Let n≥ 3 and let (M^n,g) be a complete, connected, orientable, noncompact Riemannian manifold with positive scalar curvature. If, for some (equivalently any) x_0∈ M, liminf_x→∞ d_g(x,x_0)^2 operatornameScal_g(x)>fracn-1n, then M admits a complete smooth Riemannian metric widetilde g satisfying operatornameScal_widetilde g≥ 1.

Gromov’s critical rate of decay conjecture

Let $n\ge 3$ and let $(M^n,g)$ be a complete, connected, orientable, noncompact Riemannian manifold with positive scalar curvature. If, for some (equivalently any) $x_0\in M$ , $\liminf_{x\to\infty} d_g(x,x_0)^2\operatorname{Scal}_g(x)>\frac{n-1}{n}$ , then $M$ admits a complete smooth Riemannian metric $\widetilde g$ satisfying $\operatorname{Scal}_{\widetilde g}\ge 1$ .

References

Primary source

arXiv

Additional references

Quadratic Scalar Curvature Decay and Uniform Positivity in Dimensions Four through Seven — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A new paper claims the conjecture is correct in dimensions four through seven, but the full question in every dimension remains unsettled.

Gromov’s conjecture concerns the sharp decay rate required for the relevant scalar-curvature conclusion. The latest result addresses only the conjecture’s second part and a restricted dimensional range.

September 3, 2026 development

The paper Quadratic Scalar Curvature Decay and Uniform Positivity in Dimensions Four through Seven claims that, for $4 \le n \le 7$ , an asymptotic quadratic scalar-curvature coefficient greater than $(n-1)/n$ suffices. It also claims that the threshold and the strict inequality are optimal, giving substantial progress but not a resolution in all dimensions.

Current status (as of September 2026): The criterion is claimed, but unverified, for $4 \le n \le 7$ with optimal boundary behavior; the conjecture in dimensions outside this range remains open.

Sources

No solutions have been posted yet.