Riemann mapping theorem

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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 known as a Riemann mapping.[1]

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

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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”.[2] 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.[3]

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.[4] His proof used Montel's concept of normal families, which became the standard method of proof in textbooks.[5] 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).[6]

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.[7]

Importance

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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.[8] 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.[9]

Proof via normal families

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Simple connectivity

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Theorem. For an open domain   the following conditions are equivalent:[10]

  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.

(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":[11][12] 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

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  • 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.[13]
  • 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.[14]

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.[15][16]

  • 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.[17][18]
  • 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  .
To prove existence, take   to be the family of holomorphic univalent mappings   of   into the open unit disk   with   and  . It is a normal family by Montel's theorem. By the characterization of simple-connectivity, for   there is a holomorphic branch of the square root   in  . It is univalent and   for  . Since   must contain a closed disk   with centre   and radius  , no points of   can lie in  . Let   be the unique Möbius transformation taking   onto   with the normalization   and  . By construction   is in  , so that   is non-empty. The method of Koebe is to use an extremal function to produce a conformal mapping solving the problem: in this situation it is often called the Ahlfors function of G, after Ahlfors.[19] Let   be the supremum of   for  . Pick   with   tending to  . By Montel's theorem, passing to a subsequence if necessary,   tends to a holomorphic function   uniformly on compacta. By Hurwitz's theorem,   is either univalent or constant. But   has   and  . So   is finite, equal to   and  . It remains to check that the conformal mapping   takes   onto  . If not, take   in   and let   be a holomorphic square root of   on  . The function   is univalent and maps   into  . Let
 
where  . Then   and a routine computation shows that
 
This contradicts the maximality of  , so that   must take all values in  .[20][21][22]

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

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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 x-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 x-axis.[23][24][25]

The first proof that parallel slit domains were canonical domains for in the multiply connected case was given by David Hilbert in 1909. Jenkins (1958), 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.[26] 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" (to distinguish it from "interior 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.[27][28][29]

Schiff (1993) 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  .[30][31]

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.[32][33][34]

The proof of the uniqueness of the conformal parallel slit transformation is given in Goluzin (1969) and Grunsky (1978). 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 (the images of other the inverse of the Joukowsky transform under the other parallel horizontal slits). 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 y-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 (any   will eventually be encircled by contours in   close to the  's). This contradicts the fact that the non-constant holomorphic function   is an open mapping.[35]

Sketch proof via Dirichlet problem

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Given   and a point  , we want to construct a function   which maps   to the unit disk and   to  . For this sketch, we will assume that U is bounded and its boundary is smooth, much like Riemann did. Write

 

where   is some (to be determined) holomorphic function with real part   and imaginary part  . It is then clear that   is the only zero of  . We require   for  , so we need

 

on the boundary. Since   is the real part of a holomorphic function, we know that   is necessarily a harmonic function; i.e., it satisfies Laplace's equation.

The question then becomes: does a real-valued harmonic function   exist that is defined on all of   and has the given boundary condition? The positive answer is provided by the Dirichlet principle. Once the existence of   has been established, the Cauchy–Riemann equations for the holomorphic function   allow us to find   (this argument depends on the assumption that   be simply connected). Once   and   have been constructed, one has to check that the resulting function   does indeed have all the required properties.[36]

Uniformization theorem

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The Riemann mapping theorem can be generalized to the context of Riemann surfaces: If   is a non-empty simply-connected open subset of a Riemann surface, then   is biholomorphic to one of the following: the Riemann sphere, the complex plane  , or the unit disk  . This is known as the uniformization theorem.

Smooth Riemann mapping theorem

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In the case of a simply connected bounded domain with smooth boundary, the Riemann mapping function and all its derivatives extend by continuity to the closure of the domain. This can be proved using regularity properties of solutions of the Dirichlet boundary value problem, which follow either from the theory of Sobolev spaces for planar domains or from classical potential theory. Other methods for proving the smooth Riemann mapping theorem include the theory of kernel functions[37] or the Beltrami equation.

Algorithms

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Computational conformal mapping is prominently featured in problems of applied analysis and mathematical physics, as well as in engineering disciplines, such as image processing.

