In complex analysis, the Riemann mapping theorem states that if is a non-empty simply connected open subset of the complex number plane which is not all of , then there exists a biholomorphic mapping (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from onto the open unit disk
This mapping is sometimes called the Riemann mapping from to the unit disk.The existence of f is equivalent to the existence of a Green's function.
Intuitively, the condition that be simply connected means that does not contain any "holes". The fact that is biholomorphic implies that it is a conformal map 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.
Henri Poincaré proved that the map is unique up to rotation and recentering: if is an element of and is an arbitrary angle, then there exists precisely one f as above such that and such that the argument of the derivative of at the point is equal to . This is an easy consequence of the Schwarz lemma.
As a corollary of the theorem, any two simply connected open subsets of the Riemann sphere which both lack at least two points of the sphere can be conformally mapped into each other.
History
The theorem was stated (under the assumption that the boundary of is piecewise smooth) by Bernhard Riemann in 1851 in his PhD thesis. Lars Ahlfors wrote once, concerning the original formulation of the theorem, that it was "ultimately formulated in terms which would defy any attempt of proof, even with modern methods". Riemann's flawed proof depended on the Dirichlet principle (which was named by Riemann himself), which was considered sound at the time. However, Karl Weierstrass found that this principle was not universally valid. Later, David Hilbert was able to prove that, to a large extent, the Dirichlet principle is valid under the hypothesis that Riemann was working with. However, in order to be valid, the Dirichlet principle needs certain hypotheses concerning the boundary of (namely, that it is a Jordan curve) which are not valid for simply connected domains in general.
The first rigorous proof of the theorem was given by William Fogg Osgood in 1900. He proved the existence of Green's function on arbitrary simply connected domains other than itself; this established the Riemann mapping theorem.For the original paper, see . For accounts of the history, see ; ; . Also see (acknowledging that had already proven the Riemann mapping theorem).
Constantin Carathéodory gave another proof of the theorem in 1912, which was the first to rely purely on the methods of function theory rather than potential theory., citing His proof used Montel's concept of normal families, which became the standard method of proof in textbooks. Carathéodory continued in 1913 by resolving the additional question of whether the Riemann mapping between the domains can be extended to a homeomorphism of the boundaries (see Carathéodory's theorem).
Carathéodory's proof used Riemann surfaces and it was simplified by Paul Koebe two years later in a way that did not require them. Another proof, due to Lipót Fejér and to Frigyes Riesz, was published in 1922 and it was rather shorter than the previous ones. In this proof, like in Riemann's proof, the desired mapping was obtained as the solution of an extremal problem. The Fejér–Riesz proof was further simplified by Alexander Ostrowski and by Carathéodory.
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. One such example is the Koch curve. 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 with , however there are no conformal maps between annuli except inversion and multiplication by constants so the annulus is not conformally equivalent to the annulus (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 (see Liouville's theorem). 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 analogue of the Riemann mapping theorem in several complex variables is also not true. In (), the ball and polydisk are both simply connected, but there is no biholomorphic map between them., section 8.3, p. 187
Proof via normal families
Simple connectivity
Theorem. For an open domain the following conditions are equivalent:See
is simply connected; the integral of every holomorphic function around a closed piecewise smooth curve in vanishes; every holomorphic function in is the derivative of a holomorphic function; every nowhere-vanishing holomorphic function on has a holomorphic logarithm; every nowhere-vanishing holomorphic function on has a holomorphic square root; for any , the winding number of for any piecewise smooth closed curve in is ; the complement of in the extended complex plane is connected.
(1) ⇒ (2) because any continuous closed curve, with base point , can be continuously deformed to the constant curve . So the line integral of over the curve is .
(2) ⇒ (3) because the integral over any piecewise smooth path from to can be used to define a primitive.
(3) ⇒ (4) by integrating along from to to give a branch of the logarithm.
(4) ⇒ (5) by taking the square root as where is a holomorphic choice of logarithm.
(5) ⇒ (6) because if is a piecewise closed curve and are successive square roots of for outside , then the winding number of about is times the winding number of about . Hence the winding number of about must be divisible by for all , so it must equal .
