I am interested in a proof of the following fact :

Suppose that $X$ is a Riemann surface homeomorphic to the Riemann sphere. Then $X$ is *conformally equivalent* to the Riemann sphere.

Of course, this follows from the uniformization theorem which states that every simply connected Riemann surface is conformally equivalent to either the open unit disk, the complex plane or the Riemann sphere.

However, I was wondering if it was possible to prove this without using the full classification given by the uniformization theorem.

Any relevant reference is welcome.

Thank you and best regards, Malik