Construct positive-scalar-curvature metrics realizing the subcritical quadratic decay rate

Construct, for every connected orientable noncompact n-manifold admitting a complete positive-scalar-curvature metric and every constant 0<C<C_n, a complete positive-scalar-curvature metric whose scalar curvature is eventually greater than C divided by the squared distance from a fixed basepoint, as required by part (1) of Gromov’s critical rate of decay conjecture.

Background

The paper states Gromov’s critical rate of decay conjecture, which has two parts. Part (1) asks whether every open manifold carrying some complete positive-scalar-curvature metric also admits another such metric with a prescribed subcritical quadratic lower bound on scalar curvature at infinity, for every constant below the dimensional threshold C_n.

The paper proves part (2) with the sharp proposed threshold C_n=(n-1)/n in dimensions four through seven, but explicitly leaves part (1) unresolved. This problem concerns the existence and construction of metrics with controlled quadratic scalar-curvature decay, rather than the passage from sufficiently slow decay to uniform positivity.

References

Our main theorem proves part~\textup{(2)} with $C_n=(n-1)/n$ in dimensions $4\le n\le7$. Part~\textup{(1)} remains open.

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

Gromov's critical rate of decay conjecture predicts a dimensional constant c_n>0 with two properties for open orientable n-manifolds. First, every such manifold with a complete metric of positive scalar curvature should admit, for each 0<c<c_n, a complete metric with R_g\>0 and \mathcal C_\infty(g)\ge c. Second, the existence of a complete metric with R_g>0 and \mathcal C_\infty(g)>c_n should imply the existence of a complete metric of uniformly positive scalar curvature. Chen proposed the sharp value c_n=(n-1)/n.

— Uniform positive scalar curvature from supercritical quadratic decay  (2610.05933 - Wang et al., 5 Oct 2026) in Section 1, subsection “Quadratic decay and earlier results”