Given any \documentclass[12pt]{minimal}
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${\mathbb Z}^3$\end{document}-periodic metric g on \documentclass[12pt]{minimal}
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${\mathbb R}^3$\end{document} and a plane \documentclass[12pt]{minimal}
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$P \subset{\mathbb R}^3$\end{document} through the origin, Bangert [4] shows that there exists a properly embedded surface homeomorphic to \documentclass[12pt]{minimal}
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${\mathbb R}^2$\end{document} which is homotopically area-minimizing w.r.t. g, lies in a strip of bounded width around P, and does not have self-intersections when projected to the 3-torus \documentclass[12pt]{minimal}
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$T^3 = {\mathbb R}^3 /{\mathbb Z}^3$\end{document}. For the set \documentclass[12pt]{minimal}
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${\mathbb F}_P$\end{document} of such surfaces, we show the following uniqueness theorems: If P is irrational, i.e., is not spanned by vectors in \documentclass[12pt]{minimal}
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${\mathbb Z}^3$\end{document}, the action of \documentclass[12pt]{minimal}
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${\mathbb Z}^3$\end{document} on \documentclass[12pt]{minimal}
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${\mathbb F}_P$\end{document} by translations has a unique minimal set. If P is totally irrational, i.e., \documentclass[12pt]{minimal}
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$P \cap{\mathbb Z}^3 = \{0\}$\end{document}, then the surfaces in \documentclass[12pt]{minimal}
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${\mathbb F}_P$\end{document} are pairwise disjoint.
