Weinstein exactness of nearby Lagrangians and related questions

We address the following problem: if a Hamiltonian diffeomorphism maps a Lagrangian submanifold $L$ to a small Weinstein neighborhood of $L$, is the image necessarily Hamiltonian isotopic to $L$ inside that neighborhood? On the one hand, we show that the question can have a negative answer in any symplectic manifold of dimension at least six. On the other hand, we answer an a priori weaker form of the question in the positive in various cases when $L$ satisfies a rationality condition: we prove that the image of $L$ is often exact inside the Weinstein neighborhood. We provide applications to the Lagrangian counterpart of the $C^0$ flux conjecture, to $C^0$-rigidity phenomena of Hamiltonian diffeomorphisms, and to topological properties of spaces of Lagrangians with the same rationality constraint. Moreover, we state and prove cases of an analogue of Viterbo’s spectral norm conjecture for non-exact Lagrangians; in the process, we make progress on an old question of Viterbo regarding integer difference vectors between points of Lagrangians.