Moduli of a Riemann surface

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) pp. Chapt. 10
[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)