In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parameterizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Each point in 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.
Contents
- History
- Teichmller space from complex structures
- The Teichmller space of the torus and flat metrics
- Finite type surfaces
- Teichmller spaces and hyperbolic metrics
- The topology on Teichmller space
- More examples of small Teichmller spaces
- Teichmller space and conformal structures
- Teichmller spaces as representation spaces
- A remark on categories
- Infinite dimensional Teichmller spaces
- The map to moduli space
- Action of the mapping class group
- Fixed points
- FenchelNielsen coordinates
- Shear coordinates
- Earthquakes
- Quasiconformal mappings
- Quadratic differentials and the Bers embedding
- Teichmller mappings
- The Teichmller metric
- The WeilPetersson metric
- Compactifications of Teichmller spaces
- Thurston compactification
- Bers compactification
- Teichmller compactification
- GardinerMasur compactification
- Large scale geometry of Teichmller space
- Complex geometry of Teichmller space
- Metrics coming from the complex structure
- Khler metrics on Teichmller space
- Equivalence of metrics
- References
It can also 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 6g − 6 for a surface of genus g. 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 a very rich subject of research.
Teichmüller spaces are named after Oswald Teichmüller.
History
Moduli spaces for Riemann surfaces and related Fuchsian groups have been studied since the work of Bernhard Riemann, who knew that 6g − 6 parameters were needed to describe the variations of complex structures on a surface of genus g. 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, Werner Fenchel.
The main contribution of Teichmüller to the study of moduli was the introduction of quasiconformal mappings to the subject. They allow 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 seventies, 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.
Teichmüller space from complex structures
Let
Formally it can be defined as follows. Two complex structures
Then
Another equivalent definition is as follows:
There are two simple examples that are immediately computed from the Uniformisation theorem: there is a unique complex structure on the sphere
A slightly more involved example is the open annulus, for which the Teichmüller space is the interval
The Teichmüller space of the torus and flat metrics
The next example is the torus
There is then a map
Identifying
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
Teichmüller spaces and hyperbolic metrics
Every finite type orientable surface other than the ones above admits complete Riemannian metrics 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
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
If
Let
In fact one can obtain an embedding with
More examples of small Teichmüller spaces
There is a unique complete hyperbolic metric on the three-holed sphere and so the Teichmüller space
The Teichmüller spaces
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
The map sends a marked hyperbolic structure
Note that this realises
This interpretation of Teichmüller space is generalised by higher Teichmüller theory, where the group
A remark on categories
All definitions above can be made in the topological category instead of the differentiable topology, 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
The map to moduli space
There is a map from Teichmüller space to the moduli space of Riemann surfaces diffeomorphic to
Action of the mapping class group
The mapping class group of
If
The action of the mapping class group
Fixed points
The Nielsen realisation problem asks whether any finite group 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
Fenchel–Nielsen coordinates
Fenchel–Nielsen coordinates on the space
In case of a closed surface of genus
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
Shear coordinates
If
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
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.
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
where
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
Quadratic differentials and the Bers embedding
With the definition above, if
A quadratic differential on
Teichmüller mappings
Teichmüller's theorem states that between two marked Riemann surfaces
In the geometric picture this means that for every two diffeomorphic Riemann surfaces
The Teichmüller metric
If
There is a function similarly defined, using the Lipschitz constants of maps between hyperbolic surfaces instead of the quasiconformal dilatations, on
The Weil–Petersson metric
Quadratic differentials on a Riemann surface
Compactifications of Teichmüller spaces
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. 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, Thurston (1988) 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 Bers (1970). 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
Gardiner & Masur (1991) 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 of Teichmüller space
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 of Teichmüller space
The Bers embedding gives
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 McMullen (2000) 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.