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

*Oakley, Joel*;

*Usher, Michael*;

- Publisher: MSP
- Journal: issn: 1472-2747
Related identifiers: doi: 10.2140/agt.2016.16.149 - Subject: Mathematics - Symplectic Geometry | 53D12 | Hamiltonian displaceability | Lagrangian submanifoldsarxiv: Mathematics::Differential Geometry | Mathematics::Symplectic Geometry

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Finally we show that ΨP is a symplectomorphism to its image. Given what we have already done, this will follow once we show that where α = Pin=0 x j d yj ∈ Ω1(Cn+1) and λ ∈ Ω1(D1∗Sn) is the canonical one-form, we have ( Ψ˜P)∗α = λ. Now for (p, q) ∈ D1∗Sn,

1 = X pj dqj + X pjqj p f (|p|)d p f (|p|) = X pj dqj

j j j 0 e−it

We first prove monotonicity. Since k + m + 1 ≥ 3, the boundary map ∂ : π2(T ∗Sk+m+1, Pk1,m/2) → π1(Pk1,m/2) is an isomorphism. Of course, the map ι : S1 × Sk × Sm → Pk1,m/2 embeds π1(S1 × Sk × Sm) [AF] P. Albers and U. Frauenfelder. A nondisplaceable Lagrangian torus in T ∗S2. Comm. Pure Appl. Math. 61 (2008), no. 8,

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