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Minimal foliations on a torus

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1365)

Abstract

These lectures are concerned with foliations of codimension one on a torus Tm whose leaves are minimals of a nonlinear variational problem. For m=2 such foliations correspond to an invariant torus of a Hamiltonian system of two degrees of freedom as they occur in stability theory. The recently studied invariant Aubry-Mather-sets of monotone twist mappings have as analogue a "lamination" of the torus.

This study of minimal foliations, motivated in part by the theory of dynamical systems, depends strongly on tools of variational methods and nonlinear elliptic differential equations. It is the aim to discuss the connection between the mechanical problems and the higher-dimensional variational problems. In particular, we will describe a stability theorem for foliations generalizing the invariant curve theorem to partial differential equations.

Keywords

  • Hamiltonian System
  • Euler Equation
  • Variational Problem
  • Rotation Number
  • Rotation Vector

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

I am indepted to V. Bangert for critical remarks as well as his advice and to M. Struwe and E. Zehnder for improvements of the manuscript.

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© 1989 Springer-Verlag

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Moser, J. (1989). Minimal foliations on a torus. In: Giaquinta, M. (eds) Topics in Calculus of Variations. Lecture Notes in Mathematics, vol 1365. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0089178

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  • DOI: https://doi.org/10.1007/BFb0089178

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