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The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $mathbb R^n$, $nge 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashis pseudometric on complex manifolds, which is defined in terms of holomorphic discs. They showed that on the unit ball of $mathbb R^n$, this minimal metric coincides with the classical Beltrami-Cayley-Klein metric. 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 domain having 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. Finally, we show that a convex domain is complete hyperbolic if and only if it does not contain any affine 2-plane.
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We prove that two smooth families of 2-connected domains in $cc$ are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct, for $m geq 3$, two smooth families of smoothly bounded $m$-connected domains in $cc$, and for $ngeq2$, two families of strictly pseudoconvex domains in $cc^n$, that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains.
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