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Uniformization theorem for Riemann surfaces
How does one prove that every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere, and these are not conformally equivalent to each other?
I would like to know about different ways of proving it, and appropriate references. This is not to know the best way; but to know about various possible approaches. Therefore I wouldn't be choosing a best answer.