(6) ⇒ (7) for otherwise the extended plane can be written as the disjoint union of two open and closed sets and with and bounded. Let be the shortest Euclidean distance between and and build a square grid on with length with a point of at the centre of a square. Let be the compact set of the union of all squares with distance from . Then and does not meet or : it consists of finitely many horizontal and vertical segments in forming a finite number of closed rectangular paths . Taking to be all the squares covering , then equals the sum of the winding numbers of over , thus giving . On the other hand the sum of the winding numbers of about equals . Hence the winding number of at least one of the about is non-zero.
(7) ⇒ (1) This is a purely topological argument. Let be a piecewise smooth closed curve based at . By approximation γ is in the same homotopy class as a rectangular path on the square grid of length based at ; such a rectangular path is determined by a succession of consecutive directed vertical and horizontal sides. By induction on , such a path can be deformed to a constant path at a corner of the grid. If the path intersects at a point , then it breaks up into two rectangular paths of length , and thus can be deformed to the constant path at by the induction hypothesis and elementary properties of the fundamental group. The reasoning follows a "northeast argument":, elementary proof in the non self-intersecting path there will be a corner with largest real part (easterly) and then amongst those one with largest imaginary part (northerly). Reversing direction if need be, the path go from to and then to for and then goes leftwards to . Let be the open rectangle with these vertices. The winding number of the path is for points to the right of the vertical segment from to and for points to the right; and hence inside . Since the winding number is off , lies in . If is a point of the path, it must lie in ; if is on but not on the path, by continuity the winding number of the path about is , so must also lie in . Hence . But in this case the path can be deformed by replacing the three sides of the rectangle by the fourth, resulting in two less sides (with self-intersections permitted).
Riemann mapping theorem
Weierstrass' convergence theorem. The uniform limit on compacta of a sequence of holomorphic functions is holomorphic; similarly for derivatives. This is an immediate consequence of Morera's theorem for the first statement. Cauchy's integral formula gives a formula for the derivatives which can be used to check that the derivatives also converge uniformly on compacta.
Hurwitz's theorem. If a sequence of nowhere-vanishing holomorphic functions on an open domain has a uniform limit on compacta, then either the limit is identically zero or the limit is nowhere-vanishing. If a sequence of univalent holomorphic functions on an open domain has a uniform limit on compacta, then either the limit is constant or the limit is univalent. If the limit function is non-zero, then its zeros have to be isolated. Zeros with multiplicities can be counted by the winding number for a holomorphic function . Hence winding numbers are continuous under uniform limits, so that if each function in the sequence has no zeros nor can the limit. For the second statement suppose that and set . These are nowhere-vanishing on a disk but vanishes at , so must vanish identically.
Definitions. A family of holomorphic functions on an open domain is said to be normal if any sequence of functions in has a subsequence that converges to a holomorphic function uniformly on compacta. A family is compact if whenever a sequence lies in and converges uniformly to on compacta, then also lies in . A family is said to be locally bounded if their functions are uniformly bounded on each compact disk. Differentiating the Cauchy integral formula, it follows that the derivatives of a locally bounded family are also locally bounded.
Montel's theorem. Every locally bounded family of holomorphic functions in a domain is normal. Let be a totally bounded sequence and chose a countable dense subset of . By locally boundedness and a "diagonal argument", a subsequence can be chosen so that is convergent at each point . It must be verified that this sequence of holomorphic functions converges on uniformly on each compactum . Take open with such that the closure of is compact and contains . Since the sequence is locally bounded, on . By compactness, if is taken small enough, finitely many open disks of radius are required to cover while remaining in . Since , we have that . Now for each choose some in where converges, take and so large to be within of its limit. Then for ,
Hence the sequence forms a Cauchy sequence in the uniform norm on as required.
Riemann mapping theorem. If is a simply connected domain and , there is a unique conformal mapping of onto the unit disk normalized such that and . Uniqueness follows because if and satisfied the same conditions, would be a univalent holomorphic map of the unit disk with and . But by the Schwarz lemma, the univalent holomorphic maps of the unit disk onto itself are given by the Möbius transformations
with . So must be the identity map and .