The periodic Plateau problem and its application

Given a noncompact disconnected complete periodic curve $\Gamma$ with no self intersection in $\mathbb R^3$, it is proved that there exists a noncompact simply connected periodic minimal surface spanning $\Gamma$. As an application it is shown that for any tetrahedron $T$ with dihedral angles $\leq90^\circ$ there exist four embedded minimal annuli in $T$ which are perpendicular to $\partial T$ 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.