Teichmüller space

In mathematics, the Teichmüller space ${\displaystyle T(S)}$ of a (real) topological (or differential) surface ${\displaystyle S}$ is a space that parametrizes complex structures on ${\displaystyle S}$ up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller.

Each point in a Teichmüller space ${\displaystyle T(S)}$ may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from ${\displaystyle S}$ to itself. It can be viewed as a moduli space for marked hyperbolic structure on the surface, and this endows it with a natural topology for which it is homeomorphic to a ball of dimension ${\displaystyle 6g-6}$ for a surface of genus ${\displaystyle g\geq 2}$. In this way Teichmüller space can be viewed as the universal covering orbifold of the Riemann moduli space.

The Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The study of geometric features of these various structures is an active body of research.

The sub-field of mathematics that studies the Teichmüller space is called Teichmüller theory.