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geometry-topology / Scalar curvature; simplicial volume and bounded cohomology

Gromov's integral scalar-curvature bound for simplicial volume

Theorem 1.1: for every n≥3n\ge3 there is an>0a_n>0 with ∫M(Scalg−)n/2dVg≥an∥M∥\int_M(\mathrm{Scal}_g^-)^{n/2}dV_g\ge a_n\|M\| for every closed connected oriented smooth nn-manifold MM and smooth metric gg; e.g. Scalg≥−n(n−1)\mathrm{Scal}_g\ge-n(n-1) gives Volg(M)≥an[n(n−1)]−n/2∥M∥\mathrm{Vol}_g(M)\ge a_n[n(n-1)]^{-n/2}\|M\|. The route is a Yamabe reduction to Scal=−1\mathrm{Scal}=-1 followed by a bounded-cohomology and graph-deformation estimate. No explicit value of ana_n is stated in the theorem, and dimension two is excluded.

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geometry-topologyOct 5, 2026Significance 38/100Registry: unreviewed

Gromov's integral scalar-curvature bound for simplicial volume

Prior state unknown→proved

Theorem 1.1: for every n≥3n\ge3 there is an>0a_n>0 with ∫M(Scalg−)n/2dVg≥an∥M∥\int_M(\mathrm{Scal}_g^-)^{n/2}dV_g\ge a_n\|M\| for every closed connected oriented smooth nn-manifold MM and smooth metric gg; e.g. Scalg≥−n(n−1)\mathrm{Scal}_g\ge-n(n-1) gives Volg(M)≥an[n(n−1)]−n/2∥M∥\mathrm{Vol}_g(M)\ge a_n[n(n-1)]^{-n/2}\|M\|. The route is a Yamabe reduction to Scal=−1\mathrm{Scal}=-1 followed by a bounded-cohomology and graph-deformation estimate. No explicit value of ana_n…

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Theorem 1.1: for every n≥3n\ge3 there is an>0a_n>0 with ∫M(Scalg−)n/2dVg≥an∥M∥\int_M(\mathrm{Scal}_g^-)^{n/2}dV_g\ge a_n\|M\| for every closed connected oriented smooth nn-manifold MM and smooth metric gg; e.g. Scalg≥−n(n−1)\mathrm{Scal}_g\ge-n(n-1) gives Volg(M)≥an[n(n−1)]−n/2∥M∥\mathrm{Vol}_g(M)\ge a_n[n(n-1)]^{-n/2}\|M\|. The route is a Yamabe reduction to Scal=−1\mathrm{Scal}=-1 followed by a bounded-cohomology and graph-deformation estimate. No explicit value of ana_n is stated in the theorem, and dimension two is excluded.

Theorem 1.1: for every n≥3n\ge3 there is an>0a_n>0 with ∫M(Scalg−)n/2dVg≥an∥M∥\int_M(\mathrm{Scal}_g^-)^{n/2}dV_g\ge a_n\|M\| for every closed connected oriented smooth nn-manifold MM and smooth metric gg; e.g. Scalg≥−n(n−1)\mathrm{Scal}_g\ge-n(n-1) gives Volg(M)≥an[n(n−1)]−n/2∥M∥\mathrm{Vol}_g(M)\ge a_n[n(n-1)]^{-n/2}\|M\|. The route is a Yamabe reduction to Scal=−1\mathrm{Scal}=-1 followed by a bounded-cohomology and graph-deformation estimate. No explicit value of ana_n is stated in the theorem, and dimension two is excluded.

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