Lagrangian isotopy of tori in $${S^2times S^2}$$ and $${{mathbb{C}}P^2}$$CP2

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^*{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.