Math 2.0

Functions of a complex variable Geometric function theory Theorem

Riemann mapping theorem

Builds on Holomorphic functionConformal map

Statement

Let UU be a nonempty open subset of ℂ\mathbb{C} with U≠ℂU \neq \mathbb{C}. Suppose that UU is simply connected: UU is connected, and each closed path in UU can be continuously deformed in UU to a constant path. Let 𝔻={z∈ℂ:∣z∣<1}\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}. Then there is a biholomorphic map ff from UU onto 𝔻\mathbb{D}, that is, a bijective holomorphic map with a holomorphic inverse. Moreover, for each z0∈Uz_0 \in U there is exactly one such ff with f(z0)=0f(z_0) = 0 and f′(z0)f'(z_0) a positive real number.

Remarks

A biholomorphic map is conformal, so UU and 𝔻\mathbb{D} are conformally equivalent. The case U=ℂU = \mathbb{C} is excluded: a holomorphic map ℂ→𝔻\mathbb{C} \to \mathbb{D} is bounded, so it is constant by Liouville's theorem. The uniqueness follows from the Schwarz lemma. Riemann stated the theorem in his thesis of 1851, and Osgood gave the first rigorous proof in 1900.

Reading path

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Complex numbers, object, no evidence levelComplex numbersobjectComplex plane, concept, no evidence levelComplex planeconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionConformal map, definition, no evidence levelConformal mapdefinitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionComplex differentiability, definition, no evidence levelComplex differentiabi…definitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionRiemann mapping theorem, statement, no evidence levelRiemann mapping theor…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Complex numbers object
  2. Complex plane concept
  3. Real numbers object
  4. Set bounded above definition
  5. Real and imaginary parts definition
  6. Conformal map definition
  7. Supremum definition
  8. Completeness axiom statement
  9. Existence of square roots of nonnegative reals statement
  10. Square root definition
  11. Modulus of a complex number definition
  12. Complex differentiability definition
  13. Holomorphic function definition
  14. Riemann mapping theorem statement

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Riemann_mapping_theorem, lead section
  2. Web https://en.wikipedia.org/wiki/Riemann_mapping_theorem, History
About this page
Tag02WHMEG5ZR
URIm2://02WHMEG5ZR
Aliasesriemann-mapping
Kindstatement, theorem
WritingW1 stated
MSC30C35
StatusThis copy of the site has no build data. Run m2 status 02WHMEG5ZR.
UsesHolomorphic function, Conformal map
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