Bartnik's cosmological splitting conjecture
Let $n \ge 2$ and let $(M^{n+1}, g)$ be a connected, smooth, time-oriented Lorentzian manifold without boundary, of signature $(-, +, \dots, +)$.
The Lorentzian splitting theorem of Eschenburg, Galloway and Newman gives an isometric product $\mathbb{R} \times S$, with $S$ complete, for globally hyperbolic, timelike geodesically complete spacetimes with $\operatorname{Ric}(X, X) \ge 0$ for timelike $X$ that contain a timelike line. Bartnik showed that a constant mean curvature Cauchy surface implies the splitting. Ehrlich and Galloway proved the conclusion under the stronger hypothesis of nonpositive sectional curvature on timelike planes. Galloway and Vega proved it for Ricci-flat $g$ admitting a timelike conformal Killing field.
Problem. Suppose that $M$ is globally hyperbolic with a compact smooth spacelike Cauchy hypersurface, that $\operatorname{Ric}_g(X, X) \ge 0$ for every timelike tangent vector $X$, and that $g$ is timelike geodesically complete, that is, every inextendible timelike geodesic has its proper-time parameter ranging over $\mathbb{R}$. Must there be a compact Riemannian manifold $(S^n, h)$ and an isometry $(M, g) \cong (\mathbb{R} \times S, -dt^2 + h)$?
Source: Bartnik (1988), in the form of Galloway and Vega (2018), Conjecture 1.2. Restricted to spatial dimension at least two by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, cosmological spacetimes, Lorentzian splitting theorems, timelike geodesic completeness
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{bartnik1988,
author = {Bartnik, Robert},
title = {Remarks on cosmological spacetimes and constant mean curvature surfaces},
journal = {Comm. Math. Phys.},
volume = {117},
number = {4},
year = {1988},
pages = {615--624},
doi = {10.1007/BF01218388},
}
@article{galloway2018,
author = {Galloway, Gregory J. and Vega, Carlos},
title = {Rigidity in vacuum under conformal symmetry},
journal = {Lett. Math. Phys.},
volume = {108},
number = {10},
year = {2018},
pages = {2285--2292},
doi = {10.1007/s11005-018-1079-7},
eprint = {1712.00785},
archivePrefix = {arXiv},
}
@article{eschenburg1988,
author = {Eschenburg, J.-H.},
title = {The splitting theorem for space-times with strong energy condition},
journal = {J. Differential Geom.},
volume = {27},
number = {3},
year = {1988},
pages = {477--491},
doi = {10.4310/jdg/1214442005},
}
@article{galloway1989,
author = {Galloway, Gregory J.},
title = {The {L}orentzian splitting theorem without the completeness assumption},
journal = {J. Differential Geom.},
volume = {29},
number = {2},
year = {1989},
pages = {373--387},
doi = {10.4310/jdg/1214442881},
}
@article{newman1990,
author = {Newman, Richard P. A. C.},
title = {A proof of the splitting conjecture of {S}.-{T}. {Y}au},
journal = {J. Differential Geom.},
volume = {31},
number = {1},
year = {1990},
pages = {163--184},
doi = {10.4310/jdg/1214444093},
}
@article{ehrlich1990,
author = {Ehrlich, Paul E. and Galloway, Gregory J.},
title = {Timelike lines},
journal = {Classical Quantum Gravity},
volume = {7},
number = {3},
year = {1990},
pages = {297--307},
doi = {10.1088/0264-9381/7/3/006},
}
@misc{aim-bartnik-s-cosmological-splitting-conjecture,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Bartnik's cosmological splitting conjecture}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/883-bartnik-s-cosmological-splitting-conjecture.md},
note = {Entry 883 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-01111, "Bartnik's cosmological splitting conjecture", clopen.solutions.