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Let be a quasi-conformal mapping from a Riemann surface
onto a Riemann surface
. Let
be a neighbourhood with local parameter
,
. The complex dilatation of
in terms of
is
, with
; invariantly written, it reads
. The quasi-conformal mapping mapping
is called a Teichmüller mapping if its complex dilatation is of the form
![]() |
where is an analytic quadratic differential on
, possibly with isolated singularities. (The surface is usually punctured at these points.) The norm of
is defined to be
,
; if it is finite, the singularities can be first-order poles at worst. If
, the mapping
is conformal (cf. also Conformal mapping) and there is no specific quadratic differential associated with it.
One introduces, locally and outside the set of critical points (zeros and isolated singularities of
), the function
![]() |
Since is a first-order differential, the local function elements of
are well determined up to the transformation
. In any sufficiently small neighbourhood
which does not contain a critical point, the function
is a univalent conformal mapping from
onto a neighbourhood
in the
-plane (cf. also Univalent function). Map
by the horizontal stretching
,
, onto a neighbourhood
. It is easy to see that
has the same complex dilatation as
. Therefore,
and
are related by a conformal mapping
, with
. The square of its derivative
is a holomorphic quadratic differential on
,
. The points in
are the critical points of
, and corresponding points have the same order, positive for zeros, negative for poles. Thus, to every Teichmüller mapping
there is associated a pair of quadratic differentials,
on
and
on
. The horizontal trajectories of
go over into Euclidean horizontal straight lines in the
-plane. It is immediate that they are stretched by
onto the horizontal trajectories of
, whereas the vertical trajectories of
are just shifted into those of
.
An important subclass of the class of Teichmüller mappings is the one associated with quadratic differentials of finite norm (the same is then true for
, since
). These mappings are uniquely extremal for their boundary values [a3]. Of course, they have the property that their dilatation
is constant (
). Quite recently (1998) it has been shown that there are uniquely extremal quasi-conformal mappings with non-constant dilatation [a2]. Thus, there are quasi-symmetric boundary homeomorphisms of the unit disc for which there is no extremal extension into the disc which is of Teichmüller form, contrary to an idea of O. Teichmüller in [a1], pp. 184–185.
References
[a1] | O. Teichmüller, "Extremale quasikonforme Abbildungen und quadratische Differentiale" Abh. Preuss. Akad. Wiss., Math.-naturw. Kl. 22 , 197 (1939) |
[a2] | V. Božin, N. Lakic, V. Markovič, M. Mateljevič, "Unique extremality" J. d'Anal. Math. , to appear (1999) |
[a3] | E. Reich, K. Strebel, "Extremal quasiconformal mappings with given boundary values" , Contributions to Analysis , Acad. Press (1974) pp. 375–392 |
Teichmüller mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Teichm%C3%BCller_mapping&oldid=23069