Urysohn width and macroscopic scalar curvature

Abstract

We show that the macroscopic version of Gromov's Urysohn width conjecture for scalar curvature is false in dimensions four and above. This is based on (1) a novel estimate on the codimension two Urysohn width of circle bundles over manifolds with large hypersphericity radius, and (2) a notion of ruling for Riemannian manifolds that yields circle bundles with total spaces admitting metrics of positive macroscopic scalar curvature. Along the way, we also show that Urysohn width is not continuous under Cheeger-Gromov collapsing limits. This article is a continuation of our study of metric invariants and scalar curvature for circle bundles over large Riemannian manifolds initiated in [KS25].