The Riemann Mapping Theorem

The Riemann mapping theorem states that ifis a non-empty simply connected region in andthen there exists a bijective and holomorphic functionfromonto the open unit disk

The condition thatbe simply connected means thatdoes not contain any “holes”. The fact that f is bijective and holomorphic implies that it is a conformal map so preserves angles between lines or vectors and also the shape of any sufficiently small figure, possibly rotating and scaling it.

Poincaré proved that the functionis unique: ifandis an arbitrary angle, then there exists a uniqueas above with the additional properties thatand thatThis is an easy consequence of the Schwarz lemma.

As a corollary, any simple, connected subsets ofcan be mapped to each other because giventhere exists a bijective holomorphic functionmappingtoand giventhere exists  a bijective holomorphic functionmappingtothen sinceis bijective,exists andmapsto

Further, since there exists a bijective mapping from the Riemann sphere ontoany two simply connected regions of the Riemann sphere can be mapped to each other.