Towards the C^0 flux conjecture

TL;DR

This paper advances the understanding of the $C^0$ flux conjecture by extending known results to new cases and establishing the flux homomorphism's continuity, thereby supporting the $C^0$ rigidity of Hamiltonian paths.

Contribution

It generalizes a key result related to the $C^0$ flux conjecture and proves the flux homomorphism's continuity in the $C^0$ topology, confirming a conjecture by Seyfaddini.

Findings

Confirmed the $C^0$ flux conjecture in new cases

Proved the flux homomorphism is continuous in the $C^0$ topology

Established the $C^0$ rigidity of Hamiltonian paths

Abstract

In this note, we generalise a result of Lalonde, McDuff and Polterovich concerning the $ C^0 $ flux conjecture, thus confirming the conjecture in new cases of a symplectic manifold. Also, we prove the continuity of the flux homomorphism on the space of smooth symplectic isotopies endowed with the $ C^0 $ topology, which implies the $ C^0 $ rigidity of Hamiltonian paths, conjectured by Seyfaddini.