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''of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u0952901.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u0952902.png" />)''
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A triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u0952903.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u0952904.png" /> is a system of meromorphic functions in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u0952905.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u0952906.png" />), defining a holomorphic [[Covering|covering]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u0952907.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u0952908.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u0952909.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529010.png" /> is a properly-discontinuous group of biholomorphic automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529011.png" /> whose restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529012.png" /> is the group of covering homeomorphisms of this covering, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529013.png" /> is biholomorphically equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529014.png" />.
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One may thus speak of uniformization by multi-valued analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529015.png" />, by which one understands uniformization of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529016.png" />; this corresponds to the parametrization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529017.png" /> by means of single-valued meromorphic functions.
+
''of a set $  A \subset  \mathbf C  ^ {N} $(
 +
or  $  A \subset  \mathbf C P  ^ {N} $)''
  
For example, the complex curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529018.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529019.png" /> is uniformized by the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529023.png" /> is the group of translations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529025.png" />, or the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529026.png" />, where
+
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|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} ) $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529027.png" /></td> </tr></table>
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529028.png" /></td> </tr></table>
+
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
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529029.png" /> is the trivial group. A less trivial example is the cubic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529030.png" />, which admits no rational parametrization, but which may be uniformized by means of elliptic functions (cf. [[Elliptic function|Elliptic function]]), namely by a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529033.png" /> are rational functions in the Weierstrass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529034.png" />-function and its derivative, with corresponding periods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529037.png" /> is the group generated by the translations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529039.png" />.
+
$$
 +
= \
 +
 
 +
\frac{( 1 - t  ^ {2} ) }{( 1 + t  ^ {2} ) }
 +
,\ \
 +
= \
 +
 
 +
\frac{2t }{( 1 + t  ^ {2} ) }
 +
,
 +
$$
 +
 
 +
$$
 +
= \mathbf C \setminus  \{ i, - i \} ,
 +
$$
 +
 
 +
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|Elliptic function]]), namely by a triple $  (( f _ {1} , f _ {2} ), D, G) $,  
 +
where $  f _ {1} $
 +
and $  f _ {2} $
 +
are rational functions in the Weierstrass $  {\mathcal P} $-
 +
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|algebraic curve]] defined by a general algebraic equation
 
The problem of uniformizing an arbitrary [[Algebraic curve|algebraic curve]] defined by a general algebraic equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
P ( z, w)  = \
 +
\sum _ { j,k } a _ {jk} z  ^ {j} w  ^ {k}  = 0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529041.png" /> is an irreducible algebraic polynomial over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529042.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529043.png" /> 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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529044.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529045.png" /> satisfying (*) a complex structure by means of elements of the corresponding algebraic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529046.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529047.png" />), and so obtains a compact [[Riemann surface|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.
+
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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529048.png" /> is a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529049.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529050.png" /> is a domain on the Riemann sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529052.png" /> is a regular holomorphic covering with covering group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529053.png" /> of conformal automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529054.png" />. The general problem consists in finding and describing all such triples for a given Riemann surface.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529055.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529056.png" /> (cf. [[#References|[4]]]–[[#References|[6]]]). The Klein–Poincaré uniformization theorem (proved in the general case by Poincaré, cf. [[#References|[2]]]) states: Every Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529057.png" /> is conformally equivalent to a quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529058.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529059.png" /> is one of the three canonical domains: the Riemann sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529060.png" />, the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529061.png" /> or the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529062.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529063.png" /> is a properly-discontinuous group of Möbius (fractional-linear) automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529064.png" />, defined up to conjugation in the group of all Möbius automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529065.png" />.
+
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. [[#References|[4]]]–[[#References|[6]]]). The Klein–Poincaré uniformization theorem (proved in the general case by Poincaré, cf. [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529068.png" /> are mutually exclusive. A surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529069.png" /> with such a universal holomorphic covering is called elliptic, parabolic or hyperbolic, respectively. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529070.png" /> only in the case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529071.png" /> itself is conformally equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529072.png" /> (and so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529073.png" /> is trivial); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529074.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529075.png" /> is conformally equivalent to either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529077.png" /> or the torus, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529078.png" /> is then either trivial or the group generated by the translation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529079.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529080.png" />) or the group generated by the two translations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529082.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529083.png" /> are complex numbers such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529084.png" />. In the remaining case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529085.png" /> is conformally equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529086.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529087.png" /> is a torsion-free [[Fuchsian group|Fuchsian group]]. The canonical projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529088.png" /> is an unramified covering and uniformizes all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529089.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529090.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529091.png" /> is single-valued on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529092.png" />. The Klein–Poincaré theorem also has a generalization to ramified coverings with given order of ramification.
+
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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529093.png" /> is homeomorphic to a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529094.png" /> (not necessarily simply connected), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529095.png" /> is also conformally equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529096.png" />. In the same way the uniformization problem may be reduced to the topological problem of finding all (generally speaking, ramified) flat coverings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529097.png" /> of a given Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529098.png" />. The solution of this problem is given by the following theorems of Maskit (cf. [[#References|[4]]], [[#References|[5]]]):
+
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. [[#References|[4]]], [[#References|[5]]]):
  
I) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u09529099.png" /> be an oriented surface and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290100.png" /> be a set of pairwise disjoint loops on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290101.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290102.png" /> is a regular covering with defining subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290103.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290104.png" /> are natural numbers, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290105.png" /> is a flat covering, i.e. is homeomorphic to a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290106.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290107.png" /> be a flat surface and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290108.png" /> be a regular covering of an oriented surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290109.png" /> with defining subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290110.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290111.png" /> is a surface of finite type, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290112.png" /> is finitely generated, then there exists a finite set of simple pairwise disjoint loops <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290113.png" /> and natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290114.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290115.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290116.png" /> is a flat Riemann surface and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290117.png" /> is a properly-discontinuous group of conformal automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290118.png" />, then there exists a conformal homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290119.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290120.png" /> is a [[Kleinian group|Kleinian group]] with invariant component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290121.png" />.
+
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|Kleinian group]] with invariant component $  D $.
  
Thus, every Riemann surface is uniformized by a Kleinian group. E.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290122.png" /> is a closed Riemann surface of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290123.png" />, then its fundamental group has the presentation
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290124.png" /></td> </tr></table>
+
$$
 +
\pi _ {1} ( S)  = \
 +
\left \{ {
 +
a _ {1} , b _ {1} \dots a _ {g} , b _ {g} } : {
 +
\prod _ {j = 1 } ^ { g }  [ a _ {j} , b _ {j} ] = 1 } \right \}
 +
,
 +
$$
  
and the normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290125.png" /> defined by the flat covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290126.png" /> may be taken to be the smallest normal subgroup generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290127.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290128.png" />); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290129.png" /> is now uniformized by a Schottky group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290130.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290131.png" /> — a free purely-loxodromic Kleinian group with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290132.png" /> generators (the classical Koebe theorem on cross-cuts).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290133.png" /> is a Kleinian group with invariant component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290134.png" />, then a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290135.png" /> of it is called a quotient subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290136.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290137.png" /> is a maximal subgroup such that: a) its invariant component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290138.png" /> is simply connected; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290139.png" /> does not contain random parabolic elements (i.e. parabolic elements such that for the conformal isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290140.png" /> the image under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290141.png" /> is hyperbolic); and c) every parabolic element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290142.png" /> with a fixed point in the limit set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290143.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290144.png" />. For example, in the Klein–Poincaré theorem every quotient subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290145.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290146.png" /> itself, and in Koebe's theorem on cross-cuts all quotient subgroups are trivial. A uniformization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290147.png" /> of a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290148.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290149.png" /> is the invariant component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290150.png" />, is called standard if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290151.png" /> is torsion-free and contains no random parabolic elements. For a closed surface all such uniformizations are described by the following theorem (cf. [[#References|[6]]]).
+
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. [[#References|[6]]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290152.png" /> be a closed Riemann surface of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290153.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290154.png" /> be a set of simple pairwise disjoint loops on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290155.png" />. Then there exists a standard uniformization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290156.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290157.png" />, unique up to conformal equivalence, such that every quotient subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290158.png" /> is either Fuchsian or elementary and such that the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290159.png" /> is constructed from the smallest normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290160.png" /> spanned by the loops <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095290/u095290161.png" />.
+
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|quasi-conformal mapping]] and Teichmüller spaces (cf. [[Teichmüller space|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. [[#References|[7]]]).
 
