Uniformization theorem

In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere. In particular it admits a Riemannian metric of constant curvature. 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 is a generalization of the Riemann mapping theorem from proper simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.

The uniformization theorem implies a similar result for arbitrary connected second countable surfaces: they can be given Riemannian metrics of constant curvature.

History

Felix Klein (1883) and Henri Poincaré (1882) conjectured the uniformization theorem for (the Riemann surfaces of) algebraic curves. Henri Poincaré (1883) extended this to arbitrary multivalued analytic functions and gave informal arguments in its favor. The first rigorous proofs of the general uniformization theorem were given by Poincaré (1907) and Paul Koebe (1907a, 1907b, 1907c). Paul Koebe later gave several more proofs and generalizations. The history is described in Gray (1994).

Complex classification

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.

Geometric classification of surfaces

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.

From this, a classification of metrizable surfaces follows. A connected metrizable 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. 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 −∞,0,1 of the corresponding complex algebraic curve.　For Riemann surfaces, Rado's theorem implies that the surface is automatically second countable. For general surfaces this is no longer true, so for the classification above one needs to assume that the surface is second countable (or metrizable). The Prüfer surface is an example of a surface with no (Riemannian) metric.

Connection to Ricci flow

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. Xiuxiong Chen, Peng Lu, and Gang Tian (2006) showed that it is nevertheless possible to prove the uniformization theorem via Ricci flow.

Related theorems

Koebe proved the general uniformization theorem that if a Riemann surface is homeomorphic to an open subset of the complex sphere (or equivalently if every Jordan curve separates it), then it is conformally equivalent to an open subset of the complex sphere.

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.

The simultaneous uniformization theorem of Lipman Bers shows that it is possible to simultaneously uniformize two compact Riemann surfaces of the same genus >1 with the same quasi-Fuchsian group.

The measurable Riemann mapping theorem shows more generally that the map to an open subset of the complex sphere in the uniformization theorem can be chosen to be a quasiconformal map with any given bounded measurable Beltrami coefficient.

External links

• Conformal Transformation: from Circle to Square.