On pseudospheres that are quasispheres

Abstract

We construct bounded domains $D$ not equal to a ball in $n\ge 3$ dimensional Euclidean space, ${\mathbb{R}}^n$, for which $D$ is homeomorphic to a sphere under a quasiconformal mapping of ${\mathbb{R}}^n$ and such that $n-1$ dimensional Hausdorff measure equals harmonic measure on $\partial D$.