On minimal laminations of the torus
type: Item
instance of: scholarly article
title: On minimal laminations of the torus
zbMATH Open document ID: 0678.58014
DOI: 10.1016/S0294-1449(16)30328-6
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1989-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78172
review text: 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.
Mathematics Subject Classification ID: 58E15
Mathematics Subject Classification ID: 37C85
Mathematics Subject Classification ID: 49Q20
zbMATH DE Number: 4111631
zbMATH Keywords: \(Z^ n\)-periodic variational problem
zbMATH Keywords: minimizing solutions
zbMATH Keywords: laminations
author: Victor Bangert
reviewed by: Lubomír Bakule
MaRDI profile type: Publication
cites work: The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states
cites work: Q3814339
cites work: A uniqueness theorem for \(Z^ n\)-periodic variational problems
cites work: The existence of gaps in minimal foliations
cites work: Mather sets for plane Hamiltonian systems
cites work: Geodesics on a two-dimensional Riemannian manifold with periodic coefficients
cites work: Linear and quasilinear elliptic equations
cites work: Existence of quasi-periodic orbits for twist homeomorphisms of the annulus
cites work: More Denjoy minimal sets for area preserving diffeomorphisms
cites work: Minimal solutions of variational problems on a torus
cites work: A stability theorem for minimal foliations on a torus
cites work: Three dimensional manifolds, Kleinian groups and hyperbolic geometry
Recommended article: Q3031530
Recommended article: Q4865993
Recommended article: On some minimal graphs of the torus
Recommended article: On the existence of min-max minimal torus
Recommended article: Q3473844
Recommended article: Erratum to: ``On minimal laminations of the torus''
Recommended article: Minimal measured laminations in geometric 3-manifolds
Recommended article: The minimal lamination closure theorem
Recommended article: Minimal tori in \(S^{3}\)
Recommended article: Minimal Tori in  2 ×  1
description: scientific article; zbMATH DE number 4111631