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We show that, up to Lagrangian isotopy, there is a unique Lagrangian torusinside each of the following uniruled symplectic four-manifolds: the symplecticvector space $\mathbb{R}^4$, the projective plane $\mathbb{C}P^2$, and themonotone $S^2 \times S^2$. The result is proven by studying pseudoholomorphicfoliations while performing the splitting construction from symplectic fieldtheory along the Lagrangian torus. A number of other related results are alsoshown. Notably, the nearby Lagrangian conjecture is established for$T^*\mathbb{T}^2$, i.e.~it is shown that every closed exact Lagrangiansubmanifold in this cotangent bundle is Hamiltonian isotopic to thezero-section.

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