Hyperbolic domains in real Euclidean spaces

In a recent joint work with David Kalaj, we introduced a Finsler pseudometric on any domain in the real Euclidean space $\mathbb R^n$, $n\ge 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashi's pseudometric on complex manifolds which is defined in terms of holomorphic discs. On the unit ball of $\mathbb R^n$ this minimal metric coincides with the classical Beltrami–Cayley–Klein metric. In this talk, I will describe several sufficient conditions for a domain in $\mathbb R^n$ to be (complete) hyperbolic, meaning that the minimal pseudometric is a (complete) metric.
(Joint work with Barbara Drinovec Drnovsek, University of Ljubljana.)