Determine the critical decay threshold in all dimensions

Determine whether the dimensional constant in Gromov’s critical rate of decay conjecture satisfies C_n=(n-1)/n in general, beyond the dimensions currently established, so that scalar curvature greater than ((n-1)/n) divided by the squared distance at infinity forces the existence of a complete uniformly positive-scalar-curvature metric.

Background

The paper recalls that Chen identified the higher-dimensional coefficient (n-1)/n as the relevant threshold and conjectured that this value is optimal in general. The present work proves the corresponding implication for dimensions four through seven, while the general-dimensional assertion remains a conjectural threshold statement.

This problem asks for a determination of the sharp constant governing the transition from positive scalar curvature with quadratic decay to the existence of a complete metric whose scalar curvature is uniformly bounded below by a positive constant.

References

Chen proved part~\textup{(2)} in dimension three with the sharp threshold $C_3=2/3$, and conjectured that $C_n=(n-1)/n$ in general Corollary~1.6 and the discussion following Proposition~1.7.

— Quadratic Scalar Curvature Decay and Uniform Positivity in Dimensions Four through Seven  (2609.03676 - Wang et al., 3 Sep 2026) in Section 1, Introduction, paragraph following Conjecture (Gromov)