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 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) |
Moduli of a Riemann surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moduli_of_a_Riemann_surface&oldid=53552