Persistence of Nonnegative Scalar Curvature under Intrinsic-Flat Limits

Definitions and mathematical statement

For each positive integer $j$, let $(M_j^3,g_j)$ be a smooth three-dimensional Riemannian manifold satisfying \[ \operatorname{Scal}_{g_j}(x)\geq 0\qquad\text{for every }x\in M_j. \] Let $(M_0^3,g_0)$ be a smooth three-dimensional Riemannian manifold. Here $\operatorname{Scal}_g$ denotes the scalar curvature of the metric $g$. Smoothness of the limit manifold does not mean that the metrics converge smoothly.

Sormani--Wenger intrinsic-flat convergence, abbreviated SWIF, is convergence of the associated integral current spaces in intrinsic-flat distance. To describe this distance, write $X_i=(M_i,d_{g_i},T_i)$, where $T_i$ is the integration current with the orientation and current-space structure required for this notion of convergence. Then \[ d_{\mathcal F}(X_j,X_0) =\inf\left\{\mathbf M_Z(U)+\mathbf M_Z(V): \varphi_{j\#}T_j-\varphi_{0\#}T_0=U+\partial V\right\}. \] The infimum is over common complete metric spaces $Z$, distance-preserving embeddings $\varphi_j,\varphi_0$ into $Z$, and integral currents $U$ and $V$ of dimensions three and four, respectively; $\mathbf M_Z$ denotes current mass. SWIF convergence means $d_{\mathcal F}(X_j,X_0)\to0$. This formulation presupposes appropriate integral current structures; the supplied excerpt does not specify the orientation, finiteness, completeness, compactness, or boundary assumptions needed for its intended setting.

Question 1 (SWIF formulation). Whenever the preceding current-space formulation applies, does \[ X_j\xrightarrow{\mathrm{SWIF}}X_0 \quad\text{and}\quad \operatorname{Scal}_{g_j}\geq0\ \text{for every }j \quad\Longrightarrow\quad \operatorname{Scal}_{g_0}(x)\geq0\ \text{for every }x\in M_0 \] hold? The original first question also asks about convergence "in some sense." That broader question has no specified convergence topology in the supplied excerpt; SWIF is its explicitly named case.

Question 2 (additional minimal-area hypothesis). Does the same conclusion hold if, in addition, the sequence satisfies the source's $\mathrm{MinA}$ hypothesis? The supplied reconstructed statement interprets this as a uniform positive lower bound on minimal-surface area. For a specified class $\mathcal S_j$ of minimal surfaces in $(M_j,g_j)$, this condition can be expressed as \[ \exists a>0\quad\forall j\geq1\quad\forall\Sigma\in\mathcal S_j, \qquad \operatorname{Area}_{g_j}(\Sigma)\geq a. \] A minimal surface has vanishing mean curvature. However, the excerpt does not define $\mathrm{MinA}$ or specify $\mathcal S_j$: in particular, it does not determine whether surfaces must be closed, embedded, immersed, two-sided, or subject to boundary restrictions. It also does not specify the convention when no qualifying minimal surface exists. Thus this displayed condition records the supplied interpretation with its quantifiers, while the exact additional hypothesis remains unresolved.

Short English statement

Must a smooth three-dimensional Riemannian manifold arising as a Sormani--Wenger intrinsic-flat limit of manifolds with everywhere nonnegative scalar curvature also have everywhere nonnegative scalar curvature? The source also asks this under a uniform positive lower bound on minimal-surface area. The intended current-space assumptions and the precise $\mathrm{MinA}$ hypothesis require clarification. The supplied record labels these questions open; this draft does not independently verify that status.

Sources

• Local registry: supplied UnsolvedMath source record, permanent ID $30005859$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The project identifies data/unsolvedmath/problems.json as its problem registry and catalog/problems.json as an additional registry. This draft uses only the supplied complete record.
• Original source: Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (2024), Oberwolfach Reports, pp. 483--566, DOI 10.4171/owr/2024/9; the supplied excerpt locates Open Questions 1 and 2 on p. 547. https://doi.org/10.4171/owr/2024/9. No individual proposer is identified in the supplied record.
• Supplied literature source, labeled "Sormani--Tian--Wang scalar-curvature limit report": https://arxiv.org/abs/2404.17121.
• Supplied literature source, labeled "Allen scalar-curvature stability report": https://arxiv.org/abs/2404.02662.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports stability and compactness results for particular convergence topologies with additional hypotheses, while describing preservation of nonnegative scalar curvature for smooth limits, especially under SWIF convergence, as unresolved in the stated generality. This is supplied dated curation, not an independently verified literature assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The supplied record assigns the statement status corrected_verified and says it was reconstructed and checked against the official Oberwolfach report. That verification claim is source metadata and has not been independently checked for this draft.