Riemann surfaces, conformal classes of
Classes consisting of conformally-equivalent Riemann surfaces (cf. Riemann surface). Closed Riemann surfaces have a simple topological invariant — the genus ; moreover, any two surfaces of the same genus are homeomorphic. In the simplest cases, the topological equivalence of two Riemann surfaces ensures also their membership in the same conformal class of Riemann surfaces, that is, their conformal equivalence, or, in other words, the coincidence of their conformal structures. This is true, for example, for surfaces of genus 0, i.e. homeomorphic spheres. In general, this is not the case. B. Riemann already noticed that the conformal equivalence classes of Riemann surfaces of genus
depend on
complex parameters, called the moduli of a Riemann surface; for conformally-equivalent surfaces these moduli coincide. The case when
is described below. If one considers compact Riemann surfaces of genus
with
analytic boundary components, then, in order that they be conformally equivalent, it is necessary that
real moduli-parameters (
,
,
) coincide. In particular, for
-connected plane domains
there are
of such moduli; any doubly-connected plane domain is conformally equivalent to an annulus with a certain ratio of the radii.
The above-mentioned remark of Riemann is the origin of the classical moduli problem for Riemann surfaces, which studies the nature of these parameters in order to introduce them, if possible, in such a way that they would define a complex-analytic structure on the set of Riemann surfaces of given genus . There exist two approaches to the moduli problem: an algebraic and an analytic one. The algebraic approach is connected with studies of the fields
of meromorphic functions on Riemann surfaces
. In the case of a closed surface,
is a field of algebraic functions (for
it is the field of rational functions, and for
it is the field of elliptic functions). Each closed Riemann surface
is conformally equivalent to the Riemann surface of some algebraic function defined by an equation
, where
is an irreducible polynomial over
. This equation determines a planar algebraic curve
, and the field of rational functions on
is identified with the field of meromorphic functions on
. To conformal equivalence of Riemann surfaces corresponds birational equivalence (coincidence) of their fields of algebraic functions or, which is the same, birational equivalence of the algebraic curves determined by these surfaces.
The analytic approach is based on geometric and analytic properties of Riemann surfaces. It turns out to be convenient to weaken the conformal equivalence of Riemann surfaces by imposing topological restrictions. Instead of a Riemann surface of given genus
one takes pairs
, where
is a homeomorphism of a fixed surface
of genus
onto
; two pairs
,
are considered equivalent if there exists a conformal homeomorphism
such that the mapping
![]() |
is homotopic to the identity. The set of equivalence classes is called the Teichmüller space
of the surface
. In
one can introduce a metric using quasi-conformal homeomorphisms
. Similarly, one can define the Teichmüller space for a non-compact Riemann surface, but then quasi-conformal homeomorphisms
only are accepted. For closed surfaces
of given genus
the spaces
are isometrically isomorphic, and one can speak of the Teichmüller space
of surfaces of genus
. The space
of conformal classes of Riemann surfaces of genus
is obtained by factorization of
by some countable group
of automorphisms of it, called the modular group.
The simplest is the case of surfaces of genus 1 — tori. Each torus , provided its universal covering surface has been conformally mapped onto the complex plane
can be represented as
, where
is a group of translations with two generators
such that
; here, two tori
and
are conformally equivalent if and only if the ratios
and
of the corresponding generators are related by a modular transformation
![]() |
As a (complex) modulus of the given conformal class of Riemann surfaces one can take the value of the elliptic modular function
. The Teichmüller space
coincides with the upper half-plane
,
is the elliptic modular group
, and
is a Riemann surface conformally equivalent to
. All elliptic curves (and surfaces of genus 1) admit a simultaneous uniformization by the Weierstrass function
and its derivative
(cf. Weierstrass elliptic functions).
