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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Teichmüller space T(S) of a (real) topological (or differential) surface S is a space that parametrizes complex structures on S up to the action of
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
s that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller. Each point in a Teichmüller space T(S) may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from S 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 A ball is a round object (usually spherical, but sometimes ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used for s ...
of dimension 6g-6 for a surface of
genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
g \ge 2. 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 In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
structure and a wealth of natural metrics. The study of geometric features of these various structures is an active body of research. The sub-field of mathematics that studies the Teichmüller space is called Teichmüller theory.


History

Moduli spaces for Riemann surfaces and related Fuchsian groups have been studied since the work of
Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
(1826–1866), who knew that 6g-6 parameters were needed to describe the variations of complex structures on a surface of genus g\ge 2. 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 Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
,
Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
, 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 William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurst ...
in the late 1970s, who introduced a geometric compactification which he used in his study of the
mapping class group In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Mo ...
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 S be an orientable smooth surface (a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...
of dimension 2). Informally the Teichmüller space T(S) of S is the space of Riemann surface structures on S up to isotopy. Formally it can be defined as follows. Two complex structures X, Y on S are said to be equivalent if there is a
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...
f \in \operatorname(S) such that: * It is holomorphic (the differential is complex linear at each point, for the structures X at the source and Y at the target) ; * it is isotopic to the identity of S (there is a continuous map \gamma : ,1\to \operatorname(S) such that \gamma(0)=f, \gamma(1) = \mathrm). Then T(S) is the space of equivalence classes of complex structures on S for this relation. Another equivalent definition is as follows: T(S) is the space of pairs (X, g) where X is a Riemann surface and g: S \to X a diffeomorphism, and two pairs (X, g), (Y,h) are regarded as equivalent if h \circ g^ : X \to Y is isotopic to a holomorphic diffeomorphism. Such a pair is called a ''marked Riemann surface''; the ''marking'' being the diffeomorphism; another definition of markings is by systems of curves. There are two simple examples that are immediately computed from the
Uniformization theorem In mathematics, the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generali ...
: there is a unique complex structure on the
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
\mathbb S^2 (see
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...
) and there are two on \R^2 (the complex plane and the
unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose d ...
) and in each case the group of positive diffeomorphisms is connected. Thus the Teichmüller space of \mathbb S^2 is a single point and that of \R^2 contains exactly two points. A slightly more involved example is the open annulus, for which the Teichmüller space is the interval [0, 1) (the complex structure associated to \lambda is the Riemann surface \).


The Teichmüller space of the torus and flat metrics

The next example is the torus \mathbb T^2 = \R^2/\Z^2. In this case any complex structure can be realised by a Riemann surface of the form \Complex/(\Z + \tau\Z) (a complex Elliptic curve#Elliptic curves over the complex numbers, elliptic curve) for a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
\tau \in \mathbb where : \mathbb = \, is the complex upper half-plane. Then we have a
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
: :\mathbb \longrightarrow T(\mathbb T^2) :\tau \longmapsto (\Complex/(\Z + \tau\Z), (x, y) \mapsto x + \tau y) and thus the Teichmüller space of \mathbb T^2 is \mathbb. If we identify \Complex with the
Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
then each point in Teichmüller space can also be viewed as a marked flat structure on \mathbb T^2. Thus the Teichmüller space is in bijection with the set of pairs (M,f) where M is a flat surface and f: \mathbb T^2 \to M is a diffeomorphism up to isotopy on f.


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 S is a closed surface of
genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
g then the surface obtained by removing k points from S is usually denoted S_ and its Teichmüller space by T_.


Teichmüller spaces and hyperbolic metrics

Every finite type orientable surface other than the ones above admits complete
Riemannian metric In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s of constant curvature -1. 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 2g-2+k > 0 the Teichmüller space T_ can be realised as the set of marked hyperbolic surfaces of genus g with k cusps, that is the set of pairs (M, f) where M is a hyperbolic surface and f : S \to M is a diffeomorphism, modulo the equivalence relation where (M, f) and (N, g) are identified if f \circ g^ 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 T(S), perhaps the simplest is via hyperbolic metrics and length functions. If \alpha is a
closed curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
on S and x = (M, f) a marked hyperbolic surface then one f_*\alpha is homotopic to a unique closed geodesic \alpha_x on M (up to parametrisation). The value at x of the ''length function'' associated to (the homotopy class of) \alpha is then: : \ell_\alpha(x) = \operatorname(\alpha_x). Let \mathcal S be the set of simple closed curves on S. Then the map : T(S) \to \R^ : x \mapsto \left (\ell_\alpha(x) \right )_ is an embedding. The space \R^ has the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
and T(S) is endowed with the induced topology. With this topology T(S_) is homeomorphic to \R^. In fact one can obtain an embedding with 9g-9 curves, and even 6g - 5 + 2k. 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 T(S_) is a point (this also follows from the dimension formula of the previous paragraph). The Teichmüller spaces T(S_) and T(S_) 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 or 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 S is hyperbolic, of finite type and \Gamma = \pi_1(S) is the fundamental group of S then Teichmüller space is in natural bijection with: *The set of injective representations \Gamma \to \mathrm_2(\R) with discrete image, up to conjugation by an element of \mathrm_2(\R), if S is compact ; *In general, the set of such representations, with the added condition that those elements of \Gamma which are represented by curves freely homotopic to a puncture are sent to parabolic elements of \mathrm_2(\R), again up to conjugation by an element of \mathrm_2(\R). The map sends a marked hyperbolic structure (M, f) to the composition \rho \circ f_* where \rho: \pi_1(M) \to \mathrm_2(\R) is the monodromy of the hyperbolic structure and f_*: \pi_1(S) \to \pi_1(M) is the isomorphism induced by f. Note that this realises T(S) as a closed subset of \mathrm_2(\R)^ which endows it with a topology. This can be used to see the homeomorphism T(S) \cong \R^ directly. This interpretation of Teichmüller space is generalised by higher Teichmüller theory, where the group \mathrm_2(\R) is replaced by an arbitrary semisimple
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
.


