Gromov’s simplicial volume versus lower scalar curvature bound

Establish that for each dimension n there exists a constant c(n) > 0 such that for every closed connected oriented n-dimensional Riemannian manifold M with scalar curvature bounded below by −σ, the simplicial volume satisfies ||M|| ≤ c(n) σ^n vol(M).

Background

Motivated by connections between scalar curvature and simplicial volume, the authors record Gromov’s conjecture linking lower scalar curvature bounds with an upper bound on simplicial volume proportional to σn times the Riemannian volume.

They discuss potential implications in two viewpoints: as an obstruction to volume collapse with a lower scalar curvature bound, and as a constraint for metrics of almost nonnegative scalar curvature with normalized volume. They also note partial analogs known for Ricci curvature and macroscopic scalar curvature.

References

Conjecture A.9. [16, Section 3.A] For each n ∈ Z+, there is some c > 0 so that if M is a compact connected oriented n-dimensional Riemannian manifold with R ≥ −σ then ||M|| ≤ c σn vol(M).

— Some obstructions to positive scalar curvature on a noncompact manifold  (2402.13239 - Lott, 2024) in Appendix A.2, Conjecture A.9

Gromov conjectured that a lower scalar-curvature bound and the Riemannian volume control this global topological invariant (see *{Section~28, p.~88}).

— Simplicial Volume and Scalar Curvature on Closed Kähler Surfaces  (2608.17335 - Min et al., 18 Aug 2026) in Section 1, Conjecture 1.1 (equation (1.1))