For the situation is much more complicated. In particular, the following fundamental properties of the space
have been established: 1)
is homeomorphic to
; 2)
can be biholomorphically imbedded as a bounded domain into
that is holomorphically convex; 3) the modular group
is discrete (even properly discontinuous) and for
it is the complete group of biholomorphic automorphisms of
; 4) the covering
is ramified and
is a normal complex space with non-uniformizable singularities. The same properties, apart from certain exceptions in 3), are valid for the more general case of closed Riemann surfaces with a finite number of punctures, to which correspond finite-dimensional Teichmüller spaces. The indicated biholomorphic imbedding of
in
is obtained by uniformization and using quasi-conformal mapping. The surface
can be represented as
, where
is a Fuchsian group, acting discontinuously in the upper half-plane
(defined up to conjugation in the group of all conformal automorphisms of
), and one considers quasi-conformal automorphisms
of the plane
, i.e. solutions of the Beltrami equation
, where
are forms with supports in
that are invariant under
,
. Further, suppose
leaves the points
fixed. Then
can be identified with the space of restrictions
or, which is equivalent, of restrictions
,
, and
is biholomorphically equivalent to the domain filled by the Schwarzian derivatives
![]() |
in the complex space of holomorphic solutions in
of the equation
![]() |
with the norm
![]() |
Here, . Using this imbedding one can construct the fibre space
with base
, which also admits the introduction of a complex structure and holomorphic functions
on
that make it possible to give a parametric representation of all algebraic curves of genus
in the complex projective space
,
. The above-mentioned construction related to the imbedding of
in
can be generalized to arbitrary Riemann surfaces and Fuchsian groups. In particular, for compact Riemann surfaces with analytic boundaries the Teichmüller space obtained allows the introduction of a global real-analytic structure of corresponding dimension.
Another description of conformal classes of Riemann surfaces of genus is obtained by the so-called period matrices of these surfaces. These are symmetric
-matrices with positive-definite imaginary part. The space
can be holomorphically imbedded in the set of all such matrices (the Siegel upper half-plane)
(see [4], [5]).
There are closed Riemann surfaces with a certain symmetry, the conformal classes of which depend on a smaller number of parameters. These are the hyper-elliptic surfaces equivalent to the two-sheeted Riemann surfaces of the functions , where
are polynomials of the form
. They admit a conformal involution and depend on
complex parameters. All surfaces of genus 2 are hyper-elliptic; for
such surfaces form the analytic submanifolds of dimension
in
.
The problem of the conformal automorphisms of a given Riemann surface is related to conformal classes of Riemann surfaces. Except for several particular cases, the group
of such automorphisms is discrete. In the case of closed surfaces of genus
it is finite; moreover, the order of
does not exceed
.
The existing classification of non-compact Riemann surfaces of infinite genus is based on picking out certain conformal invariants and does not define the conformal classes of Riemann surfaces completely; this is usually done in terms of the existence of analytic and harmonic functions with certain properties (cf. also Riemann surfaces, classification of).
References
[1] | R. Nevanlinna, "Uniformisierung" , Springer (1967) |
[2] | G. Springer, "Introduction to Riemann surfaces" , Chelsea, reprint (1981) |
[3] | S.L. Krushkal', "Quasi-conformal mappings and Riemann surfaces" , Winston & Wiley (1979) (Translated from Russian) |
[4] | L. Bers, "Uniformization, moduli, and Kleinian groups" Bull. London Math. Soc. , 4 (1972) pp. 257–300 |
[5] | M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954) |
[6] | W. Abikoff, "The real analytic theory of Teichmüller space" , Springer (1980) |
[7] | H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980) |
[8] | N.A. Guserkii, "Kleinian groups and uniformization in examples and problems" , Amer. Math. Soc. (1986) (Translated from Russian) |
[9] | O. Lehto, "Univalent functions and Teichmüller spaces" , Springer (1986) |
Comments
The group of connected components of the group of diffeomorphisms of the reference Riemann surface
, called the modular group above, is also frequently called the mapping class group.
References
[a1] | F.P. Gardiner, "Teichmüller theory and quadratic differentials" , Wiley (Interscience) (1987) |
[a2] | S. Nag, "The complex analytic theory of Teichmüller spaces" , Wiley (Interscience) (1988) |
[a3] | M. Schlichenmaier, "An introduction to Riemann surfaces, algebraic curves, and moduli spaces" , Springer (1989) |
Riemann surfaces, conformal classes of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_surfaces,_conformal_classes_of&oldid=12143