An integral scalar curvature bound for real simplicial volume
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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100+ reactions to 100+ solutionsProofs and PromptsOct 8, 2026
This trick and these two papers now appear in OpenAI’s deeply creative solutions of problems 207 and 335 – problems so different, they could not be farther apart! I am stunned. What do I take away from this night? We cannot defensively narrow our focus to what we currently regard as human…