Mean Curvature, Singularities and Time Functions in Cosmology

TL;DR

This paper investigates cosmological spacetimes without symmetry, proving a sharp Hawking singularity theorem for positive cosmological constant and analyzing properties of cosmological time and volume functions.

Contribution

It establishes a new, sharper Hawking singularity theorem under broader conditions and explores the regularity and geometric properties of cosmological time functions.

Findings

Proved a rigid Hawking singularity theorem for positive cosmological constant.

Identified conditions under which cosmological time functions are regular and their level sets are Cauchy hypersurfaces.

Analyzed mean curvature properties of level sets 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.