Mean Curvature, Singularities and Time Functions in Cosmology

In this contribution, we study spacetimes of cosmological interest, without making any symmetry assumptions. We prove a rigid Hawking singularity theorem for positive cosmological constant, which sharpens known results. In particular, it implies that any spacetime with $\operatorname{Ric} \geq -ng$ in timelike directions and containing a compact Cauchy hypersurface with mean curvature $H \geq n$ is timelike incomplete. We also study the properties of cosmological time and volume functions, addressing questions such as: When do they satisfy the regularity condition? When are the level sets Cauchy hypersurfaces? What can one say about the mean curvature of the level sets? This naturally leads to consideration of Hawking type singularity theorems for Cauchy surfaces satisfying mean curvature inequalities in a certain weak sense.

💡 Research Summary

This paper investigates cosmologically relevant spacetimes without imposing any symmetry assumptions, focusing on two intertwined themes: a refined Hawking‑type singularity theorem adapted to a positive cosmological constant, and the regularity and geometric properties of canonical time functions (cosmological time and cosmological volume).

The first major result is a rigid singularity theorem that replaces the classical strong energy condition (Ric ≥ 0) with the weaker Ricci lower bound Ric ≥ n g, which is compatible with a positive cosmological constant Λ = n(n‑1)/2. Assuming the existence of a smooth, spacelike, compact Cauchy hypersurface S whose mean curvature satisfies H ≥ n (the trace of the second fundamental form with respect to the future unit normal), the authors prove that the spacetime (M,g) is past‑timelike geodesically incomplete. The theorem splits into two cases:

1. If H > n at any point of S, then every past‑directed timelike geodesic is incomplete. The proof uses a modified mean‑curvature flow to produce a new Cauchy surface with strictly larger mean curvature, after which Hawking’s original theorem (adapted to the Ric ≥ n g setting) applies.

2. If H ≡ n everywhere, then either a single normal geodesic to S is already incomplete, or the past domain J⁻(S) is isometric to the warped product (‑∞,0] × S equipped with the metric –dt² ⊕ e^{2t}h, where h is the induced metric on S. In the latter situation the authors perform an explicit calculation on the 2‑dimensional submanifold generated by a non‑normal timelike geodesic, showing that the affine parameter blows up in finite time, hence the geodesic is incomplete. This warped product is precisely the past portion of de Sitter space, illustrating the rigidity of the result.

The theorem is further extended to non‑compact Cauchy hypersurfaces that admit a past S‑ray (a past‑inextendible causal curve maximizing Lorentzian distance to S). Under the same Ricci bound and the assumption that normal S‑rays are complete, the same dichotomy holds, and the spacetime again splits as a warped product with exponential warping. This generalizes earlier rigidity results of Graf and others.

The second part of the paper studies canonical time functions. The cosmological volume function τ_V is defined as the supremum of the Lorentzian volume of the causal past of a point. The authors prove that τ_V is continuous, finite‑valued, and tends to zero along every past‑inextendible causal curve, thereby satisfying the regularity condition required for a “cosmological time.” For the cosmological time function τ (the maximal proper time to the initial singularity), an additional regularity hypothesis (τ finite and τ → 0 along every past‑inextendible causal curve) is imposed. Under this hypothesis they obtain two key results:

1. If Ric ≥ ‑nκ g for some κ < 0, then each level set S_T = {τ = T} has mean curvature bounded above in the support sense by a function β_{κ,T} that depends explicitly on κ and T. This provides a quantitative control of the geometry of the τ‑slices even when they are only C⁰ hypersurfaces.

2. If the spacetime is future timelike geodesically complete and either possesses a compact Cauchy hypersurface or has a spacelike future causal boundary, then every τ‑level set S_T with T > 0 is itself a Cauchy hypersurface. Thus τ furnishes a global time foliation by Cauchy slices, a property previously known only under stronger assumptions.

Finally, the authors extend Hawking’s singularity theorem to C⁰ Cauchy hypersurfaces satisfying a mean‑curvature inequality in the support sense. By employing the geometric maximum principle for weakly regular spacelike hypersurfaces, they show that if such a hypersurface has support mean curvature ≥ n, then the spacetime must contain a past‑incomplete timelike geodesic. This result removes the smoothness requirement from the classical theorem, broadening its applicability to more realistic cosmological models where the initial singularity may be only weakly regular.

Overall, the paper provides a robust framework that unifies curvature bounds, mean‑curvature conditions, and canonical time functions to guarantee the existence of past singularities in a wide class of spacetimes, including those with a positive cosmological constant and without any symmetry assumptions. The results have significant implications for the mathematical foundations of modern cosmology, especially concerning the inevitability of a big‑bang‑type singularity and the geometric nature of cosmic time.