Riemann mapping theorem


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 from onto the open unit disk
This mapping is sometimes called the Riemann mapping from to the unit disk.
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 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 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 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 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.
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. 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.
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 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. 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 .
  • 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.
  • 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.

    Proof via normal families

Simple connectivity

Theorem. For an open domain the following conditions are equivalent:
  1. is simply connected;
  2. the integral of every holomorphic function around a closed piecewise smooth curve in vanishes;
  3. every holomorphic function in is the derivative of a holomorphic function;
  4. every nowhere-vanishing holomorphic function on has a holomorphic logarithm;
  5. every nowhere-vanishing holomorphic function on has a holomorphic square root;
  6. for any, the winding number of for any piecewise smooth closed curve in is ;
  7. the complement of in the extended complex plane is connected.
⇒ 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.
⇒ because the integral over any piecewise smooth path from to can be used to define a primitive.
⇒ by integrating along from to to give a branch of the logarithm.
⇒ by taking the square root as where is a holomorphic choice of logarithm.
⇒ 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.
⇒ 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.
⇒ 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": in the non self-intersecting path there will be a corner with largest real part and then amongst those one with largest imaginary part. 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.

Riemann mapping theorem

  • Weierstrass' convergence theorem. The uniform limit on compacta of a sequence of holomorphic functions is holomorphic; similarly for derivatives.
  • 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.
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.
Remark. As a consequence of the Riemann mapping theorem, every simply connected domain in the plane is homeomorphic to the unit disk. If points are omitted, this follows from the theorem. For the whole plane, the homeomorphism gives a homeomorphism of onto.

Parallel slit mappings

Koebe's uniformization theorem for normal families also generalizes to yield uniformizers for multiply-connected domains to finite parallel slit domains, where the slits have angle to the -axis. Thus if is a domain in containing and bounded by finitely many Jordan contours, there is a unique univalent function on with
near, maximizing and having image a parallel slit domain with angle to the -axis.
The first proof that parallel slit domains were canonical domains for in the multiply connected case was given by David Hilbert in 1909., on his book on univalent functions and conformal mappings, gave a treatment based on the work of Herbert Grötzsch and René de Possel from the early 1930s; it was the precursor of quasiconformal mappings and quadratic differentials, later developed as the technique of extremal metric due to Oswald Teichmüller. Menahem Schiffer gave a treatment based on very general variational principles, summarised in addresses he gave to the International Congress of Mathematicians in 1950 and 1958. In a theorem on "boundary variation", he derived a differential equation and inequality, that relied on a measure-theoretic characterisation of straight-line segments due to Ughtred Shuttleworth Haslam-Jones from 1936. Haslam-Jones' proof was regarded as difficult and was only given a satisfactory proof in the mid-1970s by Schober and Campbell–Lamoureux.
gave a proof of uniformization for parallel slit domains which was similar to the Riemann mapping theorem. To simplify notation, horizontal slits will be taken. Firstly, by Bieberbach's inequality, any univalent function
with in the open unit disk must satisfy. As a consequence, if
is univalent in, then. To see this, take and set
for in the unit disk, choosing so the denominator is nowhere-vanishing, and apply the Schwarz lemma. Next the function is characterized by an "extremal condition" as the unique univalent function in of the form that maximises : this is an immediate consequence of Grönwall's area theorem, applied to the family of univalent functions in.
To prove now that the multiply connected domain can be uniformized by a horizontal parallel slit conformal mapping
take large enough that lies in the open disk. For, univalency and the estimate imply that, if lies in with , then. Since the family of univalent are locally bounded in, by Montel's theorem they form a normal family. Furthermore if is in the family and tends to uniformly on compacta, then is also in the family and each coefficient of the Laurent expansion at of the tends to the corresponding coefficient of. This applies in particular to the coefficient: so by compactness there is a univalent which maximizes. To check that
is the required parallel slit transformation, suppose reductio ad absurdum that has a compact and connected component of its boundary which is not a horizontal slit. Then the complement of in is simply connected with. By the Riemann mapping theorem there is a conformal mapping
such that is with a horizontal slit removed. So we have that
and thus by the extremality of. Therefore,. On the other hand by the Riemann mapping theorem there is a conformal mapping
mapping from onto. Then
By the strict maximality for the slit mapping in the previous paragraph, we can see that, so that. The two inequalities for are contradictory.
The proof of the uniqueness of the conformal parallel slit transformation is given in and. Applying the inverse of the Joukowsky transform to the horizontal slit domain, it can be assumed that is a domain bounded by the unit circle and contains analytic arcs and isolated points. Thus, taking a fixed, there is a univalent mapping
with its image a horizontal slit domain. Suppose that is another uniformizer with
The images under or of each have a fixed -coordinate so are horizontal segments. On the other hand, is holomorphic in. If it is constant, then it must be identically zero since. Suppose is non-constant, then by assumption are all horizontal lines. If is not in one of these lines, Cauchy's argument principle shows that the number of solutions of in is zero. This contradicts the fact that the non-constant holomorphic function is an open mapping.