Lagrangian submanifolds exhibit surprising rigidity in view of their small dimension (only half that of the ambient symplectic manifold). A famous manifestation of this rigidity is that some of them cannot be displaced from themselves by any Hamiltonian diffeomorphism or even by any diffeomorphism which preserves the symplectic form. In comparison, any submanifold of the same dimension which is not Lagrangian can be displaced from itself by a Hamiltonian diffeomorphism.
Moreover, that diffeomorphism can be chosen arbitrarily C⁰-small, so that one can wonder whether a Lagrangian L (which can be displaced from itself) admits a neighborhood of Hamiltonian (resp. symplectic) non-displacement. Such a neighborhood W is defined by the property that any Lagrangian obtained from L by a Hamiltonian (resp. symplectic) diffeomorphism, and included in W, must intersect L.
On the one hand, I will give conditions which ensure the existence of such neighborhoods for a large class of Lagrangians. On the other hand, I will construct a Lagrangian which does not admit any, in any symplectic manifold of dimension at least 6. Then, I will discuss several applications of our techniques to the topology of orbits of Lagrangians. This is based on a joint work with Marcelo Attalah, Jean-Philippe Chassé, and Egor Shelukhin.
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