Invariant Metrics
			5.2.1 The Point-Pair Invariant mG
			5.2.2 The Symmetric Ratio
			5.2.3 The Generalized Hyperbolic Metric ρG
		5.3 Properties of Generalized Hyperbolic Metrics
	6 Metrics and Geometry
		6.1 Uniform Domains and Generalizations
			6.1.1 Ptolemy\'s Theorem
		6.2 Whitney Squares and (a,b,s)-Admissible Families
			6.2.1 Whitney Decomposition
			6.2.2 (a,b,s)-Admissible Families
		6.3 Harnack Functions
Part III Modulus and Capacity
	7 The Modulus of a Curve Family
		7.1 Basic Properties of the Modulus
			7.1.1 The Cylinder
			7.1.2 The Spherical Ring
			7.1.3 Constants and Dimension
		7.2 Comparison Principle for the Modulus
			7.2.1 The Modulus of a Ring
		7.3 Grötzsch and Teichmüller Rings
		7.4 Hypergeometric Functions and Elliptic Integrals
			7.4.1 Elliptic Integrals and γ2(s)
			7.4.2 The Gaussian Hypergeometric Function
			7.4.3 Landen Transformation
			7.4.4 Arithmetic-Geometric Mean
			7.4.5 Modular Equations
			7.4.6 Jacobi\'s Infinite Products
			7.4.7 Ramanujan\'s Approximation of μ(r) [29, 5.51], [64, p. 91, (2.4)]
	8 The Modulus as a Set Function
		8.1 The Construction of c(E)
			8.1.1 Proof of Theorem 8.1
		8.2 Metric Concentration of Sets
			8.2.1 Thickness and Capacity
			8.2.2 Conformal Invariants in the Plane
			8.2.3 Reduced Modulus
		8.3 Tubular Neighborhoods
	9 The Capacity of a Condenser
		9.1 Spherical Symmetrization
		9.2 Grötzsch and Teichmüller Condensers
		9.3 Hyperbolic Metric and Capacity
		9.4 Dimension Cancellation and Special Functions
	10 Conformal Invariants
		10.1 Two Conformal Metrics
		10.2 Ferrand\'s Modulus Metric
		10.3 Teichmüller\'s Function
		10.4 QED Domains
		10.5 Capacitary Geometry
Part IV Intrinsic Geometry
	11 Hyperbolic Type Metrics
		11.1 Metrics Determined by One Boundary Point
		11.2 Metrics Determined by Two Boundary Points
			11.2.1 Special Values of sBn
			11.2.2 Ptolemy-Alhazen Problem and s-Metric
	12 Comparison of Metrics
		12.1 The Unit Ball
		12.2 The Upper Half Space
		12.3 General Domains
	13 Local Convexity of Balls
		13.1 Distance Ratio Metric
		13.2 Quasihyperbolic Balls
		13.3 Apollonian Metric
		13.4 Seittenranta Metric
	14 Inclusion Results for Balls
		14.1 The Punctured Space
		14.2 The Upper Half-Space
Part V Quasiregular Mappings
	15 Basic Properties of Quasiregular Mappings
		15.1 Topological Properties of Discrete Open Mappings
			15.1.1 An Open Problem
		15.2 Path Lifting
		15.3 Analytic Properties of Quasiregular Mappings
		15.4 Curve Families and Quasiconformal Mappings
			15.4.1 Ferrand\'s Problem
			15.4.2 Open Problem
	16 Distortion Theory
		16.1 The Schwarz Lemma and Quasiregular Maps
		16.2 Bounds for Moduli of Continuity
		16.3 The Schwarz Lemma in the Planar Case
			16.3.1 Summary of Main Ideas
		16.4 Further Results
			16.4.1 An Open Problem
	17 Dimension-Free Theory
		17.1 Quasiregular Mappings and Harnack Functions
		17.2 Quasihyperbolic Metric and Quasiregular Mapping
	18 Metrics and Maps
		18.1 An Open Problem on Quasicircles
	19 Teichmüller\'s Displacement Problem
		19.1 On Krzyż\'s Constant
Part VI Solutions
	20 Solutions to Exercises
		20.1 Solution to Exercises in Part II
		20.2 Solution to Exercises in Part III
		20.3 Solution to Exercises in Part IV
		20.4 Solution to Exercises in Part V
A Some Open Problems
	Problems from [553, p. 193]
	New Problems
B Formulary
	Hyperbolic Functions and Their Inverses
Some Biographical Data
References
Index
Author Index