Dimension-free quasiconformal distortion in $n$-space

Abstract:

Most distortion theorems for $K$-quasiconformal mappings in ${{\mathbf {R}}^n}$, $n \geqslant 2$, depend on both $n$ and $K$ in an essential way, with bounds that become infinite as $n$ tends to $\infty$. The present authors obtain dimension-free versions of four well-known distortion theorems for quasiconformal mappings—namely, bounds for the linear dilatation, the Schwarz lemma, the $\Theta$-distortion theorem, and the $\eta$-quasisymmetry property of these mappings. They show that the upper estimates they have obtained in each of these four main results remain bounded as $n$ tends to $\infty$ with $K$ fixed. The proofs are based on a "dimensioncancellation" property of the function $t \mapsto {\tau ^{ - 1}}(\tau (t)/K), t > 0, K > 0$, where $\tau (t)$ is the capacity of a Teichmüller extremal ring in ${{\mathbf {R}}^n}$. The authors also prove a dimension-free distortion theorem for the absolute (cross) ratio under $K$-quasiconformal mappings of ${\overline {\mathbf {R}} ^n}$, from which several other distortion theorems follow as special cases.