Teichmüller theory

In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parameterizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Each point in T(S) may be regarded as an isomorphism class of 'marked' Riemann surfaces where a 'marking' is an isotopy class of homeomorphisms from S to itself.

It can also 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 6g − 6 for a surface of genus g. 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 a very rich subject of research.

Teichmüller spaces are named after Oswald Teichmüller.

Moduli spaces for Riemann surfaces and related Fuchsian groups have been studied since the work of Bernhard Riemann, who knew that 6g − 6 parameters were needed to describe the variations of complex structures on a surface of genus g. The early study of Teichmüller space, in the late nineteenth–early twentieth century, was geometric and founded on the interpretation of Riemann surfaces as hyperbolic surfaces. Among the main contributors were Felix Klein, Henri Poincaré, Paul Koebe, Jakob Nielsen, Robert Fricke, Werner Fenchel.

...
Wikipedia