In mathematics, the simultaneous uniformization theorem, proved by , states that it is possible to simultaneously uniformize two different

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...

s of the same

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...

using a quasi-Fuchsian group of the first kind. The quasi-Fuchsian group is essentially uniquely determined by the two Riemann surfaces, so the space of marked quasi-Fuchsian group of the first kind of some fixed genus ''g'' can be identified with the product of two copies of

Teichmüller space In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüll ...

of the same genus.

References

*{{Citation , last1=Bers , first1=Lipman , authorlink=Lipman Bers, title=Simultaneous uniformization , doi=10.1090/S0002-9904-1960-10413-2 , mr=0111834 , year=1960 , journal=

Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. I ...

, issn=0002-9904 , volume=66 , issue=2 , pages=94–97, doi-access=free Kleinian groups Riemann surfaces