The Riemann mapping theorem in complex analysis states the following: if U is a simply connected open subset of the complex number planeC which is not all of C, then there exists a bijective holomorphic conformal mapf : 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 z_{0} 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(z_{0}) = 0 and arg f '(z_{0}) = 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.

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.

Riemann wove together and generalized three crucial discoveries of the 19th century: the extension of Euclidean geometry to n dimensions; the logical consistency of geometries that are not Euclidean; and the intrinsic geometry of a surface, in terms of its metric and curvature in the neighborhood of a point.

Riemann was born on September 17, 1826 at Breselenz, Hanover, Germany, the son of a Lutheran minister.

Riemann was shy and self-effacing and recognition for his work came slowly during his lifetime; awareness of his truly striking achievements was to come later as his work was validated and as it stimulated the work of others.

The term "quasi-conformal mapping" , as a rule, presupposes the mapping to be a homeomorphism.

The general problem of constructing a quasi-conformal mapping from one simply-connected domain onto another satisfying (2) was posed and solved by M.A. Lavrent'ev [28], [31], one of the founders of the theory of quasi-conformal mappings.

Liouville's theorem is valid both in the case of Hilbert space [36] and under minimal a priori regularity conditions on the mapping [19], [38].

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