Let be a Riemann surface. Consider all pairs where is a Riemann surface and is a sense-preserving quasiconformal mapping of onto . We say if is homotopic to a conformal mapping of onto . In this case we say that and are Teichmüller equivalent. The space of equivalence classes under this relation is called the Teichmüller space and is called the initial point of . The equivalence relation is called Teichmüller equivalence.
There exists a natural Teichmüller metric on , where the distance between and is where is the smallest maximal dilatation of a mapping homotopic to .
There is also a natural isometry between and defined by a quasiconformal mapping of onto . The mapping induces an isometric mapping of onto . So we could think of as a contravariant functor from the category of Riemann surfaces with quasiconformal maps to the category of Teichmüller spaces (as a subcategory of metric spaces).
- 1 L. V. Ahlfors. . Van Nostrand-Reinhold, Princeton, New Jersey, 1966
|Date of creation||2013-03-22 14:19:48|
|Last modified on||2013-03-22 14:19:48|
|Last modified by||jirka (4157)|