#### Volume 16, issue 1 (2016)

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

### Joel Oakley and Michael Usher

Algebraic & Geometric Topology 16 (2016) 149–209
##### Abstract

We consider various constructions of monotone Lagrangian submanifolds of $ℂ{P}^{n}$, ${S}^{2}×{S}^{2}$, and quadric hypersurfaces of $ℂ{P}^{n}$. In ${S}^{2}×{S}^{2}$ 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 ${T}^{\ast }{S}^{n}$ 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.

##### Keywords
Lagrangian submanifolds, Hamiltonian displaceability
Primary: 53D12