Riemann mapping theorem: Difference between revisions

Revision as of 00:45, 27 April 2008

Statement

Any simply connected domain in $\mathbb {C}$ (i.e. any open, connected and simply connected subset of $\mathbb {C}$ ) that is not the whole of $\mathbb {C}$ , is bihomolorphically equivalent to the open unit disk.

Corollaries

Any two simply connected domains are homeomorphic (follows from the Riemann mapping theorem and the fact that the unit disc is homeomorphic to $\mathbb {C}$ ).

Proof

Proof outline

The standard proof of the Riemann mapping theorem has the following three steps:

Obtain a conformal mapping from the simply connected domain into the open unit disk (we do this by first obtaining a map into the upper half plane, and then using a fractional linear transformation to pass from the upper half plane to the open unit disk)
Maximize among all such mappings with respect to certain criteria. Prove that a maximal mapping must exist.
Prove that this maximal conformal mapping is also surjective, and hence, must be a biholomorphic mapping

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 ==Statement==
-Any [[simply connected domain]] in <math>\mathbb{C}</math> (i.e. any open, connected and simply connected subset of <math>\mathbb{C}</math>) that is not the whole of <math>\mathbb{C}</math>, is [[biholomorphically equivalent domains|bihomolorphically equivalent]] to the open unit disc.
+Any [[simply connected domain]] in <math>\mathbb{C}</math> (i.e. any open, connected and simply connected subset of <math>\mathbb{C}</math>) that is not the whole of <math>\mathbb{C}</math>, is [[biholomorphically equivalent domains|bihomolorphically equivalent]] to the [[open unit disk]].
 ==Corollaries==
 Any two simply connected domains are homeomorphic (follows from the Riemann mapping theorem and the fact that the unit disc is homeomorphic to <math>\mathbb{C}</math>).
+==Proof==
+===Proof outline===
+The standard proof of the Riemann mapping theorem has the following three steps:
+# Obtain a conformal mapping from the simply connected domain ''into'' the [[open unit disk]] (we do this by first obtaining a map into the upper half plane, and then using a [[fractional linear transformation]] to pass from the upper half plane to the open unit disk)
+# Maximize among all such mappings with respect to certain criteria. Prove that a maximal mapping must exist.
+# Prove that this maximal conformal mapping is also surjective, and hence, must be a biholomorphic mapping