Prime ends for domains in metric spaces

TL;DR

This paper introduces a new, general definition of prime ends in metric spaces, compares it with existing concepts, and explores their properties and relations to boundary accessibility and domain types.

Contribution

It proposes a novel definition of prime ends in metric spaces, relates them to existing notions, and extends the theory to almost John domains and finitely connected boundaries.

Findings

New definition of prime ends in metric spaces

Characterization of singleton prime ends and boundary accessibility

Relations between prime ends, Mazurkiewicz boundary, and domain types

Abstract

In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Carath\'eodory and N\"akki. Modulus ends and prime ends, defined by means of the \p-modulus of curve families, are also discussed and related to the prime ends. We provide characterizations of singleton prime ends and relate them to the notion of accessibility of boundary points, and introduce a topology on the prime end boundary. We also study relations between the prime end boundary and the Mazurkiewicz boundary. Generalizing the notion of John domains, we introduce almost John domains, and we investigate prime ends in the settings of John domains, almost John domains and domains which are finitely connected at the boundary.