Spectral scalar curvature and Urysohn width in dimension three
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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A note on the cube inequalityarXiv.orgYuchen BiOct 20, 2026in the index
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.