The Teichm\"uller problem for $L^p$-means of distortion

TL;DR

This paper studies extremal quasiconformal maps minimizing $L^p$-mean distortion, revealing existence of weak minimizers with specific properties and their convergence behavior as $p$ varies.

Contribution

It establishes the existence of weak $L^p$-minimizers with associated quadratic differentials and analyzes their convergence to classical extremal maps as $p$ approaches 1 and infinity.

Findings

Existence of weak minimizers with quadratic differentials having poles of order one.

No locally quasiconformal minimizer exists unless the map is the identity.

Weak $L^p$-minimisers converge to the extremal map as $p o ext{infinity}$ and to the identity as $p o 1$.

Abstract

Teichm\"uller's problem from 1944 is this: Given $x\in [0,1)$ find and describe the extremal quasiconformal map $f:\ID\to\ID$, $f|\partial \ID=identity$ and $f(0)=-x\leq 0$. 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 Teichm\"uller type with associated holomorphic quadratic differential having a pole of order one at $x$, if $x\neq 0$. For the $L^p$-norm, when $p=1$ it is known that there can be no locally quasiconformal minimiser unless $x=0$. Here we show that for $1\leq p<\infty$ there is a minimiser in a weak class and an associated Ahlfors-Hopf holomorphic quadratic differential with a pole of order $1$ at $f(0)=r$. However, this minimiser cannot be in $W^{1,2}_{loc}(\ID)$ unless $r=0$ and $f=identity$. Hence there is no locally quasiconformal minimiser. A similar statement holds for minimsers of the exponential norm of distortion. We also use our earlier work to show that as $p\to\infty$, the weak $L^p$-minimisers converge locally uniformly in $\ID$ to the extremal quasiconformal mapping, and that as $p\to 1$ the weak $L^p$-minimisers converge locally uniformly in $\ID$ to the identity.