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 \R^\N). 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 S, defined by (X, f) \mapsto X. It is a covering map, and since T(S) is
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
it is the orbifold universal cover for the moduli space.


Action of the mapping class group

The
mapping class group In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Mo ...
of S is the coset group MCG(S) of the diffeomorphism group of S 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 : g \cdot (X, f) \mapsto (X, f \circ g^). If \gamma \in MCG(S) is a mapping class and g, h two diffeomorphisms representing it then they are isotopic. Thus the classes of (X, f \circ g^) and (X, f \circ h^) are the same in Teichmüller space, and the action above factorises through the mapping class group. The action of the mapping class group MCG(S) 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 MCG(S) be realised as a group of isometries of some complete hyperbolic metric on S (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 T(S) are associated to a pants decomposition of the surface S. This is a decomposition of S 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 g there are 3g - 3 curves in a pants decomposition and we get 6g-6 parameters, which is the dimension of T(S_g). The Fenchel–Nielsen coordinates in fact define a homeomorphism T(S_g) \to \left]0, +\infty\right[^ \times \R^. 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 T(S_) \to \left]0, +\infty\right[^ \times \R^.


Shear coordinates

If k > 0 the surface S = S_ admits ideal Triangulation (geometry), triangulations (whose vertices are exactly the punctures). By the formula for the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
such a triangulation has 4g - 4 + 2k triangles. A hyperbolic structure M on S determines a (unique up to isotopy) diffeomorphism S \to M sending every triangle to a hyperbolic ideal triangle, thus a point in T(S). The parameters for such a structure are the translation lengths for each pair of sides of the triangles glued in the triangulation. There are 6g - 6 + 3k such parameters which can each take any value in \R, and the completeness of the structure corresponds to a linear equation and thus we get the right dimension 6g - 6 + 2k. 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 3g - 3 shear coordinates on T(S_g).


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 K \ge 1, called the ''dilatation'', such that : \frac \le K where f_z, f_ are the derivatives in a conformal coordinate z and its conjugate \bar z. 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 X diffeomorphic to S, and Teichmüller space is in natural bijection with the marked surfaces (Y, g) where g: X \to Y is a quasiconformal mapping, up to the same equivalence relation as above.


Quadratic differentials and the Bers embedding

With the definition above, if X = \Gamma \setminus \mathbb^2 there is a natural map from Teichmüller space to the space of \Gamma-equivariant solutions to the Beltrami differential equation. These give rise, via the Schwarzian derivative, to quadratic differentials on X. The space of those is a complex space of complex dimension 3g - 3, and the image of Teichmüller space is an open set. This map is called the Bers embedding. A quadratic differential on X can be represented by a translation surface conformal to X.


Teichmüller mappings

Teichmüller's theorem states that between two marked Riemann surfaces (X, g) and (Y, h) there is always a unique quasiconformal mapping X \to Y in the isotopy class of h \circ g^ which has minimal dilatation. This map is called a Teichmüller mapping. In the geometric picture this means that for every two diffeomorphic Riemann surfaces X, Y and diffeomorphism f: X \to Y there exists two polygons representing X, Y and an affine map sending one to the other, which has smallest dilatation among all quasiconformal maps X \to Y.


Metrics


The Teichmüller metric

If x, y \in T(S) and the Teichmüller mapping between them has dilatation K then the Teichmüller distance between them is by definition \frac12 \log K. This indeed defines a distance on T(S) 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 T(S) \times T(S), which is not symmetric.


The Weil–Petersson metric

Quadratic differentials on a Riemann surface X are identified with the cotangent space at (X, f) to Teichmüller space. The Weil–Petersson metric is the Riemannian metric defined by the L^2 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 William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurst ...
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: *Teichmüller space T(S_) contains flat subspaces of dimension 3g - 3 + k, and there are no higher-dimensional quasi-isometrically embedded flats. *In particular, if g>1 or g=1, k>1 or g=0, k>4 then T(S_) is not hyperbolic. On the other hand, Teichmüller space exhibits several properties characteristic of hyperbolic spaces, such as: *Some geodesics behave like they do in hyperbolic space. *Random walks on Teichmüller space converge almost surely to a point on the Thurston boundary. 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 T(S) a complex structure as an open subset of \Complex^.


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 In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be extended to a bigger domain. Forma ...
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.


See also

* Moduli of algebraic curves * p-adic Teichmüller theory * Inter-universal Teichmüller theory * Teichmüller modular form


References


Sources

* * * * * * * * *


Further reading

* * * * The last volume contains translations of several of Teichmüller's papers. * * * {{DEFAULTSORT:Teichmuller Space Riemann surfaces Moduli theory Differential geometry Geometric group theory