Gromov’s integral scalar-curvature bound for simplicial volume

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Proves \(\int_M(\mathrm{Scal}_g^-)^{n/2}\,dV_g\ge a_n\lVert M\rVert\) for every closed connected oriented smooth n-manifold, n ≥ 3, and every smooth metric, with \(a_n\gt 0\) depending only on dimension. Here \(\mathrm{Scal}_g^-=\max\{0,-\mathrm{Scal}_g\}\) and \(\lVert M\rVert\) 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.

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What the papers state

Theorem 1.1 (Rational inessentiality)

In Positive scalar curvature forces rational inessentiality

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Let M^n be a closed connected oriented smooth manifold of dimension n\ge2. If M admits a Riemannian metric of strictly positive scalar curvature, then

(c_M)_*[M]=0\qquad\text{in }H_n(B\pi_1(M);\Q).

Corollary 1.5 (Simplicial-volume vanishing)

In Positive scalar curvature forces rational inessentiality

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Every closed oriented smooth manifold of positive dimension admitting a Riemannian metric of nonnegative scalar curvature has zero real simplicial volume.

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Four more possible matches

Corollary 1.4 (Oriented target-map obstruction) in Positive scalar curvature forces rational inessentiality.
Corollary 1.3 (Aspherical nonnegative-scalar-curvature rigidity) in Positive scalar curvature forces rational inessentiality.
Corollary 1.2 (The Gromov–Lawson aspherical conjecture) in Positive scalar curvature forces rational inessentiality.
Theorem 1.1 in An integral scalar curvature bound for real simplicial volume, 5 October 2026.

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Papers

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An integral scalar curvature bound for real simplicial volume

5 October 2026, current edition.

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We prove the integral scalar-curvature inequality proposed by Gromov. For every dimension n ≥ 3, there is a constant \(a_n\gt 0\), depending only on n, such that \(\displaystyle \int_M(\mathop{\mathrm{Scal}}\nolimits _g^-)^{n/2}\,dV_g\ge a_n\|M\|\) for every closed connected oriented smooth n-manifold M and every smooth Riemannian metric g. Here \(\mathop{\mathrm{Scal}}\nolimits _g^-:=\max\{0,-\mathop{\mathrm{Scal}}\nolimits _g\}\), and \(\|M\|\) is real simplicial volume. The proof uses the nonnegative-scalar-curvature vanishing theorem of the companion paper on rational inessentiality.

Positive scalar curvature forces rational inessentiality

Undated, current edition.

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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.

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Jialong Deng, Sam Hopkins, Emil Jerabek, David White, Timothy Chow, Dimitri Vulis, TimH, MathOverflow: how can one contact the authors regarding corrections or missing references?, 8 October 2026. Source
The author of an arXiv paper of 2 September 2026 says this family generalises two of their results without citing it, and that OpenAI Support could not pass their message to the authors; this is their account, and a later preprint makes a separate similarity claim.

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An integral scalar curvature bound for real simplicial volume cites Positive scalar curvature forces rational inessentiality.

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Papers elsewhere in the release that cite this family’s papers

Positive scalar curvature and uniform codimension-two widthCurrent edition; no accompanying Lean statement Family Spectral scalar curvature, Urysohn width, and macroscopic dimensionCites Positive scalar curvature forces rational inessentiality.

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