Scalar Curvature Question [?59]: what is the (asymptotically forn→∞and/or fork→∞) sharp inequality for immersions of theseΣn−1 to spheres

Definitions and mathematical statement

The question is to determine the sharp inequality for immersions of the manifolds denoted $\Sigma^{n-1}$ into spheres, including the sharp asymptotic behavior as \[ n\to\infty\qquad\text{and/or}\qquad k\to\infty. \] For a smooth map $F:\Sigma^{n-1}\to S^N$, immersion means that $\operatorname{rank}(dF_x)=n-1$ at every point. The excerpt, however, does not specify the sphere dimension or radius, does not define $k$, and does not identify the class of $\Sigma^{n-1}$. It also omits the quantity to be bounded, the proposed inequality, and any intrinsic or extrinsic curvature hypotheses. Isometric immersion is suggested only by the research summary, not by the extracted statement.

The phrase "these $\Sigma^{n-1}$" refers to missing source context. Adjacent local item 6700054 discusses bands $\Sigma^{n-1}\times[-1,1]$, and item 6700056 asks about inequalities for "other exotic spheres", but neither identifies the present manifolds or the missing inequality. Thus the request for sharpness in the separate or joint large-$n$ and large-$k$ regimes is retained without assigning $k$ a codimension or choosing an unsupplied curvature bound.

Short English statement

Find the sharp spherical immersion inequality for the specified but unidentified manifolds $\Sigma^{n-1}$, especially as $n$ and/or $k$ grow. The source excerpt omits the inequality, the manifolds, and the meaning of $k$.

Sources

• Local UnsolvedMath registry, permanent ID 6700055, source code catalogue entry; Misha Gromov, 2017; update 2026-08-06. Adjacent local records 6700054 and 6700056 were checked for context but do not supply the missing definitions.
• Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?59], occurrence 58, PDF page 59: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf. Registry access date 2026-07-29; normalized PDF extraction; incomplete public author-hosted PDF; status and reuse terms require review.
• The supplied open-status summary mentions curvature-bounded isometric immersions into Euclidean spaces and spheres, but gives no precise further citation, inequality, or asymptotic constants.