## Covering Maps and Picard’s Little Theorem.

Here is a pretty wild theorem that generalizes Liouville's theorem in complex analysis: Picard's Little Theorem: If a holomorphic, entire function $f:\mathbb C \to \hat{\mathbb C}$ misses three points, it is constant. This theorem is remarkable, and I had never really heard of it, since its proof relies on a certain covering. The following argument can be made elementary (by describing a certain covering) but instead we will spend some time talking about some implications of the Uniformization Theorem, and finish with an "easy" lifting argument. Note that Picard's Little theorem works just as well for $f: \mathbb C \to \mathbb C$, since we can remove a point, and rotate to stereographically project, so we replace three points with two points. Uniformization Theorem: every simply connected Riemann surface is the…