Minimal energy of the sphere factors under the alternative five-dimensional hypothesis

Determine whether the two $S^2$ factors in the five-dimensional alternative setting, where the symplectic form on the distinguished leaf is only assumed to split as the sum of two area forms, necessarily have minimal positive $[?]$-energy among spherical classes in the filling.

Background

For the main rank-four obstruction theorem, the authors assume that the two sphere factors in the distinguished leaf have minimal positive symplectic energy among spherical classes. They then discuss an alternative hypothesis in ambient dimension five that drops this minimal-energy requirement and only specifies the split symplectic form on the product of the two spheres.

In this alternative setting, the authors explicitly state that they do not know whether the two sphere factors nevertheless have minimal energy. The subsequent argument is designed to prove the rank-four conclusion without resolving that question.

References

The only difference with respect to \ref{item:splitting_mu__thm_rank_at_least_four} is hence that we don't know whether the $S2$-factors have minimal $\Omega$-energy.

— On some rigidity results for high-dimensional confoliations  (2610.13927 - Gironella et al., 9 Oct 2026) in Section 6, subsection “Alternative proof in ambient dimension 5”