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 Näkki'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 Lindelöf theorem for principal points and the Tsuji theorem.

29 pages

References in corpus (2)

• Prime ends for domains in metric spaces
• The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary

Cited by in corpus (2)

• Prime ends in metric spaces and boundary extensions of mappings
• The prime end capacity of inaccessible prime ends, resolutivity, and the Kellogg property