View source for Teichmüller mapping

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t1200301.png" /> be a [[Quasi-conformal mapping|quasi-conformal mapping]] from a [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t1200302.png" /> onto a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t1200303.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t1200304.png" /> be a neighbourhood with local parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t1200305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t1200306.png" />. The complex dilatation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t1200307.png" /> in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t1200308.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t1200309.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003010.png" />; invariantly written, it reads <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003011.png" />. The quasi-conformal mapping mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003012.png" /> is called a Teichmüller mapping if its complex dilatation is of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003013.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003014.png" /> is an analytic [[Quadratic differential|quadratic differential]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003015.png" />, possibly with isolated singularities. (The surface is usually punctured at these points.) The norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003016.png" /> is defined to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003018.png" />; if it is finite, the singularities can be first-order poles at worst. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003019.png" />, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003020.png" /> is conformal (cf. also [[Conformal mapping|Conformal mapping]]) and there is no specific quadratic differential associated with it.

One introduces, locally and outside the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003021.png" /> of critical points (zeros and isolated singularities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003022.png" />), the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003023.png" /></td> </tr></table>

Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003024.png" /> is a first-order differential, the local function elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003025.png" /> are well determined up to the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003026.png" />. In any sufficiently small neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003027.png" /> which does not contain a critical point, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003028.png" /> is a univalent conformal mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003029.png" /> onto a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003030.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003031.png" />-plane (cf. also [[Univalent function|Univalent function]]). Map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003032.png" /> by the horizontal stretching <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003034.png" />, onto a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003035.png" />. It is easy to see that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003036.png" /> has the same complex dilatation as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003037.png" />. Therefore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003039.png" /> are related by a conformal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003040.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003041.png" />. The square of its derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003042.png" /> is a holomorphic quadratic differential on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003044.png" />. The points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003045.png" /> are the critical points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003046.png" />, and corresponding points have the same order, positive for zeros, negative for poles. Thus, to every Teichmüller mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003047.png" /> there is associated a pair of quadratic differentials, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003048.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003050.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003051.png" />. The horizontal trajectories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003052.png" /> go over into Euclidean horizontal straight lines in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003053.png" />-plane. It is immediate that they are stretched by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003054.png" /> onto the horizontal trajectories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003055.png" />, whereas the vertical trajectories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003056.png" /> are just shifted into those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003057.png" />.

An important subclass of the class of Teichmüller mappings is the one associated with quadratic differentials of finite norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003058.png" /> (the same is then true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003059.png" />, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003060.png" />). These mappings are uniquely extremal for their boundary values [[#References|[a3]]]. Of course, they have the property that their dilatation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003061.png" /> is constant (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120030/t12003062.png" />). Quite recently (1998) it has been shown that there are uniquely extremal quasi-conformal mappings with non-constant dilatation [[#References|[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 [[#References|[a1]]], pp. 184–185.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Teichmüller,   "Extremale quasikonforme Abbildungen und quadratische Differentiale"  ''Abh. Preuss. Akad. Wiss., Math.-naturw. Kl. 22'' , '''197'''  (1939)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V. Božin,   N. Lakic,   V. Markovič,   M. Mateljevič,   "Unique extremality"  ''J. d'Anal. Math.'' , '''to appear'''  (1999)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Reich,   K. Strebel,   "Extremal quasiconformal mappings with given boundary values" , ''Contributions to Analysis'' , Acad. Press  (1974)  pp. 375–392</TD></TR></table>