Macroscopic dimension under a unit-ball volume deficit

Establish that every complete, noncompact Riemannian n-manifold whose Ricci curvature is not assumed nonnegative and whose supremal unit-ball volume satisfies V(M,r)<ω_n r^n for some r≥1 has Urysohn (n−1)-width bounded by r, namely UW_{n−1}(M)≤r.

Background

The paper states a first codimension conjecture concerning macroscopic scalar curvature. For a complete noncompact Riemannian n-manifold, the quantity V(M,r)=sup_{p∈M}vol_g B(p,r) measures the largest volume of an r-ball, while UW_{n−1}(M) is the Urysohn (n−1)-width. The conjecture predicts that a deficit in the volume of balls at some scale forces codimension-one large-scale geometry, expressed through a finite bound on the Urysohn width.

The authors report substantial partial progress by Guth and Papasoglu, who obtained the same conclusion under a stronger deficit assumption with a universal constant δ(n) in place of the conjectured optimal Euclidean threshold ω_n. The exact assertion in Conjecture 1.1(1) is not proved in the paper; the paper instead proves the separate volume-growth conjectures under nonnegative Ricci curvature.