# Null geodesic incompleteness of spacetimes with no CMC Cauchy surfaces

## TL;DR

This paper proves that certain vacuum cosmological spacetimes lacking constant mean curvature Cauchy surfaces are null geodesically incomplete for large gluing parameters, indicating they are singular in both time directions.

## Contribution

It demonstrates null geodesic incompleteness in a class of vacuum spacetimes previously known to lack CMC Cauchy surfaces, extending understanding of their global structure.

## Key findings

- Spacetimes are null geodesically incomplete for large gluing parameters.
- The result applies to both future and past directions.
- Provides insight into the singularity structure of these spacetimes.

## Abstract

Chru\'sciel, Isenberg, and Pollack constructed a class of vacuum cosmological spacetimes that do not admit Cauchy surfaces with constant mean curvature. We prove that, for sufficiently large values of the gluing parameter, these examples are both future and past null geodesically incomplete. The authors are honored to dedicate this paper to Robert Bartnik on the occasion of his 60th birthday.