I am trying to understand the uniformization theorem and get some intuition about it.
The uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere.
Specifically, the Riemann mapping theorem states that every simply connected open subset of the complex plane that is different from the complex plane itself admits a conformal and bijective map to the open unit disk.
So, an open square is conformally equivalent to the open circle. But how do I find the bijection?
What about a closed square - is it conformally equivalent to the open circle, and how?
What about an L-shape, made of two orthogonal rectangles?