Uniform pinching via the best-pinching functional

Definitions and mathematical statement

Let $M$ be a closed connected smooth manifold of dimension $n\geq2$. For a Riemannian metric $g$ of strictly positive sectional curvature, define \[ K_{\min}(g)=\min_{x,\,\sigma\subseteq T_xM}\sec_g(\sigma), \qquad K_{\max}(g)=\max_{x,\,\sigma\subseteq T_xM}\sec_g(\sigma), \] where $\sigma$ ranges over tangent two-planes. Compactness ensures $0<K_{\min}(g)\leq K_{\max}(g)<\infty$. Call the metric globally $\delta$-pinched if $K_{\min}(g)/K_{\max}(g)\geq\delta$. This ratio is unchanged by multiplying the metric by a positive constant.

Does there exist, for each $n$, a number $\delta(n)>0$ such that every closed $n$-manifold admitting some positive-sectional-curvature metric also admits a globally $\delta(n)$-pinched metric? Equivalently, after rescaling the new metric, must every such manifold admit $h$ with \[ \delta(n)\leq\sec_h(\sigma)\leq1 \quad\text{for every tangent two-plane }\sigma? \] The replacement metric may depend on the manifold and need not resemble an initially supplied metric. The bound is uniform over manifold types, rather than a separate positive pinching constant for each compact metric, which exists automatically. Global pinching compares planes at different points as well as at the same point; merely pointwise pinching is a different requirement. No symmetry, connectivity beyond connectedness, or lower volume hypothesis is added. The question asks whether allowing a new metric removes arbitrarily poor optimal pinching within any fixed dimension.

Short English statement

Can every positively curved closed manifold be given a new metric with a positive global pinching ratio depending only on its dimension?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20003180 (AIM-TOPOLOGY-0268), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Manifolds with nonnegative sectional curvature, item 55. Original workshop question underlying AIM-TOPOLOGY-0268. https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf.