Higher dimensional Scherk’s hypersurfaces

In 3-dimensional Euclidean space Scherk second surfaces are singly periodic embedded minimal surfaces with 4 planar ends. In this paper, we obtain a natural generalization of Scherk’s second surfaces in higher dimensional Euclidean spaces. In particular we show that, in higher dimensional Euclidean spaces R, for n ≥ 3, there exists n−1-periodic embedded minimal hypersurfaces with 4 hyperplanar ends. The moduli space of these hypersurfaces forms 1-dimensional fibration over the moduli space of flat tori in R. A partial description of the boundary of this moduli space is given.