For every integer n≥ 2, there exists a constant C_n>0 such that, for every closed oriented n-manifold M, every Riemannian metric g on M, and every lambda≥ 0, the condition operatornameSc_g≥-lambda^2 implies lVert M rVert≤ C_n lambda^n operatornamevol_g(M), where lVert M rVert denotes the simplicial volume of M.

Gromov’s quantitative scalar-curvature–simplicial-volume conjecture

For every integer $n\geq 2$ , there exists a constant $C_n>0$ such that, for every closed oriented $n$ -manifold $M$ , every Riemannian metric $g$ on $M$ , and every $\lambda\geq 0$ , the condition $\operatorname{Sc}_g\geq-\lambda^2$ implies $\lVert M\rVert\leq C_n\lambda^n\operatorname{vol}_g(M)$ , where $\lVert M\rVert$ denotes the simplicial volume of $M$ .

Progress summary

Open

A new preprint reports the conjectured curvature–topology inequality for closed Kähler surfaces and some related four-dimensional examples, while the general case remains open.

Gromov’s conjecture links scalar curvature and topological complexity through a quantitative inequality. The retrieved sources give no original formulation date beyond its attribution to Gromov.

Known results

The qualitative positive-scalar-curvature analogue is known in dimension $3$ and under certain analytic hypotheses; this does not settle the quantitative statement.

August 2026 restricted advances

Nicolas V. Ivanov and Mikhail Katz report the inequality for closed Kähler surfaces, with further examples among non-Kähler symplectic four-manifolds.
A separate 2026 preprint claims vanishing of simplicial volume under spin universal-cover and Ingham-decay assumptions, substantially weaker than classical rapid decay. Both are restricted results, and the retrieved sources provide no independent verification of the newer Kähler-surface claim.

Current status (as of August 2026): restricted cases are reported for closed Kähler surfaces, some symplectic four-manifolds, and selected fundamental groups, but the unrestricted conjecture remains open.

Sources

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Simplicial Volume and Scalar Curvature on Closed Kähler Surfaces — arXiv — Nicolas V. Ivanov, Mikhail Katz

differential-geometry geometric-topology

2 days ago