Vacuum cosmological spacetimes without CMC Cauchy surfaces

TL;DR

This paper constructs a broad class of vacuum cosmological spacetimes lacking constant mean curvature (CMC) Cauchy surfaces, using initial data methods, which may shed light on Bartnik's cosmological splitting conjecture.

Contribution

It extends previous constructions to include new vacuum spacetimes with specific topologies that do not admit CMC Cauchy surfaces, using initial data techniques.

Findings

Constructed a large class of vacuum cosmological spacetimes without CMC Cauchy surfaces.

Allowed spatial topologies are connected sums of non-spherical, closed, oriented, irreducible 3-manifolds.

Provides insights into the structure of spacetimes relevant to Bartnik's conjecture.

Abstract

In this article, we extend a construction of [6] to obtain a large class of vacuum cosmological spacetimes that do not contain any CMC Cauchy surfaces. The allowed spatial topologies for these examples are of the form $M \# M$, where $M$ is any closed, oriented, irreducible $3$-manifold which is not spherical. This complements the recent results of [19], where, instead of initial data methods, global spacetime gluing arguments were used. The study of such examples is sure to yield insight into Bartnik's cosmological splitting conjecture [1].