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 quasiconformalmapping 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 initialpoint of $T(S_0)$. The equivalence relation is called Teichmüller equivalence.

Definition.

There exists a natural Teichmüller metric on $T(S_0)$, where the distancebetween $(S_1,f_1)$ 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 bya 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 fromthe category of Riemann surfaces with quasiconformal maps to the category ofTeichmüller spaces (as a subcategory of metric spaces).