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

TL;DR

This paper extends the theory of prime ends to the Heisenberg group, establishing boundary behavior results for quasiconformal mappings analogous to classical Euclidean theorems.

Contribution

It introduces a prime end theory in the Heisenberg group and proves boundary extension and limit theorems for quasiconformal maps in this setting.

Findings

Prime end theory is successfully extended to $ abla_{1}$.

Boundary extension theorems for quasiconformal maps are established.

Classical boundary behavior results are adapted to the Heisenberg group.

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.