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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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