In my self-study of Riemann surfaces ([1]), I am stuck at seeing the connection between the uniformisation theorem and the following statement;
Uniformisation theorem
Every simply connected Riemann surface is conformally equivalent to just one of $\Sigma$ (Riemann sphere), $\mathbb{C}$ (Complex plane), $\mathcal{D}$ (Open unit disc).
Statement
The process of representing many-valued functions in terms of single-valued functions is called uniformisation.
The book gave the following example.
Example
The uniformisation of $x^2 + y^2 = 1$ can be done by $x = \sin t, y = \cos t$.
I can see that as $x^2 + y^2 = 1$ is a unit circle, we can represent x and y coordinates by trig-functions.
But still, it's unclear to me how the theorem relates to the above statement as well as the example above... Can somebody help me?
Reference
[1] G. A. JONES and D. SINGERMAN. Complex Function Theory: an Algebraic and Geometric Viewpoint (Cambridge University Press, 1987).