Foliated Gromov band-width conjecture
Determine whether every compact foliated band with positive leafwise scalar curvature satisfies the foliated analogue of Gromov's sharp band-width estimate, bounding its leafwise width in terms of the leafwise scalar-curvature lower bound and . The supplied sources do not state the unrestricted hypotheses or the exact inequality, so a more precise formalization is not supported here.
References
Primary source
Additional references
- Positive Scalar Curvature on Foliations: Leafwise Band Width — arXiv — Hengyu Chen
Progress summary
A new preprint claims the conjectured width bound in a restricted foliated setting, while the full conjecture remains open.
The conjecture seeks a foliated analogue of Gromov’s band-width estimate. The reported result addresses only a compact spin foliated-band case subject to cowaist hypotheses.
September 2026 special-case proof
Hengyu Chen’s preprint Positive Scalar Curvature on Foliations: Leafwise Band Width claims to prove the sharp estimate under the stated spin and cowaist assumptions. This is substantial progress for that regime, but it does not settle the unrestricted conjecture; the claim has not been independently verified in the retrieved sources.
Current status (as of September 2026): A sharp width bound is claimed for the specified compact spin foliated-band case, while the full foliated Gromov band-width conjecture remains open.
Sources
Solutions 0
No solutions have been posted yet.