Teichmüller space
In mathematics, the Teichmüller space formula_1 of a (real) topological (or differential) surface formula_2, is a space that parametrizes complex structures on formula_2 up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller. Each point in a Teichmüller space formula_1 may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from formula_2 to itself. It can be viewed as a moduli space for marked hyperbolic structure on the surface, and this endows it with a natural topology for  which it is homeomorphic to a ball of dimension formula_6 for a surface of genus formula_7. In this way Teichmüller space can be viewed as the universal covering orbifold of the Riemann moduli space. The Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The study of geometric features of these various structures is an active body of research. History. Moduli spaces for Riemann surfaces and related Fuchsian groups have been studied since the work of Bernhard Riemann (1826-1866), who knew that formula_6 parameters were needed to describe the variations of complex structures on  a surface of genus formula_9. The early study of Teichmüller space, in the late nineteenth–early twentieth century, was geometric and founded on the interpretation of Riemann surfaces as hyperbolic surfaces. Among the main contributors were Felix Klein, Henri Poincaré, Paul Koebe, Jakob Nielsen, Robert Fricke and Werner Fenchel. The main contribution of Teichmüller to the study of moduli was the introduction of quasiconformal mappings to the subject. They allow us to give much more depth to the study of moduli spaces by endowing them with additional features that were not present in the previous, more elementary works. After World  War II the subject was developed further in this analytic vein, in particular by Lars Ahlfors and Lipman Bers. The theory continues to be active, with numerous studies of the complex structure of Teichmüller space (introduced by Bers). The geometric vein in the study of Teichmüller space was revived following the work of William Thurston in the late 1970s, who introduced a geometric compactification which he used in his study of the mapping class group of a surface. Other more combinatorial objects associated to this group (in particular the curve complex) have also been related to Teichmüller space, and this  is a very active subject of research in geometric group theory. Definitions. Teichmüller space from complex structures. Let formula_2 be an orientable smooth surface (a differentiable manifold of dimension 2). Informally the Teichmüller space formula_1 of formula_2 is the space of Riemann surface structures on formula_2 up to isotopy. Formally it can be defined as follows. Two complex structures formula_14 on formula_2 are said to be equivalent if there is a diffeomorphism formula_16 such that: Then formula_1 is the space of equivalence classes of complex structures on formula_2 for this relation. Another equivalent definition is as follows: formula_1 is the  space of pairs formula_25 where formula_17 is a Riemann surface and formula_27 a diffeomorphism, and two pairs formula_28 are regarded as equivalent if formula_29 is isotopic to a holomorphic diffeomorphism. Such a pair is called a "marked Riemann surface"; the "marking" being the diffeomeorphism; another definition of markings is by systems of curves. There are two simple examples that are immediately computed from the Uniformization theorem: there is a unique complex structure on the sphere formula_30 (see Riemann sphere) and there are two on formula_31 (the complex plane and the unit disk) and in each case the group of positive  diffeomorphisms is contractible. Thus the Teichmüller space of formula_30 is a single point and that of formula_31 contains exactly two points. A slightly more involved example is the open annulus, for which the Teichmüller space is the interval formula_34 (the complex structure associated to formula_35 is the Riemann surface formula_36). The Teichmüller space of the torus and flat metrics. The next example is the torus formula_37 In this case any complex structure can be realised by a Riemann surface of the form formula_38 (a complex elliptic curve) for a complex number formula_39 where is the complex upper half-plane. Then  we have a bijection: and thus the Teichmüller space of formula_43 is formula_44 If we identify formula_45 with the Euclidean plane then each point in Teichmüller space can also be viewed as a marked flat structure on formula_46 Thus the Teichmüller space is in bijection with the set of pairs formula_47 where formula_48 is a flat surface and formula_49 is a diffeomorphism up to isotopy on formula_50. Finite type surfaces. These are the surfaces for which Teichmüller space is most often studied, which include closed surfaces. A surface is of finite type if it is diffeomorphic to a compact surface  minus a finite set. If