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) |
Moduli of a Riemann surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moduli_of_a_Riemann_surface&oldid=17726