- Uniformization theorem
In
mathematics , the uniformization theorem forsurface s says that any surface admits aRiemannian metric of constantGaussian 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)
#theEuclidean plane (curvature 0)
#thehyperbolic plane (curvature −1).The first case includes all surfaces with positive
Euler characteristic : thesphere and the realprojective plane . The second includes all surfaces with vanishing Euler characteristic: theEuclidean plane , cylinder,Möbius strip ,torus , andKlein 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 theGauss-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 inducesanalmost 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 anyalmost complex structure is integrable, thus turns the given surface into aRiemann 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 adiscrete 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, andGang 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 byGrigori 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.
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