Scalar Curvature Question [?56]: What are possible values of the co-s+areas of the complements ofr-balls in compact spin manifoldsX withSc(X) ≥

Definitions and mathematical statement

Let $(X^n,g)$ be a compact spin Riemannian manifold satisfying the pointwise lower bound \[ \operatorname{Scal}_g(x)\ge n(n-1)\qquad(x\in X). \] For a center $p\in X$ and radius $r>0$, put \[ B_g(p,r)=\{x:d_g(x,p)<r\},\qquad C_{p,r}=X\setminus B_g(p,r). \] The problem asks for the possible values of the source's "co-s+area" invariant of such complements. If $\mathcal A_{\mathrm{source}}$ denotes that as-yet-undefined invariant, the intended range question is to determine \[ \{\mathcal A_{\mathrm{source}}(C_{p,r}): (X^n,g)\text{ compact and spin},\ \operatorname{Scal}_g\ge n(n-1),\ p\in X\}, \] with the dimension and radius fixed or with their dependence made explicit.

The excerpt supplies no definition or normalization of co-s+area. It must not be identified with a particular spherical mapping, K-area, or coarea quantity solely from the summary. The dimension range, boundary convention for $X$, open versus closed removed ball, intrinsic versus restricted complement metric, and admissible boundary regularity are not specified. Metric balls can meet the cut locus and need not have smooth boundary; no injectivity-radius restriction is inserted. For large $r$ the complement may be empty, so the invariant's domain and empty-set convention also matter.

The summary calls this an open quantitative question in a spherical/coarea program and reports no dedicated resolution located. It supplies no exact theorem or auxiliary definition; the status is a qualified source attribution without independent current verification.

Short English statement

What values can the intended co-s+area take on complements of radius-$r$ balls in compact spin manifolds with scalar curvature at least $n(n-1)$? The invariant and complement conventions are missing, so no sharp numerical range can be specified from this excerpt.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6700053$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Misha Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?56], occurrence 55, PDF page 58: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf. Supplied normalized PDF-text extraction accessed 2026-07-29; described as incomplete.
• No separate co-s+area definition or resolution reference is supplied.