Teichmüller space

Definition.

Let $S_0$ be a Riemann surface. Consider all pairs $(S,f)$ where $S$ is a Riemann surface and $f$ is a sense-preserving quasiconformal mapping of $S_0$ onto $S$. We say $(S_1,f_1)\sim (S_2,f_2)$ if $f_2\circ f_1^{-1}$ is homotopic to a conformal mapping of $S_1$ onto $S_2$. In this case we say that $(S_1,f_1)$ and $(S_2,f_2)$ are Teichmüller equivalent. The space of equivalence classes under this relation is called the Teichmüller space $T(S_0)$ and $(S_0,I)$ is called the initial point of $T(S_0)$. The equivalence relation is called Teichmüller equivalence.

Definition.

There exists a natural Teichmüller metric on $T(S_0)$, where the distance between $(S_1,f_1)$ and $(S_2,f_2)$ is $\log K$ where $K$ is the smallest maximal dilatation of a mapping homotopic to $f_2\circ f_1^{-1}$.

There is also a natural isometry between $T(S_0)$ and $T(S_1)$ defined by a quasiconformal mapping of $S_0$ onto $S_1$. The mapping $(S,f)\mapsto (S,f\circ g)$ induces an isometric mapping of $T(S_1)$ onto $T(S_0)$. So we could think of $T(\cdot )$ 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).

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