Geometry and topology · Oct 6, 2026

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.

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