# 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.

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)
How to Cite This Entry:
Moduli of a Riemann surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moduli_of_a_Riemann_surface&oldid=47875
This article was adapted from an original article by G.V. Kuz'minaE.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article