Geroch Stability Without Bubbles and Drawstrings

Definitions and mathematical statement

For an oriented compact Riemannian manifold $(M,g)$ of dimension $n$, write $d_g$ for its Riemannian distance, $T_M$ for its current of integration, and $\operatorname{Vol}_g(M)$ for its volume. The Sormani--Wenger intrinsic-flat distance between two such integral current spaces is \[ d_{\mathcal F}(M_1,M_2) =\inf\left\{\mathbf M(A)+\mathbf M(B): \varphi_{1\#}T_{M_1}-\varphi_{2\#}T_{M_2} =A+\partial B\right\}, \] where the infimum ranges over complete metric spaces $Z$, distance-preserving embeddings $\varphi_i:(M_i,d_{g_i})\to Z$, and integral currents $A$ and $B$ in $Z$ of dimensions $n$ and $n+1$, respectively; $\mathbf M$ denotes current mass. A flat $n$-torus is ${\mathbb{R}}^n/\Lambda$ with its induced Euclidean metric, where $\Lambda\subset{\mathbb{R}}^n$ is a full-rank lattice.

Requested formulation (incomplete in the supplied source). Formulate and prove a Geroch stability theorem with hypotheses that exclude bubbling and drawstring degeneration, and with a volume-preserving intrinsic-flat convergence conclusion. In notation, the desired conclusion for the sequence covered by such a theorem is convergence to some flat torus $(T^n,g_\infty)$ with \[ d_{\mathcal F}\left((M_j,d_{g_j},T_{M_j}), (T^n,d_{g_\infty},T_{T^n})\right)\longrightarrow0, \qquad \operatorname{Vol}_{g_j}(M_j) \longrightarrow\operatorname{Vol}_{g_\infty}(T^n). \] These equations specify the requested type of conclusion, not a theorem asserted for an identified class of sequences. The dimension $n$, topology and boundary conditions of the approximating manifolds, orientations, scalar-curvature assumptions, normalization and other geometric bounds, precise exclusion conditions, and whether passage to a subsequence is permitted are not specified in the supplied excerpt.

The original Question 1 says that an entropy bound in its Theorem 2 excludes splines and bubbles along a sequence, then asks for a Geroch stability formulation excluding bubbles and drawstrings. Theorem 2 and its entropy bound are not supplied. The accompanying figure description characterizes splines as becoming arbitrarily thin, a bubble neck as pinching to a point, and a drawstring as pulling a circle almost to a point, with that circle collapsing to a point along the sequence. These descriptions do not supply quantitative definitions of the excluded degenerations. Consequently, a fully quantified stability conjecture cannot be recovered without additional source information or an explicitly proposed formulation.

Short English statement

The problem asks for hypotheses preventing bubbles and drawstrings under which a Geroch stability theorem yields convergence to a flat torus in Sormani--Wenger intrinsic-flat distance, together with convergence of total volumes. The supplied excerpt leaves the admissible sequences and exclusion hypotheses unspecified, so this is a request to formulate a theorem rather than a complete conjecture. The supplied literature status is partially solved; it is dated curation requiring independent review.

Sources

• Local UnsolvedMath registry, permanent ID 30005848, title Geroch Stability Without Bubbles and Drawstrings, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified in the supplied record; the proposed year is 2024.
• Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (2024), Oberwolfach Reports, pp. 483--566, DOI 10.4171/owr/2024/9. The supplied original excerpt identifies Question 1 and an accompanying Figure 1 on p. 491. https://doi.org/10.4171/owr/2024/9.
• Stability of the positive mass theorem and torus rigidity. Supplied literature reference: https://arxiv.org/abs/2210.04340.
• Conjectures on convergence and scalar curvature. Supplied literature reference: https://arxiv.org/abs/2103.10093.
• Literature assessment checked at 2026-08-22: this is supplied dated curation, not independently verified here. It labels the problem partially solved and reports that intrinsic-flat stability results cover important torus and positive-mass regimes under curvature, diameter, or integral bounds, while the proposed bubble-free Geroch stability statement in its full generality remains unproved. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that it was reconstructed and checked against the official Oberwolfach report. These are supplied provenance claims, not independent source verification performed for this draft.