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 (2)

An integral scalar curvature bound for real simplicial volume

October 5, 2026 · source · cite as \cite{OAI:An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026}

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

September 23, 2026 · source · cite as \cite{OAI:Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026}

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.

Result 335 · Differential geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.