Teichmüller space


Let S0 be a Riemann surface. Consider all pairs (S,f) where S is a Riemann surface and f is a sense-preserving quasiconformal mapping of S0 onto S. We say (S1,f1)(S2,f2) if f2f1-1 is homotopicMathworldPlanetmath to a conformal mappingMathworldPlanetmathPlanetmath of S1 onto S2. In this case we say that (S1,f1) and (S2,f2) are Teichmüller equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath. The space of equivalence classesMathworldPlanetmath under this relationMathworldPlanetmathPlanetmath is called the Teichmüller space T(S0) and (S0,I) is called the initial point of T(S0). The equivalence relation is called Teichmüller equivalence.


There exists a natural Teichmüller metric on T(S0), where the distance between (S1,f1) and (S2,f2) is logK where K is the smallest maximal dilatation of a mapping homotopic to f2f1-1.

There is also a natural isometry between T(S0) and T(S1) defined by a quasiconformal mapping of S0 onto S1. The mapping (S,f)(S,fg) induces an isometric mapping of T(S1) onto T(S0). So we could think of T() as a contravariant functorMathworldPlanetmath from the categoryMathworldPlanetmath of Riemann surfaces with quasiconformal maps to the category of Teichmüller spaces (as a subcategoryMathworldPlanetmath of metric spaces).


  • 1 L. V. Ahlfors. . Van Nostrand-Reinhold, Princeton, New Jersey, 1966
Title Teichmüller space
Canonical name TeichmullerSpace
Date of creation 2013-03-22 14:19:48
Last modified on 2013-03-22 14:19:48
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 8
Author jirka (4157)
Entry type Definition
Classification msc 30F60
Defines Teichmüller metric
Defines Teichmüller equivalence
Defines Teichmüller equivalent