Teichmüller space
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
The Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics.
parameters were needed to describe the variations of complex structures on a surface of genus
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).
[1] There are two simple examples that are immediately computed from the Uniformization theorem: there is a unique complex structure on the sphere
(the complex plane and the unit disk) and in each case the group of positive diffeomorphisms is connected.
A slightly more involved example is the open annulus, for which the Teichmüller space is the interval
with the Euclidean plane then each point in Teichmüller space can also be viewed as a marked flat structure on
Every finite type orientable surface other than the ones above admits complete Riemannian metrics of constant curvature
For a given surface of finite type there is a bijection between such metrics and complex structures as follows from the uniformisation theorem.
then Teichmüller space is in natural bijection with: The map sends a marked hyperbolic structure
Surfaces which are not of finite type also admit hyperbolic structures, which can be parametrised by infinite-dimensional spaces (homeomorphic to
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).
[10] The Fenchel–Nielsen coordinates (so named after Werner Fenchel and Jakob Nielsen) on the Teichmüller space
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.
[12] 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.
The parameters for such a structure are the translation lengths for each pair of sides of the triangles glued in the triangulation.
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[14]).
A theorem of Thurston then states that two points in Teichmüller space are joined by a unique earthquake path.
There are quasi-conformal mappings in every isotopy class and so an alternative definition for The Teichmüller space is as follows.
There is a function similarly defined, using the Lipschitz constants of maps between hyperbolic surfaces instead of the quasiconformal dilatations, on
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.
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.
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.
The Bers embedding realises Teichmüller space as a domain of holomorphy and hence it also carries a Bergman metric.
Teichmüller space also carries a complete Kähler metric of bounded sectional curvature introduced by McMullen (2000) that is Kähler-hyperbolic.
