View source for Moduli of a Riemann surface

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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]].

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====
<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>