In 1990, Hoffman and Meeks proved that any connected, proper, possibly branched minimal surface in three-dimensional Euclidean space that is contained in a halfspace must be a plane. This result does not extend to higher dimensions, since the higher-dimensional catenoid is contained in a slab (and hence in a halfspace).
We consider the corresponding nonlocal problem: whether a set in Euclidean space with zero s-mean curvature, contained in a halfspace, must itself be a halfspace. We prove that if the boundary of the set is continuously differentiable, or if it satisfies universal volumetric density estimates at all scales, then the set must indeed be a halfspace. In contrast with the classical case, our theorem holds in all dimensions. This is a joint work with Matteo Cozzi.