Continuum families of non-displaceable Lagrangian tori in (CP¹)²ᵐ

الأصل · EN

We construct a family of Lagrangian tori Θⁿₛ ⊂ (CP¹)ⁿ, s ∈ (0,1), where Θⁿ₁/₂ = Θⁿ, is the monotone twist Lagrangian torus described by Chekanov-Schlenk. We show that for n = 2m and s ≥ 1/2 these tori are non-displaceable. Then by considering Θᵏ¹ₛ₁ × × Θᵏˡₛₗ × (S²eq)ⁿ ⁻ ∑ⁱ ᵏⁱ ⊂ (CP¹)ⁿ, with sᵢ ∈ [1/2,1) and kᵢ ∈ 2Z>₀, ∑ᵢ kᵢ ≤ n we get several l-dimensional families of non-displaceable Lagrangian tori. We also show that there exists partial symplectic quasi-states ζᵇˢₑₛ and linearly independent homogeneous Calabi quasimorphims μᵇˢₑₛ or which Θ²ᵐₛ are ζᵇˢₑₛ-superheavy and μᵇˢₑₛ-superheavy. We also prove a similar result for (CP² 3CP², ωε), where {ωε; 0 < ε< 1} is a family of symplectic forms in CP² 3CP², for which ω₁/₂ is monotone.