2. Moduli of Families of Curves and Extremal Partitions

Abstract

• 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 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