Riemann mapping theorem

E47349

mathematical theorem theorem in complex analysis

The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.

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All labels observed (1)

Label	Occurrences
Riemann mapping theorem canonical	13

How this entity was disambiguated

This entity first appeared as the object of triple T373781 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Riemann mapping theorem
Context triple: [Bernhard Riemann, knownFor, Riemann mapping theorem]

A. Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
B. Israel–Carter–Robinson uniqueness theorems
The Israel–Carter–Robinson uniqueness theorems are a set of results in general relativity showing that stationary, asymptotically flat black holes in four-dimensional spacetime are completely characterized by just their mass, charge, and angular momentum.
C. Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
D. Theorema Egregium
Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
E. Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Riemann mapping theorem
Target entity description: The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.

A. Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
B. Israel–Carter–Robinson uniqueness theorems
The Israel–Carter–Robinson uniqueness theorems are a set of results in general relativity showing that stationary, asymptotically flat black holes in four-dimensional spacetime are completely characterized by just their mass, charge, and angular momentum.
C. Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
D. Theorema Egregium
Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
E. Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
F. None of above. chosen

Statements (50)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in complex analysis ⓘ
appliesTo	bounded simply connected planar domains not equal to the plane ⓘ simply connected Jordan domains ⓘ
category	result about planar domains ⓘ
codomainCondition	open unit disk in the complex plane ⓘ
conclusion	Any two such domains are conformally equivalent ⓘ There exists a biholomorphic map from the given domain onto the open unit disk ⓘ
consequence	classification of simply connected Riemann surfaces of genus 0 as sphere, plane, or disk (with additional results) ⓘ
doesNotApplyTo	multiply connected domains ⓘ the entire complex plane ⓘ
domainCondition	non-empty open subset of the complex plane ⓘ simply connected subset of the complex plane ⓘ subset of the complex plane not equal to the whole complex plane ⓘ
excludes	entire complex plane ⓘ
field	complex analysis ⓘ
generalizationOf	existence of conformal maps from simply connected domains to canonical domains ⓘ
guaranteesExistenceOf	holomorphic bijection with holomorphic inverse onto the unit disk ⓘ
hasCanonicalTarget	open unit disk ⓘ
historicalPeriod	19th century mathematics ⓘ
implies	All simply connected proper domains in the complex plane are conformally equivalent ⓘ The unit disk is a universal model for simply connected proper planar domains ⓘ
importance	cornerstone of geometric function theory ⓘ fundamental classification result for simply connected planar domains ⓘ
language	complex variable theory ⓘ
mapType	biholomorphic map ⓘ conformal map ⓘ
namedAfter	Bernhard Riemann ⓘ
normalizationCondition	derivative at the chosen point is real and positive ⓘ map sends a chosen point to 0 in the unit disk ⓘ
proofMethod	Dirichlet principle (historically) ⓘ Montel theorem ⓘ extremal problems for holomorphic functions ⓘ normal families ⓘ
relatedTo	Carathéodory theorem ⓘ linked to: Carathéodory’s theorem in convex geometry Koebe quarter theorem ⓘ Schwarz lemma ⓘ conformal mapping theory ⓘ uniformization theorem ⓘ
statement	Every non-empty simply connected open subset of the complex plane that is not the entire plane is conformally equivalent to the open unit disk ⓘ
typicalApplication	boundary value problems in two dimensions ⓘ construction of conformal coordinates ⓘ
uniquenessCondition	The conformal map is unique if one fixes the image of a point and the argument of the derivative at that point ⓘ The conformal map is unique up to post-composition with a conformal automorphism of the unit disk ⓘ
usesConcept	biholomorphism ⓘ conformal equivalence ⓘ holomorphic function ⓘ open set in the complex plane ⓘ simply connected domain ⓘ unit disk ⓘ

How these facts were elicited

Referenced by (13)

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

Bernhard Riemann → knownFor → Riemann mapping theorem ⓘ

Friedrich Bernhard Riemann → notableConcept → Riemann mapping theorem ⓘ

subject linked to: Friedrich

uniformization theorem → relatesTo → Riemann mapping theorem ⓘ

uniformization theorem → generalizes → Riemann mapping theorem ⓘ

Schwarz lemma → usedInProofOf → Riemann mapping theorem ⓘ

Koebe quarter theorem → relatedTo → Riemann mapping theorem ⓘ

Montel's theorem → relatedTo → Riemann mapping theorem ⓘ

subject linked to: Montel theorem

Complex Analysis (Ahlfors) → topic → Riemann mapping theorem ⓘ

subject linked to: Complex Analysis

Lectures on Quasiconformal Mappings → influencedBy → Riemann mapping theorem ⓘ

Functions of One Complex Variable → topic → Riemann mapping theorem ⓘ

subject linked to: Functions of One Complex Variable by John B. Conway

Schwarz–Pick theorem → relatedTo → Riemann mapping theorem ⓘ

Schwarz–Christoffel mapping → relatedTo → Riemann mapping theorem ⓘ

Christoffel–Schwarz formula → relatedTo → Riemann mapping theorem ⓘ

All labels observed (1)

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Statements (50)

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Referenced by (13)

How this entity was disambiguated

How these facts were elicited