Existence of maximal (zero mean curvature) spacelike hypersurfaces in spacetimes under natural energy, completeness, and asymptotic hypotheses

This problem concerns the foundational question flagged in the Introduction of Schoen and Yau's "Proof of the positive mass theorem. II" (1981): the classical route to the positive mass theorem required the spacetime to admit a maximal slice - a spacelike hypersurface whose mean curvature (the trace of the second fundamental form, $\mathrm{tr}\,p=\sum_i p_i^i$) vanishes - the existence of such slices was unknown in general, and a spacetime admitting one was expected to be subject to additional restrictive conditions. The unrestricted existence claim is now false: Bartnik (1988), Chruściel-Isenberg-Pollack (2005) and Ling-Ohanyan (2024-2025) constructed globally hyperbolic spacetimes - satisfying the strong energy condition, some of them vacuum - with no constant-mean-curvature Cauchy surfaces at all, and asymptotically flat spacetimes are subject to positive-scalar-curvature topological obstructions (Brill's example). The surviving open core is precise and actively studied: (A) whether every asymptotically flat spacetime satisfying the dominant energy condition, free of the topological obstruction, contains a maximal Cauchy hypersurface (known only under near-flatness, barrier, or stationarity hypotheses), and (B) the conjecture of Galloway and Ling that every future timelike geodesically complete cosmological spacetime with the strong energy condition contains a CMC Cauchy surface.

Background

The positive mass theorem and maximal slices

In general relativity, an initial data set for the Einstein equations is a Riemannian 3-manifold $(N,g)$ together with a symmetric 2-tensor $p$ (the second fundamental form, encoding how $N$ sits as a spacelike hypersurface of the ambient spacetime) and matter data $(\mu,J)$, subject to the Einstein constraint equations; the dominant energy condition (DEC) is the physical requirement that matter energy-momentum be transported no faster than light. $N$ is asymptotically flat if, outside a compact set, it is a finite union of ends diffeomorphic to the complement of a compact set in $\mathbb{R}^3$, with the metric approaching the Euclidean metric at a prescribed rate. Each end carries a total mass $M$ (the ADM mass, of Arnowitt, Deser and Misner), defined by a flux integral of the metric falloff over large spheres. The positive mass theorem states that each end of an asymptotically flat data set satisfying the DEC has $M\ge 0$, with $M=0$ only for flat (Minkowski) data. Schoen and Yau proved the theorem in two steps: their 1979 paper established it in the important case when the spacetime admits a maximal slice, and the 1981 paper "Proof of the positive mass theorem. II" removed that assumption.

A maximal slice is a spacelike hypersurface whose mean curvature vanishes; the mean curvature is the trace of the second fundamental form along the hypersurface, $\mathrm{tr}\,p=\sum_i p_i^i$ in Schoen-Yau's notation, and measures the expansion rate of the hypersurface along its future unit normal. Maximal slices are the Lorentzian analogues of minimal surfaces (they are critical for hypersurface volume), and the maximal-slice case of the positive mass theorem was the tractable case because $\mathrm{tr}\,p=0$ turns the Hamiltonian constraint into a scalar-curvature inequality amenable to minimal-surface comparison arguments.

The flagged question and why it survived

In the Introduction of the 1981 paper, Schoen and Yau explain why they could not simply reduce the general theorem to the maximal case: the classical strategy (advocated, e.g., in Brill-Deser's 1968 variational approach) was to first obtain a maximal slice in the spacetime and then apply the maximal-case theorem, but "Not only is the existence of such a slice unknown, but also the space-time is expected to be more restrictive if such a slice does exist." Schoen and Yau bypassed the issue with Jang's equation - a nonlinear elliptic equation for a graph over the initial data in the product $N\times\mathbb{R}$, whose solvability is obstructed exactly at apparent horizons (boundaries of trapped regions, where outgoing light rays stop expanding) - and the spacetime positive mass theorem is now known in considerable generality without any maximal-slice assumption (Eichmair-Huang-Lee-Schoen settled it in spacetime dimensions less than eight). The existence question for maximal slices therefore no longer gates the positive mass theorem; it survives as an independent structural problem of mathematical relativity, with its own consumers: constant-mean-curvature (CMC) hypersurfaces - maximal being the CMC case of value zero - decouple the standard conformal method for the Einstein constraint equations, CMC gauge underlies the main global a priori-estimate results for the Einstein evolution, and maximal slicing is a standard gauge choice in numerical relativity. Whether a given spacetime contains such a slice determines whether these methods apply to it at all.

