# 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 $\mathrm{log}K$ 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 |
---|---|

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 |