The theory of [[Quasi-conformal mapping|quasi-conformal mapping]] and Teichmüller spaces (cf. [[Teichmüller space|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. [[#References|[7]]]).
Line 49: Line 230:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Klein,  "Neue Beiträge zur Riemannschen Funktionentheorie"  ''Math. Ann.'' , '''21'''  (1883)  pp. 141–218</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Poincaré,  "Sur l'uniformisation des fonctions analytiques"  ''Acta Math.'' , '''31'''  (1907)  pp. 1–64</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  P. Koebe,  "Ueber die Uniformisierung beliebiger analytischer Kurven"  ''Nachr. K. Ges. Wissenschaft. Göttinger Math. Phys. Kl.''  (1907)  pp. 191–210</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  P. Koebe,  "Ueber die Uniformisierung beliebiger analytischer Kurven II"  ''Nachr. K. Ges. Wissenschaft. Göttinger Math. Phys. Kl.''  (1907)  pp. 177–198</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top">  P. Koebe,  "Ueber die Uniformisierung beliebiger analytischer Kurven III"  ''Nachr. K. Ges. Wissenschaft. Göttinger Math. Phys. Kl.''  (1908)  pp. 337–358</TD></TR><TR><TD valign="top">[3d]</TD> <TD valign="top">  P. Koebe,  "Ueber die Uniformisierung beliebiger analytischer Kurven IV"  ''Nachr. K. Ges. Wissenschaft. Göttinger Math. Phys. Kl.''  (1909)  pp. 324–361</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B. Maskit,  "A theorem on planar covering surfaces with applications to 3-manifolds"  ''Ann. of Math.'' , '''81''' :  2  (1965)  pp. 341–355</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  B. Maskit,  "The conformal group of a plane domain"  ''Amer. J. Math.'' , '''90''' :  3  (1968)  pp. 718–722</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B. Maskit,  L.V. Ahlfors (ed.)  et al. (ed.) , ''Contributions to Analysis. Uniformization of Riemann surfaces'' , Acad. Press  (1974)  pp. 293–312</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L. Bers,  "Uniformization. Moduli and Kleinian groups"  ''Bull. London Math. Soc.'' , '''4'''  (1972)  pp. 257–300</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  S.L. Krushkal',  B.N. Apanasov,  N.A. Gusevskii,  "Kleinian groups and uniformization in examples and problems" , Amer. Math. Soc.  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  R. Nevanlinna,  "Uniformisierung" , Springer  (1953)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  L.R. Ford,  "Automorphic functions" , Chelsea, reprint  (1957)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Klein,  "Neue Beiträge zur Riemannschen Funktionentheorie"  ''Math. Ann.'' , '''21'''  (1883)  pp. 141–218</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Poincaré,  "Sur l'uniformisation des fonctions analytiques"  ''Acta Math.'' , '''31'''  (1907)  pp. 1–64</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  P. Koebe,  "Ueber die Uniformisierung beliebiger analytischer Kurven"  ''Nachr. K. Ges. Wissenschaft. Göttinger Math. Phys. Kl.''  (1907)  pp. 191–210</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  P. Koebe,  "Ueber die Uniformisierung beliebiger analytischer Kurven II"  ''Nachr. K. Ges. Wissenschaft. Göttinger Math. Phys. Kl.''  (1907)  pp. 177–198</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top">  P. Koebe,  "Ueber die Uniformisierung beliebiger analytischer Kurven III"  ''Nachr. K. Ges. Wissenschaft. Göttinger Math. Phys. Kl.''  (1908)  pp. 337–358</TD></TR><TR><TD valign="top">[3d]</TD> <TD valign="top">  P. Koebe,  "Ueber die Uniformisierung beliebiger analytischer Kurven IV"  ''Nachr. K. Ges. Wissenschaft. Göttinger Math. Phys. Kl.''  (1909)  pp. 324–361</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B. Maskit,  "A theorem on planar covering surfaces with applications to 3-manifolds"  ''Ann. of Math.'' , '''81''' :  2  (1965)  pp. 341–355</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  B. Maskit,  "The conformal group of a plane domain"  ''Amer. J. Math.'' , '''90''' :  3  (1968)  pp. 718–722</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B. Maskit,  L.V. Ahlfors (ed.)  et al. (ed.) , ''Contributions to Analysis. Uniformization of Riemann surfaces'' , Acad. Press  (1974)  pp. 293–312</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L. Bers,  "Uniformization. Moduli and Kleinian groups"  ''Bull. London Math. Soc.'' , '''4'''  (1972)  pp. 257–300</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  S.L. Krushkal',  B.N. Apanasov,  N.A. Gusevskii,  "Kleinian groups and uniformization in examples and problems" , Amer. Math. Soc.  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  R. Nevanlinna,  "Uniformisierung" , Springer  (1953)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  L.R. Ford,  "Automorphic functions" , Chelsea, reprint  (1957)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Gunning,  "On uniformization of complex manifolds: the role of connections" , Princeton Univ. Press  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Springer,  "Introduction to Riemann surfaces" , Addison-Wesley  (1957)  pp. Chapt.10</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.N. Apanasov,  "Discrete groups in space and uniformization problems" , Kluwer  (1991)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Gunning,  "On uniformization of complex manifolds: the role of connections" , Princeton Univ. Press  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Springer,  "Introduction to Riemann surfaces" , Addison-Wesley  (1957)  pp. Chapt.10</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.N. Apanasov,  "Discrete groups in space and uniformization problems" , Kluwer  (1991)  (Translated from Russian)</TD></TR></table>

Revision as of 08:27, 6 June 2020


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 $ {\mathcal P} $- 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)

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)
How to Cite This Entry:
Uniformization. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniformization&oldid=49073
This article was adapted from an original article by N.A. Gusevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article