Gromov’s integral scalar-curvature bound for simplicial volume

What OpenAI claims

Proves ∫_M(Scal_g^-)^n/2 dV_g ≥ a_n‖ M‖ for every closed connected oriented smooth n-manifold, n ≥ 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.

Papers (2)

1. An integral scalar curvature bound for real simplicial volume

   5 October 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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

2. Positive scalar curvature forces rational inessentiality

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We prove that every closed connected oriented smooth manifold admitting strictly positive scalar curvature is rationally inessential: its rational fundamental class maps to zero under the classifying map. No spin or fundamental-group hypothesis is needed, and there is no upper dimension bound. This proves the Gromov–Lawson aspherical conjecture; moreover, every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat. We also show that every closed oriented smooth manifold of positive dimension with nonnegative scalar curvature has zero real simplicial volume, proving the qualitative vanishing consequence of Gromov's conjectural comparison.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.