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.
KeywordsRiemann Surface Conformal Mapping Quasiconformal Mapping Fuchsian Group Quasi Conformal Mapping
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