formula_2 is a closed surface of genus formula_52 then the surface obtained by removing formula_53 points from formula_2 is usually denoted formula_55 and its Teichmüller space by formula_56 Teichmüller spaces and hyperbolic metrics. Every finite type orientable surface other than the ones above admits complete Riemannian metrics of constant curvature formula_57. For a given surface of finite type there is a bijection between such metrics and complex structures as follows from the uniformisation theorem. Thus if formula_58 the Teichmüller space formula_59 can be realised as the set of marked hyperbolic surfaces of genus formula_52 with  formula_53 cusps, that is the set of pairs formula_62 where formula_48 is an hyperbolic surface and formula_64 is a diffeomorphism, modulo the equivalence relation where formula_62 and formula_66 are identified if formula_67 is isotopic to an isometry. The topology on Teichmüller space. In all cases computed above there is an obvious topology on Teichmüller space. In the general case there are many natural ways to topologise formula_1, perhaps the simplest is via hyperbolic metrics and length functions. If formula_69 is a closed curve on formula_2 and formula_71 a marked hyperbolic surface then one formula_72 is homotopic to a unique closed  geodesic formula_73 on formula_48 (up to parametrisation). The value at formula_75 of the "length function" associated to (the homotopy class of) formula_69 is then: Let formula_78 be the set of simple closed curves on formula_2. Then the map is an embedding. The space formula_82 has the product topology and formula_1 is endowed with the induced topology. With this topology formula_84 is homeomorphic to formula_85 In fact one can obtain an embedding with formula_86 curves, and even formula_87. In both case one can use the embedding to give a geometric proof of the homeomorphism above. More examples of small Teichmüller spaces . There is a unique complete hyperbolic metric of finite volume on the three-holed sphere and so the Teichmüller space of finite-volume complete metrics of constant curvature formula_88 is a point (this also follows from the dimension formula of the previous paragraph). The Teichmüller spaces formula_89 and formula_90 are naturally realised as the upper half-plane, as can be seen using Fenchel–Nielsen coordinates. Teichmüller space and conformal structures. Instead of complex structures of hyperbolic metrics one can define Teichmüller space using conformal structures. Indeed, conformal structures are the same as complex structures in two (real) dimensions. Moreover, the  Uniformisation Theorem also implies that in each conformal class of Riemannian metrics on a surface there is a unique metric of constant curvature. Teichmüller spaces as representation spaces. Yet another interpretation of Teichmüller space is as a representation space for surface groups. If formula_2 is hyperbolic, of finite type and formula_92 is the fundamental group of formula_2 then Teichmüller space is in natural bijection with: The map sends a marked hyperbolic structure formula_62 to the composition formula_101 where formula_102 is the monodromy of the hyperbolic structure and formula_103 is the isomorphism induced by formula_50. Note that this realises formula_1 as  a closed subset of formula_106 which endows it with a topology. This can be used to see the homeomorphism formula_107 directly. This interpretation of Teichmüller space is generalised by higher Teichmüller theory, where the group formula_95 is replaced by an arbitrary semisimple Lie group. A remark on categories. All definitions above can be made in the topological category instead of the category of differentiable manifolds, and this does not change the objects. Infinite-dimensional Teichmüller spaces. Surfaces which are not of finite type also admit hyperbolic structures, which can be parametrised by infinite-dimensional spaces (homeomorphic to formula_109). Another example  of infinite-dimensional space related to Teichmüller theory is the Teichmüller space of a lamination by surfaces. Action of the mapping class group and relation to moduli space. The map to moduli space. There is a map from Teichmüller space to the moduli space of Riemann surfaces diffeomorphic to formula_2, defined by formula_111. It is a covering map, and since formula_1 is simply connected it is the orbifold universal cover for the moduli space. Action of the mapping class group. The mapping class group of formula_2 is the coset group formula_114 of the diffeomorphism group of formula_2 by the normal  subgroup of those that are isotopic to the identity (the same definition can be made with homeomorphisms instead of diffeomorphisms and, for surfaces, this does not change the resulting group). The group of diffeomorphisms acts naturally on Teichmüller space by If formula_117 is a mapping class and formula_118 two diffeomorphisms