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.

Background

Within the discussion of cotangent bundles, the author recalls Arnold’s nearby Lagrangian conjecture, a central open problem in symplectic topology. Despite strong constraints proved via Fukaya-category techniques (e.g., Nadler–Zaslow and Abouzaid–Kragh), the classification up to Hamiltonian isotopy remains open in general.

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.

— Lagrangian Floer theory, from geometry to algebra and back again  (2510.22476 - Auroux, 26 Oct 2025) in Subsection “Cotangent bundles and the nearby Lagrangian conjecture,” Section 2.3

Such a result relies on the Nearby Lagrangian Conjecture for T*S1, a conjecture of Arnol'd that states that any closed, exact Lagrangian submanifold of the cotangent bundle of a closed manifold is Hamiltonian isotopic to the zero section. The statement is known to be true for only a handful of spaces, one of which is T* S1.

— The Legendrian Hopf Link has exactly two Lagrangian fillings  (2506.15111 - Thomson, 18 Jun 2025) in Section 4.2 (The Vertical Isotopy)

This is a significant partial result on Arnold's 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.

— Lagrangian Floer theory, from geometry to algebra and back again  (2510.22476 - Auroux, 26 Oct 2025) in Section 2.3, “Cotangent bundles and the nearby Lagrangian conjecture.”