In the early 1980s an elementary algorithm for computing conformal maps was discovered. Given points   in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve   with   This algorithm converges for Jordan regions[38] in the sense of uniformly close boundaries. There are corresponding uniform estimates on the closed region and the closed disc for the mapping functions and their inverses. Improved estimates are obtained if the data points lie on a   curve or a K-quasicircle. The algorithm was discovered as an approximate method for conformal welding; however, it can also be viewed as a discretization of the Loewner differential equation.[39]

The following is known about numerically approximating the conformal mapping between two planar domains.[40]

Positive results:

  • There is an algorithm A that computes the uniformizing map in the following sense. Let   be a bounded simply-connected domain, and  .   is provided to A by an oracle representing it in a pixelated sense (i.e., if the screen is divided to   pixels, the oracle can say whether each pixel belongs to the boundary or not). Then A computes the absolute values of the uniformizing map   with precision   in space bounded by   and time  , where   depends only on the diameter of   and   Furthermore, the algorithm computes the value of   with precision   as long as   Moreover, A queries   with precision of at most   In particular, if   is polynomial space computable in space   for some constant   and time   then A can be used to compute the uniformizing map in space   and time  
  • There is an algorithm A′ that computes the uniformizing map in the following sense. Let   be a bounded simply-connected domain, and   Suppose that for some     is given to A′ with precision   by   pixels. Then A′ computes the absolute values of the uniformizing map   within an error of   in randomized space bounded by   and time polynomial in   (that is, by a BPL(n)-machine). Furthermore, the algorithm computes the value of   with precision   as long as  

Negative results:

  • Suppose there is an algorithm A that given a simply-connected domain   with a linear-time computable boundary and an inner radius   and a number   computes the first   digits of the conformal radius   then we can use one call to A to solve any instance of a #SAT(n) with a linear time overhead. In other words, #P is poly-time reducible to computing the conformal radius of a set.
  • Consider the problem of computing the conformal radius of a simply-connected domain   where the boundary of   is given with precision   by an explicit collection of   pixels. Denote the problem of computing the conformal radius with precision   by   Then,   is AC0 reducible to   for any  

See also

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Notes

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  1. ^ The existence of f is equivalent to the existence of a Green’s function.
  2. ^ Ahlfors, Lars (1953), L. Ahlfors; E. Calabi; M. Morse; L. Sario; D. Spencer (eds.), "Developments of the Theory of Conformal Mapping and Riemann Surfaces Through a Century", Contributions to the Theory of Riemann Surfaces: 3–4
  3. ^ For the original paper, see Osgood 1900. For accounts of the history, see Walsh 1973, pp. 270–271; Gray 1994, pp. 64–65; Greene & Kim 2017, p. 4. Also see Carathéodory 1912, p. 108, footnote ** (acknowledging that Osgood 1900 had already proven the Riemann mapping theorem).
  4. ^ Gray 1994, pp. 78–80, citing Carathéodory 1912
  5. ^ Greene & Kim 2017, p. 1
  6. ^ Gray 1994, pp. 80–83
  7. ^ "What did Riemann Contribute to Mathematics? Geometry, Number Theory and Others" (PDF).
  8. ^ Lakhtakia, Akhlesh; Varadan, Vijay K.; Messier, Russell (August 1987). "Generalisations and randomisation of the plane Koch curve". Journal of Physics A: Mathematical and General. 20 (11): 3537–3541. doi:10.1088/0305-4470/20/11/052.
  9. ^ Remmert 1998, section 8.3, p. 187
  10. ^ See
  11. ^ Gamelin 2001, pp. 256–257, elementary proof
  12. ^ Berenstein & Gay 1991, pp. 86–87
  13. ^ Gamelin 2001
  14. ^ Gamelin 2001
  15. ^ Duren 1983
  16. ^ Jänich 1993
  17. ^ Duren 1983
  18. ^ Jänich 1993
  19. ^ Gamelin 2001, p. 309
  20. ^ Duren 1983
  21. ^ Jänich 1993
  22. ^ Ahlfors 1978
  23. ^ Jenkins 1958, pp. 77–78
  24. ^ Duren 1980
  25. ^ Schiff 1993, pp. 162–166
  26. ^ Jenkins 1958, pp. 77–78
  27. ^ Schober 1975
  28. ^ Duren 1980
  29. ^ Duren 1983
  30. ^ Schiff 1993
  31. ^ Goluzin 1969, pp. 210–216
  32. ^ Schiff 1993
  33. ^ Goluzin 1969, pp. 210–216
  34. ^ Nehari 1952, pp. 351–358
  35. ^ Goluzin 1969, pp. 214−215
  36. ^ Gamelin 2001, pp. 390–407
  37. ^ Bell 1992
  38. ^ A Jordan region is the interior of a Jordan curve.
  39. ^ Marshall, Donald E.; Rohde, Steffen (2007). "Convergence of a Variant of the Zipper Algorithm for Conformal Mapping". SIAM Journal on Numerical Analysis. 45 (6): 2577. CiteSeerX 10.1.1.100.2423. doi:10.1137/060659119.
  40. ^ Binder, Ilia; Braverman, Mark; Yampolsky, Michael (2007). "On the computational complexity of the Riemann mapping". Arkiv för Matematik. 45 (2): 221. arXiv:math/0505617. Bibcode:2007ArM....45..221B. doi:10.1007/s11512-007-0045-x. S2CID 14545404.

References

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