The uniformisation theorem is usually stated as: every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. But why is that a theorem about uniformisation? What's being made uniform?
As I understand it, there are two related theorems that could be called uniformisation theorems, which I'll call I and II:
Uniformisation Theorem I. Given a multivalued analytic function $y$ of a variable $x$, find a variable $z$ such that $y$ and $x$ are single-valued functions of $z$.
Uniformisation Theorem II. Every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere.
My understanding is that you can use II to prove I. Is that right? If so, is that the (or: a?) reason to call II uniformisation, since a single-valued function is, in French, une fonction uniforme? Or is there some more direct reason to see II as making something uniform?