Here is a pretty wild theorem that generalizes Liouville’s theorem in complex analysis:
Picard’s Little Theorem: If a holomorphic, entire function 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 , 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 sphere, plane, or disk (up to bilomorphism).
Of course, the remarkable fact here is that we can now obtain every Riemann surface as the quotient of each one of these, which tells us a lot about the possible structures. In particular, note that deck transformations are also biholomorphisms, since they just permute sections each having a perfectly good complex structure on them.
Hence, we can see that since self-biholomorphisms of are just Mobius transformations, they also have fixed points, so they cannot be deck transformations, so the sphere is not covering any other Riemann surfaces.
Similarly, any holomorphic bijection of diverges as , and hence extends to a bijection of that fixes , so it is a mobius transformation that is equivalent to a pure translation , so any group of deck transformations must also be abelian, and we see that can cover (topologically!) a cylinder, itself, or the genus surface.
Hence, all the rest are covered by the disk (so they are hyperbolic.) In particular, automorphisms that map the disk to itself are precisely of the form , and the transformations are either rotations around a point in the disk, limit rotations (on the boundary) or hyperbolic translations (loxodromies.) We can see that rotations around a point on the interior have an obvious fixed point, so cannot be deck transformations. Hence loxodromies and limit rotations (although this wont be importatant.)
Proof of Picard’s Little Theorem:
If misses three points, it is a map to the thriced punctured sphere, which evidently is covered by the hyperbolic plane, so factors , and by Liouvilles theorem, the first map is constant.
Haha, what a good time that proof is. The covering is famous by the way: it is called the elliptic modular function.