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Hamiltonian disjunction and limits of Lagrangian submanifolds. (English) Zbl 0812.53031
The authors prove that for a closed connected submanifold \(L^ n\) of a symplectic manifold \((M^{2n}, \omega)\) which is non-Lagrangian and the normal fibre bundle of \(L^ n\) in \(M^{2n}\) has a nonvanishing section, there exists a Hamiltonian vector field on \(M^{2n}\) which is nowhere tangent to \(L^ n\). They also prove that a closed submanifold \(L^ n \subset (M^{2n},\omega)\) which is a \(C^ 0\)-limit of Lagrangian submanifolds is itself Lagrangian in both of the cases: if \((M^{2n}, \omega)\) is “tame” in Gromov sense, \(\pi_ 2(M^{2n}, L^ n) = 0\) and \((M^{2n},\omega) =(\mathbb{R}^{2n}, \omega_ 0)\) where \(\omega_ 0\) is the canonical symplectic form on \(\mathbb{R}^{2n}\).

MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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