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 $ g $; 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 $ g> 1 $ depend on $ 3g- 3 $ complex parameters, called the moduli of a Riemann surface; for conformally-equivalent surfaces these moduli coincide. The case when $ g= 1 $ is described below. If one considers compact Riemann surfaces of genus $ g $ with $ n $ analytic boundary components, then, in order that they be conformally equivalent, it is necessary that $ 6g- 6+ 3n $ real moduli-parameters ( $ g \geq 0 $, $ n \geq 0 $, $ 6g- 6+ 3n > 0 $) coincide. In particular, for $ n $- connected plane domains $ ( n \geq 3) $ there are $ 3n- 6 $ 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 $ g $. There exist two approaches to the moduli problem: an algebraic and an analytic one. The algebraic approach is connected with studies of the fields $ K( S) $ of meromorphic functions on Riemann surfaces $ S $. In the case of a closed surface, $ K( S) $ is a field of algebraic functions (for $ g= 0 $ it is the field of rational functions, and for $ g= 1 $ it is the field of elliptic functions). Each closed Riemann surface $ S $ is conformally equivalent to the Riemann surface of some algebraic function defined by an equation $ P( z, w) = 0 $, where $ P $ is an irreducible polynomial over $ \mathbf C $. This equation determines a planar algebraic curve $ X $, and the field of rational functions on $ X $ is identified with the field of meromorphic functions on $ S $. 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 $ S $ of given genus $ g \geq 1 $ one takes pairs $ ( S, f ) $, where $ f $ is a homeomorphism of a fixed surface $ S _ {0} $ of genus $ g $ onto $ S $; two pairs $ ( S, f ) $, $ ( S ^ \prime , f ^ { \prime } ) $ are considered equivalent if there exists a conformal homeomorphism $ h: S \rightarrow S ^ \prime $ such that the mapping

$$ ( f ^ { \prime } ) ^ {-} 1 \circ h \circ f: S _ {0} \rightarrow S _ {0} $$

is homotopic to the identity. The set of equivalence classes $ \{ ( S, f ) \} $ is called the Teichmüller space $ T( S _ {0} ) $ of the surface $ S _ {0} $. In $ T( S _ {0} ) $ one can introduce a metric using quasi-conformal homeomorphisms $ S \rightarrow S ^ \prime $. Similarly, one can define the Teichmüller space for a non-compact Riemann surface, but then quasi-conformal homeomorphisms $ f $ only are accepted. For closed surfaces $ S _ {0} $ of given genus $ g $ the spaces $ T( S _ {0} ) $ are isometrically isomorphic, and one can speak of the Teichmüller space $ T _ {g} $ of surfaces of genus $ g $. The space $ R _ {g} $ of conformal classes of Riemann surfaces of genus $ g $ is obtained by factorization of $ T _ {g} $ by some countable group $ \Gamma _ {g} $ of automorphisms of it, called the modular group.

The simplest is the case of surfaces of genus 1 — tori. Each torus $ S $, provided its universal covering surface has been conformally mapped onto the complex plane $ \mathbf C $ can be represented as $ \mathbf C /G $, where $ G $ is a group of translations with two generators $ \omega _ {1} , \omega _ {2} $ such that $ \mathop{\rm Im} ( \omega _ {2} / \omega _ {1} ) > 0 $; here, two tori $ S $ and $ S ^ \prime $ are conformally equivalent if and only if the ratios $ \tau = \omega _ {2} / \omega _ {1} $ and $ \tau ^ \prime = \omega _ {2} ^ \prime / \omega _ {1} ^ \prime $ of the corresponding generators are related by a modular transformation

$$ \tau ^ \prime = \frac{a \tau + b }{c \tau + d } ,\ ad - bc = 1; \ \ a, b, c, d \in \mathbf Z . $$

As a (complex) modulus of the given conformal class of Riemann surfaces $ \{ S \} $ one can take the value of the elliptic modular function $ J( \tau ) $. The Teichmüller space $ T _ {1} $ coincides with the upper half-plane $ H = \{ {\tau \in \mathbf C } : { \mathop{\rm Im} \tau > 0 } \} $, $ \Gamma _ {1} $ is the elliptic modular group $ \mathop{\rm SL} ( 2, \mathbf Z ) / \{ \pm 1 \} $, and $ R _ {1} = T _ {1} / \Gamma _ {1} $ is a Riemann surface conformally equivalent to $ \mathbf C $. All elliptic curves (and surfaces of genus 1) admit a simultaneous uniformization by the Weierstrass function $ {\mathcal p} ( z; 1, \tau ) $ and its derivative $ {\mathcal p} ^ \prime ( z; 1, \tau ) $( cf. Weierstrass elliptic functions).

