Weinstein exactness of nearby Lagrangians and related questions

TL;DR

This paper investigates when Lagrangian submanifolds that are mapped close to themselves by Hamiltonian diffeomorphisms are actually Hamiltonian isotopic to their original, providing both negative examples and positive results under rationality conditions.

Contribution

It demonstrates that the Weinstein exactness property can fail in high dimensions but holds under rationality assumptions, and connects these results to conjectures and rigidity phenomena in symplectic topology.

Findings

Negative examples in dimensions ≥6 showing failure of Weinstein exactness.

Positive results for Lagrangians satisfying rationality conditions.

Progress on Viterbo's spectral norm conjecture and integer difference vectors.

Abstract

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.