Editing Riemann mapping theorem (section)

==Importance==
The following points detail the uniqueness and power of the Riemann mapping theorem:

* Even relatively simple Riemann mappings (for example a map from the interior of a circle to the interior of a square) have no explicit formula using only [[elementary function]]s.
* Simply connected open sets in the plane can be highly complicated, for instance, the [[boundary (topology)|boundary]] can be a nowhere-[[Differentiable function|differentiable]] [[fractal curve]] of infinite length, even if the set itself is bounded. One such example is the [[Koch curve]].<ref name="koch">{{cite journal |last1=Lakhtakia |first1=Akhlesh |last2=Varadan |first2=Vijay K. |last3=Messier |first3=Russell |title=Generalisations and randomisation of the plane Koch curve |journal=Journal of Physics A: Mathematical and General |date=August 1987 |volume=20 |issue=11 |pages=3537–3541 |doi=10.1088/0305-4470/20/11/052}}</ref> The fact that such a set can be mapped in an ''angle-preserving'' manner to the nice and regular unit disc seems counter-intuitive.
* The analog of the Riemann mapping theorem for more complicated domains is not true. The next simplest case is of doubly connected domains (domains with a single hole).  Any doubly connected domain except for the punctured disk and the punctured plane is conformally equivalent to some annulus <math>\{z:r<|z|<1\}</math> with <math>0<r<1</math>, however  there are no conformal maps between [[Annulus (mathematics)|annuli]] except inversion and multiplication by constants so the annulus <math>\{z:1<|z|<2\}</math> is not conformally equivalent to the annulus <math>\{z:1<|z|<4\}</math> (as can be [[extremal length#Some applications of extremal length|proven using extremal length]]).
* The analogue of the Riemann mapping theorem in three or more real dimensions is not true. The family of conformal maps in three dimensions is very poor, and essentially contains only [[Möbius transformation]]s (see [[Liouville's theorem (conformal mappings) | Liouville's theorem]]).
* Even if arbitrary [[homeomorphism]]s in higher dimensions are permitted, [[contractible]] [[manifold]]s can be found that are not homeomorphic to the [[Ball (mathematics) | ball]] (e.g., the [[Whitehead continuum]]).
* The analogue of the Riemann mapping theorem in [[Function of several complex variables|several complex variables]] is also not true. In <math>\mathbb{C}^n</math> (<math>n \ge 2</math>), the ball and [[polydisk]] are both simply connected, but there is no biholomorphic map between them.<ref>{{harvnb|Remmert|1998}}, section 8.3, p. 187</ref>