Domains without parabolic minimal submanifolds and weakly hyperbolic domains

Abstract

AbstractWe show that if is an ‐convex domain in for some whose boundary has a tubular neighbourhood of positive radius and is not ‐flat near infinity, then does not contain any immersed parabolic minimal submanifolds of dimension . In particular, if is a properly embedded non‐flat minimal hypersurface in with a tubular neighbourhood of positive radius, then every immersed parabolic hypersurface in intersects . In dimension , this holds if has bounded Gaussian curvature function. We also introduce the class of weakly hyperbolic domains in , characterised by the property that every conformal harmonic map is constant, and we elucidate their relationship with hyperbolic domains, and domains without parabolic minimal surfaces.