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Gromov’s integral scalar-curvature bound for simplicial volume

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.

Manuscripts in this family · 2

An integral scalar curvature bound for real simplicial volume

Oct 5, 2026·v126-proof.pdf

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.

Full abstract

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Positive scalar curvature forces rational inessentiality

Sep 23, 2026·paper.pdf

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.

Full abstract

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