Macroscopic scalar curvature and codimension 2 width

TL;DR

This paper establishes conditions under which a complete 3D Riemannian manifold has macroscopic dimension 1, based on volume bounds and homological properties of loops, linking curvature assumptions to large-scale geometric structure.

Contribution

It introduces new macroscopic curvature conditions that imply a reduction in the manifold's large-scale dimension, connecting local geometric constraints to global topological features.

Findings

Manifolds with specified volume and homology conditions have macroscopic dimension 1.

The results relate local curvature bounds to global geometric properties.

Provides criteria for macroscopic dimension reduction in 3D manifolds.

Abstract

We show that a complete $3$-dimensional Riemannian manifold $M$ with finitely generated first homology has macroscopic dimension $1$ if it satisfies the following "macroscopic curvature" assumptions: every ball of radius $10$ in $M$ has volume at most $4$, and every loop in every ball of radius $1$ in $M$ is null-homologous in the concentric ball of radius $2$.