Uniformization

of a set $ A \subset \mathbf C ^ {N} $ (or $ A \subset \mathbf C P ^ {N} $)

A triple $ ( f, D, G) $, where $ f = ( f _ {1} \dots f _ {N} ) $ is a system of meromorphic functions in a domain $ D \subset \mathbf C ^ {N} $ (respectively, $ D \subset \mathbf C P ^ {N} $), defining a holomorphic covering $ D _ {0} \rightarrow f ( D _ {0} ) $, where $ f ( D _ {0} ) $ is dense in $ A $, and $ G $ is a properly-discontinuous group of biholomorphic automorphisms of $ D $ whose restriction to $ D _ {0} $ is the group of covering homeomorphisms of this covering, i.e. $ D _ {0} /G $ is biholomorphically equivalent to $ f ( D _ {0} ) $.

One may thus speak of uniformization by multi-valued analytic functions $ w = F ( z): \mathbf C ^ {n} \rightarrow \mathbf C ^ {m} $, by which one understands uniformization of the set $ A = \{ ( z, w) \} $; this corresponds to the parametrization of $ F $ by means of single-valued meromorphic functions.

For example, the complex curve $ z ^ {2} + w ^ {2} = 1 $ in $ \mathbf C ^ {2} $ is uniformized by the triple $ (( z, w), \mathbf C , G) $, where $ z = \cos t $, $ w = \sin t $, $ G $ is the group of translations $ t \rightarrow t + 2k \pi $, $ k \in \mathbf Z $, or the triple $ (( z, w), D, G) $, where

$$ z = \ \frac{( 1 - t ^ {2} ) }{( 1 + t ^ {2} ) } ,\ \ w = \ \frac{2t }{( 1 + t ^ {2} ) } , $$

$$ D = \mathbf C \setminus \{ i, - i \} , $$

and $ G $ is the trivial group. A less trivial example is the cubic curve $ w ^ {2} = a _ {0} z ^ {3} + a _ {1} z ^ {2} + a _ {2} z + a _ {3} $, which admits no rational parametrization, but which may be uniformized by means of elliptic functions (cf. Elliptic function), namely by a triple $ (( f _ {1} , f _ {2} ), D, G) $, where $ f _ {1} $ and $ f _ {2} $ are rational functions in the Weierstrass $ \wp $-function and its derivative, with corresponding periods $ \omega _ {1} $, $ \omega _ {2} $, and $ G $ is the group generated by the translations $ t \rightarrow t + \omega _ {1} $, $ t \rightarrow t + \omega _ {2} $.

The problem of uniformizing an arbitrary algebraic curve defined by a general algebraic equation

$$ \tag{* } P ( z, w) = \ \sum _ { j,k } a _ {jk} z ^ {j} w ^ {k} = 0, $$

where $ P $ is an irreducible algebraic polynomial over $ \mathbf C $, arose already in the first half of the 19th century, particularly in connection with the integration of algebraic functions. H. Poincaré raised the question of the uniformization of the set of solutions of an arbitrary analytic equation of the form (*), when $ P $ is a convergent power series in two variables, considered with all possible analytic continuations of it. The uniformization of algebraic and arbitrary analytic varieties constituted Hilbert's twenty-second problem. A complete solution of the uniformization problem has so far (1992) not been obtained, with the exception of the one-dimensional case.

One introduces on the set of pairs $ ( z, w) $ in $ \mathbf C ^ {2} $ satisfying (*) a complex structure by means of elements of the corresponding algebraic function $ w ( z) $ (or $ z ( w) $), and so obtains a compact Riemann surface; the coordinates of points of the curve (*) are meromorphic functions on this surface. Furthermore, all compact Riemann surfaces, up to conformal equivalence, are obtained in this way. Therefore the problem of uniformization of algebraic curves is contained in the problem of uniformization of Riemann surfaces.

A uniformization of an arbitrary Riemann surface $ S $ is a triple $ ( D, \pi , G) $ where $ D $ is a domain on the Riemann sphere $ \overline{\mathbf C}\; $ and $ \pi : D \rightarrow S $ is a regular holomorphic covering with covering group $ G $ of conformal automorphisms of $ D $. The general problem consists in finding and describing all such triples for a given Riemann surface.

The possibility of uniformizing an arbitrary Riemann surface $ S $, giving in principle the solution of the problem, was achieved in the classical papers of P. Koebe, Poincaré and F. Klein; a complete solution was obtained, giving a description of all possible uniformizations of the surface $ S $ (cf. [4]–[6]). The Klein–Poincaré uniformization theorem (proved in the general case by Poincaré, cf. [2]) states: Every Riemann surface $ S $ is conformally equivalent to a quotient space $ D/G $, where $ D $ is one of the three canonical domains: the Riemann sphere $ \overline{\mathbf C}\; $, the complex plane $ \mathbf C $ or the unit disc $ \Delta $, while $ G $ is a properly-discontinuous group of Möbius (fractional-linear) automorphisms of $ D $, defined up to conjugation in the group of all Möbius automorphisms of $ D $.

