Scalar Curvature Question [?61]: Is then every immersion fromXj to the unit ball in RN satisfies supcurv(Xj↪BN(1) ⊂RN)) ≥ √ k for allN ≥n1+

Definitions and mathematical statement

Let $B^N(1)\subset{\mathbb{R}}^N$ denote the Euclidean unit ball. An immersion $F:X\to{\mathbb{R}}^N$ is a differentiable map whose differential has rank $\dim X$ at every point, with sufficient further regularity needed to define curvature.

The source asks whether every immersion of its space $X_j$ into the unit ball satisfies the displayed sharp estimate \[ \operatorname{supcurv}\left(F:X_j\looparrowright B^N(1)\subset{\mathbb{R}}^N\right)\ge\sqrt{k} \qquad\text{for all }N\ge n_1+\cdots+n_j+1. \] This preserves the indices and dimension threshold of the extracted statement. However $X_j$, the integers or dimensions $n_1,\ldots,n_j$, the parameter $k$, and $\operatorname{supcurv}$ are not defined in the record. In particular the text does not state that $X_j$ is a product of manifolds or identify $k$ with $j$ or with a dimension, so no such correction is inferred.

The curvature functional could involve maximal normal curvature, an operator norm or another norm of the second fundamental form, or curvature of selected image curves; the exact choice affects a sharp constant. The source does not fix differentiability, the manifold's metric and completeness, or whether the immersion is required to be isometric. The research summary adds the word "isometric," but the supplied statement says only immersion; that missing hypothesis must be confirmed. Open-versus-closed unit ball and boundary conditions are likewise unspecified. These source dependencies prevent a standalone quantified curvature theorem beyond the retained formula.

Short English statement

Does every immersion of the source's $X_j$ into every permitted-dimensional unit ball have supremal curvature at least $\sqrt k$? The original manifold construction, parameter $k$, curvature norm, and isometry requirement are missing.

The supplied summary describes a curvature-bounded-immersion theme, mentions Gromov, Conway's "large solutions," and tangle/immersion theory, but gives no precise applicable result or reference. It reports the sharp inequality unresolved. The OPEN-TRIAGE label and 2017 passage are retained with current status marked for review.

Sources

• Local UnsolvedMath registry, permanent ID 6700057, source code catalogue entry, scalar-curvature Question [?61], proposed by Misha Gromov, 2017. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• Misha Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?61], occurrence 60, PDF page 59, https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf; normalized pdf-text extraction, accessed 2026-07-29.
• The summary's Gromov/Conway and immersion-theory context has no complete primary references. Source-specific provenance: the author-hosted PDF is incomplete and publicly available; current status and statement-reuse terms NEEDS_REVIEW, separately from the dataset licence.