Topological Bernstein Theorems for Minimal Hypersurfaces in $\mathbb{R}^4$ confined in space

Abstract

The three-dimensional catenoid in $\mathbb{R}^4$ is a complete embedded minimal hypersurface contained in a slab, showing that the half-space theorem does not extend directly to higher dimensions. We show that this obstruction is topological in $\mathbb{R}^4$. Specifically, we show that a complete, properly embedded minimal hypersurface $Σ^3\subset\mathbb{R}^4$ with bounded curvature, diffeomorphic to $\mathbb{R}^3$, and contained in a slab must be a hyperplane. Under the additional assumption of cubic volume growth, the same conclusion holds for minimal hypersurfaces contained in a half-space.