Every **simply connected** Riemann surface must comformally equivalent to one of the followings:

- The Riemann sphere,
- The complex plane,
- The unit disk.

Every Riemann surface is conformally equivalent to where is one of standard types, , , . is a subgroup of that acts freely discontinuously on . Furthermore .

Two Riemann surfaces are conformally equivalent if and only if they have the same and their are conjugate in .

Basic facts:

= Mobius transformations==

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