{"id":"openai-math/Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three","pdf":"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","url":"https://github.com/openai/math/tree/main/preprints/Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026","tags":["scalar-curvature","riemannian-geometry","urysohn-width","spectral-geometry"],"score":0,"title":"Spectral scalar curvature and Urysohn width in dimension three","domain":"Differential geometry","family":"336","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DG","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-10-05","reactions":{},"discussion":[{"url":"https://arxiv.org/abs/2610.28390","date":"2026-10-20","kind":"web","site":"arXiv.org","text":"In this note, we adapt ideas from No. 336 of OpenAI's October 6, 2026 mathematics release to prove the weighted cube inequality in all dimensions.","added":"2026-10-24","title":"A note on the cube inequality","author":"Yuchen Bi","record":"arxiv/2610.28390"}],"formalized":false,"family_title":"Spectral scalar curvature, Urysohn width, and macroscopic dimension"}