Nearby Lagrangian conjecture in cotangent bundles
Determine whether every compact exact Lagrangian submanifold L in the cotangent bundle T*N of a smooth manifold N is Hamiltonian isotopic to the zero section; equivalently, resolve Arnold’s nearby Lagrangian conjecture for compact exact Lagrangians in T*N.
References
Arnold's {\bf nearby Lagrangian conjecture}, which asks whether every compact exact Lagrangian submanifold in $T*N$ is Hamiltonian isotopic to the zero section. (By the Weinstein neighborhood theorem, a tubular neighborhood of a Lagrangian submanifold $N\subset M$ is symplectomorphic to a neighborhood of the zero section in $T*N$, so Arnold's conjecture indeed constrains nearby Lagrangians.) Arnold's question remains open in general (though it has been answered positively in a few cases), essentially because, even though Hamiltonian isotopic exact Lagrangian submanifolds are Fukaya isomorphic, it is not clear that the converse should hold.