Uniformization Theorem

Definition

Given surface $\Sigma$, there exists unique smooth covering $\Phi :X\to \Sigma$. Here, $X=\{\begin{array}{ll}{S}^2,& \text{if}\mathrm{genus}(\Sigma )=0\\ {\mathbb{R}}^2,& \text{if}\mathrm{genus}(\Sigma )=1\\ {\mathbb{H}}^2,& \text{if}\mathrm{genus}(\Sigma )\ge 2\end{array}$