The intrinsic geometry of a Jordan domain

TL;DR

This paper demonstrates that the length metric space of a Jordan domain in the plane forms a CAT(0) and Gromov hyperbolic space, with a boundary at infinity topologically equivalent to the original domain.

Contribution

It establishes the intrinsic geometric properties of Jordan domains, showing they are CAT(0) and Gromov hyperbolic, and describes their boundary at infinity.

Findings

The length metric space of a Jordan domain is CAT(0).

The space is Gromov hyperbolic.

The boundary at infinity is topologically equivalent to the original domain.

Abstract

For a Jordan domain in the plane the length metric space of points connected to an interior point by a curve of finite length is a CAT(0)space and Gromov hyperbolic. With respect to the cone topology, that space plus its boundary at infinity is topologically the same as the original Jordan domain.