Mean Curvature, Singularities and Time Functions in Cosmology

Abstract

In this contribution, we study spacetimes of cosmological interest, withoutmaking any symmetry assumptions. We prove a rigid Hawking singularity theoremfor positive cosmological constant, which sharpens known results. Inparticular, it implies that any spacetime with $\operatorname{Ric} \geq -ng$ intimelike directions and containing a compact Cauchy hypersurface with meancurvature $H \geq n$ is timelike incomplete. We also study the properties ofcosmological time and volume functions, addressing questions such as: When dothey satisfy the regularity condition? When are the level sets Cauchyhypersurfaces? What can one say about the mean curvature of the level sets?This naturally leads to consideration of Hawking type singularity theorems forCauchy surfaces satisfying mean curvature inequalities in a certain weak sense