Tags: #todo #expository

Uniformization Theorem

The uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. In particular it implies that every Riemann surface admits a Riemannian metric of constant curvature.

For compact Riemann surface,

• Those with universal cover the unit disk are precisely the hyperbolic surfaces of genus greater than 1, all with non-abelian fundamental group;
• Those with universal cover the complex plane are the Riemann surfaces of genus 1, namely the complex tori or elliptic curves with fundamental group \({\mathbf{Z}}^{\ast 2}\);
• Those with universal cover the Riemann sphere are those of genus 0, namely the Riemann sphere itself, with trivial fundamental group.

The uniformization theorem is a generalization of the Riemann mapping theorem in Qual Complex Analysis from proper simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.

Every Riemann surface is the quotient of a free, proper and holomorphic action of a discrete group on its universal covering and this universal covering is holomorphically isomorphic (one also says: “conformally equivalent” or “biholomorphic”) to one of the following:

• the Riemann sphere
• the complex plane
• the unit disk in the complex plane.

In 3 dimensions, there are 8 geometries. The geometrization proved by Grigori Perelman states that every 3-manifold can be cut into pieces that are geometrizable.

Smooth Category: Uniformization

Generally expect things to split into more classes.

• Dimension 0: The point (terminal object)
• Dimension 1: \({\mathbb{S}}^1, {\mathbf{R}}^1\)
• Dimension 2: \(\left\langle{{\mathbb{S}}^2, {\mathbb{T}}^2, {\mathbf{RP}}^2 {~\mathrel{\Big\vert}~}{\mathbb{S}}^2 = 0,\,\,3{\mathbf{RP}}^2 = {\mathbf{RP}}^2 + {\mathbb{T}}^2 }\right\rangle\).
  • Classified by \(\pi_1\) (orientability and “genus”). Riemann, Poincare, Klein.
  • Every surface admits a complex structure and a metric. Thus always orientable.
  • Uniformization: Holomorphically equivalent to a quotient of one of three spaces/geometries:
    • \({\mathbf{CP}}^1\), positive curvature (spherical)\
    • \({\mathbf{C}}\), zero curvature (flat, Euclidean)
    • \({\mathbb{H}}\) (equiv. \({\mathbb{D}}^\circ\)), negative curvature (hyperbolic)
  • Stratified by genus:
    • Genus 0: Only \({\mathbf{CP}}^1\)
    • Genus 1: All of the form \({\mathbf{C}}/\Lambda\), with a distinguished point \([0]\), i.e. an elliptic curve.
      • Has a topological group structure!
    • Genus \(\geq 2\): Complicated?