Uniformization theorem

In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gaussian curvature. In fact, one can find a metric with constant Gaussian curvature in any given conformal class.

As a special case, every simply connected Riemann surface is conformally equivalent to one of the three canonical domains: the open unit disk, the complex plane, or the Riemann sphere, and these are not conformally equivalent to each other. This classifies Riemannian surfaces as elliptic (positively curved – rather, admitting a constant positively curved metric), parabolic (flat), and hyperbolic (negatively curved) according to their universal cover.

The uniformization theorem was proved by the work of Felix Klein in the 1880s, then Paul Koebe and Henri Poincaré in the 1900s, with the general result proven in (Poincaré 1907); accordingly, the theorem is sometimes called the Klein–Poincaré uniformization theorem.

Geometric classification of surfaces

From this, a classification of surfaces follows. A surface is a quotient of one of the following by a free action of a discrete subgroup of an isometry group:

1. the sphere (curvature +1)
2. the Euclidean plane (curvature 0)
3. the hyperbolic plane (curvature −1).

The first case includes all surfaces with positive Euler characteristic: the sphere and the real projective plane. The second includes all surfaces with vanishing Euler characteristic: the Euclidean plane, cylinder, Möbius strip, torus, and Klein bottle. The third case covers all surfaces with negative Euler characteristic: almost all surfaces are hyperbolic. Note that, for closed surfaces, this classification is consistent with the Gauss-Bonnet Theorem, which implies that for a closed surface with constant curvature, the sign of that curvature must match the sign of the Euler characteristic.

The positive/flat/negative classification corresponds in algebraic geometry to Kodaira dimension -1,0,1 of the corresponding complex algebraic curve.

Complex classification

On an oriented surface, a Riemannian metric naturally induces an almost complex structure as follows: For a tangent vector v we define J(v) as the vector of the same length which is orthogonal to v and such that (v, J(v)) is positively oriented. On surfaces any almost complex structure is integrable, so this turns the given surface into a Riemann surface. Therefore the above classification of orientable surfaces of constant Gauss curvature is equivalent to the following classification of 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") to one of the following:

1. the Riemann sphere
2. the complex plane
3. the unit disk in the complex plane.

Connection to Ricci flow

Further information: Ricci flow § Relationship_to_uniformization_and_geometrization, and this relationship

In introducing the Ricci flow, Richard Hamilton showed that the Ricci flow on a closed surface uniformizes the metric (i.e., the flow converges to a constant curvature metric). However, his proof relied on the uniformization theorem. In 2006, it was pointed out by Xiuxiong Chen, Peng Lu, and Gang Tian that it is nevertheless possible to prove the uniformization theorem via Ricci flow.

3-manifold analog

In 3 dimensions, there are 8 geometries, called the eight Thurston geometries. Not every 3-manifold admits a geometry, but Thurston's geometrization conjecture proved by Grigori Perelman states that every 3-manifold can be cut into pieces that are geometrizable.

Examples

Uniformization of complement of Mandelbrot set

Uniformization of complement of Julia set

• Boettcher function - uniformizing map of basin of attraction of infinity (superattracting)
• Koenigs function - uniformizing map of basin of attraction of finite attractor (attracting but not superattracting)

See also

• Carathéodory's theorem

References

• Poincaré, Henri (1907), "Sur l'uniformisation des fonctions analytiques", Acta Math., 31: 1–64
• Xiuxiong Chen, Peng Lu, and Gang Tian, A note on uniformization of Riemann surfaces by Ricci flow, Proceedings of the AMS. vol. 134, no. 11 (2006) 3391--3393.

External links

• Uniformization, N.A. Gusevskii, Encyclopaedia of Mathematics – ISBN 1-40200609-8