Nearby Lagrangian Conjecture in cotangent bundles

Prove that for every closed manifold M, any closed exact Lagrangian submanifold of the cotangent bundle T* M is Hamiltonian isotopic to the zero section.

Background

The authors use the Nearby Lagrangian Conjecture (NLC) in the special case of T* S^1 to construct a vertical isotopy that fiberwise brings an exact Lagrangian to the zero section in each fiber of the Lefschetz fibration. While the conjecture is known in a few cases, including T* S^1 (as used here), its general validity for arbitrary closed manifolds M remains open and is a fundamental question in symplectic topology.

This conjecture provides the key mechanism to standardize fiberwise intersections in their classification argument, underscoring its central role and broad significance beyond the specific setting addressed in the paper.