Domains without parabolic minimal submanifolds and weakly hyperbolic domains

Abstract

We show that if $\Omega$ is an $m$-convex domain in $\mathbb R^n$ for some $2\le m<n$ whose boundary $b\Omega$ has a tubular neighbourhood of positive radius and is not $m$-flat near infinity, then $\Omega$ 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 $\Omega$ in $\mathbb R^n$ characterised by the property that every conformal harmonic map $\mathbb C\to\Omega$ is constant, and we elucidate their relationship with hyperbolic domains and domains without parabolic minimal surfaces.