On minimal laminations of the torus
Consider the problem \(\int F(x,u(x),u_ x(x))dx\to \min\) where \(u: {\mathbb{R}}^ n\to {\mathbb{R}}\), \(F: {\mathbb{R}}^ n\times {\mathbb{R}}\times {\mathbb{R}}^ n\to {\mathbb{R}}\) is periodic in \((x,u)\in {\mathbb{R}}^{n+1}\) and uniformly convex in \(u_ x\in {\mathbb{R}}^ n\). Non-selfintersecting minimizers u are investigated, i.e. the hypersurface graph \((u)\subseteq {\mathbb{R}}^{n+1}\) has no selfinteractions when projected into \(T^{n+1}\), where \(T^{n+1}\) denotes the torus \({\mathbb{R}}^{n+1}/{\mathbb{Z}}^{n+1}\). There exists a ``rotation vector'' \(\alpha =\alpha (u)\in {\mathbb{R}}^ n\) for such u so that \(u(x)-\alpha\) is bounded uniformly for all \(x\in {\mathbb{R}}^ n\). The structure of the set \({\mathcal M}_{\alpha}={\mathcal M}_{\alpha}(F)\) of non-selfintersecting F-minimal solutions with fixed rotation vector \(\alpha\) is determined for rationally dependent \({\bar \alpha}=(-\alpha,1)\). These investigations are primarily topological. \(u\in {\mathcal M}_{\alpha}\) are classified by secondary invariants. The proved uniqueness results mean that the graphs of functions in \({\mathcal M}_{\alpha}\) with the same secondary invariants do not intersect. Using these results and the results by \textit{J. Moser} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 229-272 (1986; Zbl 0609.49029)], the existence of minimal solutions \(u\in {\mathcal M}_{\alpha}\) with prescribed secondary invariants is proved, particularly the existence of secondary laminations in the gaps between the functions in \({\mathcal M}_{\alpha}\) with maximal periodicity. Further, two open problems are mentioned.
scientific article; zbMATH DE number 4111631
scientific article