Future timelike completeness and a CMC Cauchy hypersurface
Galloway and Ling proved that a spacetime with compact Cauchy surfaces which is future timelike geodesically complete contains a Cauchy surface of constant mean curvature if its timelike sectional curvatures are everywhere nonpositive (2018), and also if it satisfies the strong energy condition and has a smooth spacelike Cauchy surface with mean curvature $H = \operatorname{tr}_h(\nabla \nu) \ge 0$ for the future unit normal $\nu$ (2025). A positive answer to the problem below would imply Bartnik's splitting conjecture (Galloway and Ling): a timelike geodesically complete spacetime with compact Cauchy surfaces satisfying the strong energy condition splits isometrically as a product $(\mathbb{R} \times V, -dt^2 \oplus h)$ with $(V, h)$ compact Riemannian.
Problem. Let $d \ge 4$ and let $(M^d, g)$ be a connected, smooth, time-oriented Lorentzian manifold without boundary which is globally hyperbolic with a compact Cauchy hypersurface, satisfies the strong energy condition ($\operatorname{Ric}_g(X, X) \ge 0$ for every timelike $X$) and is future timelike geodesically complete. Must $M$ contain a compact smooth spacelike Cauchy hypersurface with constant mean curvature?
Source: Galloway and Ling (2018), the conjecture of the introduction; restated as Conjecture 2 of Galloway (2019), p. 675, in spacetime dimension at least 4, as AIM — Open Applied Problems states it.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, constant mean curvature hypersurfaces, cosmological spacetimes, strong energy condition
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{gallowayling2018,
author = {Galloway, Gregory J. and Ling, Eric},
title = {Existence of {CMC} {C}auchy surfaces from a spacetime curvature condition},
journal = {Gen. Relativity Gravitation},
volume = {50},
number = {9},
year = {2018},
pages = {108},
doi = {10.1007/s10714-018-2428-7},
eprint = {1804.05949},
archivePrefix = {arXiv},
}
@article{galloway2019,
author = {Galloway, Gregory J.},
title = {Existence of {CMC} {C}auchy surfaces and spacetime splitting},
journal = {Pure Appl. Math. Q.},
volume = {15},
number = {2},
year = {2019},
pages = {667--682},
doi = {10.4310/pamq.2019.v15.n2.a2},
eprint = {1902.08803},
archivePrefix = {arXiv},
}
@article{gallowayling2025,
author = {Galloway, Gregory J. and Ling, Eric},
title = {A {CMC} existence result for expanding cosmological spacetimes},
journal = {Trans. Amer. Math. Soc.},
volume = {378},
number = {11},
year = {2025},
pages = {8055--8069},
doi = {10.1090/tran/9471},
eprint = {2410.16619},
archivePrefix = {arXiv},
}
@misc{aim-future-timelike-completeness-and-a-cmc-cauchy-hypersurface,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Future timelike completeness and a CMC Cauchy hypersurface}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/877-future-timelike-completeness-and-a-cmc-cauchy-hypersurface.md},
note = {Entry 877 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-01116, "Future timelike completeness and a CMC Cauchy hypersurface", clopen.solutions.