Long Neck Foliated Conjecture

Establish the full Long Neck Foliated Conjecture by determining scalar-curvature lower bounds and leafwise neck-length conditions for a compact foliated manifold with boundary and a smooth area-non-increasing map to a sphere that force the map to have degree zero, beyond the spin-case estimates proved in the paper.

Background

The paper studies foliated analogues of Gromov’s long neck problem, which asks when curvature bounds and a sufficiently long region between the support of a map’s differential and the boundary force the map to have degree zero. For foliated manifolds, the relevant curvature is the leafwise scalar curvature and the relevant neck length is the leafwise distance to the boundary.

The authors state that Gromov proposed the Long Neck Foliated Conjecture and that their results provide estimates for it in spin cases. Thus, the broader conjecture remains an explicitly identified problem beyond the cases addressed by these estimates.

References

For foliated manifolds with boundary, Gromov pp.~258--259 proposes the Long Neck Foliated Conjecture and suggests that combining and may lead to a proof for spin manifolds or spin foliations. Combining the methods from , we will prove estimates for this conjecture in the spin cases (see \cref{t1.2}).

— Positive Scalar Curvature on Foliations: Long Neck Problem  (2610.10344 - Chen et al., 7 Oct 2026) in Section 1, subsection “Long neck problem”