Non-existence of prescribable conformally equivariant dilatation in space

Abstract:

In this paper, we study the prescribable conformally equivariant dilatations for orientation preserving quasiconformal homeomorphisms. The complex dilatation is a prescribable conformally equivariant dilatation in $\mathbb {R}^{2}$. A Schottky set is a subset of the unit sphere $\mathbb {S}^n$ whose complement is the union of at least three disjoint open balls. By using the result of Bonk, Kleiner, and Merenkov that there are rigid Schottky sets of positive measure in each dimension at least $3$, we prove that it is not possible to have a prescribable conformally equivariant dilatation in $\mathbb {R}^{n}$, where $n \geq 3$.