Domains without parabolic minimal submanifolds and weakly hyperbolic domains

We show that if $\mathrm{\Omega }$ is an $m$-convex domain in ${\mathbb{R}}^n$ for some $2\le m<n$ whose boundary $b\mathrm{\Omega }$ has a tubular neighbourhood of positive radius and is not $m$-flat near infinity, then $\mathrm{\Omega }$ does not contain any immersed parabolic minimal submanifold of dimension $\ge m$. In particular, if $M$ is a properly embedded nonflat minimal hypersurface in ${\mathbb{R}}^n$ with a tubular neighbourhood of positive radius then every immersed parabolic hypersurface in ${\mathbb{R}}^n$ intersects $M$. In dimension $n=3$ this holds if $M$ has bounded Gaussian curvature. We also introduce the class of weakly hyperbolic domains $\mathrm{\Omega }$ in ${\mathbb{R}}^n$ characterised by the property that every conformal harmonic map $\mathbb{C}\to \mathrm{\Omega }$ is constant, and we elucidate their relationship with hyperbolic domains and domains without parabolic minimal surfaces.