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Teichmüller Spaces

In: An Introduction to Teichmüller Spaces

Author

Listed:
  • Yoichi Imayoshi

    (Osaka University, Department of Mathematics, College of General Education)

  • Masahiko Taniguchi

    (Kyoto University, Department of Mathematics, Faculty of Science)

Abstract

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.

Suggested Citation

  • Yoichi Imayoshi & Masahiko Taniguchi, 1992. "Teichmüller Spaces," Springer Books, in: An Introduction to Teichmüller Spaces, chapter 0, pages 119-145, Springer.
    • Handle: RePEc:spr:sprchp:978-4-431-68174-8_5
    DOI: 10.1007/978-4-431-68174-8_5
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