Scalar-Curvature Rigidity on Wrapped Sphere Complements

Definitions and mathematical statement

Let $S^n=\{x\in{\mathbb{R}}^{n+1}:\lVert x\rVert=1\}$ be the unit sphere with its round metric, and let $\Sigma\subseteq S^n$. Set $X=S^n\setminus\Sigma$, and denote the restricted round metric by $g_0$. The source requires $\Sigma$ to have the "wrapping property," but does not define this property or specify the allowed dimensions $n$. It also does not state hypotheses on $\Sigma$ ensuring that $X$ is a smooth domain on which the asserted Riemannian metrics and scalar curvatures are defined. These are unresolved qualifications of the formulation below.

For Riemannian metrics on such a domain, interpret $g\geq g_0$ as pointwise domination of quadratic forms: \[ g_x(v,v)\geq (g_0)_x(v,v) \qquad\text{for every }x\in X\text{ and }v\in T_xX. \] Write $\operatorname{Sc}_g(x)$ for scalar curvature, with the convention that the unit round sphere has scalar curvature $n(n-1)$.

Rigidity question, subject to the unresolved source qualifications. For every admissible dimension $n$, every subset $\Sigma\subseteq S^n$ with the source's wrapping property, and every possibly incomplete Riemannian metric $g$ on $X=S^n\setminus\Sigma$, do the conditions \[ g_x(v,v)\geq (g_0)_x(v,v) \quad(x\in X,\ v\in T_xX), \qquad \operatorname{Sc}_g(x)\geq n(n-1)=\operatorname{Sc}_{g_0}(x) \quad(x\in X) \] imply \[ g_x=(g_0)_x\qquad\text{for every }x\in X? \] The conclusion is equality of metrics on the given domain. Completeness is not assumed. The source supplies no further regularity, boundary, or asymptotic conditions near $\Sigma$; none are added here. A fully specified quantified statement requires the missing wrapping definition and admissibility hypotheses.

Short English statement

Does every possibly incomplete metric on a sphere complement with the source's wrapping property equal the restricted round metric whenever it dominates that metric and has scalar curvature at least $n(n-1)$? The wrapping property, dimension range, and admissibility conditions on the removed set are unspecified, so this remains a qualified statement-only draft.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30004715$, source problem code catalogue entry, title "Scalar-Curvature Rigidity on Wrapped Sphere Complements," in the Oberwolfach Reports Open Problems collection. The record lists proposed year 2021 and names no individual proposer.
• Supplied report citation: Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/30, pp. 1579--1649. https://doi.org/10.4171/owr/2021/30. The supplied background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports; no problem-specific page or named contributor is supplied.
• Supplied literature reference: Gromov, scalar curvature lectures. https://www.ihes.fr/~gromov/wp-content/uploads/2019/08/scalar-lectures-IHES-2019-4.pdf.
• Supplied literature reference: Li, Su and Wang, scalar curvature rigidity. https://arxiv.org/abs/2306.06906.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem partially solved and reports rigidity in restricted singular-set and codimension settings, including wrapping hypotheses, while describing the stated general rigidity problem on a punctured sphere as unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says that set-difference, scalar-curvature, and comparison notation were normalized without changing the rigidity question. These are supplied registry assertions, not independent mathematical verification of this draft.