Geometry and topology · Oct 6, 2026
Spectral scalar curvature, Urysohn width, and macroscopic dimension
Every complete connected smooth boundaryless n-manifold, n ≥ 3, satisfying as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most whose entire fibers have diameter bounded only by n in the original metric. This strengthens Gromov's width conclusion to spectral scalar curvature. Universal covers of closed positive-scalar-curvature manifolds also have continuous macroscopic dimension at most for every n ≥ 2.
Contributors
Unreleased OpenAI model
AI systems: Unreleased OpenAI model