uniformization theorem

E259768

mathematical theorem theorem in complex analysis

The uniformization theorem is a fundamental result in complex analysis stating that every simply connected Riemann surface is conformally equivalent to either the Riemann sphere, the complex plane, or the unit disk.

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All labels observed (1)

Label	Occurrences
uniformization theorem canonical	3

Statements (49)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in complex analysis ⓘ
appliesTo	connected 1-dimensional complex manifolds ⓘ open Riemann surfaces ⓘ
classifiesAs	Riemann sphere ⓘ complex plane ⓘ unit disk ⓘ
concerns	conformal equivalence classes of Riemann surfaces ⓘ simply connected Riemann surfaces ⓘ universal covering surfaces ⓘ
describes	classification of simply connected Riemann surfaces ⓘ
field	Riemann surface theory ⓘ complex analysis ⓘ differential geometry ⓘ geometric function theory ⓘ
generalizes	Riemann mapping theorem ⓘ
hasConsequence	existence of universal covering Riemann surface of canonical type ⓘ trichotomy of Riemann surfaces into elliptic, parabolic, and hyperbolic types ⓘ
hasModelType	elliptic type (Riemann sphere) ⓘ hyperbolic type (unit disk) ⓘ parabolic type (complex plane) ⓘ
historicallyAssociatedWith	Henri Poincaré ⓘ Paul Koebe ⓘ
implies	every Riemann surface is a quotient of the sphere, plane, or disk by a group of automorphisms ⓘ every simply connected Riemann surface is conformally equivalent to a canonical model surface ⓘ existence of constant curvature metrics on simply connected Riemann surfaces ⓘ
isConsidered	cornerstone of Riemann surface theory ⓘ cornerstone of modern complex analysis ⓘ
isFundamentalIn	Teichmüller theory ⓘ classification theory of Riemann surfaces ⓘ theory of Fuchsian groups ⓘ
provedIndependentlyBy	Henri Poincaré ⓘ Paul Koebe ⓘ
relatedConcept	Fuchsian group ⓘ surface form: Fuchsian groups Kleinian group ⓘ surface form: Kleinian groups automorphism group of the unit disk ⓘ covering space theory ⓘ
relatesTo	Poincaré metric ⓘ Riemann mapping theorem ⓘ elliptic geometry ⓘ hyperbolic geometry ⓘ parabolic geometry ⓘ
statement	Every simply connected Riemann surface is conformally equivalent to the Riemann sphere, the complex plane, or the unit disk. ⓘ
usesConcept	Riemann surfaces ⓘ surface form: Riemann surface conformal map ⓘ holomorphic function ⓘ simply connectedness ⓘ universal covering map ⓘ
yearProvedApprox	1907 ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Riemann surfaces → hasTheorem → uniformization theorem ⓘ

subject surface form: Riemann surface

Riemann mapping theorem → relatedTo → uniformization theorem ⓘ

Hilbert’s twenty-second problem → relatedConcept → uniformization theorem ⓘ