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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
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<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>
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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.
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Let $f : R \rightarrow R ^ { \prime }$ be a [[Quasi-conformal mapping|quasi-conformal mapping]] from a [[Riemann surface|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
  
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
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\begin{equation*} \mu ( z ) = k \frac { \overline { \varphi } ( z ) } { | \varphi ( z ) | } , 0 < k < 1, \end{equation*}
  
<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>
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where $\varphi$ is an analytic [[Quadratic differential|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|Conformal mapping]]) and there is no specific quadratic differential associated with it.
  
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" />.
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One introduces, locally and outside the set $E$ of critical points (zeros and isolated singularities of $\varphi$), the function
  
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.
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\begin{equation*} \zeta = \xi + i \eta = \Phi ( z ) = \int ^ { z } \sqrt { \varphi ( z ) } d z. \end{equation*}
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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|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 $\psi$.
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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 [[#References|[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 [[#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====
 
====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>
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<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>
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<tr><td valign="top">[a2]</td> <td valign="top"> V. Božin, N. Lakic, V. Markovič, M. Mateljevič, "Unique extremality"  ''J. Anal. Math.'' , '''75''' (1998) {{ZBL|0929.30017}}</td></tr>
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<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>
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</table>

Latest revision as of 19:05, 22 January 2024

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
How to Cite This Entry:
Teichmüller mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Teichm%C3%BCller_mapping&oldid=23544
This article was adapted from an original article by Kurt Strebel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article