Riemann mapping theorem
E47349
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Riemann mapping theorem canonical | 7 |
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]
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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.
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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.
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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.
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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.
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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.
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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.
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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’s theorem in convex geometry
ⓘ
surface form:
Carathéodory theorem
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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Riemann mapping theorem Description of subject: 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.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Friedrich Bernhard Riemann
subject surface form:
Montel's theorem