Moduli of Families of Curves for Conformal and Quasiconformal Mappings [Paperback]

The monograph is concerned with the modulus of families of curves on Riemann surfaces and its applications to extremal problems for conformal, quasiconformal mappings, and the extension of the modulus onto Teichm?ller spaces. The main part of the monograph deals with extremal problems for compact classes of univalent conformal and quasiconformal mappings. Many of them are grouped around two-point distortion theorems. Montel's functions and functions with fixed angular derivatives are also considered. The last portion of problems is directed to the extension of the modulus varying the complex structure of the underlying Riemann surface that sheds some new light on the metric problems of Teichm?ller spaces.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 Gr?tzsch 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 rangl£Ý