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.

To appear in Bull. London Math. Soc. Research supported by the European Union (ERC Advanced grant HPDR, 101053085 to Franc Forstneric and by grants P1-0291, J1-3005, and N1-0237 from ARRS, Republic of Slovenia. arXiv admin note: text overlap with arXiv:2206.06045

References in corpus (3)

• A barrier principle at infinity for varifolds with bounded mean curvature
• Flexible domains for minimal surfaces in Euclidean spaces
• Every smoothly bounded p-convex domain in R^n admits a p-plurisubharmonic defining function