Spectral scalar curvature, Urysohn width, and macroscopic dimension

For every n ≥ 4, a complete connected smooth Riemannian n-manifold without boundary satisfying -4\Delta+\mathop{Scal}\nolimits \ge1 as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most n-2 whose entire fibers have diameter bounded only in terms of n. The bound is measured in the original metric. This extends the uniform Urysohn width theorem from a pointwise scalar-curvature lower bound to a spectral lower bound.

Every connected complete smooth Riemannian three-manifold without boundary satisfying -4\Delta+\mathop{Scal}\nolimits ≥\lambda> 0 as a quadratic-form inequality admits a continuous map to a graph whose entire fibers have diameter at most 500/\sqrt\lambda in the original metric. No orientability, spin, compactness, or bounded-geometry assumption is required.

We prove the quantitative continuous form of Gromov's scalar-curvature conjecture in every dimension n ≥ 4. Every complete connected smooth boundaryless n-manifold with scalar curvature at least one admits a continuous map to a simplicial complex of dimension at most n-2 whose entire fibers have diameter bounded only in terms of n. We also obtain the continuous macroscopic-dimension conclusion for universal covers of closed manifolds with positive scalar curvature in every dimension n ≥ 2.