Scalar-flat realization of extendible Bartnik boundary data
Let $\gamma$ be a smooth Riemannian metric on $\mathbb{S}^2$ and let $H : \mathbb{S}^2 \to (0, \infty)$ be smooth. An extension of $(\gamma, H)$ is a smooth connected Riemannian three-manifold $(M, g)$, complete as a manifold with boundary, with $\partial M$ identified with $\mathbb{S}^2$, inducing the metric $\gamma$ and the mean curvature $H$ on it (the trace of the second fundamental form for the unit normal pointing toward the end), and with exactly one asymptotically flat end: outside a compact set there are coordinates in which, for some $\tau > 1/2$, $$\partial^\alpha (g_{ij} - \delta_{ij}) = O(r^{-\tau - |\alpha|}), \qquad |\alpha| \le 2.$$ Shi and Tam proved that if $\gamma$ has positive Gauss curvature, then every positive $H$ is realized by an asymptotically flat metric of zero scalar curvature; Miao and Xie built such metrics, on $\mathbb{S}^2 \times [1, \infty)$, with ADM mass at most $\sqrt{|\mathbb{S}^2|_\gamma / 16\pi} + \epsilon$ for any $\epsilon > 0$. McCormick proved two separate results on conformal changes to zero scalar curvature, under the assumption that the Sobolev quotient is positive: one keeps the boundary metric, and the other prescribes a boundary mean curvature that is not too large.
Problem. Suppose $(\gamma, H)$ has an extension $(M, g)$ with scalar curvature $R_g \ge 0$ and $R_g \in L^1(M, dV_g)$. Must $(\gamma, H)$ have an extension $(\widetilde{M}, \widetilde{g})$ with $R_{\widetilde{g}} \equiv 0$?
Source: Anderson (2019), Conjecture 2.4. Stated by AIM — Open Applied Problems for strictly mean-convex data on the sphere, with explicit asymptotics.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, Bartnik data, scalar-flat extensions
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{anderson2019,
author = {Anderson, Michael T.},
title = {Recent progress and problems on the {B}artnik quasi-local mass},
journal = {Pure Appl. Math. Q.},
volume = {15},
number = {3},
year = {2019},
pages = {851--873},
doi = {10.4310/PAMQ.2019.v15.n3.a4},
eprint = {1903.03822},
archivePrefix = {arXiv},
}
@article{shitam2002,
author = {Shi, Yuguang and Tam, Luen-Fai},
title = {Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature},
journal = {J. Differential Geom.},
volume = {62},
number = {1},
year = {2002},
pages = {79--125},
doi = {10.4310/jdg/1090425530},
eprint = {math/0301047},
archivePrefix = {arXiv},
}
@article{miaoxie2019,
author = {Miao, Pengzi and Xie, Naqing},
title = {Bartnik mass via vacuum extensions},
journal = {Internat. J. Math.},
volume = {30},
number = {13},
year = {2019},
pages = {1940006},
doi = {10.1142/S0129167X19400068},
eprint = {1907.02039},
archivePrefix = {arXiv},
}
@article{mccormick2017,
author = {McCormick, Stephen},
title = {The asymptotically flat scalar-flat {Y}amabe problem with boundary},
journal = {J. Geom. Anal.},
volume = {27},
number = {3},
year = {2017},
pages = {2269--2277},
doi = {10.1007/s12220-017-9760-0},
eprint = {1603.05318},
archivePrefix = {arXiv},
}
@misc{aim-scalar-flat-realization-of-extendible-bartnik-boundary-data,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Scalar-flat realization of extendible Bartnik boundary data}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/873-scalar-flat-realization-of-extendible-bartnik-boundary-data.md},
note = {Entry 873 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-01124, "Scalar-flat realization of extendible Bartnik boundary data", clopen.solutions.