representing it then they are isotopic. Thus the classes of formula_119 and formula_120 are the same in Teichmüller space, and the action above factorises through the mapping class group. The action of the mapping class group formula_114 on the Teichmüller space is properly discontinuous, and the quotient is the moduli  space. Fixed points. The Nielsen realisation problem asks whether any finite subgroup of the mapping class group has a global fixed point (a point fixed by all group elements) in Teichmüller space. In more classical terms the question is: can every finite subgroup of formula_114 be realised as a group of isometries of some complete hyperbolic metric on formula_2 (or equivalently as a group of holomorphic diffeomorphisms of some complex structure). This was solved by Steven Kerckhoff. Coordinates. Fenchel–Nielsen coordinates. The Fenchel–Nielsen coordinates (so named after Werner Fenchel and Jakob Nielsen) on the Teichmüller space formula_1 are associated  to a pants decomposition of the surface formula_2. This is a decomposition of formula_2 into pairs of pants, and to each curve in the decomposition is associated its length in the hyperbolic metric corresponding to the point in Teichmüller space, and another real parameter called the twist which is more involved to define. In case of a closed surface of genus formula_52 there are formula_128 curves in a pants decomposition and we get formula_6 parameters, which is the dimension of formula_130. The Fenchel–Nielsen coordinates in fact define a homeomorphism formula_131. In the case of a surface with punctures some  pairs of pants are "degenerate" (they have a cusp) and give only two length and twist parameters. Again in this case the Fenchel–Nielsen coordinates define a homeomorphism formula_132. Shear coordinates. If formula_133 the surface formula_134 admits ideal triangulations (whose vertices are exactly the punctures). By the formula for the Euler characteristic such a triangulation has formula_135 triangles. An hyperbolic structure formula_48 on formula_2 determines a (unique up to isotopy) diffeomorphism formula_138 sending every triangle to an hyperbolic ideal triangle, thus a point in formula_1. The parameters for such a structure are the translation lengths for each pair of sides  of the triangles glued in the triangulation. There are formula_140 such parameters which can each take any value in formula_141, and the completeness of the structure corresponds to a linear equation and thus we get the right dimension formula_142. These coordinates are called "shear coordinates". For closed surfaces, a pair of pants can be decomposed as the union of two ideal triangles (it can be seen as an incomplete hyperbolic metric on the three-holed sphere). Thus we also get formula_128 shear coordinates on formula_130. Earthquakes. A simple "earthquake path" in Teichmüller space is a path determined by varying a  single shear or length Fenchel–Nielsen coordinate (for a fixed ideal triangulation of a surface). The name comes from seeing the ideal triangles or the pants as tectonic plates and the shear as plate motion. More generally one can do earthquakes along geodesic laminations. A theorem of Thurston then states that two points in Teichmüller space are joined by a unique earthquake path. Analytic theory. Quasiconformal mappings. A quasiconformal mapping between two Riemann surfaces is a homeomorphism which deforms the conformal structure in a bounded manner over the surface. More precisely it is differentiable almost everywhere and there is a  constant formula_145, called the "dilatation", such that where formula_147 are the derivatives in a conformal coordinate formula_148 and its conjugate formula_149. There are quasi-conformal mappings in every isotopy class and so an alternative definition for The Teichmüller space is as follows. Fix a Riemann surface formula_17 diffeomorphic to formula_2, and Teichmüller space is in natural bijection with the marked surfaces formula_152 where formula_153 is a quasiconformal mapping, up to the same equivalence relation as above. Quadratic differentials and the Bers embedding. With the definition above, if formula_154 there is a natural map from Teichmüller space to the space of  formula_97-equivariant solutions to the Beltrami differential equation. These give rise, via the Schwarzian derivative, to quadratic differentials on formula_17. The space of those is a complex space of complex dimension formula_128, and the image of Teichmüller space is an open set. This map is called the Bers embedding. A quadratic differential on formula_17 can be represented by a translation surface conformal to formula_17. Teichmüller mappings. Teichmüller's theorem states that between two marked Riemann surfaces formula_25 and formula_161 there is always a unique quasiconformal mapping formula_162 in the isotopy class of formula_163 which has minimal