Macroscopic scalar curvature and local collapsing

TL;DR

This paper establishes a link between macroscopic scalar curvature and local collapsing in Riemannian manifolds, showing that small-volume metrics have points with large-volume balls in the universal cover, extending to polyhedral spaces.

Contribution

It introduces a new perspective on scalar curvature via macroscopic properties and generalizes recent metric geometry results to polyhedral length spaces.

Findings

Existence of points with large-volume balls in the universal cover for small-volume metrics

Interpretation of results in terms of macroscopic scalar curvature

Extension of concepts to polyhedral length spaces

Abstract

Consider a closed Riemannian $n$-manifold $M$ admitting a negatively curved Riemannian metric. We show that for every Riemannian metric on $M$ of sufficiently small volume, there is a point in the universal cover of $M$ such that the volume of every ball of radius $r \geq 1$ centered at this point is greater or equal to the volume of the ball of the same radius in the hyperbolic $n$-space. We also give an interpretation of this result in terms of macroscopic scalar curvature. This result, which holds more generally in the context of polyhedral length spaces, is related to a question of Guth. Its proof relies on a generalization of recent progress in metric geometry about the Alexandrov/Urysohn width involving the volume of balls of radius in a certain range with collapsing at different scales.