Volume-Above Distance-Below Without Fixed Topology

Definitions and mathematical statement

Let $(M_j,g_j)$, $j\in{\mathbb{N}}$, and $(M_\infty,g_\infty)$ denote the compact oriented Riemannian manifolds in the supplied formulation. A common dimension $n$ is implicit, but the admissible dimensions and boundary conditions are not specified. Write $d_j$ and $d_\infty$ for the induced distances, and $T_j$ and $T_\infty$ for integration with the given orientations, whenever these data define integral current spaces. Denote these spaces by \[ \mathcal M_j=(M_j,d_j,T_j),\qquad \mathcal M_\infty=(M_\infty,d_\infty,T_\infty). \] Here $d_{\mathcal F}$ denotes intrinsic flat distance. Interpret the requested volume-preserving intrinsic flat convergence as \[ d_{\mathcal F}(\mathcal M_j,\mathcal M_\infty)\longrightarrow0, \qquad \operatorname{Vol}_{g_j}(M_j)\longrightarrow \operatorname{Vol}_{g_\infty}(M_\infty). \] The explicit numerical hypotheses in the supplied reconstruction are that there exists $D<\infty$ such that \[ \operatorname{diam}(M_j,d_j)\le D\quad\text{for every }j\in{\mathbb{N}}, \qquad \operatorname{Vol}_{g_j}(M_j)\longrightarrow \operatorname{Vol}_{g_\infty}(M_\infty). \] It also requires metric tensors to be bounded below by factors tending to one relative to the limit metric. On a common smooth manifold, a tensor comparison of this kind can be expressed as \[ g_j(v,v)\ge a_j g_\infty(v,v) \quad\text{for every tangent vector }v, \qquad a_j>0,\quad a_j\longrightarrow1. \] This expression has no defined meaning for tangent vectors on different manifolds without additional comparison data. The supplied record specifies neither such data nor their required properties.

Intended question, with an unresolved hypothesis. Establish an appropriate extension of the report's Theorem 8 that yields the displayed volume-preserving intrinsic flat convergence under the diameter, volume, and suitably formulated lower metric-comparison hypotheses, without requiring $M_j$ to be diffeomorphic to $M_\infty$. The complete statement of Theorem 8 is absent. Consequently, the supplied evidence does not determine a complete quantified theorem: the comparison hypothesis, applicable dimensions, boundary assumptions, and orientation compatibility must be recovered or specified before this target can be made precise.

The original excerpt also contains neighboring material whose relationship to this target is not specified. It attributes to Hirsch and Zhang two other versions of Llarull stability: one involving an $L^p$ assumption on the negative part of scalar curvature below $n(n-1)$, and another using a good-set/bad-set decomposition similar to Theorem 4. No exponent range or full hypotheses are supplied. Its Question 4 asks whether spinors can establish Geroch stability and positive mass theorem stability in dimensions $n\ge3$. Its Question 5 asks whether Theorem 7 generalizes to the results of Goette and Semmelmann, cited there as reference [23]. These neighboring questions are retained as source qualifications; their missing theorem statements and reference details do not define additional precise claims here.

Short English statement

The intended problem asks for an extension of a volume-above, distance-below convergence theorem to manifolds whose topology may differ from that of the limit. Uniform diameter control, volume convergence, and an appropriate lower metric comparison should imply volume-preserving intrinsic flat convergence. The supplied record does not define the comparison between different manifolds or provide Theorem 8, so essential hypotheses remain unresolved.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30005852$, titled Volume-Above Distance-Below Without Fixed Topology; exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection, gives proposed year 2024, and supplies no proposer attribution.
• Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (2024), Oberwolfach Reports, pp. 483--566, DOI 10.4171/owr/2024/9: https://doi.org/10.4171/owr/2024/9. This is the authoritative report citation supplied by the record; its contents have not been independently consulted for this draft.
• Supplied literature source: Scalar curvature under weak limits of manifolds, https://arxiv.org/abs/2605.03136.
• Supplied literature source: Analysis, Geometry and Topology of Positive Scalar Curvature Metrics, https://ems.press/content/serial-article-files/48658?nt=1.
• The literature assessment dated exactly 2026-08-22 is supplied dated curation, not independently verified. It labels the problem partially solved and reports that a 2026 weak-limit theorem treats non-diffeomorphic three-manifolds under almost-Lipschitz maps and volume convergence, answering a version of the question but not the full all-dimensional VADB theorem requested. The record expressly calls for independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its verification note says that the reconstruction was checked against the official Oberwolfach report. This is a claim in the supplied registry, not independent verification performed for this draft. The excerpt names Hirsch and Zhang and Goette and Semmelmann, but supplies no complete bibliographic entries for those attributions or for reference [23].