Mean Curvature, Singularities and Time Functions in Cosmology

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 $\mathrm{Ric}\ge -ng$ in timelike directions and containing a compact Cauchy hypersurface with mean curvature $H\ge n$ 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.

15 pages, 2 figures. Changes in v2: typos fixed and small improvements made

References in corpus (6)

• The Cosmological Time Function
• Volume Comparison for Hypersurfaces in Lorentzian Manifolds and Singularity Theorems
• Topology and singularities in cosmological spacetimes obeying the null energy condition
• Hausdorff closed limits and rigidity in Lorentzian geometry
• Volume Singularities in General Relativity
• CMC foliations of open spacetimes asymptotic to open Robertson-Walker spacetimes

Cited by in corpus (2)

• Isometries of spacetimes without observer horizons
• Differentiability and Other Properties of the Cosmological Volume Function