Teichmüller space
From Wikipedia, the free encyclopedia
In mathematics, given a Riemann surface X, the Teichmüller space of X, notated TX or Teich(X), is a complex manifold whose points represent all complex structures of Riemann surfaces whose underlying topological structure is the same as that of X. It is named after the German mathematician Oswald Teichmüller.
Contents |
[edit] Relation to moduli space
The Teichmüller space of a surface is related to its moduli space, but preserves more information about the surface. More precisely, the surface X (or its underlying topological structure) provides a marking X → Y of each Riemann surface Y represented in TX: whereas moduli space identifies all surfaces which are isomorphic, TX only identifies those surfaces which are isomorphic via a biholomorphic map f that is isotopic to the identity (with respect to the marking, hence its need). The automorphisms of X, up to isotopy, form a discrete group (the Teichmüller modular group, or mapping class group of X) that acts on TX. The action is as follows: if [g] is an element of the mapping class group of X, then [g] sends the point represented by the marking h: X → Y to the point with the marking hg : X → X → Y. The quotient of TX by this action is precisely the moduli space of X.
[edit] Properties of TX
The Teichmüller space of X is a complex manifold. Its complex dimension depends on topological properties of X. If X is obtained from a compact surface of genus g (take g greater than 1) by removing n points, then the dimension of TX is 3g-3+n. These are the cases of "finite type".
Note that, even though a compact surface with a point removed and the same surface with a disc removed are topologically the same, a complex structure on the surface behaves very differently around a point and around a removed disc. In particular, the boundary of the removed disc becomes an "ideal boundary" for the Riemann surface, and isomorphisms between surfaces with non-empty ideal boundary must take this ideal boundary into account. Varying the structure quasiconformally along the ideal boundary shows that the Teichmüller space of a Riemann surface with nonempty ideal boundary must be infinite-dimensional.
[edit] Teichmüller metric
There is, in general, no isomorphism from one Riemann surface to another of the same topological type that is isotopic to the identity. There is, however, always a quasiconformal map from one to the other that is isotopic to the identity, and the measure of how far such a map is from being conformal (i.e., holomorphic) gives a metric on TX, called the Teichmüller metric.
[edit] Examples of Teichmüller spaces
The sphere with four points removed, the torus, and the torus with one point removed all have identical Teichmüller spaces. These Teichmüller spaces are naturally identified with the complex upper half plane. These exhaust the parabolic Riemann surfaces, and the spherical case is trivial: hence the only interesting Teichmuller theory is that of the hyperbolic surfaces.
