Teichmüller mapping

Let $f : R \rightarrow R ^ { \prime }$ be a quasi-conformal mapping from a Riemann surface $R$ onto a Riemann surface $R ^ { \prime }$. Let $U \subset R$ be a neighbourhood with local parameter $z$, $U ^ { \prime } = f ( U ) \subset R ^ { \prime }$. The complex dilatation of $f$ in terms of $z$ is $\mu ( z ) = f _ { z^- } / f _ { z }$, with $\| \mu \| _ { \infty } < 1$; invariantly written, it reads $\mu ( z ) ( d \overline{z} / d z )$. The quasi-conformal mapping mapping $f$ is called a Teichmüller mapping if its complex dilatation is of the form

\begin{equation*} \mu ( z ) = k \frac { \overline { \varphi } ( z ) } { | \varphi ( z ) | } , 0 < k < 1, \end{equation*}

where $\varphi$ is an analytic quadratic differential on $R$, possibly with isolated singularities. (The surface is usually punctured at these points.) The norm of $\varphi$ is defined to be $\| \varphi \| = \int \int _ { R } | \varphi ( z ) | d x d y$, $z = x + i y$; if it is finite, the singularities can be first-order poles at worst. If $k = 0$, the mapping $f$ is conformal (cf. also Conformal mapping) and there is no specific quadratic differential associated with it.

One introduces, locally and outside the set $E$ of critical points (zeros and isolated singularities of $\varphi$), the function

\begin{equation*} \zeta = \xi + i \eta = \Phi ( z ) = \int ^ { z } \sqrt { \varphi ( z ) } d z. \end{equation*}

Since $\sqrt { \varphi ( z ) } d z$ is a first-order differential, the local function elements of $\Phi$ are well determined up to the transformation $\Phi _ { 2 } = \pm \Phi _ { 1 } + \text{const}$. In any sufficiently small neighbourhood $U \subset R$ which does not contain a critical point, the function $\Phi$ is a univalent conformal mapping from $U$ onto a neighbourhood $V = \Phi ( U )$ in the $\zeta = \xi + i \eta$-plane (cf. also Univalent function). Map $V$ by the horizontal stretching $F _ { K } : \xi + i \eta \rightarrow K \xi + i \eta$, $K = ( 1 + k ) / ( 1 - k )$, onto a neighbourhood $V ^ { \prime }$. It is easy to see that $F _ { K } \circ \Phi$ has the same complex dilatation as $f$. Therefore, $U ^ { \prime } = f ( U )$ and $V ^ { \prime } = F _ { K } \circ \Phi ( V )$ are related by a conformal mapping $\Psi : U ^ { \prime } \rightarrow V ^ { \prime }$, with $\Psi \circ f = F _ { K } \circ \Phi$. The square of its derivative $\psi = \Psi ^ { \prime 2}$ is a holomorphic quadratic differential on $R ^ { \prime } \backslash E ^ { \prime }$, $E ^ { \prime } = f ( E )$. The points in $E ^ { \prime }$ are the critical points of $\psi$, and corresponding points have the same order, positive for zeros, negative for poles. Thus, to every Teichmüller mapping $f$ there is associated a pair of quadratic differentials, $\varphi$ on $R$ and $\psi$ on $R ^ { \prime } = f ( R )$. The horizontal trajectories of $\varphi$ go over into Euclidean horizontal straight lines in the $\zeta$-plane. It is immediate that they are stretched by $f$ onto the horizontal trajectories of $\psi$, whereas the vertical trajectories of $\varphi$ are just shifted into those of $\psi$.

An important subclass of the class of Teichmüller mappings is the one associated with quadratic differentials of finite norm $\| \varphi \| < \infty$ (the same is then true for $\psi$, since $\| \psi \| = K \| \varphi \|$). These mappings are uniquely extremal for their boundary values [a3]. Of course, they have the property that their dilatation $D$ is constant ($\equiv K$). 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. Anal. Math. , 75 (1998) Zbl 0929.30017
[a3]	E. Reich, K. Strebel, "Extremal quasiconformal mappings with given boundary values" , Contributions to Analysis , Acad. Press (1974) pp. 375–392