How many Riemann surfaces homeomorphic to the sphere are there?

Question

In a previous question, I learned that there exist infinitely many non biholomorphic Riemann surfaces homeomorphic to the torus.

Is it also true for the sphere?

Answer 1

As stated in comments, the answer turns out to be an immediate consequence of the uniformization theorem:

Theorem: Every simply connected Riemann surface is biholomorphic to the open unit disk, the complex plane, or the Riemann sphere.

So if $S$ is a Riemann surface homeomorphic to $\mathbb{S}^2$, it is simply connected, so biholomorphic to the open unit disk, the complex plane or the Riemann sphere. But a biholomorphism is a homeomorphism and neither the open disk nor the complex plane are homeomorphic to $\mathbb{S}^2$. Therefore, $S$ is biholomorphic to the Riemann sphere.