## Uniformization of Riemann Surface and their automorphisms

$\mathbf{Thm1:}$ Every simply connected Riemann surface must comformally equivalent to one of the followings:

• The Riemann sphere, $\hat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$
• The complex plane, $\mathbb{C}$
• The unit disk. $\mathbb{D}=\{z\in \mathbb{C}||z|<1\}$

$\mathbf{Thm2:}$ Every Riemann surface $M$ is conformally equivalent to $\Sigma/G$ where $\Sigma$ is one of standard types, $\hat{\mathbb{C}}$, $\mathbb{C}$, $\mathbb{D}$. $G$ is a subgroup of $\text{Aut}(\Sigma)$ that acts freely discontinuously on $\Sigma$. Furthermore $G\cong \pi_1(M)$.

$\mathbf{Thm3:}$ Two Riemann surfaces are conformally equivalent if and only if they have the same $\Sigma$ and their $G$ are conjugate in $\text{Aut}(\Sigma)$.

Basic facts:

$\text{Aut}(\hat{\mathbb{C}})$= Mobius  transformations=$PSL(2, \mathbb{C})$= $SL(2,\mathbb{C})/{\pm I}$

$\text{Aut}({\mathbb{C}})$=$\{az+b|a, b\in \mathbb{C}, a\neq 0\}=\text{Aff}(1,\mathbb{C})$

$\text{Aut}({\mathbb{D}})\cong PSL(2,\mathbb{R})$