# 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).

## References

• 1 L. V. Ahlfors. . Van Nostrand-Reinhold, Princeton, New Jersey, 1966
Title Teichmüller space TeichmullerSpace 2013-03-22 14:19:48 2013-03-22 14:19:48 jirka (4157) jirka (4157) 8 jirka (4157) Definition msc 30F60 Teichmüller metric Teichmüller equivalence Teichmüller equivalent