The Teichmüller problem for L^p-means of distortion

Abstract

Teichmüller's problem from 1944 is this: Given $x\in [0,1)$ find and describe the extremal quasiconformal map $f:\mathbb{D}\to \mathbb{D}$, $f|\partial \mathbb{D}=$ identity and $f(0)=-x\le 0$. We consider this problem in the setting of minimisers of $L^p$-mean distortion. The classical result is that there is an extremal map of Teichmüller type with associated holomorphic quadratic differential having a pole of order one at 0, if $x\ne 0$. For the $L^1$-norm it is known that there can be no locally quasiconformal minimiser unless $x=0$. Here we show that for $1\le p<\mathrm{\infty }$ there is a minimiser in a weak class and an associated Ahlfors-Hopf holomorphic quadratic differential with a pole of order 1 at $f(0)=-x$. However, this minimiser cannot be in $W_{loc}^{1,2}(\mathbb{D})$ unless $x=0$ and $f=$ identity. Hence no minimiser for the $L^p$-Teichmüller problem can be locally quasiconformal other than the identity.   Similar statements holds for minimisers of the exponential norm of distortion. We also use our earlier work to show that as $p\to \mathrm{\infty }$, the weak $L^p$-minimisers converge locally uniformly in $\mathbb{D}$ to the extremal quasiconformal Teichmüller mapping, and that as $p\to 1$ the weak $L^p$-minimisers converge locally uniformly in $\mathbb{D}$ to the identity.