Quasiconformality and hyperbolic skew
We prove that if $f:\mathbb{B}^n \to \mathbb{B}^n$, for $n\geq 2$, is a homeomorphism with bounded skew over all equilateral hyperbolic triangles, then $f$ is in fact quasiconformal. Conversely, we show that if $f:\mathbb{B}^n \to \mathbb{B}^n$ is quasiconformal then $f$ is $η$-quasisymmetric in the hyperbolic metric, where $η$ depends only on $n$ and $K$. We obtain the same result for hyperbolic $n$-manifolds. Analogous results in $\mathbb{R}^n$, and metric spaces that behave like $\mathbb{R}^n$, are known, but as far as we are aware, these are the first such results in the hyperbolic setting, which is the natural metric to use on $\mathbb{B}^n$.
math.CV↗