Normalized obstructions to pointwise quarter-pinched collapse

Definitions and mathematical statement

For a Riemannian $n$-manifold define the extremal sectional curvatures at a point by $K_{\min}(p)$ and $K_{\max}(p)$. Positive pointwise strict quarter pinching means \[ 0<K_{\min}(p),\qquad K_{\max}(p)<4K_{\min}(p) \quad\text{for every }p. \] This compares planes at the same point; global quarter pinching compares extrema over the entire manifold. Problem 8 asks whether there is a collapsing sequence of simply connected $(M_i^n,g_i)$ of fixed dimension $n>2$ with pointwise strict quarter pinching. The supplied verification repairs the OCR "14" to $1/4$ and $M\,ni$ to $M_i^n$.

The canonical sentence omits closedness, scale normalization and the meaning of collapse. Literally, round spheres $(S^n,r_i^2g_{\mathrm{round}})$ with $r_i\to0$ collapse to a point and remain strictly quarter pinched, while their curvature diverges. The aid's substantive interpretation instead uses closed manifolds with \[ 0<\sec_{g_i}\leq1,\qquad\operatorname{diam}(M_i,g_i)\leq D, \] and bounded-curvature collapse, for example volume tending to zero or the corresponding lower-dimensional Gromov--Hausdorff collapse. These are explicit extra conventions, not source hypotheses silently assumed.

The aid reports that Brendle--Schoen and Hamilton identify the simply connected pointwise strictly quarter-pinched examples with $S^n$ in the respective dimensional ranges. It also reports classical injectivity-radius obstructions to normalized collapse in even dimensions and under the stronger global quarter-pinching assumption. These statements require their exact closedness, curvature and dimensional hypotheses; the full references are not supplied.

For a stable normalized collapse in the report's setting, it gives the necessary condition \[ \kappa_i=\min\sec_{g_i}\longrightarrow0. \] If $p_i$ and a plane at $p_i$ realize $\kappa_i$, pointwise pinching then forces every sectional curvature at $p_i$ to lie between $\kappa_i$ and $4\kappa_i$. Accordingly \[ |\operatorname{Rm}_{g_i}|(p_i)\leq C(n)\kappa_i\longrightarrow0. \] This is existence of flattening points, not uniform convergence of curvature to zero over $M_i$. The word "stable" and the theorem excluding collapse with a uniform positive lower sectional bound are not defined or cited in detail in the aid, so the precise scope of that necessary condition requires verification.

The complex Hopf--Berger family is a reported excluded collapse model. In its standard fiber-shrinking regime $0<t\leq1$, with fiber lengths scaled by $t$, the source gives \[ K_{\min}=t^2,\qquad K_{\max}=4-3t^2. \] Thus strict quarter pinching in this regime is equivalent to $t^2>4/7$, excluding collapse as $t\to0$. The summary's unrestricted wording "exactly $t^2>4/7$" needs this parameter-range qualification: outside the shrinking regime the ordering of the displayed extrema changes and the same one-sided criterion cannot be used without further analysis. This family obstruction and the necessary flattening condition do not construct or exclude every normalized odd-dimensional sequence. The aid leaves the normalized general question open.

Short English statement

Without a curvature scale, shrinking round spheres answer the literal question. The intended bounded-curvature problem is different. The aid reports normalized noncollapse cases and necessary flattening points, and excludes the usual shrinking Hopf--Berger model within its correct parameter range.

Sources

• Local UnsolvedMath registry, permanent ID 20003133, source code catalogue entry; aim-topology-notes.json, notes[220].
• AIM, Manifolds with nonnegative sectional curvature, Section 2, "Collapse and Alexandrov Geometry," Problem 8: https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf. The supplied verification reports checking the fraction and indices in the PDF; this draft does not independently repeat it.
• Brendle--Schoen, Hamilton, classical injectivity-radius estimates and Hopf--Berger curvature formulas, named in the supplied aid without complete bibliography or theorem-level hypotheses.
• Machine-generated research attempt 1: partial result, open at the research run, rigor labelled "proved," low novelty confidence; registry partially solved and corrected verified statement. Not peer reviewed; independently verify before citation. Registry updated 14 August 2026.