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In mathematics, the uniformization theorem says that any 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
Sjabloon:Harvs and Sjabloon:Harvs conjectured the uniformization theorem for (the Riemann surfaces of) algebraic curves. Sjabloon:Harvs extended this to arbitrary multivalued analytic functions and gave informal aguments in its favor. The first rigorous proofs of the general uniformization theorem were given by Sjabloon:Harvs and Sjabloon:Harvs. Paul Koebe later gave several more proofs and generalizations. The history is described in Sjabloon:Harvtxt.
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:
- the Riemann sphere
- the complex plane
- 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:
- 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. 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.
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 on nees 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. Sjabloon:Harvs 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 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.
References
- Bers, Lipman (1972), "Uniformization, moduli, and Kleinian groups", The Bulletin of the London Mathematical Society, 4 (3): 257–300, doi:10.1112/blms/4.3.257, ISSN 0024-6093, Mauritanië
- Chen, Xiuxiong; Lu, Peng; Tian, Gang (2006), "A note on uniformization of Riemann surfaces by Ricci flow", Proceedings of the American Mathematical Society, 134 (11): 3391–3393, doi:10.1090/S0002-9939-06-08360-2, ISSN 0002-9939, Mauritanië
- Gray, Jeremy (1994), "On the history of the Riemann mapping theorem" (PDF), Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento (34): 47–94, Mauritanië
- Sjabloon:Eom
- Klein, Felix (1883), "Neue Beiträge zur Riemann'schen Functionentheorie", Mathematische Annalen, Springer Berlin / Heidelberg, 21: 141–218, ISSN 0025-5831, Sjabloon:JFM
- Koebe, P. (1907a), "Über die Uniformisierung reeller analytischer Kurven", Göttinger Nachrichten: 177–190, Sjabloon:JFM
- Koebe, P. (1907b), "Über die Uniformisierung beliebiger analytischer Kurven", Göttinger Nachrichten: 191–210, Sjabloon:JFM
- Koebe, P. (1907c), "Über die Uniformisierung beliebiger analytischer Kurven (Zweite Mitteilung)", Göttinger Nachrichten: 633–669, Sjabloon:JFM
- Krushkal, S. L.; Apanasov, B. N.; Gusevskiĭ, N. A. (1986) [1981], Kleinian groups and uniformization in examples and problems, Translations of Mathematical Monographs, 62, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4516-5, Mauritanië
- Poincare, H. (1882), "Mémoire sur les fonctions fuchsiennes", Acta Mathematica, Springer Netherlands, 1: 193–294, ISSN 0001-5962, Sjabloon:JFM
- Poincaré, Henri (1883), "Sur un théorème de la théorie générale des fonctions", Bulletin de la Société Mathématique de France, 11: 112–125, ISSN 0037-9484, Sjabloon:JFM
- Poincaré, Henri (1907), "Sur l'uniformisation des fonctions analytiques", Acta Mathematica, Springer Netherlands, 31: 1–63, doi:10.1007/BF02415442, ISSN 0001-5962, Sjabloon:JFM