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Numerical characteristics (parameters) which are one and the same for all conformally-equivalent Riemann surfaces, and in their totality characterize the conformal equivalence class of a given Riemann surface. Here two Riemann surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m0644901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m0644902.png" /> are called conformally equivalent if there is a conformal mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m0644903.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m0644904.png" />. For example, the conformal classes of compact Riemann surfaces of topological genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m0644905.png" /> are characterized by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m0644906.png" /> real moduli; a Riemann surface of torus type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m0644907.png" /> is characterized by 2 moduli; an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m0644908.png" />-connected plane domain, considered as a Riemann surface with boundary, is characterized by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m0644909.png" /> moduli for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449010.png" />. About the structure of the moduli space of a Riemann surface see [[Riemann surfaces, conformal classes of|Riemann surfaces, conformal classes of]].
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A necessary condition for the conformal equivalence of two plane domains is that they have the same connectivity. According to the [[Riemann theorem|Riemann theorem]], all simply-connected domains with more than one boundary point are conformally equivalent to each other; each such domain can be conformally mapped onto one canonical domain, usually taken to be the unit disc. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449011.png" />-connected domains, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449012.png" />, a precise equivalent of this Riemann mapping theorem does not exist: It is impossible to give any fixed domain whatever onto which it is possible to univalently and conformally map all domains of a given order of connectivity. This has led to a more flexible definition of a canonical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449013.png" />-connected domain, which reflects the general geometric structure of this domain, but does not fix its moduli (see [[Conformal mapping|Conformal mapping]]).
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Each doubly-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449014.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449015.png" />-plane with non-degenerate boundary continua can be conformally mapped onto some circular annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449017.png" />. The ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449018.png" /> of the radii of the boundary circles of this annulus is a conformal invariant and is called the modulus of the doubly-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449019.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449020.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449021.png" />-connected domain, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449022.png" />, with a non-degenerate boundary. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449023.png" /> can be conformally mapped onto some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449024.png" />-connected circular domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449025.png" />, which is a circular annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449026.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449027.png" /> discs with bounding circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449029.png" />, removed; the circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449031.png" />, lie in the annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449032.png" /> and pairwise do not have points in common. Here it can be assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449034.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449035.png" /> depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449036.png" /> real parameters: the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449037.png" /> numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449038.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449039.png" /> real parameters defining the centres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449040.png" /> of the circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449042.png" />. These <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449043.png" /> real parameters can be taken as moduli of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449045.png" />-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449046.png" /> in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449047.png" />.
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Numerical characteristics (parameters) which are one and the same for all conformally-equivalent Riemann surfaces, and in their totality characterize the conformal equivalence class of a given Riemann surface. Here two Riemann surfaces  $  R _ {1} $
 +
and  $  R _ {2} $
 +
are called conformally equivalent if there is a conformal mapping from  $  R _ {1} $
 +
onto $  R _ {2} $.  
 +
For example, the conformal classes of compact Riemann surfaces of topological genus  $  g > 1 $
 +
are characterized by  $  6 g - 6 $
 +
real moduli; a Riemann surface of torus type  $  ( g = 1 ) $
 +
is characterized by 2 moduli; an $  n $-
 +
connected plane domain, considered as a Riemann surface with boundary, is characterized by  $  3 n - 6 $
 +
moduli for  $  n \geq  3 $.  
 +
About the structure of the moduli space of a Riemann surface see [[Riemann surfaces, conformal classes of|Riemann surfaces, conformal classes of]].
  
As moduli of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449048.png" />-connected domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449049.png" /> it is also possible to take any other <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449050.png" /> real parameters (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449051.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449052.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449053.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449054.png" />) which determine a conformal mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449055.png" /> onto some canonical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064490/m06449056.png" />-connected domain of another shape.
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A necessary condition for the conformal equivalence of two plane domains is that they have the same connectivity. According to the [[Riemann theorem|Riemann theorem]], all simply-connected domains with more than one boundary point are conformally equivalent to each other; each such domain can be conformally mapped onto one canonical domain, usually taken to be the unit disc. For  $  n $-
 +
connected domains,  $  n \geq  2 $,
 +
a precise equivalent of this Riemann mapping theorem does not exist: It is impossible to give any fixed domain whatever onto which it is possible to univalently and conformally map all domains of a given order of connectivity. This has led to a more flexible definition of a canonical  $  n $-
 +
connected domain, which reflects the general geometric structure of this domain, but does not fix its moduli (see [[Conformal mapping|Conformal mapping]]).
 +
 
 +
Each doubly-connected domain  $  D $
 +
of the  $  z $-
 +
plane with non-degenerate boundary continua can be conformally mapped onto some circular annulus  $  r < | w | < R $,
 +
0 < r < R < \infty $.
 +
The ratio  $  R / r $
 +
of the radii of the boundary circles of this annulus is a conformal invariant and is called the modulus of the doubly-connected domain  $  D $.  
 +
Let  $  D $
 +
be an  $  n $-
 +
connected domain,  $  n \geq  3 $,
 +
with a non-degenerate boundary. $  D $
 +
can be conformally mapped onto some  $  n $-
 +
connected circular domain  $  \Delta $,
 +
which is a circular annulus  $  r < | w | < R $
 +
with  $  n- 2 $
 +
discs with bounding circles  $  C _ {k} = \{ {w } : {| w - w _ {k} | = r _ {k} } \} $,
 +
$  k = 1 \dots n - 2 $,
 +
removed; the circles  $  C _ {k} $,
 +
$  k = 1 \dots n - 2 $,
 +
lie in the annulus  $  r < | w | < R $
 +
and pairwise do not have points in common. Here it can be assumed that  $  R = 1 $
 +
and  $  w _ {1} > 0 $.
 +
Then  $  \Delta $
 +
depends on  $  3 n - 6 $
 +
real parameters: the  $  n - 1 $
 +
numbers  $  r , r _ {1} \dots r _ {n-} 2 $
 +
and the  $  2 n - 5 $
 +
real parameters defining the centres  $  w _ {k} $
 +
of the circles  $  C _ {k} $,
 +
$  k = 1 \dots n - 2 $.  
 +
These  $  3 n - 6 $
 +
real parameters can be taken as moduli of the  $  n $-
 +
connected domain  $  D $
 +
in the case  $  n \geq  3 $.
 +
 
