Spectral scalar curvature, Urysohn width, and macroscopic dimension

3 papers

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.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 3

Spectral scalar curvature and uniform Urysohn width

OpenAI · 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.

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@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}
}

Spectral scalar curvature and Urysohn width in dimension three

OpenAI · 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.

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@misc{OAI:Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026,
  author = {{OpenAI}},
  title = {{Spectral scalar curvature and Urysohn width in dimension three}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026/spectral-urysohn-three-manifolds.pdf}{OAI:Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026}},
  year = {2026}
}

Positive scalar curvature and uniform codimension-two width

OpenAI · 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.

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@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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.