On certain Lagrangian submanifolds of $S^2\times S^2$ and $\mathbb{C}\mathrm{P}^n$

Preprint, Other literature type English OPEN
Oakley, Joel ; Usher, Michael (2016)
  • Publisher: MSP
  • Journal: (issn: 1472-2747)
  • Related identifiers: doi: 10.2140/agt.2016.16.149
  • Subject: Mathematics - Symplectic Geometry | 53D12 | Hamiltonian displaceability | Lagrangian submanifolds
    arxiv: Mathematics::Differential Geometry | Mathematics::Symplectic Geometry

We consider various constructions of monotone Lagrangian submanifolds of $C P^n, S^2\times S^2$, and quadric hypersurfaces of $C P^n$. In $S^2\times S^2$ and $C P^2$ we show that several different known constructions of exotic monotone tori yield results that are Hamiltonian isotopic to each other, in particular answering a question of Wu by showing that the monotone fiber of a toric degeneration model of $C P^2$ is Hamiltonian isotopic to the Chekanov torus. Generalizing our constructions to higher dimensions leads us to consider monotone Lagrangian submanifolds (typically not tori) of quadrics and of $C P^n$ which can be understood either in terms of the geodesic flow on $T^*S^n$ or in terms of the Biran circle bundle construction. Unlike previously-known monotone Lagrangian submanifolds of closed simply connected symplectic manifolds, many of our higher-dimensional Lagrangian submanifolds are provably displaceable.
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