  The second named author and David Kalaj introduced a 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. They showed that on
the unit ball of $\mathbb R^n$, the minimal metric coincides with the classical
Beltrami-Cayley-Klein metric, one of the models of hyperbolic geometry. In the
present paper we investigate properties of the minimal pseudometric and give
sufficient conditions for a domain to be (complete) hyperbolic, meaning that
the minimal pseudometric is a (complete) metric. We show in particular that a
convex domain is complete hyperbolic if and only if it does not contain any
affine 2-planes. One of our main results is that a domain with a negative
minimal plurisubharmonic exhaustion function is hyperbolic, and a bounded
strongly minimally convex domain is complete hyperbolic. We also prove a
localization theorem for the minimal pseudometric.