dilatation. This map is  called a Teichmüller mapping. In the geometric picture this means that for every two diffeomorphic Riemann surfaces formula_14 and diffeomorphism formula_165 there exists two polygons representing formula_14 and an affine map sending one to the other, which has smallest dilatation among all quasiconformal maps formula_162. Metrics. The Teichmüller metric. If formula_168 and the Teichmüller mapping between them has dilatation formula_169 then the Teichmüller distance between them is by definition formula_170. This indeed defines a distance on formula_1 which induces its topology, and for which it is complete. This is the metric most commonly used for the study of the metric  geometry of Teichmüller space. In particular it is of interest to geometric group theorists. There is a function similarly defined, using the Lipschitz constants of maps between hyperbolic surfaces instead of the quasiconformal dilatations, on formula_172, which is not symmetric. The Weil–Petersson metric. Quadratic differentials on a Riemann surface formula_17 are identified with the tangent space at formula_174 to Teichmüller space. The Weil–Petersson metric is the Riemannian metric defined by the formula_175 inner product on quadratic differentials. Compactifications. There are several inequivalent compactifications of Teichmüller spaces that have been studied. Several of the earlier compactifications depend on the  choice of a point in Teichmüller space so are not invariant under the modular group, which can be inconvenient. William Thurston later found a compactification without this disadvantage, which has become the most widely used compactification. Thurston compactification. By looking at the hyperbolic lengths of simple closed curves for each point in Teichmüller space and taking the closure in the (infinite-dimensional) projective space, introduced a compactification whose points at infinity correspond to projective measured laminations. The compactified space is homeomorphic to a closed ball. This Thurston compactification is acted on continuously by the modular group. In particular any element  of the modular group has a fixed point in Thurston's compactification, which Thurston used in his classification of elements of the modular group. Bers compactification. The Bers compactification is given by taking the closure of the image of the Bers embedding of Teichmüller space, studied by . The Bers embedding depends on the choice of a point in Teichmüller space so is not invariant under the modular group, and in fact the modular group does not act continuously on the Bers compactification. Teichmüller compactification. The "points at infinity" in the Teichmüller compactification consist of geodesic rays (for the Teichmüller metric ) starting at a fixed basepoint. This compactification depends on the choice of basepoint so is not acted on by the modular group, and in fact Kerckhoff showed that the action of the modular group on Teichmüller space does not extend to a continuous action on this compactification. Gardiner–Masur compactification. considered a compactification similar to the Thurston compactification, but using extremal length rather than hyperbolic length. The modular group acts continuously on this compactification, but they showed that their compactification has strictly more points at infinity. Large-scale geometry. There has been an extensive study of the geometric properties of  Teichmüller space endowed with the Teichmüller metric. Known large-scale properties include: On the other hand, Teichmüller space exhibits several properties characteristic of hyperbolic spaces, such as: Some of these features can be explained by the study of maps from Teichmüller space to the curve complex, which is known to be hyperbolic. Complex geometry. The Bers embedding gives formula_1 a complex structure as an open subset of formula_183 Metrics coming from the complex structure. Since Teichmüller space is a complex manifold it carries a Carathéodory metric. Teichmüller space is Kobayashi hyperbolic and its Kobayashi metric coincides with the Teichmüller metric . This latter result is used in Royden's proof that the mapping class group is the full group of isometries for the Teichmüller metric. The Bers embedding realises Teichmüller space as a domain of holomorphy and hence it also carries a Bergman metric. Kähler metrics on Teichmüller space. The Weil–Petersson metric is Kähler but it is not complete. Cheng and Yau showed that there is a unique complete Kähler–Einstein metric on Teichmüller space. It has constant negative scalar curvature. Teichmüller space also carries a complete Kähler metric of bounded sectional curvature introduced by that is Kähler-hyperbolic. Equivalence  of metrics. With the exception of the incomplete Weil–Petersson metric, all metrics on Teichmüller space introduced here are quasi-isometric to each other.