Teichmüller Spaces

  • Yoichi Imayoshi
  • Masahiko Taniguchi


In this chapter, we shall construct Teichmüller spaces alternatively by using quasiconformal mappings. First, in Section 1, we give a new definition of the Teichmüller space of an arbitrary Riemann surface by using quasiconformal mappings. In Sections 2 and 3, we investigate the case of closed Riemann surfaces of genus g (≥ 2), and prove Teichmüller’s theorem, which states that the Teichmüller space of a closed Riemann surface of genus g (≥ 2) is homeomorphic to the open unit ball in the real (6g – 6)-dimensional Euclidean space. The key of the proof is the existence and uniqueness of the extremal quasiconformal mappings, called Teichmüller mappings.


Riemann Surface Conformal Mapping Quasiconformal Mapping Fuchsian Group Quasi Conformal Mapping 
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Copyright information

© Springer-Verlag Tokyo 1992

Authors and Affiliations

  • Yoichi Imayoshi
    • 1
  • Masahiko Taniguchi
    • 2
  1. 1.Department of Mathematics, College of General EducationOsaka UniversityToyonaka, Osaka 560Japan
  2. 2.Department of Mathematics, Faculty of ScienceKyoto UniversitySakyo-ku, Kyoto 606Japan

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