OpenAI · 335-01 · 2026-10-05 · Differential geometry

An integral scalar curvature bound for real simplicial volume

preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026

Family 335. Gromov's integral scalar-curvature inequality.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves $\int_{M} (Scal_{g}^{-})^{n /2} d V_{g} \geq a_{n} ∥ M ∥$ for every closed connected oriented smooth $n$ -manifold, $n \geq 3$ , and every smooth metric, with $a_{n} > 0$ depending only on dimension. Here $Scal_{g}^{-} = max {0, - Scal_{g}}$ and $∥ M ∥$ is real simplicial volume. Also proves rational inessentiality under positive scalar curvature, resolving the Gromov–Lawson conjecture; every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat.

We prove the integral scalar-curvature inequality proposed by Gromov. For every dimension n ≥ 3, there is a constant $a_{n} > 0$ , depending only on n, such that $\int_{M} (Scal_{g}^{-})^{n /2} d V_{g} \geq a_{n} ∥ M ∥$ for every closed connected oriented smooth n-manifold M and every smooth Riemannian metric g. Here $Scal_{g}^{-} := max {0, - Scal_{g}}$ , and $∥ M ∥$ is real simplicial volume. The proof uses the nonnegative-scalar-curvature vanishing theorem of the companion paper on rational inessentiality.

Preprint source and Lean scope

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