Editing Uniformization theorem (section)

==Generalizations==

Koebe proved the '''general uniformization theorem''' that if a Riemann surface is homeomorphic to an open subset of the complex sphere (or equivalently if every Jordan curve separates it), then it is conformally equivalent to an open subset of the complex sphere.

In 3 dimensions, there are 8 geometries, called the [[Geometrization conjecture#The eight Thurston geometries|eight Thurston geometries]]. Not every 3-manifold admits a geometry, but Thurston's [[geometrization conjecture]] proved by [[Grigori Perelman]] states that every 3-manifold can be cut into pieces that are geometrizable.

The [[simultaneous uniformization theorem]] of [[Lipman Bers]] shows that it is possible to simultaneously uniformize two compact Riemann surfaces of the same genus >1 with the same [[quasi-Fuchsian group]].

The [[measurable Riemann mapping theorem]] shows more generally that the map to an open subset of the complex sphere in the uniformization theorem can be chosen to be a [[quasiconformal map]] with any given bounded measurable Beltrami coefficient.