Mean Curvature, Singularities and Time Functions in Cosmology

TL;DR

The article establishes a rigid Hawking-type singularity theorem in spacetimes with positive cosmological constant by assuming a timelike Ricci bound $\operatorname{Ric} \ge n g$ and a compact Cauchy hypersurface with $H \ge n$, proving past timelike incompleteness and identifying a warped-product obstruction when $H = n$. It also analyzes cosmological time and volume functions as canonical time measures, proving finiteness under these curvature assumptions and exploring regularity and level-set Cauchy properties. The work connects these time functions to Hawking’s theorem, showing that regular cosmological time level sets carry uniform mean-curvature bounds in a support sense and can be Cauchy under suitable completeness and boundary conditions. Additionally, it extends Hawking-type incompleteness results to $C^0$ Cauchy hypersurfaces with mean-curvature bounds in the support sense, broadening the applicability of singularity theorems in cosmological spacetimes.

Abstract

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.

Figures (2)

• Figure 1: An illustration of the construction of $\gamma_\infty$ in the proof of Theorem \ref{['thm:taualmostcauchy']}.
• Figure 2: The spacetime in Example \ref{['ex:cyl']}, with a Cauchy and a non-Cauchy level set. The dashed lines represent the end of the spacetime. The dotted vertical lines can be either identified, or infinitely many copies of the same picture can be attached.

Theorems & Definitions (24)

• Theorem 1
• Remark 1
• Theorem 2
• Theorem 3
• Remark 2
• proof
• Theorem 4
• Lemma 1
• proof : Proof of the lemma
• proof : Proof of Theorem \ref{['thm:newHawking2']}