Geometric Noncollapse and Hybrid Convergence

Definitions and mathematical statement

The supplied record combines Open Questions 5 and 6 from the cited workshop report. They are requests for geometric hypotheses and a convergence framework; the supplied excerpt does not specify enough data to give a single fully quantified conjecture.

For a smooth surface $\Sigma$ in a setting admitting a chosen unit normal, the constant-mean-curvature condition is \[ H_\Sigma\equiv c \] for some constant $c\in{\mathbb{R}}$. Given a Riemannian manifold $(M,g)$, an admissible class of regions, and a perimeter convention $P_g$, an isoperimetric region of prescribed volume $v$ is an admissible region $\Omega$ satisfying \[ \operatorname{Vol}_g(\Omega)=v, \qquad P_g(\Omega)=\inf\{P_g(E): E\text{ admissible},\ \operatorname{Vol}_g(E)=v\}. \] The admissible volumes, boundary convention, and existence or regularity assumptions are not specified in the record.

Open Question 5: geometric noncollapse. Formulate noncollapsing hypotheses for sequences of Riemannian manifolds using constant-mean-curvature surfaces or isoperimetric regions. The source does not specify a quantitative meaning of noncollapse, the geometric quantities to control, or the conclusion those hypotheses should imply. Consequently, no particular volume lower bound or other noncollapse criterion is asserted here.

Open Question 6: convergence retaining Sobolev information. For a sequence of varying Riemannian manifolds $(M_j,g_j)$, construct a notion of convergence that implies volume-preserving intrinsic-flat convergence and also captures information encoded in a $W^{1,p}$ limit. The supplied reconstructed statement describes retaining this information on "good regions." Determine additionally whether a distance between Riemannian manifolds can capture the proposed convergence.

To make the intrinsic-flat requirement explicit, suppose the manifolds and a candidate limit $(M_\infty,g_\infty)$ determine integral current spaces $X_j$ and $X_\infty$, respectively, in a setting where intrinsic-flat distance $d_{\mathcal F}$ and finite total Riemannian volume are defined. In this setting, the intended volume-preserving requirement is \[ d_{\mathcal F}(X_j,X_\infty)\longrightarrow 0, \qquad \operatorname{Vol}_{g_j}(M_j)\longrightarrow \operatorname{Vol}_{g_\infty}(M_\infty). \] Here intrinsic-flat distance is defined by comparing the pushed-forward integral currents in common complete metric spaces using flat distance. This notation describes the requirement when the limit is represented by a Riemannian manifold; the source does not specify whether more general limit spaces are intended.

The notation $W^{1,p}$ refers to Sobolev regularity: the relevant functions or tensor coefficients and their first weak derivatives have the prescribed $L^p$ integrability in chosen coordinates or identifications. On varying manifolds, a precise convergence assertion additionally requires identification maps, specified regions, specified objects to compare, an exponent $p$, and a choice of strong or weak convergence. None of these is supplied. In particular, "good regions" has no supplied definition. Likewise, the class of manifolds and the equivalence relation on which the requested distance should be a metric remain unspecified.

The report title supplies a positive-scalar-curvature context, but the excerpt gives no precise scalar-curvature hypothesis, dimension, compactness assumption, boundary condition, or other uniform geometric bounds. Such hypotheses must be recovered from the surrounding source before either question can be made fully precise.

Short English statement

The record asks for noncollapse conditions expressed through constant-mean-curvature surfaces or isoperimetric regions, and for convergence of varying Riemannian manifolds that preserves volume under intrinsic-flat convergence while retaining $W^{1,p}$ information. It also asks whether a distance can describe that convergence. The geometric setting, Sobolev comparisons, and meaning of "good regions" remain unspecified, so this is a qualified research-program statement requiring human mathematical review.

Sources

• Local UnsolvedMath registry: permanent ID $30005861$, title Geometric Noncollapse and Hybrid Convergence, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied record combines Open Questions 5 and 6 and lists the proposed year as 2024; no individual proposer is identified.
• Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (2024), Oberwolfach Reports, pp. 483--566, DOI 10.4171/owr/2024/9. The supplied excerpt identifies p. 549. Original report: https://doi.org/10.4171/owr/2024/9.
• 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 checked at 2026-08-22: the supplied dated curation labels the problem open and reports no established general construction or equivalence theorem simultaneously encoding volume-preserving SWIF convergence and $W^{1,p}$ information in the literature checked. This is supplied dated curation, not an independently verified assessment; the record requests independent verification of time-sensitive status and cited claims.
• The registry labels its statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. These are supplied provenance assertions; no source visit or independent mathematical verification is claimed here.