View source for Riemann surfaces, conformal classes of

Classes consisting of conformally-equivalent Riemann surfaces (cf. [[Riemann surface|Riemann surface]]). Closed Riemann surfaces have a simple topological invariant — the genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r0820601.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r0820602.png" /> depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r0820603.png" /> complex parameters, called the [[Moduli of a Riemann surface|moduli of a Riemann surface]]; for conformally-equivalent surfaces these moduli coincide. The case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r0820604.png" /> is described below. If one considers compact Riemann surfaces of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r0820605.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r0820606.png" /> analytic boundary components, then, in order that they be conformally equivalent, it is necessary that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r0820607.png" /> real moduli-parameters (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r0820608.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r0820609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206010.png" />) coincide. In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206011.png" />-connected plane domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206012.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206013.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206014.png" />. There exist two approaches to the moduli problem: an algebraic and an analytic one. The algebraic approach is connected with studies of the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206015.png" /> of meromorphic functions on Riemann surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206016.png" />. In the case of a closed surface, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206017.png" /> is a field of algebraic functions (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206018.png" /> it is the field of rational functions, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206019.png" /> it is the field of elliptic functions). Each closed Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206020.png" /> is conformally equivalent to the Riemann surface of some algebraic function defined by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206022.png" /> is an irreducible polynomial over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206023.png" />. This equation determines a planar [[Algebraic curve|algebraic curve]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206024.png" />, and the field of rational functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206025.png" /> is identified with the field of meromorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206026.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206027.png" /> of given genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206028.png" /> one takes pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206030.png" /> is a homeomorphism of a fixed surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206031.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206032.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206033.png" />; two pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206035.png" /> are considered equivalent if there exists a conformal homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206036.png" /> such that the mapping

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206037.png" /></td> </tr></table>

is homotopic to the identity. The set of equivalence classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206038.png" /> is called the Teichmüller space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206039.png" /> of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206040.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206041.png" /> one can introduce a metric using quasi-conformal homeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206042.png" />. Similarly, one can define the Teichmüller space for a non-compact Riemann surface, but then quasi-conformal homeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206043.png" /> only are accepted. For closed surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206044.png" /> of given genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206045.png" /> the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206046.png" /> are isometrically isomorphic, and one can speak of the Teichmüller space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206047.png" /> of surfaces of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206048.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206049.png" /> of conformal classes of Riemann surfaces of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206050.png" /> is obtained by factorization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206051.png" /> by some countable group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206052.png" /> of automorphisms of it, called the [[Modular group|modular group]].

The simplest is the case of surfaces of genus 1 — tori. Each torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206053.png" />, provided its [[Universal covering|universal covering]] surface has been conformally mapped onto the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206054.png" /> can be represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206055.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206056.png" /> is a group of translations with two generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206057.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206058.png" />; here, two tori <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206060.png" /> are conformally equivalent if and only if the ratios <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206062.png" /> of the corresponding generators are related by a modular transformation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206063.png" /></td> </tr></table>

As a (complex) modulus of the given conformal class of Riemann surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206064.png" /> one can take the value of the elliptic [[Modular function|modular function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206065.png" />. The Teichmüller space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206066.png" /> coincides with the upper half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206068.png" /> is the elliptic modular group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206069.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206070.png" /> is a Riemann surface conformally equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206071.png" />. All elliptic curves (and surfaces of genus 1) admit a simultaneous uniformization by the Weierstrass function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206072.png" /> and its derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206073.png" /> (cf. [[Weierstrass elliptic functions|Weierstrass elliptic functions]]).

