Uniformization theorem
Statement
There are two components to the statement:
- Any simply connected Riemann surface is conformally equivalent to one of these three: the open unit disk, the complex plane, the Riemann sphere.
- Any Riemann surface can be expressed as the quotient of its universal cover (which is one of these) by the action of the fundamental group (the quotient is in the sense of Riemann surfaces).
Related facts
Particular cases/applications
- Genus zero Riemann surface is conformally equivalent to Riemann sphere: States that any compact, simply connected Riemann surface is conformally equivalent to the Riemann sphere.
- Riemann mapping theorem