Uniqueness of least area surfaces in the 3-torus (Q5949446)

Let \(g\) be a \(\mathbb{Z}^3\)-periodic metric on \(\mathbb{R}^3\) and let \(\mathcal F\) be the set of all properly embedded surfaces in \(\mathbb{R}^3\) homeomorphic to the plane \(\mathbb{R}^2\) which are homotopically area minimizing with respect to \(g\) and without self-intersection. From [\textit{V. Bangert}, Comment. Math. Helv. 62, 511-531 (1987; Zbl 0634.49018)], every \(F\in {\mathcal F}\) lies in a strip between two parallel affine planes and for every affine plane \(P\) there exists a surface \(F\in{\mathcal F}\) which lies between two planes parallel to \(P\). Let \({\mathcal F}_P\) denote the set of all \(F\in {\mathcal F}\) corresponding to the plane \(P\in G(3,2)\) through the origin. The following uniqueness theorems are proved here: If \(P\) is not spanned by vectors in \(\mathbb{Z}^3\), the action of \(\mathbb{Z}^3\) on \({\mathcal F}_P\) by translations has a unique minimal set. If \(P\cap \mathbb{Z}^3=\{0\}\), then the surfaces in \({\mathcal F}_P\) are pairwise disjoint.