Volume-preserving intrinsic-flat compactness with minimal boundary
Fix positive constants $D, V, A$. Consider sequences $(M_j, g_j)$ of smooth compact connected oriented Riemannian three-manifolds with smooth nonempty boundary such that $$\operatorname{Scal}_{g_j} \ge 0, \quad \operatorname{diam}_{g_j}(M_j) \le D, \quad \operatorname{Vol}_{g_j}(M_j) \le V, \quad \operatorname{Area}_{g_j}(\partial M_j) \le A,$$ every boundary component is a weakly stable minimal surface (a zero eigenvalue of the stability operator is allowed), and the interior of $M_j$ contains no smooth closed embedded minimal surface, orientable or not. Regard $M_j$ as the integral current space $(M_j, d_j, T_j)$, with $d_j$ the length distance using curves in $M_j$ and $T_j$ integration with the chosen orientation. Write $d_{\mathcal{F}}$ for the Sormani–Wenger intrinsic flat distance and $\mathbf{M}$ for Ambrosio–Kirchheim mass. By Wenger's compactness theorem, a subsequence converges for $d_{\mathcal{F}}$ to an integral current space, possibly the zero space, and $\mathbf{M}$ is lower semicontinuous under this convergence.
Problem. Does every such sequence have a subsequence $(M_{j_k})$ and a three-dimensional integral current space $M_\infty = (X_\infty, d_\infty, T_\infty)$, possibly the zero current space, with $$d_{\mathcal{F}}\bigl((M_{j_k}, d_{j_k}, T_{j_k}), M_\infty\bigr) \to 0 \quad \text{and} \quad \operatorname{Vol}_{g_{j_k}}(M_{j_k}) \to \mathbf{M}(T_\infty)?$$
Source: Gromov; Sormani and participants at the IAS Emerging Topics Workshop (2023), Conjecture 5.1 (NoMin Boundary Scalar Compactness), formulated in 2018. Stated by AIM — Open Applied Problems as its volume-preservation part.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, integral currents, intrinsic flat convergence, scalar curvature
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@incollection{sormani2023,
author = {Sormani, Christina and {Participants at the IAS Emerging Topics Workshop on Scalar Curvature and Convergence}},
title = {Conjectures on convergence and scalar curvature},
booktitle = {Perspectives in Scalar Curvature},
volume = {2},
editor = {Gromov, Mikhail L. and Lawson, H. Blaine},
publisher = {World Scientific},
year = {2023},
pages = {645--722},
doi = {10.1142/9789811273230_0018},
eprint = {2103.10093},
archivePrefix = {arXiv},
}
@article{wenger2011,
author = {Wenger, Stefan},
title = {Compactness for manifolds and integral currents with bounded diameter and volume},
journal = {Calc. Var. Partial Differential Equations},
volume = {40},
number = {3-4},
year = {2011},
pages = {423--448},
doi = {10.1007/s00526-010-0346-y},
}
@article{sormaniwenger2011,
author = {Sormani, Christina and Wenger, Stefan},
title = {The intrinsic flat distance between {R}iemannian manifolds and other integral current spaces},
journal = {J. Differential Geom.},
volume = {87},
number = {1},
year = {2011},
pages = {117--199},
doi = {10.4310/jdg/1303219774},
}
@article{ambrosiokirchheim2000,
author = {Ambrosio, Luigi and Kirchheim, Bernd},
title = {Currents in metric spaces},
journal = {Acta Math.},
volume = {185},
number = {1},
year = {2000},
pages = {1--80},
doi = {10.1007/BF02392711},
}
@misc{aim-volume-preserving-intrinsic-flat-compactness-with-minimal-boundary,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Volume-preserving intrinsic-flat compactness with minimal boundary}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/876-volume-preserving-intrinsic-flat-compactness-with-minimal-boundary.md},
note = {Entry 876 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01130, "Volume-preserving intrinsic-flat compactness with minimal boundary", clopen.solutions.