## Teichmüller's theorem

Teichmüller's extremal problem can be stated as follows: given a homeomorphism between Riemann surfaces, find a qc homeomorphism , homotopic to , such that is infimum in this class. We can state the problem in terms of groups as well: given , find with and infimum in this class. Existence is given by normality properties of quasiconformal mapping (proposition 2.12).

As usual, we assume is of type with . Let be the space of integrable holomorphic quadratic differentials on .

Lemma 2.20   For and , set . Then belongs to the unit ball of and is essentially constant on , with . Moreover, if and only if and , for some . is called a Teichmüller-Beltrami differential form on .

Theorem 2.21 (Teichmüller's Theorem)   Given in , there exists a unique extremal mapping solving Teichmüller's extremal problem. The complex dilatation of is a Teichmüller-Beltrami differential for a unique and a unique, up to a positive multiple, element , if is not the origin of (for the origin, is conformal so ).

One can construct explicitly Teichmüller mappings for once-punctured tori as follows. First of all, if is a torus given by , the group of translations on induces a group of biholomorphic mappings on that acts transitively. So one can assume that has a puncture at (the projection of) the origin. A mapping is given by the affine mapping taking the parallelogram of (with vertices at 0, , and ) to the parallelogram of (with vertices at 0, , and ):

To show this mapping is extremal, it is enough to show that is a Teichmüller-Beltrami differential. The lift of to the complex plane is given by , so

which shows that is a Teichmüller-Beltrami differential. Teichmüller distance can then be easily computed:

which is precisely the Poincaré distance between and in the upper half plane.

Theorem 2.22   Let denote the unit ball of the space of quadratic differentials on ; the mapping given by is a homeomorphism, called Teichmüller's homeomorphism.

Corollary 2.23   ( a compact surface with punctures and universal covering space the upper half plane) and the Teichmüller metric is simply connected (more than that, it is contractible).

The boundary of is Teichmüller embedding is a sphere of dimension .

Pablo Ares Gastesi 2005-08-31