# Three Important Riemann Surfaces

/But before I do, let's have

## A Little Motivation

Perhaps you're wondering,

*Why should I care what the automorphisms of these four spaces look like? (Er, besides the fact that I need it for tonight's homework!)*

*But why is this nice? *

Because, from a topological viewpoint, it tells us that the universal cover of any Riemann surface will - up to conformal equivalence - either be $\hat{\mathbb{C}}$ or $\mathbb{C}$ or $\Delta$!

*Okay, so?*

*Soooo,* the universal cover is a very important object in mathematics. And the fact that there are only *three* options (complex-analytically speaking) for the universal cover of any given Riemann surface is quite nice!

*And why are universal covers so important?!*

This - the idea of lifting your (mathematical) problems to the universal cover - is just one technique that mathematicians use to gain information about a space $X$. Another well-known technique is to study functions *on* or *to* $X$. (For example, one may want to study functions $f$ from a circle $S^1$ into $X$. This results in a very powerful tool called the fundamental group!) In particular, one can study maps from $X$ to itself - and those are precisely the automorphisms of the space!

So you see? We really do care about the automorphisms of $\hat{\mathbb{C}}$, $\mathbb{C}$, and $\Delta\cong\mathcal{U}$! (To my more knowledgeable readers: feel free to provide us with more motivation in the comments!)

So far I've just rambled on a bit, but I hope this gives you at least a little motivation/background for the proofs in this series. Admittedly, the next few posts will be mostly computational and thus - in my opinion - not terribly exciting. But even so, I want to include them on the blog just in case a student or two may find them helpful.

Next time, I'll start by proving that all automorphisms of the unit disc $\Delta$ can be expressed in the form $f(z)=\frac{az+b}{\bar bz+\bar a}$ where $a,b\in\mathbb{C}$ satisfy $|a|^2-|b|^2=1$.

Until then!