Moduli of Families of Curves for Conformal and Quasiconformal Mappings

ISBN-13: 9783540438465 / Angielski / Miękka / 2002 / 214 str.

Alexander Vasil'ev

Opis

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 Teichmuller 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 Teichmuller spaces.

Spis treści

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 '''