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 and are called conformally equivalent if there is a conformal mapping from onto . For example, the conformal classes of compact Riemann surfaces of topological genus are characterized by real moduli; a Riemann surface of torus type is characterized by 2 moduli; an -connected plane domain, considered as a Riemann surface with boundary, is characterized by moduli for . 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 -connected domains, , 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 -connected domain, which reflects the general geometric structure of this domain, but does not fix its moduli (see Conformal mapping).

Each doubly-connected domain of the -plane with non-degenerate boundary continua can be conformally mapped onto some circular annulus , . The ratio of the radii of the boundary circles of this annulus is a conformal invariant and is called the modulus of the doubly-connected domain . Let be an -connected domain, , with a non-degenerate boundary. can be conformally mapped onto some -connected circular domain , which is a circular annulus with discs with bounding circles , , removed; the circles , , lie in the annulus and pairwise do not have points in common. Here it can be assumed that and . Then depends on real parameters: the numbers and the real parameters defining the centres of the circles , . These real parameters can be taken as moduli of the -connected domain in the case .

As moduli of -connected domains it is also possible to take any other real parameters ( if , and if ) which determine a conformal mapping of onto some canonical -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)