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

Abstract

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space ${\mathbb{R}}^4$, the projective plane $\mathbb{C}{P}^2$, and the monotone $S^2\times S^2$. 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 $T^{\ast }{\mathbb{T}}^2$, i.e.~it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.

66 pages, 10 figures. Final version. Added Lemmas 4.1 and 6.8, added details in the proof of Corollary 3.7, and altered the statement and proof of Lemma 5.14