The periodic Plateau problem and its application

Given a noncompact disconnected complete periodic curve $\mathrm{\Gamma }$ with no self intersection in ${\mathbb{R}}^3$, it is proved that there exists a noncompact simply connected periodic minimal surface spanning $\mathrm{\Gamma }$. As an application it is shown that for any tetrahedron $T$ with dihedral angles $\le {90}^{\circ }$ there exist four embedded minimal annuli in $T$ which are perpendicular to $\mathrm{\partial }T$ along their boundary. It is also proved that every Platonic solid of ${\mathbb{R}}^3$ contains five types of free boundary embedded minimal surfaces of genus zero.