Riemann mapping theorem

In <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, the Riemann mapping theorem states that if 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is a <a href="/facts/Non-empty/ffEk3eA6">non-empty</a> <a href="/facts/Simply_connected_space/xFlta63J">simply connected</a> <a href="/facts/Open_set/QfTCIqMu">open subset</a> of the <a href="/facts/Complex_plane/vE0QrxJp">complex number plane</a> 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 which is not all of 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
, then there exists a <a href="/facts/Biholomorphy/3g5kL1RD">biholomorphic</a> mapping 
 
 
 
 f
 
 
 {\displaystyle f}
 
 (i.e. a <a href="/facts/Bijective_function/j4gTuTmW">bijective</a> <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic</a> mapping whose inverse is also holomorphic) from 
 
 
 
 U
 
 
 {\displaystyle U}
 
 onto the <a href="/facts/Open_unit_disk/phrUu7yC">open unit disk</a>

D
 =
 {
 z
 ∈
 
 C
 
 :
 
 |
 
 z
 
 |
 
 <
 1
 }
 .
 
 
 {\displaystyle D=\{z\in \mathbb {C} :|z|<1\}.}

This mapping is known as a Riemann mapping.
Intuitively, the condition that 
 
 
 
 U
 
 
 {\displaystyle U}
 
 be simply connected means that 
 
 
 
 U
 
 
 {\displaystyle U}
 
 does not contain any “holes”. The fact that 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is biholomorphic implies that it is a <a href="/facts/Conformal_map/cEpQnMad">conformal map</a> and therefore angle-preserving. Such a map may be interpreted as preserving the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it.
<a href="/facts/Henri_Poincar%C3%A9/AazL2j4X">Henri Poincaré</a> proved that the map 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is unique up to rotation and recentering: if 
 
 
 
 
 z
 
 0
 
 
 
 
 {\displaystyle z_{0}}
 
 is an element of 
 
 
 
 U
 
 
 {\displaystyle U}
 
 and 
 
 
 
 ϕ
 
 
 {\displaystyle \phi }
 
 is an arbitrary angle, then there exists precisely one f as above such that 
 
 
 
 f
 (
 
 z
 
 0
 
 
 )
 =
 0
 
 
 {\displaystyle f(z_{0})=0}
 
 and such that the <a href="/facts/Complex_number/2w7mqI7e">argument</a> of the derivative of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 at the point 
 
 
 
 
 z
 
 0
 
 
 
 
 {\displaystyle z_{0}}
 
 is equal to 
 
 
 
 ϕ
 
 
 {\displaystyle \phi }
 
. This is an easy consequence of the <a href="/facts/Schwarz_lemma/70DeQvPJ">Schwarz lemma</a>.
As a corollary of the theorem, any two simply connected open subsets of the <a href="/facts/Riemann_sphere/4XgBzplC">Riemann sphere</a> which both lack at least two points of the sphere can be conformally mapped into each other.

Riemann mapping theorem open-in-new

Riemann mapping theorem