Browse › Differential geometry › 336 Spectral scalar curvature, Urysohn width, and macroscopic dimension
Spectral scalar curvature and uniform Urysohn width
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Abstract
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.
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Follow-up works
- Yuchen Bi, A note on the cube inequality (arXiv:2610.28390, v1 20 Oct 2026). Its abstract says it adapts ideas from "No. 336" of the OpenAI release, but the only manuscript it cites is the companion paper Positive scalar curvature and uniform codimension-two width, not this one. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Gioacchino Antonelli, Kai Xu, New spectral Bishop–Gromov and Bonnet–Myers theorems and applications to isoperimetry (arXiv:2405.08918). Proves a spectral Bonnet–Myers diameter bound under a lower bound for $-\frac{n-1}{n-2}\Delta+\mathrm{Ric}$, using warped μ-bubbles; a Ricci-type analogue of the paper's spectral hypothesis $-4\Delta+\mathrm{Scal}\ge1$ controlling metric size.
- Xiaoxiang Chai and Yukai Sun, Band width estimates with lower spectral curvature bounds (arXiv:2504.10142, 2025; revised 2026-04-07). Its warped μ-bubble estimates for torical bands under spectral curvature hypotheses provide a comparison for the manuscript's spectral-to-width claim, although band width and Urysohn width are different conclusions.
Digests and expositions
- Latent Seminar, Spectral scalar curvature and codimension-two width (2026-10-07). This exposition distinguishes the three manuscripts, explains the imported partition theorem and the three-dimensional weighted-surface construction, and states that it does not independently verify the complete proofs. [family-level]
Elsewhere
- Emergent Mind: family 336
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026,
author = {{OpenAI}},
title = {{Spectral scalar curvature and uniform Urysohn width}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026/main.pdf}{OAI:Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026}},
year = {2026}
}Family 336: 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.
Other papers in this family:
- Spectral scalar curvature and Urysohn width in dimension three · October 5, 2026
- Positive scalar curvature and uniform codimension-two width · September 23, 2026