Domains without parabolic minimal submanifolds and weakly hyperbolic domains

TL;DR

This paper investigates the geometric properties of certain domains in Euclidean space, showing conditions under which they cannot contain parabolic minimal submanifolds and introducing weakly hyperbolic domains with unique harmonic map properties.

Contribution

It establishes new geometric criteria preventing the existence of parabolic minimal submanifolds in specific convex domains and introduces the concept of weakly hyperbolic domains.

Findings

Domains with positive radius tubular neighborhoods lack parabolic minimal submanifolds of certain dimensions.

Properly embedded nonflat minimal hypersurfaces with tubular neighborhoods intersect all parabolic hypersurfaces.

In dimension 3, bounded Gaussian curvature ensures the non-existence of certain parabolic minimal surfaces.

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.