Gromov’s integral scalar-curvature bound for simplicial volume

2 · Going further

The precise main claim is that for every dimension $n\ge 3$, there is a constant $a_n>0$, depending only on $n$, such that ${\int }_M({\mathrm{Scal}}_g^{-}{)}^{n/2}d{V}_g\ge a_n\Vert M\Vert$ for every closed connected oriented smooth $n$-manifold $M$ and every smooth Riemannian metric $g$. Here ${\mathrm{Scal}}_g^{-}:=\max \{0,-{\mathrm{Scal}}_g\}$, $d{V}_g$ is the metric’s volume element, and $\Vert M\Vert$ is real simplicial volume. The source identifies this as the integral scalar-curvature inequality proposed by Gromov.

The argument’s stated dependency is the nonnegative-scalar-curvature vanishing theorem in the companion manuscript. That theorem claims that every closed oriented smooth manifold of positive dimension with nonnegative scalar curvature has zero real simplicial volume. This is the qualitative vanishing consequence of Gromov’s conjectural comparison: when ${\mathrm{Scal}}_g^{-}=0$ everywhere, the main inequality would force $\Vert M\Vert =0$. The abstracts do not explain the additional steps that turn this vanishing result into the quantitative integral bound.

The companion manuscript also claims 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. It explicitly requires no spin or fundamental-group hypothesis and imposes no upper dimension bound. The manuscript describes this as resolving the Gromov–Lawson aspherical conjecture and additionally claims that every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat.

Useful notions

• Smooth manifolds and Riemannian metrics
• Scalar curvature and its negative part
• Integration with the metric volume element
• Real simplicial volume
• Fundamental classes and classifying maps

What was known before / what is new

Limits and assumptions

The integral bound is stated only for closed connected oriented smooth manifolds of dimension $n\ge 3$ with smooth Riemannian metrics; its positive constant depends only on dimension, and no explicit value is supplied. The rational-inessentiality claim assumes strictly positive scalar curvature, whereas the simplicial-volume vanishing claim allows nonnegative scalar curvature and is stated in positive dimension. The flatness claim requires a closed aspherical manifold. The main result is not formalized in Lean and has not been formally verified; these are manuscript claims, and the abstracts provide only the dependency between the arguments, not their detailed justification. The source identifies no further specific open problem.

Mini glossary

Negative part of scalar curvature

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The function ${\mathrm{Scal}}_g^{-}=\max \{0,-{\mathrm{Scal}}_g\}$ records the magnitude of negative scalar curvature and is zero wherever scalar curvature is nonnegative.

Real simplicial volume

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The manifold invariant denoted $\Vert M\Vert$ is the complexity measure appearing on the right-hand side of the claimed inequality.

Rational inessentiality

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A manifold is rationally inessential when its rational fundamental class maps to zero under the classifying map.

Vanishing theorem

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Here this means the companion manuscript’s claim that nonnegative scalar curvature forces real simplicial volume to be zero under its stated hypotheses.

Keywords : Scalar curvature · Real simplicial volume · Rational inessentiality

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family