Global Prescribed Mean Curvature foliations in cosmological spacetimes with matter, Part I

TL;DR

This paper demonstrates that certain symmetric cosmological spacetimes with matter can be globally foliated by prescribed mean curvature surfaces, extending towards singularities and generalizing previous constant mean curvature results.

Contribution

It establishes the existence and uniqueness of a global foliation by prescribed mean curvature surfaces in symmetric cosmological spacetimes with matter, overcoming topological obstructions.

Findings

Spacetimes admit a global foliation by prescribed mean curvature surfaces.

Foliation extends at least towards a crushing singularity.

Time function is geometrically defined and unique up to initial surface.

Abstract

This work investigates some global questions about cosmological spacetimes with two dimensional spherical, plane and hyperbolic symmetry containing matter. The result is, that these spacetimes admit a global foliation by prescribed mean curvature surfaces, which extends at least towards a crushing singularity. The time function of the foliation is geometrically defined and unique up to the choice of an initial Cauchy surface. This work generalizes a similar analysis on constant mean curvature foliations and avoids the topological obstructions arising from the existence problem.