The two settings in which the question lives

A spacetime is globally hyperbolic when it is strongly causal and all causal diamonds are compact; this is the standard well-posedness condition, equivalent to the existence of a Cauchy surface - a hypersurface met exactly once by every inextendible causal curve. A cosmological spacetime is a globally hyperbolic spacetime with compact Cauchy surfaces satisfying the strong energy condition, $\mathrm{Ric}(v,v)\ge 0$ for all timelike $v$ (the timelike convergence condition, satisfied by vacuum solutions). The two settings behave differently:

• In asymptotically flat spacetimes, an asymptotically flat CMC Cauchy surface is necessarily maximal ($H\equiv 0$). Moreover, a maximal Cauchy surface in a spacetime satisfying the DEC has nonnegative scalar curvature (by the Hamiltonian constraint), so its topology is restricted: one-point compactifying its ends must yield a manifold that carries a positive-scalar-curvature metric (Brill's example, and the topological restrictions reviewed by Dilts and Holst, show this obstruction is real).
• In cosmological spacetimes, a maximal Cauchy surface is itself highly restrictive: by results going back to Bartnik (as reviewed by Dilts and Holst), a cosmological spacetime satisfying the strong energy condition that contains a maximal Cauchy surface is static (admitting a timelike Killing field $T$, that is, an infinitesimal isometry, with $\mathrm{Ric}(T,T)=0$), and under full timelike geodesic completeness - meaning every inextendible timelike geodesic has infinite proper time in both directions - Bartnik proved that existence of a CMC Cauchy surface is equivalent to the spacetime splitting isometrically as a metric product $\mathbb{R}\times S$. An expanding cosmological spacetime therefore has no maximal slice, and the natural question there is the existence of a CMC Cauchy surface for some constant.

What is known around the open core

Negative results: unrestricted existence fails. Bartnik (1988) constructed a globally hyperbolic, spatially compact ($T^3\#T^3$, the connected sum of two 3-tori) spacetime satisfying the strong energy condition with no CMC - hence no maximal - Cauchy surfaces (Tolman-Bondi gluing, joining regions of spherically symmetric dust - pressureless matter - to Schwarzschild cylinders and Friedmann-Lemaître-Robertson-Walker (FLRW) cosmologies, the standard homogeneous models), and proved a general criterion: a cosmological spacetime containing a future ray and a past ray (infinite-length maximizing timelike geodesics) but no line (no timelike geodesic that maximizes proper time between any two of its points) has no CMC Cauchy surfaces. Chruściel-Isenberg-Pollack (2005) produced vacuum analogues by initial-data gluing; Ling and Ohanyan (2024-2025) generalized both constructions - their Tolman-Bondi examples attain spatial topologies $Q\#\tilde{Q}$ for compact quotients $Q,\tilde{Q}$ of $\mathbb{R}^3$ or of hyperbolic 3-space, and their vacuum examples attain $M\#M$ for any closed, connected, oriented, irreducible (every embedded 2-sphere bounds a 3-ball) 3-manifold $M$ that is not spherical (not a quotient of the 3-sphere). All known examples are timelike geodesically incomplete.

Positive results: existence is known under additional hypotheses. In the asymptotically flat class: Cantor-Fischer-Marsden-Ō Murchadha-York (1976) proved that maximal slicings exist in the development of vacuum data on $\mathbb{R}^3$ that can be joined to trivial data by a sufficiently bounded curve; Bartnik (1984) proved existence under an interior barrier condition (comparison hypersurfaces trapping the desired one); Bartnik-Chruściel-Ō Murchadha (1990) showed every vacuum evolution of asymptotically flat data can be foliated by slices maximal outside a spatially compact set, and that strictly stationary (admitting an everywhere timelike Killing field) asymptotically flat spacetimes admit maximal foliations; Chruściel-Wald (1994) extended the stationary theory to spacetimes containing black- and white-hole regions. In the cosmological class, besides the classical barrier results for crushing singularities (sequences of slices whose mean curvature diverges uniformly) surveyed by Dilts and Holst: Ecker-Huisken (1991) constructed spacelike slices of prescribed constant mean curvature by mean curvature flow (a parabolic deformation of a hypersurface along its mean-curvature vector); Galloway-Ling (2018) proved CMC existence for future timelike geodesically complete spacetimes (every inextendible timelike geodesic has infinite proper time to the future) with everywhere nonpositive timelike sectional curvature (the sectional curvature of timelike two-planes); and Galloway-Ling (2025) proved that a future timelike geodesically complete cosmological spacetime with the strong energy condition that contains an expanding Cauchy surface (one with mean curvature $H\ge 0$ everywhere) contains a CMC Cauchy surface, and that a maximal Cauchy surface exists under both future and past timelike completeness together with a ray-to-ray condition (a future ray and a past ray from some Cauchy surface whose timelike past and future overlap). What is not known is existence under the natural hypotheses alone - the two open components stated in the problem.

Problem Statement

The problem is to settle the maximal-hypersurface existence question that Schoen and Yau flagged in the Introduction of "Proof of the positive mass theorem. II": under the natural physical hypotheses of the positive-mass setting, must a spacetime admit a maximal (zero mean curvature) spacelike hypersurface? The two components below are the two canonical forms in which this question survives after the unrestricted version was refuted by counterexamples.

