Riemann mapping theorem
The Riemann mapping theorem in complex analysis states the following: if U is a simply connected open subset of the complex number plane C which is not all of C, then there exists a bijective holomorphic conformal map f : U -> D, where D = { z in C : |z| < 1 } denotes the open disk. Intuitively, the condition that U be simply connected means that U does not contain any "holes"; the conformality of f means that f maintains the shape of small figures.
The map f is essentially unique: if z0 is an element of U and φ in (-π, π] is an arbitrary angle, then there exists precisely one f as above with the additional properties f(z0) = 0 and arg f '(z0) = φ.
As a corollary, any two such simply connected open sets (which are different from C and C U {∞}) can be conformally mapped into each other.
The theorem was proved by Bernhard Riemann in 1851, but his proof depended on a statement in the calculus of variations which was only later proven by David Hilbert.
Why is this theorem impressive?
To better understand how unique and powerful the Riemann mapping theorem is, consider the following facts:
- This analog of the Riemann mapping theorem for doubly connected domains is not true. In fact, there are no conformal maps between annuli except multiplication by constants, so the annulus
is not conformally equivalent to the annulus 
. However, any doubly connected domain is conformally equivalent to some annulus.
- The analog of the Riemann mapping theorem in three dimensions or above is not even remotely true. In fact, the family of conformal maps in three dimensions is very poor, and contains, essentially, only Möbius transformations.
- Even if we allow arbitrary homeomorphisms in higher dimensions, we can find contractable spaces that are not homeomorphic to the ball, such as the Whitehead continuum.
- The Riemann mapping theorem is the easiest way to prove that any two simply connected domains in the plane are homeomorphic. Even though the class of continuous functions is infinitely larger than that of conformal maps, it is not easy to construct a one-to-one function onto the disk knowing only that the domain is simply connected.
- Even relatively simple Riemann mappings, say a map of the circle to the square, have no explicit formula using only elementary functions.