Prime ends in the Heisenberg group $\mathbb{H}_{1}$ and the boundary behavior of quasiconformal mappings

Abstract

We investigate prime ends in the Heisenberg group $\mathbb{H}_{1}$ extending N\"akki's construction for collared domains in Euclidean spaces. The corresponding class of domains is defined via uniform domains and the Loewner property. Using prime ends we show the counterpart of Caratheodory's extension theorem for quasiconformal mappings, the Koebe theorem on arcwise limits, the Lindel\"of theorem for principal points and the Tsuji theorem.