The cases $ D = \overline{\mathbf C}\; $, $ \mathbf C $ and $ \Delta $ are mutually exclusive. A surface $ S $ with such a universal holomorphic covering is called elliptic, parabolic or hyperbolic, respectively. Moreover, $ D = \overline{\mathbf C}\; $ only in the case that $ S $ itself is conformally equivalent to $ \overline{\mathbf C}\; $ (and so $ G $ is trivial); $ D = \mathbf C $ when $ S $ is conformally equivalent to either $ \mathbf C $, $ \mathbf C \setminus \{ 0 \} $ or the torus, and $ G $ is then either trivial or the group generated by the translation $ z \rightarrow z + \omega $ ($ \omega \in \mathbf C \setminus \{ 0 \} $) or the group generated by the two translations $ z \rightarrow z + \omega _ {1} $, $ z \rightarrow z + \omega _ {2} $, where $ \omega _ {1} , \omega _ {2} \neq 0 $ are complex numbers such that $ \mathop{\rm Im} ( \omega _ {2} / \omega _ {1} ) \neq 0 $. In the remaining case $ S $ is conformally equivalent to $ \Delta /G $, where $ G $ is a torsion-free Fuchsian group. The canonical projection $ \pi : D \rightarrow S $ is an unramified covering and uniformizes all functions $ f $ on $ S $ such that $ f \circ \pi $ is single-valued on $ D $. The Klein–Poincaré theorem also has a generalization to ramified coverings with given order of ramification.

Another approach to the uniformization problem relies on the following principle: If a Riemann surface $ \widetilde{S} $ is homeomorphic to a domain $ D \subset \overline{\mathbf C}\; $ (not necessarily simply connected), then $ \widetilde{S} $ is also conformally equivalent to $ D $. In the same way the uniformization problem may be reduced to the topological problem of finding all (generally speaking, ramified) flat coverings $ \widetilde{S} \rightarrow S $ of a given Riemann surface $ S $. The solution of this problem is given by the following theorems of Maskit (cf. [4], [5]):

I) Let $ S $ be an oriented surface and let $ v _ {1} \dots v _ {n} \dots $ be a set of pairwise disjoint loops on $ S $. If $ \widetilde{S} \rightarrow S $ is a regular covering with defining subgroup $ N = \langle v _ {1} ^ {\alpha _ {1} } \dots v _ {n} ^ {\alpha _ {n} } , . . . \rangle $, where $ \alpha _ {1} \dots \alpha _ {n} \dots $ are natural numbers, then $ \widetilde{S} $ is a flat covering, i.e. is homeomorphic to a domain in $ \overline{\mathbf C}\; $.

II) Let $ \widetilde{S} $ be a flat surface and let $ \widetilde{S} \rightarrow S $ be a regular covering of an oriented surface $ S $ with defining subgroup $ N $. If $ S $ is a surface of finite type, i.e. $ \pi _ {1} ( S) $ is finitely generated, then there exists a finite set of simple pairwise disjoint loops $ v _ {1} \dots v _ {n} $ and natural numbers $ \alpha _ {1} \dots \alpha _ {n} $ such that $ \langle v _ {1} ^ {\alpha _ {1} } \dots v _ {n} ^ {\alpha _ {n} } \rangle = N $.

III) If $ \widetilde{S} $ is a flat Riemann surface and $ \overline{G}\; $ is a properly-discontinuous group of conformal automorphisms of $ \widetilde{S} $, then there exists a conformal homeomorphism $ h: \widetilde{S} \rightarrow D \subset \overline{\mathbf C}\; $ such that $ hGh ^ {-} 1 $ is a Kleinian group with invariant component $ D $.

Thus, every Riemann surface is uniformized by a Kleinian group. E.g., if $ S $ is a closed Riemann surface of genus $ g \geq 1 $, then its fundamental group has the presentation

$$ \pi _ {1} ( S) = \ \left \{ { a _ {1} , b _ {1} \dots a _ {g} , b _ {g} } : { \prod _ {j = 1 } ^ { g } [ a _ {j} , b _ {j} ] = 1 } \right \} , $$

and the normal subgroup $ N $ defined by the flat covering $ \widetilde{S} $ may be taken to be the smallest normal subgroup generated by $ a _ {1}, \dots, a _ {g} $ (or $ b _ {1}, \dots, b _ {g} $); $ S $ is now uniformized by a Schottky group $ G $ of genus $ g $ — a free purely-loxodromic Kleinian group with $ g $ generators (the classical Koebe theorem on cross-cuts).

