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.

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 → 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 →

Referenced by (2)

Subject (surface form when different)	Predicate
Bernhard Riemann →	knownFor
Friedrich Bernhard Riemann →	notableConcept