Topological Half-Space Theorems for Minimal Hypersurfaces in $\mathbb{R}^4$
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 $\mathbb{R}^4$. More precisely, we prove that a connected, complete embedded minimal hypersurface $Σ^3\subset\mathbb{R}^4$ contained in a half-space with bounded curvature and trivial second homology must be a hyperplane. We obtain further rigidity when a connected, complete embedded minimal hypersurface is confined to a slab. In particular, we show that if in addition the minimal hypersurface is proper and has trivial second homology then it must be a hyperplane, thus obtaining flatness without the bounded curvature assumption. If instead the minimal hypersurface has bounded curvature and finite second Betti number then it must have finite total curvature or finite index. As a corollary, we also obtain a new characterization of the three-dimensional catenoid as a minimal hypersurface confined in a slab with bounded curvature and second Betti number equal to one.