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206074.png" /> the situation is much more complicated. In particular, the following fundamental properties of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206075.png" /> have been established: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206076.png" /> is homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206077.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206078.png" /> can be biholomorphically imbedded as a bounded domain into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206079.png" /> that is holomorphically convex; 3) the modular group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206080.png" /> is discrete (even properly discontinuous) and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206081.png" /> it is the complete group of biholomorphic automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206082.png" />; 4) the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206083.png" /> is ramified and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206084.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206085.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206086.png" /> is obtained by [[Uniformization|uniformization]] and using [[Quasi-conformal mapping|quasi-conformal mapping]]. The surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206087.png" /> can be represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206088.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206089.png" /> is a [[Fuchsian group|Fuchsian group]], acting discontinuously in the upper half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206090.png" /> (defined up to conjugation in the group of all conformal automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206091.png" />), and one considers quasi-conformal automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206092.png" /> of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206093.png" />, i.e. solutions of the Beltrami equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206094.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206095.png" /> are forms with supports in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206096.png" /> that are invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206098.png" />. Further, suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r08206099.png" /> leaves the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060100.png" /> fixed. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060101.png" /> can be identified with the space of restrictions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060102.png" /> or, which is equivalent, of restrictions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060104.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060105.png" /> is biholomorphically equivalent to the domain filled by the Schwarzian derivatives

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060106.png" /></td> </tr></table>

in the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060107.png" /> of holomorphic solutions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060108.png" /> of the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060109.png" /></td> </tr></table>

with the norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060110.png" /></td> </tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060111.png" />. Using this imbedding one can construct the fibre space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060112.png" /> with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060113.png" />, which also admits the introduction of a complex structure and holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060114.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060115.png" /> that make it possible to give a parametric representation of all algebraic curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060116.png" /> in the complex projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060118.png" />. The above-mentioned construction related to the imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060119.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060120.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060121.png" /> is obtained by the so-called period matrices of these surfaces. These are symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060122.png" />-matrices with positive-definite imaginary part. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060123.png" /> can be holomorphically imbedded in the set of all such matrices (the Siegel upper half-plane) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060124.png" /> (see [[#References|[4]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060125.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060126.png" /> are polynomials of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060127.png" />. They admit a conformal involution and depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060128.png" /> complex parameters. All surfaces of genus 2 are hyper-elliptic; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060129.png" /> such surfaces form the analytic submanifolds of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060130.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060131.png" />.

The problem of the conformal automorphisms of a given Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060132.png" /> is related to conformal classes of Riemann surfaces. Except for several particular cases, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060133.png" /> of such automorphisms is discrete. In the case of closed surfaces of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060134.png" /> it is finite; moreover, the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060135.png" /> does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060136.png" />.

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|Riemann surfaces, classification of]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Nevanlinna, "Uniformisierung" , Springer (1967) {{MR|0228671}} {{ZBL|0152.27401}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G. Springer, "Introduction to Riemann surfaces" , Chelsea, reprint (1981) {{MR|0122987}} {{MR|1530201}} {{MR|0092855}} {{ZBL|0501.30039}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S.L. Krushkal', "Quasi-conformal mappings and Riemann surfaces" , Winston &amp; Wiley (1979) (Translated from Russian) {{MR|536488}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> L. Bers, "Uniformization, moduli, and Kleinian groups" ''Bull. London Math. Soc.'' , '''4''' (1972) pp. 257–300 {{MR|0348097}} {{ZBL|0257.32012}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954) {{MR|0065652}} {{ZBL|0059.06901}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> W. Abikoff, "The real analytic theory of Teichmüller space" , Springer (1980) {{MR|0590044}} {{ZBL|0452.32015}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980) {{MR|0583745}} {{ZBL|0475.30001}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> N.A. Guserkii, "Kleinian groups and uniformization in examples and problems" , Amer. Math. Soc. (1986) (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> O. Lehto, "Univalent functions and Teichmüller spaces" , Springer (1986) {{MR|0867407}} {{ZBL|0606.30001}} </TD></TR></table>



====Comments====
The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060137.png" /> of connected components of the group of diffeomorphisms of the reference Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082060/r082060138.png" />, called the modular group above, is also frequently called the mapping class group.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F.P. Gardiner, "Teichmüller theory and quadratic differentials" , Wiley (Interscience) (1987) {{MR|0903027}} {{ZBL|0629.30002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Nag, "The complex analytic theory of Teichmüller spaces" , Wiley (Interscience) (1988) {{MR|0927291}} {{ZBL|0667.30040}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Schlichenmaier, "An introduction to Riemann surfaces, algebraic curves, and moduli spaces" , Springer (1989) {{MR|0981595}} {{ZBL|0683.14007}} </TD></TR></table>