Sharp Dimension Threshold for Ricci-Flow Curvature Conditions

Definitions and mathematical statement

For $n\geq4$, let $\mathcal R_n$ denote the space of algebraic curvature operators on ${\mathbb{R}}^n$: self-adjoint operators on $\Lambda^2{\mathbb{R}}^n$ satisfying the first Bianchi identity. Use the orthogonal decomposition \[ R=R_I+R_{\operatorname{Ric}_0}+R_W, \] into scalar-curvature, trace-free-Ricci, and Weyl components. For $c>0$, the cone in the restored source statement is \[ \mathcal C_{n,c}=\{R\in\mathcal R_n: \operatorname{scal}(R)\geq0,\quad \|R_W\|^2\leq c\|R_I\|^2\}. \] No bound on $R_{\operatorname{Ric}_0}$ is included in the supplied condition. Norms are understood as the usual induced operator-space norms; their precise source normalization and the convention for the quadratic operation $R^\#$ should be confirmed. Forward invariance means that a solution of \[ \frac{{\,\mathrm{d}} R}{{\,\mathrm{d}} t}=R^2+R^\# \] starting in $\mathcal C_{n,c}$ remains in it at all later times in its interval of existence.

Let $\mathcal T(n)$ mean that the Böhm--Wilking classification referred to in the report gives the exact set of $c>0$ for which $\mathcal C_{n,c}$ is invariant. The question concerns the least integer threshold \[ n_0=\min\{N:\mathcal T(n)\text{ holds for every integer }n\geq N\}. \] The source reports $n_0\geq12$ and asks whether $n_0=12$. This is an eventual threshold over all larger dimensions, not only a question in dimension twelve. The supplied record does not reproduce the classification theorem or its proposed set of admissible $c$. Thus $\mathcal T(n)$, and hence the threshold, remains a reference-dependent predicate until that missing theorem is supplied; the existence of a finite threshold is part of the reported theorem context, not established in this draft.

Short English statement

Is twelve the sharp dimension from which the cited Böhm--Wilking classification gives precisely the invariant cones defined by nonnegative scalar curvature and $\|R_W\|^2\leq c\|R_I\|^2$? The exact classification theorem is missing from the extraction.

Sources

• Local UnsolvedMath registry, permanent ID $30000789$, source code catalogue entry, "Sharp Dimension Threshold for Ricci-Flow Curvature Conditions." The cleaned statement attributes the classification to Böhm--Wilking; no individual proposer is supplied.
• Differentialgeometrie im Grossen (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/32, pp. 1865--1912: https://doi.org/10.4171/owr/2007/32. This is the sole supplied external reference.
• The literature triage checked 2026-08-21 reports the lower bound n_0 at least 12 and no located authoritative proof of equality or counterexample. Neither the bound nor present-day status is certified by this draft.