Title: Construction of embedded periodic surfaces in $\mathbb{R}^n$,
Abstract: We construct embedded minimal surfaces which are $n$-periodic in
$\mathbb{R}^n$. They are new for codimension $n-2\ge 2$. We start with a Jordan
curve of edges of the $n$-dimensional cube. It bounds a Plateau minimal disk
which Schwarz reflection extends to a complete minimal surface. Studying the
group of Schwarz reflections, we can characterize those Jordan curves for which
the complete surface is embedded. For example, for $n=4$ exactly five such
Jordan curves generate embedded surfaces. Our results apply to surface classes
other than minimal as well, for instance polygonal surfaces.