Zero vacuum Bartnik mass without a no-horizon condition
Let $(\Omega, g_\Omega)$ be a smooth compact connected orientable Riemannian three-ball with nonnegative scalar curvature and strictly positive outward boundary mean curvature $H$ (the trace of the second fundamental form), and write $\gamma = g_\Omega|_{\partial\Omega}$. Let $\mathcal{C}(\gamma, H)$ consist of the smooth connected orientable Riemannian three-manifolds $(M, g)$, complete up to their boundary, with $\partial M$ identified with $\partial\Omega$, $$R_g \equiv 0, \qquad g|_{\partial M} = \gamma, \qquad H_{\partial M} = H,$$ for the boundary normal pointing toward the end, and with exactly one asymptotically flat end: outside a compact set there are coordinates in which, for some $\tau > 1/2$, $$\partial^\alpha (g_{ij} - \delta_{ij}) = O(r^{-\tau - |\alpha|}), \qquad |\alpha| \le 2.$$
Define $$m_B^0(\gamma, H) = \inf_{(M, g) \in \mathcal{C}(\gamma, H)} m_{\mathrm{ADM}}(M, g), \qquad \inf \varnothing = +\infty.$$ The positive mass theorem with corners gives $m_B^0 \ge 0$, and gives the implication below when a zero-mass extension attains the infimum (Anderson). For Bartnik's mass, whose extensions have nonnegative scalar curvature and no closed minimal surfaces in the interior, Huisken and Ilmanen proved positivity unless $\Omega$ is locally flat, and Dong and Song proved positivity unless the interior of $\Omega$ embeds isometrically in $\mathbb{R}^3$. Following a suggestion of Jauregui, Anderson describes how a metric with data $(\gamma, H)$, $R_g \ge 0$ and equal to a Schwarzschild metric of small mass outside a compact set can be deformed and conformally changed to a scalar-flat metric with boundary metric $\gamma$, smaller mass, and mean curvature at most $H$.
Problem. Does $$m_B^0(\gamma, H) = 0 \quad \Longrightarrow \quad \operatorname{Riem}(g_\Omega) \equiv 0$$ always hold?
Source: Anderson (2019), Conjecture 2.2, with Definition 2.1. Stated by AIM — Open Applied Problems for strictly mean-convex three-balls.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, Bartnik mass, positive mass theorems, 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{huiskenilmanen2001,
author = {Huisken, Gerhard and Ilmanen, Tom},
title = {The inverse mean curvature flow and the {R}iemannian {P}enrose inequality},
journal = {J. Differential Geom.},
volume = {59},
number = {3},
year = {2001},
pages = {353--437},
doi = {10.4310/jdg/1090349447},
}
@article{dongsong2025,
author = {Dong, Conghan and Song, Antoine},
title = {Stability of {E}uclidean 3-space for the positive mass theorem},
journal = {Invent. Math.},
volume = {239},
number = {1},
year = {2025},
pages = {287--319},
doi = {10.1007/s00222-024-01302-z},
eprint = {2302.07414},
archivePrefix = {arXiv},
}
@misc{aim-zero-vacuum-bartnik-mass-without-a-no-horizon-condition,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Zero vacuum Bartnik mass without a no-horizon condition}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/878-zero-vacuum-bartnik-mass-without-a-no-horizon-condition.md},
note = {Entry 878 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-01131, "Zero vacuum Bartnik mass without a no-horizon condition", clopen.solutions.