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Positive scalar curvature and uniform codimension-two width
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Abstract
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.
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Follow-up works
- Yuchen Bi, A note on the cube inequality (arXiv:2610.28390, v1 20 Oct 2026). Proves a weighted cube inequality in all dimensions by adapting ideas from this manuscript, which it cites as [OAI], stating that its treatment of singularities "largely follows [OAI, Appendices C–E]"; its AI disclosure says the note arose from discussions with ChatGPT about [OAI].
Comments and discussion
- Jialong Deng, personal homepage (undated page). Deng says his earlier locally-conformally-flat result (arXiv:2609.02267) gives a sharper macroscopic-dimension bound in that setting than the related results announced in Families 335 and 336, and that "Neither manuscript mentions my earlier work."
Related prior literature mentioned by others
- Jialong Deng, Locally Conformally Flat Manifolds with Positive Scalar Curvature: Kleinian Groups, Moduli Spaces, and Euclidean Rigidity (arXiv:2609.02267, 2 Sep 2026). Cited by its author in the homepage comment above; it bounds the macroscopic dimension of universal covers of closed oriented LCF PSC manifolds, which relates to Corollary 1.3 and is not in this paper's bibliography.
Possibly relevant literature (found by the agent, not checked)
- Hannah Alpert, Alexey Balitskiy, Larry Guth, Macroscopic scalar curvature and codimension 2 width (arXiv:2112.04594, Dec 2021). An earlier macroscopic-curvature width result for 3-manifolds, relevant to the paper's discussion (via Kumar–Sen) of whether a fixed-scale macroscopic curvature hypothesis can replace the pointwise one.
- Otis Chodosh, Chao Li, Yevgeny Liokumovich, Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions (arXiv:2105.07306; Geom. Topol. 27 (2023)). Proves that 4- and 5-manifolds with a nonzero-degree map to a closed aspherical manifold carry no PSC metric, which relates to Corollary 1.4; the companion spectral manuscript in this family cites its Theorem 4.
- Junyu Ma, Urysohn 1-width for 4 and 5 manifolds with positive biRicci curvature (arXiv:2402.18141, v3 Mar 2025). Bounds Urysohn 1-width for closed 4- and 5-manifolds under the stronger positive biRicci hypothesis, a comparison point for Theorem 1.1 in dimensions 4 and 5.
- Hannah Alpert, Arka Banerjee, Panos Papasoglu, 1-Uryson width and covers (arXiv:2505.21126, v3 Jul 2026). Compares the Urysohn width of a space with that of its universal cover, relevant to the passage from complete manifolds to universal covers in Corollary 1.3.
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
- VibeMathed: Problem record
- Emergent Mind: family 336
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026,
author = {{OpenAI}},
title = {{Positive scalar curvature and uniform codimension-two width}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/paper.pdf}{OAI:Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026}},
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 uniform Urysohn width · October 5, 2026
- Spectral scalar curvature and Urysohn width in dimension three · October 5, 2026