Intrinsic-flat stability of scalar sphere rigidity
For a smooth Riemannian metric $g$ on the standard smooth three-sphere $\mathbb{S}^3$ let $$\operatorname{MinA}(\mathbb{S}^3, g) = \inf\{\operatorname{Area}_g(\Sigma) : \Sigma \subset \mathbb{S}^3 \text{ a nonempty smooth closed embedded minimal surface}\}.$$ Let $d_g$ be the intrinsic Riemannian distance of $g$, $T_g$ its integration current for the standard orientation, and $d_{\mathcal{F}}$ the Sormani-Wenger intrinsic flat distance; write $g_{\mathrm{rd}}$ for the unit round metric.
Marques and Neves proved that a metric on $\mathbb{S}^3$ with positive Ricci curvature and scalar curvature at least $6$ has width at most $4\pi$, with equality only for the unit round metric. Bamler and Maximo proved that positively curved metrics on $\mathbb{S}^3$ with almost maximal width are nearly round. Stufflebeam and Sweeney proved the statement below for rotationally symmetric metrics, without a volume bound. Sweeney constructed metrics on $\mathbb{S}^3$ with the scalar curvature bound below, diameter and volume bounded by some constants, and width at least $4\pi$ whose volume-preserving intrinsic flat limit is a disjoint union of two three-spheres; on these $\operatorname{MinA} \to 0$.
Problem. Fix constants $D, V > 0$, and let $g_j$, $j = 1, 2, \dots$, be any sequence of smooth Riemannian metrics on $\mathbb{S}^3$ with $$\operatorname{Scal}_{g_j} \ge 6 - \frac{1}{j}, \qquad \operatorname{MinA}(\mathbb{S}^3, g_j) \ge 4\pi - \frac{1}{j}, \qquad \operatorname{diam}_{g_j}(\mathbb{S}^3) \le D, \qquad \operatorname{Vol}_{g_j}(\mathbb{S}^3) \le V.$$ Must the whole sequence converge to the unit round sphere in the volume-preserving intrinsic flat sense, that is, $$d_{\mathcal{F}}\bigl((\mathbb{S}^3, d_{g_j}, T_{g_j}), (\mathbb{S}^3, d_{g_{\mathrm{rd}}}, T_{g_{\mathrm{rd}}})\bigr) \to 0 \qquad \text{and} \qquad \operatorname{Vol}_{g_j}(\mathbb{S}^3) \to 2\pi^2?$$
Source: Marques and Neves (proposed 2018), recorded as Conjecture 9.1, equations (124)-(125), with Remark 9.3, in Sormani (arXiv:2103.10093v1; 2023).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- MG, MP
- Keywords
- AIM Open Applied Problems, intrinsic flat convergence, minimal surfaces, scalar curvature
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@incollection{sormani2023,
author = {Sormani, Christina},
title = {Conjectures on convergence and scalar curvature},
booktitle = {Perspectives in Scalar Curvature},
editor = {Gromov, Misha and Lawson, Jr., H. Blaine},
volume = {2},
publisher = {World Scientific},
year = {2023},
pages = {645--722},
doi = {10.1142/9789811273230_0018},
eprint = {2103.10093},
archivePrefix = {arXiv},
}
@article{marques2012,
author = {Marques, Fernando C. and Neves, Andr{\'e}},
title = {Rigidity of min-max minimal spheres in three-manifolds},
journal = {Duke Math. J.},
volume = {161},
number = {14},
year = {2012},
pages = {2725--2752},
doi = {10.1215/00127094-1813410},
}
@article{bamler2017,
author = {Bamler, Richard H. and Maximo, Davi},
title = {Almost-rigidity and the extinction time of positively curved {R}icci flows},
journal = {Math. Ann.},
volume = {369},
number = {1-2},
year = {2017},
pages = {899--911},
doi = {10.1007/s00208-016-1494-y},
eprint = {1506.03421},
archivePrefix = {arXiv},
}
@article{stufflebeam2025,
author = {Stufflebeam, Hunter and Sweeney, Jr., Paul},
title = {Width stability of rotationally symmetric metrics},
journal = {J. Geom. Anal.},
volume = {35},
number = {8},
year = {2025},
pages = {238},
doi = {10.1007/s12220-025-02020-5},
eprint = {2409.13646},
archivePrefix = {arXiv},
}
@article{sweeney2025,
author = {Sweeney, Jr., Paul},
title = {Examples for scalar sphere stability},
journal = {Calc. Var. Partial Differential Equations},
volume = {64},
number = {6},
year = {2025},
pages = {188},
doi = {10.1007/s00526-025-03036-z},
eprint = {2301.01292},
archivePrefix = {arXiv},
}
@misc{aim-intrinsic-flat-stability-of-scalar-sphere-rigidity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Intrinsic-flat stability of scalar sphere rigidity}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/882-intrinsic-flat-stability-of-scalar-sphere-rigidity.md},
note = {Entry 882 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-01118, "Intrinsic-flat stability of scalar sphere rigidity", clopen.solutions.