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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.
Manuscripts in this family · 3
Spectral scalar curvature and uniform Urysohn width
Oct 5, 2026·main.pdf
For every n ≥ 4, a complete connected smooth Riemannian n-manifold without boundary 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 in terms of n. The bound is measured in the original metric. This extends the uniform Urysohn width theorem from a pointwise scalar-curvature lower bound to a spectral lower bound.
Spectral scalar curvature and Urysohn width in dimension three
Oct 5, 2026·spectral-urysohn-three-manifolds.pdf
Every connected complete smooth Riemannian three-manifold without boundary satisfying as a quadratic-form inequality admits a continuous map to a graph whose entire fibers have diameter at most in the original metric. No orientability, spin, compactness, or bounded-geometry assumption is required.
Positive scalar curvature and uniform codimension-two width
Sep 23, 2026·paper.pdf
We prove the quantitative continuous form of Gromov's scalar-curvature conjecture in every dimension n ≥ 4. Every complete connected smooth boundaryless n-manifold with scalar curvature at least one admits a continuous map to a simplicial complex of dimension at most whose entire fibers have diameter bounded only in terms of n. We also obtain the continuous macroscopic-dimension conclusion for universal covers of closed manifolds with positive scalar curvature in every dimension n ≥ 2.