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A stability theorem for minimal foliations on a torus

Published online by Cambridge University Press:  10 December 2009

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Abstract

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This paper is concerned with minimal foliations; these are foliations whose leaves are extremals of a prescribed variational problem, as for example foliations consisting of minimal surfaces. Such a minimal foliation is called stable if for any small perturbation of the variational problem there exists a minimal foliation conjugate under a smooth diffeomorphism to the original foliation. In this paper the stability of special foliations of codimension 1 on a higher-dimensional torus is established. This result requires small divisor assumptions similar to those encountered in dynamical systems. This theorem can be viewed as a generalization of the perturbation theory of invariant tori for Hamiltonian systems to elliptic partial differential equations for which one obtains quasi-periodic solutions.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Volume 8: Charles Conley Memorial Issue , December 1988 , pp. 251 - 281
Copyright
Copyright © Cambridge University Press 1988

References

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