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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 .

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