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An integral scalar curvature bound for real simplicial volume

OpenAI · October 5, 2026

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Abstract

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.

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Comments and discussion

  • Roman Sauer, contribution to 100+ reactions to 100+ solutions (Proofs and Prompts, 8 October 2026). Sauer identifies the replacement of manifolds by measurable foliated objects, used in his work with Sabine Braun and subsequent work by Hannah Alpert, as a method appearing in result families 207 and 335. [family-level]
  • Sabine Braun and Roman Sauer, Volume and macroscopic scalar curvature (arXiv:2012.08999, 2020; revised 2021). Sauer's family-specific reaction identifies this work as a methodological antecedent; it proves analogues of the simplicial-volume bound and rational-essentiality obstruction with upper bounds on universal-cover unit-ball volumes replacing scalar-curvature lower bounds. [family-level]

Possibly relevant literature (found by the agent, not checked)

  • Q. Ma and G. Yu, Small scale index theory, scalar curvature, and Gromov's simplicial norms (arXiv:2508.14791, v1 Aug 2025, v3 Nov 2025). For manifolds with spin universal cover, bounds the simplicial norm of the Poincaré dual of the Â-class using a scalar curvature lower bound, a volume upper bound and an injectivity radius lower bound; an earlier quantitative scalar-curvature/simplicial-norm estimate under extra hypotheses, to compare with Theorem 1.1.

Digests and expositions

  • Latent Seminar, Integral scalar curvature and simplicial volume (7 October 2026). This exposition covers both manuscripts, including the integral estimate's dependence on the companion's vanishing theorem, the Yamabe reduction, measured deformations and degree estimates, while explicitly disclaiming independent verification of the complete proofs. [family-level]

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026,
  author = {{OpenAI}},
  title = {{An integral scalar curvature bound for real simplicial volume}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026/v126-proof.pdf}{OAI:An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026}},
  year = {2026}
}

Family 335: 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.

Other papers in this family: