Higher-dimensional Scherk's hypersurfaces

In three-dimensional Euclidean space, Scherk second surfaces are singly periodic embedded minimal surfaces with four planar ends. In this paper, we obtain a natural generalization of these minimal surfaces in any higher-dimensional Euclidean space ℝⁿ⁺¹, for n ≥ 3. More precisely, we show that there exist (n - 1)-periodic embedded minimal hypersurfaces with four hyperplanar ends. The moduli space of these hypersurfaces forms a one-dimensional fibration over the moduli space of flat tori in ℝ^n-1. A partial description of the boundary of this moduli space is also given.