Higher-dimensional adaptation of the low-dimensional Reeb-stability approach

Determine whether a more direct adaptation of the low-dimensional approach of Eliashberg and Thurston can establish an analogous Reeb-stability result for higher-dimensional c-symplectic confoliations without relying on the stricter symplectic filling or cobordism assumptions used in the paper.

Background

The paper proves a higher-dimensional Reeb-stability-type result for c-symplectic confoliations using pseudo-holomorphic curve techniques. These methods require additional hypotheses, namely weak symplectic semi-fillability or a suitable symplectic cobordism from a hypertight stable Hamiltonian structure. The authors note that the resulting theorem is therefore weaker than the three-dimensional theorem of Eliashberg and Thurston, whose proof uses a more direct low-dimensional analysis.

The unresolved issue is whether that low-dimensional strategy can be adapted to the higher-dimensional setting while avoiding the stronger symplectic assumptions. The authors identify the main difficulty as the greater complexity of manipulating symplectic forms in ambient dimensions at least four.

References

It is unclear to the authors whether a more direct adaptation of the low-dimensional approach in the style of is possible in our higher dimensional setting; the main underlying complication is the fact that, as is typical in symplectic topology, manipulating symplectic forms in ambient dimension at least $4$ is harder than on surfaces.

— On some rigidity results for high-dimensional confoliations  (2610.13927 - Gironella et al., 9 Oct 2026) in Section 1, subsection “A Reeb-stability result for high-dimensional confoliations”