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

Abstract

Abstract. We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic fourmanifolds: the symplectic vector space R 4 , the projective plane CP 2 , and the monotone S 2 × S 2 . The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for T * T 2 , i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.

Squeezing Lagrangian tori in dimension 4

“…We construct a Hamiltonian diffeomorphism with support near the sphere ∂B(3) displacing L(1, 2) ∪ L(2, 1) from a neighborhood of the cylinder T . This is equivalent to constructing a Hamiltonian isotopy of L(1, 2) ∪ L(2, 1), where the Lagrangian tori remain close to ∂B (3). It is convenient to use symplectic polar coordinates (R 1 , θ 1 , R 2 , θ 2 ) where…”

“…Our proof concludes as follows. First, by [3], Theorem B, we can apply a Hamiltonian isotopy of T * T 2 mapping L(1, 2) to a constant section. Composing with a translation in the fiber we get a global symplectomorphism φ of T * T 2 with φ(L(1, 2)) = O, the zero section.…”

Exact Lagrangians in four‐dimensional symplectisations

“…Notably, Shevchishin [31] and Nemirovski [29] have shown that the standard symplectic vector space $$(\mathbb {C}^2,\omega _0)$$ does not admit a Lagrangian embedding of the Klein bottle. The author together with Goodman–Ivrii showed that any Lagrangian torus inside the symplectic vector space must be unknotted [10].…”

The classification of Lagrangians nearby the Whitney immersion

“…Notably, Shevchishin [She09] and Nemirovski [Nem09] have shown that the standard symplectic vector-space (C 2 , ω 0 ) does not admit a Lagrangian embedding of the Klein bottle. The author together with Goodman-Ivrii showed that any Lagrangian torus inside the symplectic vector space must be unknotted [DRGI16].…”