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 σ^n 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.