Reference for Complex Manifolds for Great Picard Theorem and Julia Sets

Question

I’ve recently been looking to understand the proofs of some theorems about complex analysis and Julia sets that require results on complex manifolds, especially The Great Picard Theorem. I need The Uniformization Theorem, then I need to understand why the plane with two points removed can’t have the entire complex plane as its cover. I know basic complex analysis, real differential topology and results on coverings from algebraic topology. Where can I find a text with the sort of results I need?

Answer 1

Once one knows the uniformization theorem, the universal cover $\widetilde{X}$ of $X =\mathbb{C} - \{p_1, p_2\}$ is a simply connected noncompact Riemann surface and so is uniformized to either $\mathbb{C}$ or $\mathbb{D}$. As the universal cover of $X$, $\pi_1(X) \cong \mathbb{Z} *\mathbb{Z}$ acts on $\widetilde{X}$ by holomorphic automorphisms. Any holomorphic automorphism of the plane is affine complex-linear, i.e. $z \mapsto az + b, a \neq 0$. One can check that the group of affine complex-linear maps of $\mathbb{C}$ contains no $\mathbb{Z} * \mathbb{Z}$ subgroup, which rules out the possibility.

Proofs of the uniformization theorem all require substantial machinery on Riemann surfaces. Many people starting studying subjects that use the uniformization theorem as a tool treat it as a black box for a while and come back to it once they have decided to specialize in the field. Learning a proof of it is an investment of energy that does not give a particularly good feeling if one wants to pursue a PhD in complex dynamics, for instance.

If you want to read a proof of the uniformization theorem, I am partial to Donaldson's proof in his book Riemann Surfaces. It uses differential forms and some techniques that are more broadly useful. Its main tool is a "main theorem" for Riemann surfaces about inversion of the Laplacian. This theorem can be read in Donaldson. I learned it from Royden's article Function Theory on Riemann Surfaces, which is a great paper if one is interested in Riemann surfaces and has some background on analysis and differential geometry (e.g. Stokes' theorem, ${L}^2$).