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Urysohn width and macroscopic scalar curvature

Aditya Kumar, Balarka Sen

TL;DR

The paper refutes the macroscopic version of Gromov's Urysohn width conjecture for scalar curvature in dimensions $d\ge4$ by constructing total spaces $E_n$ of circle bundles over carefully built base manifolds $M_n$ with large $\mathbf{Z}_2$-hypersphericity yet admitting adapted metrics yielding arbitrarily large codimension-2 Urysohn width, while maintaining large macroscopic scalar curvature. The core method combines a fiber contraction lemma with mod-2 Hopf-invariant obstructions to force lower bounds on $\mathrm{UW}_{d-2}$ of circle-bundle total spaces, and introduces the soft notion of ruling to guarantee macroscopic curvature on the bundle total space. The base manifolds are produced via two constructions (corrugation of enlargeable manifolds and the Alpert-Balitskiy-Guth example) that yield $\mathrm{HS}(M_n;\mathbf{Z}_2)\gtrsim n$ and admit $3/2$-small rulings, enabling the synthesis into $E_n$ with $\mathrm{mscal}(E_n)\ge\sigma$ and $\mathrm{UW}_{d-2}(E_n)\gtrsim n$. These results demonstrate that macroscopic curvature control does not impose uniform codimension-2 width bounds, and that Urysohn width can be unstable under Cheeger-Gromov collapsing.

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].

Urysohn width and macroscopic scalar curvature

TL;DR

The paper refutes the macroscopic version of Gromov's Urysohn width conjecture for scalar curvature in dimensions by constructing total spaces of circle bundles over carefully built base manifolds with large -hypersphericity yet admitting adapted metrics yielding arbitrarily large codimension-2 Urysohn width, while maintaining large macroscopic scalar curvature. The core method combines a fiber contraction lemma with mod-2 Hopf-invariant obstructions to force lower bounds on of circle-bundle total spaces, and introduces the soft notion of ruling to guarantee macroscopic curvature on the bundle total space. The base manifolds are produced via two constructions (corrugation of enlargeable manifolds and the Alpert-Balitskiy-Guth example) that yield and admit -small rulings, enabling the synthesis into with and . These results demonstrate that macroscopic curvature control does not impose uniform codimension-2 width bounds, and that Urysohn width can be unstable under Cheeger-Gromov collapsing.

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].
Paper Structure (17 sections, 27 theorems, 36 equations)

This paper contains 17 sections, 27 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Comparison with previous work
  5. Organization
  6. Fiber contraction and Hopf invariant
  7. Fiber contraction lemma
  8. Generalized Hopf invariant
  9. Hopf invariant for circle bundles
  10. Width of circle bundles
  11. Urysohn width and Cheeger-Gromov collapsing
  12. Singular ruling and macroscopic scalar curvature
  13. Two constructions of base manifolds
  14. Corrugation of enlargeable manifolds
  15. The Alpert-Balitskiy-Guth example
  16. ...and 2 more sections

Key Result

Theorem 1.1

There exists a constant $\delta_d >0$ such that for any closed Riemannian manifold $(M^d,g)$, if every ball of radius $1$ in $M$ has volume at most $\delta_d$, then $\mathrm{UW}_ {d-1}(M) \leq 1$.

Theorems & Definitions (77)

  • Definition 1.1: Urysohn width
  • Definition 1.2: Scalar curvature
  • Example 1.1
  • Theorem 1.1: Guth
  • Example 1.2
  • Definition 1.3: Macroscopic scalar curvature
  • Remark 1.1
  • Theorem 1: Theorem \ref{['thm-final']}
  • Remark 1.2
  • Theorem 2: Theorem \ref{['cor-fcl3']}
  • ...and 67 more