Uniformization theorem

Remarks

The three surfaces are pairwise not biholomorphic: $\hat\C$ is compact and the others are not, and a holomorphic map $\C \to \mathbb{D}$ is constant by Liouville's theorem. Applied to the universal cover, the theorem shows that every Riemann surface $X$ is biholomorphic to a quotient $\tilde X / \Gamma$. Here $\tilde X$ is one of the three surfaces and $\Gamma$ is a group of automorphisms of $\tilde X$ acting freely and properly discontinuously. Only $\hat\C$ itself is covered by $\hat\C$, and only $\C$, $\C \setminus \set{0}$ and the tori $\C / \Lambda$ are covered by $\C$. Every other Riemann surface is a quotient $\mathbb{D} / \Gamma$ with $\Gamma$ a Fuchsian group. For simply connected open subsets of $\C$ the theorem contains the Riemann mapping theorem.

The theorem was proved in 1907, independently by Koebe and by Poincaré. One proof constructs a harmonic Green's function or a harmonic dipole function on $X$ and uses it to build a holomorphic injection into $\hat\C$.

Example: $\C \setminus \set{0}$ is not simply connected; its universal cover is $\exp \colon \C \to \C \setminus \set{0}$, and $\C \setminus \set{0} \cong \C / 2\pi i \Z$.

Going deeper

For a student

The theorem gives every Riemann surface a metric of constant curvature $+1$, $0$ or $-1$ compatible with its complex structure. Compact surfaces of genus $0$ are spherical, genus $1$ flat, and genus at least $2$ hyperbolic, a fact that links complex analysis to hyperbolic geometry. It is the starting point of Teichmüller theory, because a complex structure on a surface of genus $g \ge 2$ becomes a Fuchsian group up to conjugation.