Half-Space Theorem for Minimal Hypersurfaces in ${\mathbb{R}}^4$

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 ${\mathbb{R}}^4$. More precisely, we prove that a connected, complete, embedded minimal hypersurface ${\Sigma }^3\subset {\mathbb{R}}^4$ contained in a half-space with bounded curvature and trivial second homology must be a hyperplane.

11 pages, improved the main results in the previous versions