Riemann Mapping Theorem - Importance

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 functions.
• Simply connected open sets in the plane can be highly complicated, for instance the boundary can be a nowhere-differentiable fractal curve of infinite length, even if the set itself is bounded. 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 { : r < < 1 } with 0 < r < 1, however there are no conformal maps between annuli except inversion and multiplication by constants so the annulus { : 1 < < 2 } is not conformally equivalent to the annulus { : 1 < < 4 } (as can be 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 transformations.
• Even if arbitrary homeomorphisms in higher dimensions are permitted, contractible manifolds can be found that are not homeomorphic to the ball (e.g., 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 vastly 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.