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 vers ...

s of the same

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

using a

quasi-Fuchsian group In the mathematical theory of Kleinian groups, a quasi-Fuchsian group is a Kleinian group whose limit set is contained in an invariant Jordan curve. If the limit set is equal to the Jordan curve the quasi-Fuchsian group is said to be of type one, a ...

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. Teichmülle ...

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