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2. Moduli of Families of Curves and Extremal Partitions

  • Alexander Vasil’evEmail author
Chapter
  • 289 Downloads
Part of the Lecture Notes in Mathematics book series (LNM, volume 1788)

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

Mathematics Subject Classification (2000):

30C35 30C55 30C62 30C75 30F10 30F60 

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Copyright information

© Springer-Verlag Berlin/Heidelberg 2002

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidad Técnica Federico Santa María Casilla110-V ValparaísoChile

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