Urysohn width and macroscopic scalar curvature

TL;DR

This paper disproves a macroscopic version of Gromov's Urysohn width conjecture for scalar curvature in dimensions four and higher, introducing new estimates and concepts related to Riemannian manifolds and their scalar curvature properties.

Contribution

It presents a novel estimate on Urysohn width for circle bundles and introduces a ruling concept that affects scalar curvature, challenging previous conjectures.

Findings

Macroscopic Gromov width conjecture is false in dimensions ≥4.

Urysohn width is not continuous under Cheeger-Gromov collapsing.

New estimates relate circle bundles' geometry to 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].