(A) Asymptotically flat spacetimes (the original setting of the question). Determine whether every globally hyperbolic spacetime satisfying the dominant energy condition that contains an asymptotically flat Cauchy surface whose spatial topology carries no positive-scalar-curvature obstruction - for example data on $\mathbb{R}^3$, or more generally ends whose one-point compactifications (one point added per end) admit positive-scalar-curvature metrics - must contain a maximal Cauchy hypersurface, or whether some such spacetime contains no maximal Cauchy hypersurface; note that in this class an asymptotically flat Cauchy surface of constant mean curvature is necessarily maximal ($H\equiv 0$), its constant mean curvature tending to zero along the ends.

(B) Cosmological spacetimes (spatially compact; the natural target here is CMC rather than maximal, since a maximal Cauchy surface in a cosmological spacetime forces staticity). Determine whether every globally hyperbolic spacetime with compact Cauchy surfaces satisfying the strong energy condition that is future timelike geodesically complete must contain a constant mean curvature Cauchy surface (the conjecture of Galloway and Ling), or whether some such spacetime contains no CMC Cauchy surface.

References

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2. Richard Schoen and Shing-Tung Yau, On the proof of the positive mass conjecture in general relativity, Communications in Mathematical Physics 65 (1979) 45-76. DOI: 10.1007/BF01940959. https://doi.org/10.1007/BF01940959
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13. James Dilts and Michael Holst, When Do Spacetimes Have Constant Mean Curvature Slices?, expository preprint (2017). arXiv:1710.03209. https://arxiv.org/abs/1710.03209
14. Gregory J. Galloway and Eric Ling, Existence of CMC Cauchy surfaces from a spacetime curvature condition, General Relativity and Gravitation 50 (2018), Article 108. DOI: 10.1007/s10714-018-2428-7. https://doi.org/10.1007/s10714-018-2428-7
15. Greg Galloway, Existence of CMC Cauchy surfaces and spacetime splitting, Pure and Applied Mathematics Quarterly 15 (2019) 667-682. DOI: 10.4310/pamq.2019.v15.n2.a2. https://doi.org/10.4310/pamq.2019.v15.n2.a2
16. Gregory J. Galloway and Eric Ling, Remarks on the existence of CMC Cauchy surfaces, in: Developments in Lorentzian Geometry, Springer Proceedings in Mathematics and Statistics 389, Springer, Cham (2022) 93-104. DOI: 10.1007/978-3-031-05379-5_6, arXiv:2104.02136. https://doi.org/10.1007/978-3-031-05379-5_6
17. Gregory J. Galloway and Eric Ling, A CMC existence result for expanding cosmological spacetimes, Transactions of the American Mathematical Society 378 (2025), no. 11, 8055-8069. DOI: 10.1090/tran/9471, arXiv:2410.16619. https://doi.org/10.1090/tran/9471
18. Eric Ling and Argam Ohanyan, Examples of cosmological spacetimes without CMC Cauchy surfaces, Letters in Mathematical Physics 114 (2024), Article 96. DOI: 10.1007/s11005-024-01843-7, arXiv:2311.13715. https://doi.org/10.1007/s11005-024-01843-7
19. Eric Ling and Argam Ohanyan, Vacuum Cosmological Spacetimes Without CMC Cauchy Surfaces, in: Progress in Lorentzian Geometry, Springer Proceedings in Mathematics and Statistics 512, Springer, Cham (2025) 231-238. DOI: 10.1007/978-3-031-99212-4_13, arXiv:2407.21524. https://doi.org/10.1007/978-3-031-99212-4_13
20. Dieter Brill, On spacetimes without maximal surfaces, in: Proceedings of the Third Marcel Grossmann Meeting on General Relativity (Shanghai, September 8-11, 1982), Hu Ning (ed.), North-Holland Publishing Company, Amsterdam (1983), p. 1025. NASA ADS bibcode 1983mmg..conf.1025B, https://ui.adsabs.harvard.edu/abs/1983mmg..conf.1025B/abstract. No DOI or arXiv ID exists for this proceedings paper.
21. Robert J. McCann and Argam Ohanyan, Positive resolution of Bartnik's cosmological splitting conjecture, preprint arXiv:2606.03873 (June 2026); withdrawn by the authors in June 2026. The withdrawal note states that the starting point of the claimed proof (the paper's Proposition 2.2 and the reference on which it is based) is refuted by a counterexample of Ehrlich and Galloway (1990). https://arxiv.org/abs/2606.03873
22. Gregory J. Galloway, Robert J. McCann and Argam Ohanyan, Failure of local equi-Lipschitzness for families of Lorentz distances to Cauchy surface foliations, preprint arXiv:2606.21997 (June 2026). https://arxiv.org/abs/2606.21997