Moduli of Families of Curves for Conformal and Quasiconformal Mappings

1. Introduction 2. Moduli of Families of Curves and Extremal Partitions 2.1 Simple definition and properties of the modulus 2.1.1 Definition 2.1.2 Properties 2.1.3 Examples 2.1.4 Grötzsch lemmas 2.1.5 Exercises 2.2 Reduced moduli and capacity 2.2.1 Reduced modulus 2.2.2 Capacity and the transfinite diameter 2.2.3 Digons, triangles and their reduced moduli 2.3 Elliptic functions and integrals 2.3.1 Elliptic functions 2.3.2 Elliptic integrals and Jacobi's functions 2.4 Some frequently used moduli 2.4.1 Moduli of doubly connected domains 2.4.2 Moduli of quadrilaterals 2.4.3 Reduced moduli 2.4.4 Reduced moduli of digons 2.5 Symmetrization and polarization 2.5.1 Circular symmetrization 2.5.2 Polarization 2.6 Quadratic differentials on Riemann surfaces 2.6.1 Riemann surfaces 2.6.2 Quadratic differentials 2.6.3 Local trajectory structure 2.6.4 Trajectory structure in the large 2.7 Free families of homotopy classes of curves and extremal partitions 2.7.1 The case of ring domains and quadrangles 2.7.2 The case of circular, strip domains, and triangles 2.7.3 Continuous and differentiable moduli 3. Moduli in Extremal Problems for Conformal Mapping 3.1 Classical extremal problems for univalent functions 3.1.1 Koebe set, growth, distortion 3.1.2 Lower boundary curve for the range of (/f(z)/,/f(z)/) 3.1.3 Special moduli 3.1.4 Upper boundary curve for the range of (/f(z)/,/f(z)/) 3.2 Two-point distortion for univalent functions 70 3.2.1 Lower boundary curve for the range of (/f(r 1)/,/f(r 2)/) in S R 70 3.2.2 Special moduli 3.2.3 Upper boundary curve for the range of (/f(r 1)/,/f(r 2)/) in S R 3.2.4 Upper boundary curve for the range of (/f(r 1)/, /f(r 2)/) in S 3.3 Bounded univalent functions 3.3.1 Elementary estimates 3.3.2 Boundary curve for the range of (/f(z)/,/f(z)/) in B s(b) 3.4 Montel functions 3.4.1 Covering theorems 3.4.2 Distortion at the points of normalization 3.4.3 The range of (/f(r)/,/f(r)/) in M R(w) 3.5 Univalent functions with the angular derivatives 3.5.1 Estimates of the angular derivatives)(120) 3.5.2 The range of (/f(r)/, /f(0)/) 4. Moduli in Extremal Problems for Quasiconformal Mapping 4.1 General information and simple extremal problems 4.1.1 Quasiconformal mappings of Riemann surfaces 4.1.2 Growth and Hölder continuity 4.1.3 Quasiconformal motion of a quadruple of points 4.2 Two-point distortion for quasiconformal maps of the plane 4.2.1 Special differentials and extremal partitions 4.2.2 Quasisymmetric functions and the extremal maps 4.2.3 Boundary parameterization 4.2.4 The class Q K. Estimations of functionals 4.2.5 Conclusions and unsolved problems 166 4.3 Two-point distortion for quasiconformal maps of the unit disk 4.3.1 Special differentials and extremal partitions 4.3.2 Extremal problems 5. Moduli on Teichmüller Spaces 5.1 Some information on Teichmüller spaces 5.2 Moduli on Teichmüller spaces 5.2.1 Variational formulae 5.2.2 Three lemmas 5.3 Harmonic properties of the moduli 5.4 Descriptions of the Teichmüller metric 5.5 Invariant metrics References List of symbols Index '''