About: Uniformization theorem

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In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. In particular it implies that every Riemann surface admits a Riemannian metric of constant curvature. For compact Riemann surfaces, those with universal cover the unit disk are precisely the hyperbolic surfaces of genus greater than 1, all with non-abelian fundamental group; those with universal cover the complex plane are the Riemann surfaces of genus 1, namely the complex tori or elliptic curves with fundamental group Z2; and those with universal cover the Riemann sphere are those of genus zero, namely the Riemann sphere itself, with trivial fundamental group.

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rdf:type	Thing yago:Proposition106750804 yago:Statement106722453 yago:Theorem106752293 yago:WikicatMathematicalTheorems yago:WikicatTheorems yago:WikicatTheoremsInDifferentialGeometry yago:WikicatTheoremsInGeometry yago:Abstraction100002137 yago:Communication100033020 yago:Message106598915
rdfs:label	Théorème d'uniformisation de Riemann Uniformization theorem Teorema di uniformizzazione di Riemann 一意化定理 Uniformeringsstelling 균일화 정리 单值化定理
rdfs:comment	En mathématiques, le théorème d'uniformisation de Riemann est un résultat de base dans la théorie des surfaces de Riemann, c'est-à-dire des variétés complexes de dimension 1. Il assure que toute surface de Riemann simplement connexe peut être mise en correspondance biholomorphe avec l'une des trois surfaces suivantes : le plan complexe C, le disque unité de ce plan, ou la sphère de Riemann, c'est-à-dire la droite projective complexe P1(C). 一意化定理(uniformization theorem)とは、すべての単連結リーマン面は、開円板、複素平面、リーマン球面の 3つのうちのひとつに共形同値であるという定理である。特に、単連結リーマン面は(constant curvature)のリーマン計量を持つ。この定理は普遍被覆リーマン面を楕円型（正の曲率、正の曲がった曲率をもつ）、放物型（平坦）、双曲型(負曲率）として分類する。一意化定理はリーマンの写像定理の平面の固有な単連結開部分集合から、任意の単連結はリーマン面への一般化である。一意化定理は、任意の連結である第二可算の面の同様な結果、定数曲率のリーマン計量を与えることができることを意味している。 Il teorema di uniformizzazione di Riemann è un importante teorema di analisi complessa, dimostrato dal matematico Bernhard Riemann. Il teorema descrive un forte collegamento fra l'analisi complessa e la geometria differenziale per le superfici. 복소해석학에서, 균일화 정리(均一化定理, uniformization theorem)는 단일 연결 리만 곡면이 열린 단위 원판이나 복소평면, 리만 구 가운데 하나로 전단사 이 존재한다는 정리다. 数学上，曲面的单值化定理是说任何曲面上都有一个常高斯曲率的度量。事实上，在每一个给定的中我们都可以找到一个常高斯曲率的度量。等价的說，用复分析的语言，任何单连通的黎曼曲面都共形等价於复平面、单位圆盘和黎曼球面三者之一。 In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. In particular it implies that every Riemann surface admits a Riemannian metric of constant curvature. For compact Riemann surfaces, those with universal cover the unit disk are precisely the hyperbolic surfaces of genus greater than 1, all with non-abelian fundamental group; those with universal cover the complex plane are the Riemann surfaces of genus 1, namely the complex tori or elliptic curves with fundamental group Z2; and those with universal cover the Riemann sphere are those of genus zero, namely the Riemann sphere itself, with trivial fundamental group. In de riemann-meetkunde, een deelgebied van de wiskunde, zegt de uniformeringsstelling dat elk enkelvoudig samenhangende riemann-oppervlak hoekgetrouw equivalent is aan een van de drie domeinen: de open eenheidsschijf, het complexe vlak of de riemann-sfeer. In het bijzonder staat het een riemann-metriek met constante kromming toe. Dit classificeert riemann-oppervlakken als elliptisch (positief gekromd - of beter een constante positieve metriek toelatend), parabolisch (vlak) of hyperbolisch (negatief gekromd) op basis van hun universele overdekking.
rdfs:seeAlso	Planar Riemann surface Geometrization Ricci flow
foaf:isPrimaryTopicOf	wikipedia-en:Uniformization_theorem
dct:subject	Riemann surfaces Theorems in differential geometry
Wikipage page ID	288270(xsd:integer)
Wikipage revision ID	975572500(xsd:integer)
Link from a Wikipage to another Wikipage	Constant curvature Teichmüller space Green's function Grigori Perelman Group action Quasiconformal mapping Mathematische Annalen Harmonic Hyperbolic space Quotient space (topology) Subset Riemann surfaces Dirichlet's principle Dirichlet problem Riemann mapping theorem Mathematische Zeitschrift Acta Mathematica Elliptic operator Harmonic function Quasi-Fuchsian group Liouville's equation Lipman Bers Proceedings of the American Mathematical Society Geometrization conjecture Simultaneous uniformization theorem Isometry group Gauss–Bonnet theorem Sobolev space Sphere Harmonic map Hodge star operator Schwarz alternating method Euler characteristic Hodge theory American Mathematical Society Open set Ricci flow Richard S. Hamilton Riemann sphere Riemann surface Riemannian manifold Second-countable space Conformal geometry Torus Triangulation (topology) Measurable Riemann mapping theorem Complex plane L. E. J. Brouwer Laplace operator Unit disk David Hilbert Beltrami equation Paul Koebe W. V. D. Hodge Exterior derivative Felix Klein Theorems in differential geometry Nicolas Bourbaki Plane (geometry) Discrete group Perron method Radó's theorem (Riemann surfaces) Weyl's lemma (Laplace equation) Elliptic curve Hermann Weyl Hilbert space Simply connected space

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