A CMC existence result for expanding cosmological spacetimes

We establish a new CMC (constant mean curvature) existence result for cosmological spacetimes, i.e., globally hyperbolic spacetimes with compact Cauchy surfaces satisfying the strong energy condition. If the spacetime contains an expanding Cauchy surface and is future timelike geodesically complete, then the spacetime contains a CMC Cauchy surface. This result settles, under certain circumstances, a conjecture of the authors and a conjecture of Dilts and Holst. Our proof relies on the construction of barriers in the support sense, and the CMC Cauchy surface is found as the asymptotic limit of mean curvature flow. Analogous results are also obtained in the case of a positive cosmological constant $Λ> 0$. Lastly, we include some comments concerning the future causal boundary for cosmological spacetimes which pertain to the CMC conjecture of the authors.

💡 Research Summary

The paper addresses the long‑standing problem of whether a globally hyperbolic spacetime with compact Cauchy surfaces (a “cosmological spacetime”) necessarily contains a constant‑mean‑curvature (CMC) Cauchy hypersurface. Earlier results required strong curvature assumptions such as non‑positive timelike sectional curvature. The authors prove a much weaker criterion: if the spacetime satisfies the strong energy condition (Ric ≥ 0 on timelike vectors), is future timelike geodesically complete, and contains at least one spacelike Cauchy surface whose mean curvature is non‑negative (i.e., an expanding slice), then a CMC Cauchy surface must exist.

The proof proceeds by constructing “support‑sense” barriers. For any compact Cauchy surface S, the Lorentzian distance function d(S,·) defines level sets S_τ = {q | d(S,q)=τ}. Under the strong energy condition and future completeness, each S_τ is itself a compact Cauchy surface whose mean curvature satisfies H ≤ n/τ in the support sense. This uses the Raychaudhuri equation along maximizing timelike geodesics and a comparison argument.

Next, the authors invoke the mean‑curvature flow (MCF) of spacelike hypersurfaces as developed by Ecker and Huisken. Given two barriers S₁ (with H ≥ a) and S₂ (with H ≤ b) and a constant c with b < c < a, they run the MCF with prescribed constant speed c. The support‑sense barrier condition guarantees that the flow never crosses S₁ or S₂, so it remains in a compact region and exists for all flow time. By standard parabolic regularity, the flow converges (along a subsequence) to a smooth spacelike Cauchy surface with constant mean curvature H = c.

Applying this scheme, the authors start from the given expanding slice V (H ≥ 0). If H is identically zero, V itself is CMC. Otherwise they perturb V slightly to obtain a strictly positive mean curvature. Choosing τ large enough, the level surface V_τ provides a lower‑curvature barrier (H ≤ b). Selecting a constant c between the two barrier curvatures, the MCF yields a CMC hypersurface with H = c. The future completeness hypothesis can be weakened: it suffices that the “future existence time” of V exceeds n c, which is automatically satisfied in many cosmological models (e.g., big‑bang spacetimes with a crushing singularity).

The authors also treat the case of a positive cosmological constant Λ > 0. By replacing the strong energy condition with Ric − Λg ≥ 0, the same Raychaudhuri comparison works, and the entire argument carries over unchanged, establishing CMC existence for Λ‑positive expanding cosmologies.

Finally, the paper discusses the relationship between CMC existence and the future causal boundary. Roughly, if the future causal boundary is non‑trivial (i.e., not a single point), the construction of barriers is possible and a CMC slice exists. Conversely, spacetimes whose future boundary collapses to a point (as in certain singular big‑bang models) can fail to admit CMC slices, illustrating the sharpness of the hypotheses.

In summary, the authors provide a robust existence theorem for CMC Cauchy surfaces in expanding cosmological spacetimes under minimal geometric assumptions, using a novel combination of support‑sense barrier techniques and mean‑curvature flow. The result advances the understanding of the CMC gauge in general relativity and offers new tools for analyzing the global geometry of realistic cosmological models, both with and without a positive cosmological constant.