Urysohn width and macroscopic scalar curvature

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].

💡 Research Summary

The paper disproves the macroscopic version of Gromov’s Urysohn‑width conjecture for scalar curvature in dimensions four and higher. Gromov’s original conjecture asserts that a closed d‑dimensional Riemannian manifold admitting a metric with scalar curvature bounded below by a positive constant σ² must have its codimension‑2 Urysohn width bounded by C_d/σ. Guth later showed that a macroscopic analogue holds for codimension‑1 width: if every unit ball has sufficiently small volume, then UW_{d‑1} ≤ 1. However, the authors demonstrate that an analogous statement for codimension‑2 width fails.

The authors first present two intuitive counter‑examples (a product S¹(ε)×S²(R) and a Berger‑metric Hopf fibration S³) where unit‑ball volumes are arbitrarily small while the codimension‑2 width grows like R. The second example survives even after imposing the stronger “macroscopic scalar curvature” condition defined via volumes of balls in the universal cover of a slightly larger ball.

To obtain genuine counter‑examples they develop two new tools:

1. ℤ₂‑hypersphericity (HS): a quantitative invariant measuring how large a map to a sphere can be made without collapsing ℤ₂‑homology. For a closed (d‑1)‑manifold M admitting a Pin⁻ structure, they prove (Theorem B) that any principal S¹‑bundle E→M equipped with an “adapted metric” (π* g_M + ε²θ⊗θ) satisfies \