Remark: Family 335. Gromov's integral scalar-curvature inequality

Proves ${\int }_M({\mathrm{Scal}}_g^{-}{)}^{n/2}d{V}_g\ge a_n\Vert M\Vert$ for every closed connected oriented smooth $n$-manifold, $n\ge 3$, and every smooth metric, with $a_n>0$ depending only on dimension. Here ${\mathrm{Scal}}_g^{-}=\max \{0,-{\mathrm{Scal}}_g\}$ and $\Vert M\Vert$ 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.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.