Spectral scalar curvature, Urysohn width, and macroscopic dimension

Every complete connected smooth boundaryless n-manifold, n ≥ 3, satisfying -4\Delta+\mathrm{Scal}\ge1 as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most n-2 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 n-2 for every n ≥ 2.

Manuscripts (3)

Spectral scalar curvature and uniform Urysohn width

October 5, 2026 · source · cite as \cite{OAI:Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026}

For every n ≥ 4, a complete connected smooth Riemannian n-manifold without boundary satisfying -4\Delta+\mathop{\mathrm{Scal}}\nolimits \ge1 as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most n-2 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

October 5, 2026 · source · cite as \cite{OAI:Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026}

Every connected complete smooth Riemannian three-manifold without boundary satisfying -4\Delta+\mathop{\mathrm{Scal}}\nolimits \ge\lambda\gt 0 as a quadratic-form inequality admits a continuous map to a graph whose entire fibers have diameter at most 500/\sqrt\lambda in the original metric. No orientability, spin, compactness, or bounded-geometry assumption is required.

Positive scalar curvature and uniform codimension-two width

September 23, 2026 · source · cite as \cite{OAI:Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026}

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 n-2 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.

Result 336 · Differential geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.