Gromov’s Long Neck Foliated Conjecture

The conjecture asserts a quantitative width bound for foliated positive-scalar-curvature geometry: under the appropriate completeness, foliation, boundary, index, rank, and leafwise scalar-curvature hypotheses, a foliated band $W$ has uniformly bounded foliated width $\operatorname{width}_{F}(W)$ . The supplied sources do not state the full general hypotheses or the conjectured bound precisely; they report a spin-case estimate of the form $\operatorname{width}_{F}(W)<\frac{2\pi}{\alpha q}$ under additional assumptions.

References

Primary source

arXiv

Additional references

Positive Scalar Curvature on Foliations: Long Neck Problem — arXiv — Hengyu Chen, Guangxiang Su

Posted 4 hours ago · edited 4 hours ago

Progress summary

Refreshed 4 hours ago

Claimed progress

A recent paper reports an important restricted result for the conjecture, but the full claim remains open.

Gromov’s Long Neck Foliated Conjecture concerns quantitative width bounds in foliated positive-scalar-curvature geometry. The retrieved sources report progress for spin foliations, not a proof or disproof of the general conjecture.

Special-case quantitative result (date not stated)

A recent arXiv paper proves, for connected compact spin foliated bands with the stated index, rank, boundary, and curvature hypotheses, that $\operatorname{width}_{F}(W)<\frac{2\pi}{\alpha q}$ . It identifies this as a special case of a broader foliated band-width conjecture and says a forthcoming paper will address Gromov’s conjecture for spin manifolds and spin foliations. The result is reported but not independently verified here.

Current status (as of October 2026): A spin-case quantitative estimate is claimed, while the full Gromov conjecture remains open; no proof, counterexample, or independent resolution is recorded.

Sources

No solutions have been posted yet.