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

TL;DR

This paper proves the uniqueness of Lagrangian tori in certain symplectic four-manifolds up to isotopy, using pseudoholomorphic foliations and symplectic field theory techniques, and also confirms the nearby Lagrangian conjecture for the cotangent bundle of a torus.

Contribution

It establishes the uniqueness of Lagrangian tori in specific symplectic four-manifolds and verifies the nearby Lagrangian conjecture for $T^*\mathbb{T}^2$, advancing understanding of Lagrangian submanifolds.

Findings

Unique Lagrangian tori in $\mathbb{R}^4$, $\mathbb{C}P^2$, and $S^2 \times S^2$ up to isotopy.

Proof of the nearby Lagrangian conjecture for $T^*\mathbb{T}^2$.

Application of pseudoholomorphic foliations and symplectic field theory techniques.

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 $T^*\mathbb{T}^2$, i.e.~it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.