Dice Question Streamline Icon: https://streamlinehq.com

Does Fukaya isomorphism imply Hamiltonian isotopy for exact Lagrangians in T* N?

Ascertain whether, for compact exact Lagrangian submanifolds in a cotangent bundle T* N, being isomorphic as objects in the Fukaya category implies that the submanifolds are Hamiltonian isotopic.

Information Square Streamline Icon: https://streamlinehq.com

Background

This question arises in the discussion of the nearby Lagrangian conjecture. While Hamiltonian isotopic exact Lagrangians are known to be isomorphic in the Fukaya category, the converse direction is not established. The authors point out that this gap is a key reason why the nearby Lagrangian conjecture remains open in general.

References

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.”