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

The three-dimensional catenoid in $\R^4$ 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 $\R^4$.

Specifically, we show that a complete, properly embedded minimal hypersurface $\Sigma^3\subset\R^4$ with bounded curvature, diffeomorphic to $\R^3$, 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.