For $ g > 1 $ the situation is much more complicated. In particular, the following fundamental properties of the space $ T _ {g} $ have been established: 1) $ T _ {g} $ is homeomorphic to $ R _ {6g-} 6 $; 2) $ T _ {g} $ can be biholomorphically imbedded as a bounded domain into $ \mathbf C ^ {3g-} 3 $ that is holomorphically convex; 3) the modular group $ \Gamma _ {g} $ is discrete (even properly discontinuous) and for $ g > 2 $ it is the complete group of biholomorphic automorphisms of $ T _ {g} $; 4) the covering $ T _ {g} \rightarrow T _ {g} / \Gamma _ {g} $ is ramified and $ T _ {g} / \Gamma _ {g} = R _ {6g-} 6 $ 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 $ T _ {g} $ in $ \mathbf C ^ {3g-} 3 $ is obtained by uniformization and using quasi-conformal mapping. The surface $ S _ {0} $ can be represented as $ S _ {0} = H/ \Gamma _ {0} $, where $ \Gamma _ {0} $ is a Fuchsian group, acting discontinuously in the upper half-plane $ H $( defined up to conjugation in the group of all conformal automorphisms of $ H $), and one considers quasi-conformal automorphisms $ w = f ^ { \mu } ( z) $ of the plane $ \overline{\mathbf C}\; = \mathbf C \cup \{ \infty \} $, i.e. solutions of the Beltrami equation $ w _ \overline{ {z}}\; = \mu ( z) w _ {z} $, where $ \mu ( z) d \overline{z}\; / dz $ are forms with supports in $ H $ that are invariant under $ \Gamma _ {0} $, $ \| \mu \| _ {L _ \infty } < 1 $. Further, suppose $ f ^ { \mu } $ leaves the points $ 0, 1, \infty $ fixed. Then $ T _ {g} $ can be identified with the space of restrictions $ f ^ { \mu } \mid _ {R} $ or, which is equivalent, of restrictions $ f ^ { \mu } \mid _ {L} $, $ L = \{ {z \in \mathbf C } : { \mathop{\rm Im} z < 0 } \} $, and $ T _ {g} $ is biholomorphically equivalent to the domain filled by the Schwarzian derivatives

$$ \{ f ^ { \mu } , z \} = \frac{f ^ { \prime\prime\prime } ( z) }{f ^ { \prime } ( z) } - \frac{3}{2} \left [ \frac{f ^ { \prime\prime } ( z) }{f ^ { \prime } ( z) } \right ] ^ {2} ,\ \ f = f ^ { \mu } ,\ \ z \in L , $$

in the complex space $ B( L, \Gamma _ {0} ) $ of holomorphic solutions in $ L $ of the equation

$$ \phi ( \gamma ( z)) \gamma ^ \prime 2 ( z) = \phi ( z),\ \ \gamma \in \Gamma _ {0} , $$

with the norm

$$ \| \phi \| = \sup _ { L } | \mathop{\rm Im} z | ^ {2} | \phi ( z) | . $$

Here, $ \mathop{\rm dim} B( L, \Gamma _ {0} ) = 3g- 3 $. Using this imbedding one can construct the fibre space $ \widetilde{T} _ {g} $ with base $ T _ {g} $, which also admits the introduction of a complex structure and holomorphic functions $ \psi _ {1} \dots \psi _ {5g-} 5 $ on $ \widetilde{T} _ {g} $ that make it possible to give a parametric representation of all algebraic curves of genus $ g $ in the complex projective space $ \mathbf C P ^ {n} $, $ n \geq 2 $. The above-mentioned construction related to the imbedding of $ T _ {g} $ in $ B( L, \Gamma _ {0} ) $ 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 $ g > 1 $ is obtained by the so-called period matrices of these surfaces. These are symmetric $ ( g \times g ) $- matrices with positive-definite imaginary part. The space $ T _ {g} $ can be holomorphically imbedded in the set of all such matrices (the Siegel upper half-plane) $ H _ {g} $( 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 $ w = \sqrt {p( z) } $, where $ p( z) $ are polynomials of the form $ ( z- z _ {1} ) \dots ( z- z _ {2g+} 2 ) $. They admit a conformal involution and depend on $ 2g- 1 $ complex parameters. All surfaces of genus 2 are hyper-elliptic; for $ g> 2 $ such surfaces form the analytic submanifolds of dimension $ 2g- 1 $ in $ T _ {g} $.

The problem of the conformal automorphisms of a given Riemann surface $ S $ is related to conformal classes of Riemann surfaces. Except for several particular cases, the group $ \mathop{\rm Aut} S $ of such automorphisms is discrete. In the case of closed surfaces of genus $ g > 1 $ it is finite; moreover, the order of $ \mathop{\rm Aut} S $ does not exceed $ 84( g- 1) $.

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) MR0228671 Zbl 0152.27401
[2]	G. Springer, "Introduction to Riemann surfaces" , Chelsea, reprint (1981) MR0122987 MR1530201 MR0092855 Zbl 0501.30039
[3]	S.L. Krushkal', "Quasi-conformal mappings and Riemann surfaces" , Winston & Wiley (1979) (Translated from Russian) MR536488
[4]	L. Bers, "Uniformization, moduli, and Kleinian groups" Bull. London Math. Soc. , 4 (1972) pp. 257–300 MR0348097 Zbl 0257.32012
[5]	M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954) MR0065652 Zbl 0059.06901
[6]	W. Abikoff, "The real analytic theory of Teichmüller space" , Springer (1980) MR0590044 Zbl 0452.32015
[7]	H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980) MR0583745 Zbl 0475.30001
[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) MR0867407 Zbl 0606.30001

Comments

The group $ \Gamma _ {g} = ( \mathop{\rm Diff} ( S _ {0} ) ) / ( \mathop{\rm Diff} _ {0} ( S _ {0} )) $ of connected components of the group of diffeomorphisms of the reference Riemann surface $ S _ {0} $, 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) MR0903027 Zbl 0629.30002
[a2]	S. Nag, "The complex analytic theory of Teichmüller spaces" , Wiley (Interscience) (1988) MR0927291 Zbl 0667.30040
[a3]	M. Schlichenmaier, "An introduction to Riemann surfaces, algebraic curves, and moduli spaces" , Springer (1989) MR0981595 Zbl 0683.14007