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

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 ${\mathrm{\Sigma }}^3\subset {\mathbb{R}}^4$ 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.