C0-flux conjecture: closure of Ham(M, ω) in Symp0(M, ω)

Determine whether, for every closed symplectic manifold (M, ω), the Hamiltonian diffeomorphism group Ham(M, ω) is closed in the C0-topology inside the identity component Symp0(M, ω) of the symplectomorphism group Symp(M, ω).

Background

The paper studies C0-topology questions for symplectomorphism groups and mentions the foundational result of Gromov–Eliashberg that Symp(M, ω) is C0-closed in Diff(M). The next central problem is whether the Hamiltonian subgroup is also C0-closed in Symp0(M, ω). This question is commonly referred to as the C0-flux conjecture and has seen partial progress in prior work (e.g., Lalonde–McDuff–Polterovich, Buhovsky), but remains unsettled in general.

The conjecture is closely related to the continuity of the flux homomorphism, and its resolution would have significant implications in C0 symplectic topology, including potential extensions of the flux to larger subgroups and strengthening the rigidity phenomena in the Hamiltonian context.