Symplectic rigidity: Lagrangian submanifolds

Lagrangian non-intersections

“…Note that statement (1) is a special case of statement (2), since for a subcritical polarization P we have Λ P = ∅. We therefore prove statement (2).…”

“…See [6,7,4] for more details and precise statements. (2) Note that the cohomology of the Lagrangian L 1 ⊂ Q n (taken from (2) above with r = 1) is precisely A * Q . Can L 1 be Hamiltonianly isotoped to lie in the complement of Λ Q ?…”

“…The Maslov index of the loop of Lagrangian subspaces {τ u(t) } t∈∂D ⊂ u * π * P N Σ , u * π * P σ is 0 because the bundle τ is globally defined, hence this loop extends to the disc D. Thus by (2)…”

“…Let (M, ω) be a tame symplectic manifold (see [2], chapter 10), and let L ⊂ (M, ω) be a monotone Lagrangian submanifold with N L ≥ 2. In this situation one can define the Floer homology of L which is an invariant of the Hamiltonian isotopy class of L.…”

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

“…We now recall the Lagrangian circle bundle construction due to M. Audin, F. Lalonde, and L. Polterovich in [3] in the special case of a four-dimensional symplectic manifold. This construction enables us to create a smooth family of Lagrangian tori associated to any smooth family of isotropic curves.…”