Riemann minimal surfaces in higher dimensions

Abstract

We prove the existence of a one parameter family of minimal embedded hypersurfaces in $R^{n+1}$, for $n\ge 3$, which generalize the well known 2 dimensional "Riemann minimal surfaces". The hypersurfaces we obtain are complete, embedded, simply periodic hypersurfaces which have infinitely many parallel hyperplanar ends. By opposition with the 2-dimensional case, they are not foliated by spheres.