Scalar-flat apparent-horizon extensions with prescribed Schwarzschild mass
Let $\gamma$ be a smooth Riemannian metric on $\mathbb{S}^2$ with Gauss curvature $K_\gamma$ such that the lowest eigenvalue of $-\Delta_\gamma + K_\gamma$ is positive, and let $A$ be its area. Write $$G_m = \left(1 - \frac{2m}{r}\right)^{-1} dr^2 + r^2 g_{\mathbb{S}^2}, \qquad r > 2m,$$ for the mass-$m$ Schwarzschild metric, $g_{\mathbb{S}^2}$ being the unit round metric.
Mantoulidis and Schoen proved that for every $m > \sqrt{A/(16\pi)}$ there is an asymptotically flat three-manifold of nonnegative scalar curvature whose boundary is minimal and isometric to $(\mathbb{S}^2, \gamma)$, which is isometric to the mass-$m$ Schwarzschild metric outside a compact set, and which is foliated by mean-convex spheres eventually coinciding with the Schwarzschild coordinate spheres; their metric has positive scalar curvature somewhere. Miao and Xie proved that if $K_\gamma > 0$, then for every $\varepsilon > 0$ there is an asymptotically flat scalar-flat extension with minimal boundary isometric to $(\mathbb{S}^2, \gamma)$, its interior foliated by surfaces of positive mean curvature, and mass at most $\sqrt{A/(16\pi)} + \varepsilon$; it is not required to equal Schwarzschild outside a compact set.
Problem. For every $m > \sqrt{A/(16\pi)}$, does there exist a smooth Riemannian metric $G$ on $M = \mathbb{S}^2 \times [0, \infty)$ such that $(M, G)$ is complete as a manifold with boundary and has one asymptotically flat end; $R_G = 0$ everywhere, and $\partial M$ is minimal with induced metric isometric to $\gamma$; outside a compact set $G$ is isometric to an exterior region of $G_m$; and $M$ has a global smooth foliation by spheres, starting at $\partial M$, whose interior leaves have strictly positive mean curvature toward the end and which eventually coincides with the standard Schwarzschild spheres?
Source: Carlotto (2021), Open Problem 3.21, after Mantoulidis and Schoen (2015), who say they do not know whether their construction can be modified to produce vacuum initial data.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, apparent horizons, Bartnik mass, scalar-flat metrics
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{carlotto2021,
author = {Carlotto, Alessandro},
title = {The general relativistic constraint equations},
journal = {Living Rev. Relativ.},
volume = {24},
number = {1},
year = {2021},
pages = {2},
doi = {10.1007/s41114-020-00030-z},
}
@article{mantoulidis2015,
author = {Mantoulidis, Christos and Schoen, Richard},
title = {On the {B}artnik mass of apparent horizons},
journal = {Classical Quantum Gravity},
volume = {32},
number = {20},
year = {2015},
pages = {205002},
doi = {10.1088/0264-9381/32/20/205002},
eprint = {1412.0382},
archivePrefix = {arXiv},
}
@article{miao2019,
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},
}
@misc{aim-scalar-flat-apparent-horizon-extensions-with-prescribed-schwarzschild-mass,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Scalar-flat apparent-horizon extensions with prescribed Schwarzschild mass}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/880-scalar-flat-apparent-horizon-extensions-with-prescribed-schwarzschild-mass.md},
note = {Entry 880 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-01123, "Scalar-flat apparent-horizon extensions with prescribed Schwarzschild mass", clopen.solutions.