Riemann Mapping Theorem

In complex analysis, the Riemann mapping theorem states that if is a non-empty simply connected open subset of the complex number plane which is not all of, then there exists a biholomorphic (bijective and holomorphic) mapping from onto the open unit disk

This mapping is known as a Riemann mapping.

Intuitively, the condition that be simply connected means that does not contain any “holes”. The fact that is biholomorphic implies that it is a conformal map and therefore angle-preserving. Intuitively, such a map preserves the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it.

Henri Poincaré proved that the map is essentially unique: if is an element of and φ is an arbitrary angle, then there exists precisely one as above with the additional properties that maps into and that the argument of the derivative of at the point is equal to φ. This is an easy consequence of the Schwarz lemma.

As a corollary of the theorem, any two simply connected open subsets of the Riemann sphere which both lack at least two points of the sphere can be conformally mapped into each other (because conformal equivalence is an equivalence relation).