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On minimal laminations of the torus. (English) Zbl 0678.58014
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 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.
Reviewer: L.Bakule

MSC:
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
49Q20 Variational problems in a geometric measure-theoretic setting
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References:
[1] Aubry, S.; Le Daeron, P. Y., The discrete Frenkel-Kontorova model and its extensions I. exact results for the ground states, Physica, 8D, 381-422, (1983) · Zbl 1237.37059
[2] Bangert, V., Mather sets for twist maps and geodesics on tori, dynamics reported, (Kirchgraber, U.; Walther, H. O., Vol. 1, (1988), Stuttgart-chichester, B. G. Teubner-John Wiley), 1-56 · Zbl 0664.53021
[3] Bangert, V., A uniqueness theorem for Z^{n}-periodic variational problems, Comment. Math. Helv., Vol. 62, 511-531, (1987) · Zbl 0634.49018
[4] Bangert, V., The existence of gaps in minimal foliations, Aequationes Math., Vol. 34, 153-166, (1987) · Zbl 0645.58017
[5] G. D. Birkhoff, Dynamical Systems, Amer. Math. Soc. Colloq. Publ., Vol. IX, New York, Amer. Math. Soc., 1927.
[6] Denzler, J., Mather sets for plane Hamiltonian systems, Z. Angew. Math. Phys. (ZAMP), Vol. 38, 791-812, (1987) · Zbl 0641.70014
[7] Hedlund, G. A., Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. of Math., Vol. 33, 739, 719, (1932) · Zbl 0006.32601
[8] Ladyzhenskaya, O. A.; Ural’tseva, N. N., Linear and quasilinear elliptic equations, (1968), Academic Press New York-London · Zbl 0164.13002
[9] Mather, J. N., Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology, Vol. 21, 457-467, (1982) · Zbl 0506.58032
[10] Mather, J. N., More Denjoy minimal sets for area preserving diffeomorphisms, Comment. Math. Helv., Vol. 60, 508-557, (1985) · Zbl 0597.58015
[11] Morse, M., A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Amer. Math. Soc., Vol. 26, 25-60, (1924) · JFM 50.0466.04
[12] Moser, J., Minimal solutions of variational problems on a torus, Ann. Inst. Henri-Poincaré (Analyse non linéaire), Vol. 3, 229-272, (1986) · Zbl 0609.49029
[13] Moser, J., A stability theorem for minimal foliations on a torus, Ergod. Th. Dynam. Sys., Vol. 8, 251-281, (1988) · Zbl 0632.57018
[14] Thurston, W. P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. (N.S.) Amer. Math. Soc., Vol. 6, 357-381, (1982) · Zbl 0496.57005
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