In the uniformization of Riemann surfaces of finite type, the possible Kleinian groups may be classified. For this purpose one introduces the notion of a quotient subgroup. If $ G $ is a Kleinian group with invariant component $ D ( G) $, then a subgroup $ H $ of it is called a quotient subgroup of $ G $ if $ H $ is a maximal subgroup such that: a) its invariant component $ D ( H) \supset D ( G) $ is simply connected; b) $ H $ does not contain random parabolic elements (i.e. parabolic elements such that for the conformal isomorphism $ b: D ( H) \rightarrow \Delta $ the image under $ h \circ g \circ h ^ {-} 1 $ is hyperbolic); and c) every parabolic element of $ G $ with a fixed point in the limit set of $ H $ belongs to $ H $. For example, in the Klein–Poincaré theorem every quotient subgroup of $ G $ coincides with $ G $ itself, and in Koebe's theorem on cross-cuts all quotient subgroups are trivial. A uniformization $ ( D, \pi , G) $ of a Riemann surface $ S $, where $ D $ is the invariant component of $ G $, is called standard if $ G $ is torsion-free and contains no random parabolic elements. For a closed surface all such uniformizations are described by the following theorem (cf. [6]).

Let $ S $ be a closed Riemann surface of genus $ g > 0 $ and let $ \{ v _ {1} \dots v _ {n} \} $ be a set of simple pairwise disjoint loops on $ S $. Then there exists a standard uniformization $ ( D, \pi , G) $ of $ S $, unique up to conformal equivalence, such that every quotient subgroup $ G $ is either Fuchsian or elementary and such that the covering $ \pi : D \rightarrow S $ is constructed from the smallest normal subgroup of $ \pi _ {1} ( S) $ spanned by the loops $ v _ {1} \dots v _ {n} $.

The theory of quasi-conformal mapping and Teichmüller spaces (cf. Teichmüller space) allows one to prove the possibility of simultaneous uniformization of several Riemann surfaces by a single Kleinian group, as well as that of all Riemann surfaces of a given type (cf. [7]).

References

[1]	F. Klein, "Neue Beiträge zur Riemannschen Funktionentheorie" Math. Ann. , 21 (1883) pp. 141–218
[2]	H. Poincaré, "Sur l'uniformisation des fonctions analytiques" Acta Math. , 31 (1907) pp. 1–64
[3a]	P. Koebe, "Ueber die Uniformisierung beliebiger analytischer Kurven" Nachr. K. Ges. Wissenschaft. Göttinger Math. Phys. Kl. (1907) pp. 191–210
[3b]	P. Koebe, "Ueber die Uniformisierung beliebiger analytischer Kurven II" Nachr. K. Ges. Wissenschaft. Göttinger Math. Phys. Kl. (1907) pp. 177–198
[3c]	P. Koebe, "Ueber die Uniformisierung beliebiger analytischer Kurven III" Nachr. K. Ges. Wissenschaft. Göttinger Math. Phys. Kl. (1908) pp. 337–358
[3d]	P. Koebe, "Ueber die Uniformisierung beliebiger analytischer Kurven IV" Nachr. K. Ges. Wissenschaft. Göttinger Math. Phys. Kl. (1909) pp. 324–361
[4]	B. Maskit, "A theorem on planar covering surfaces with applications to 3-manifolds" Ann. of Math. , 81 : 2 (1965) pp. 341–355
[5]	B. Maskit, "The conformal group of a plane domain" Amer. J. Math. , 90 : 3 (1968) pp. 718–722
[6]	B. Maskit, L.V. Ahlfors (ed.) et al. (ed.) , Contributions to Analysis. Uniformization of Riemann surfaces , Acad. Press (1974) pp. 293–312
[7]	L. Bers, "Uniformization. Moduli and Kleinian groups" Bull. London Math. Soc. , 4 (1972) pp. 257–300
[8]	S.L. Krushkal', B.N. Apanasov, N.A. Gusevskii, "Kleinian groups and uniformization in examples and problems" , Amer. Math. Soc. (1986) (Translated from Russian)
[9]	R. Nevanlinna, "Uniformisierung" , Springer (1953)
[10]	L.R. Ford, "Automorphic functions" , Chelsea, reprint (1957)

References

[a1]	R.C. Gunning, "On uniformization of complex manifolds: the role of connections" , Princeton Univ. Press (1978)
[a2]	G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10
[a3]	B.N. Apanasov, "Discrete groups in space and uniformization problems" , Kluwer (1991) (Translated from Russian)