 +
As moduli of  $  n $-
 +
connected domains  $  D $
 +
it is also possible to take any other $  \mu $
 +
real parameters ( $  \mu = 1 $
 +
if $  n = 2 $,  
 +
and $  \mu = 3 n - 6 $
 +
if $  n \geq  3 $)  
 +
which determine a conformal mapping of $  D $
 +
onto some canonical $  n $-
 +
connected domain of another shape.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Springer,   "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt. 10</TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Courant,  "Dirichlet's principle, conformal mapping, and minimal surfaces" , Interscience  (1950)  (With appendix by M. Schiffer: Some recent developments in the theory of conformal mapping)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> G. Springer, "Introduction to Riemann surfaces", Addison-Wesley (1957) Chapt. 10 {{ZBL|0078.06602}}</TD></TR>
 +
<TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Courant,  "Dirichlet's principle, conformal mapping, and minimal surfaces" , Interscience  (1950)  (With appendix by M. Schiffer: Some recent developments in the theory of conformal mapping)</TD></TR>
 +
</table>

Latest revision as of 06:30, 31 March 2023


Numerical characteristics (parameters) which are one and the same for all conformally-equivalent Riemann surfaces, and in their totality characterize the conformal equivalence class of a given Riemann surface. Here two Riemann surfaces $ R _ {1} $ and $ R _ {2} $ are called conformally equivalent if there is a conformal mapping from $ R _ {1} $ onto $ R _ {2} $. For example, the conformal classes of compact Riemann surfaces of topological genus $ g > 1 $ are characterized by $ 6 g - 6 $ real moduli; a Riemann surface of torus type $ ( g = 1 ) $ is characterized by 2 moduli; an $ n $- connected plane domain, considered as a Riemann surface with boundary, is characterized by $ 3 n - 6 $ moduli for $ n \geq 3 $. About the structure of the moduli space of a Riemann surface see Riemann surfaces, conformal classes of.

A necessary condition for the conformal equivalence of two plane domains is that they have the same connectivity. According to the Riemann theorem, all simply-connected domains with more than one boundary point are conformally equivalent to each other; each such domain can be conformally mapped onto one canonical domain, usually taken to be the unit disc. For $ n $- connected domains, $ n \geq 2 $, a precise equivalent of this Riemann mapping theorem does not exist: It is impossible to give any fixed domain whatever onto which it is possible to univalently and conformally map all domains of a given order of connectivity. This has led to a more flexible definition of a canonical $ n $- connected domain, which reflects the general geometric structure of this domain, but does not fix its moduli (see Conformal mapping).

Each doubly-connected domain $ D $ of the $ z $- plane with non-degenerate boundary continua can be conformally mapped onto some circular annulus $ r < | w | < R $, $ 0 < r < R < \infty $. The ratio $ R / r $ of the radii of the boundary circles of this annulus is a conformal invariant and is called the modulus of the doubly-connected domain $ D $. Let $ D $ be an $ n $- connected domain, $ n \geq 3 $, with a non-degenerate boundary. $ D $ can be conformally mapped onto some $ n $- connected circular domain $ \Delta $, which is a circular annulus $ r < | w | < R $ with $ n- 2 $ discs with bounding circles $ C _ {k} = \{ {w } : {| w - w _ {k} | = r _ {k} } \} $, $ k = 1 \dots n - 2 $, removed; the circles $ C _ {k} $, $ k = 1 \dots n - 2 $, lie in the annulus $ r < | w | < R $ and pairwise do not have points in common. Here it can be assumed that $ R = 1 $ and $ w _ {1} > 0 $. Then $ \Delta $ depends on $ 3 n - 6 $ real parameters: the $ n - 1 $ numbers $ r , r _ {1} \dots r _ {n-} 2 $ and the $ 2 n - 5 $ real parameters defining the centres $ w _ {k} $ of the circles $ C _ {k} $, $ k = 1 \dots n - 2 $. These $ 3 n - 6 $ real parameters can be taken as moduli of the $ n $- connected domain $ D $ in the case $ n \geq 3 $.

As moduli of $ n $- connected domains $ D $ it is also possible to take any other $ \mu $ real parameters ( $ \mu = 1 $ if $ n = 2 $, and $ \mu = 3 n - 6 $ if $ n \geq 3 $) which determine a conformal mapping of $ D $ onto some canonical $ n $- connected domain of another shape.

References

[1] G. Springer, "Introduction to Riemann surfaces", Addison-Wesley (1957) Chapt. 10 Zbl 0078.06602
[2] L. Bers, "Uniformization, moduli, and Kleinian groups" Bull. London Math. Soc. , 4 (1972) pp. 257–300
[3] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[4] R. Courant, "Dirichlet's principle, conformal mapping, and minimal surfaces" , Interscience (1950) (With appendix by M. Schiffer: Some recent developments in the theory of conformal mapping)
How to Cite This Entry:
Moduli of a Riemann surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moduli_of_a_Riemann_surface&oldid=17726
This article was adapted from an original article by G.V. Kuz'minaE.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article