In mathematics, the simultaneous uniformization theorem, proved by Bers (1960), states that it is possible to simultaneously uniformize two different Riemann surfaces of the same genus 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 of the same genus.
References
- Bers, Lipman (1960), "Simultaneous uniformization", Bulletin of the American Mathematical Society, 66 (2): 94–97, doi:10.1090/S0002-9904-1960-10413-2, ISSN 0002-9904, MR 0111834