Uniformization
of a set (or
)
A triple , where
is a system of meromorphic functions in a domain
(respectively,
), defining a holomorphic covering
, where
is dense in
, and
is a properly-discontinuous group of biholomorphic automorphisms of
whose restriction to
is the group of covering homeomorphisms of this covering, i.e.
is biholomorphically equivalent to
.
One may thus speak of uniformization by multi-valued analytic functions , by which one understands uniformization of the set
; this corresponds to the parametrization of
by means of single-valued meromorphic functions.
For example, the complex curve in
is uniformized by the triple
, where
,
,
is the group of translations
,
, or the triple
, where
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and is the trivial group. A less trivial example is the cubic curve
, which admits no rational parametrization, but which may be uniformized by means of elliptic functions (cf. Elliptic function), namely by a triple
, where
and
are rational functions in the Weierstrass
-function and its derivative, with corresponding periods
,
, and
is the group generated by the translations
,
.
The problem of uniformizing an arbitrary algebraic curve defined by a general algebraic equation
![]() | (*) |
where is an irreducible algebraic polynomial over
, 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
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 in
satisfying (*) a complex structure by means of elements of the corresponding algebraic function
(or
), 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 is a triple
where
is a domain on the Riemann sphere
and
is a regular holomorphic covering with covering group
of conformal automorphisms of
. The general problem consists in finding and describing all such triples for a given Riemann surface.
The possibility of uniformizing an arbitrary Riemann surface , 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
(cf. [4]–[6]). The Klein–Poincaré uniformization theorem (proved in the general case by Poincaré, cf. [2]) states: Every Riemann surface
is conformally equivalent to a quotient space
, where
is one of the three canonical domains: the Riemann sphere
, the complex plane
or the unit disc
, while
is a properly-discontinuous group of Möbius (fractional-linear) automorphisms of
, defined up to conjugation in the group of all Möbius automorphisms of
.
The cases ,
and
are mutually exclusive. A surface
with such a universal holomorphic covering is called elliptic, parabolic or hyperbolic, respectively. Moreover,
only in the case that
itself is conformally equivalent to
(and so
is trivial);
when
is conformally equivalent to either
,
or the torus, and
is then either trivial or the group generated by the translation
(
) or the group generated by the two translations
,
, where
are complex numbers such that
. In the remaining case
is conformally equivalent to
, where
is a torsion-free Fuchsian group. The canonical projection
is an unramified covering and uniformizes all functions
on
such that
is single-valued on
. 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 is homeomorphic to a domain
(not necessarily simply connected), then
is also conformally equivalent to
. In the same way the uniformization problem may be reduced to the topological problem of finding all (generally speaking, ramified) flat coverings
of a given Riemann surface
. The solution of this problem is given by the following theorems of Maskit (cf. [4], [5]):
I) Let be an oriented surface and let
be a set of pairwise disjoint loops on
. If
is a regular covering with defining subgroup
, where
are natural numbers, then
is a flat covering, i.e. is homeomorphic to a domain in
.
II) Let be a flat surface and let
be a regular covering of an oriented surface
with defining subgroup
. If
is a surface of finite type, i.e.
is finitely generated, then there exists a finite set of simple pairwise disjoint loops
and natural numbers
such that
.
III) If is a flat Riemann surface and
is a properly-discontinuous group of conformal automorphisms of
, then there exists a conformal homeomorphism
such that
is a Kleinian group with invariant component
.
Thus, every Riemann surface is uniformized by a Kleinian group. E.g., if is a closed Riemann surface of genus
, then its fundamental group has the presentation
![]() |
and the normal subgroup defined by the flat covering
may be taken to be the smallest normal subgroup generated by
(or
);
is now uniformized by a Schottky group
of genus
— a free purely-loxodromic Kleinian group with
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 is a Kleinian group with invariant component
, then a subgroup
of it is called a quotient subgroup of
if
is a maximal subgroup such that: a) its invariant component
is simply connected; b)
does not contain random parabolic elements (i.e. parabolic elements such that for the conformal isomorphism
the image under
is hyperbolic); and c) every parabolic element of
with a fixed point in the limit set of
belongs to
. For example, in the Klein–Poincaré theorem every quotient subgroup of
coincides with
itself, and in Koebe's theorem on cross-cuts all quotient subgroups are trivial. A uniformization
of a Riemann surface
, where
is the invariant component of
, is called standard if
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 be a closed Riemann surface of genus
and let
be a set of simple pairwise disjoint loops on
. Then there exists a standard uniformization
of
, unique up to conformal equivalence, such that every quotient subgroup
is either Fuchsian or elementary and such that the covering
is constructed from the smallest normal subgroup of
spanned by the loops
.
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) |
Comments
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) |
Uniformization. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniformization&oldid=13136