Teichmüller mapping

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