The periodic Plateau problem and its application

Given a noncompact disconnected periodic curve Γ of infinite length with two components and no self-intersection in R3, it is proved that there exists a noncompact simply connected periodic minimal surface spanning Γ. As an application, it is shown that for any tetrahedron T with dihedral angles ≤90∘, there exist four embedded minimal annuli in T, which are perpendicular to ∂T along their boundary. It is also proved that every Platonic solid of R3 contains a free boundary embedded minimal surface of genus zero. © The Author(s), under exclusive licence to Springer Nature B.V. 2025.