Relationship between volume-growth order and macroscopic dimension

Characterize the relationship between polynomial volume-growth order and macroscopic dimension for noncompact Riemannian manifolds with nonnegative Ricci curvature.

Background

The paper studies two related manifestations of Gromov’s large-scale codimension philosophy: polynomial volume-growth bounds under nonnegative Ricci curvature, and Urysohn-width or macroscopic-dimension bounds under volume deficit or positive scalar curvature. Although these notions are connected in some settings, the authors explicitly state that their relationship in the noncompact, nonnegative-Ricci-curvature setting is unresolved.

The remark contrasts this unresolved noncompact situation with Perelman’s result for closed nonnegatively curved spaces, where volume is comparable to the product of all Urysohn widths. The paper does not provide a general characterization for complete noncompact manifolds.