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.In other words every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere. [ [http://www.math.harvard.edu/theses/senior/chan/fulldraft7.pdf Uniformization of Riemann Surfaces. Kevin Timothy Chan. Thesis from Harvard Mathematics Department ,April 5, 2004] ]

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:

#the sphere (curvature +1)
#the Euclidean plane (curvature 0)
#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 inducesan 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, thus 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:

#the Riemann sphere
#the complex plane
#the unit disc in the complex plane.

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. 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.

References

*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.