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Nearby Lagrangian conjecture in cotangent bundles

Determine whether every compact exact Lagrangian submanifold L⊂T* N is Hamiltonian isotopic to the zero section of T* N.

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Background

This conjecture, due to Arnold, asks for a strong classification of compact exact Lagrangians in cotangent bundles. The paper reviews major progress such as Nadler–Zaslow, Fukaya–Seidel–Smith, and Abouzaid–Kragh, which imply strong homotopy-theoretic constraints and categorical equivalences, yet do not settle Hamiltonian isotopy classification.

The authors emphasize that despite partial results, the conjecture remains widely open and identify a key obstruction: it is not known whether isomorphism in the Fukaya category implies Hamiltonian isotopy, even for exact Lagrangians in cotangent bundles.

References

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.”