# Collapsing in the Einstein flow

## TL;DR

This paper studies the long-term behavior of expanding vacuum spacetimes with specific geometric bounds, showing they resemble flat or Kasner spacetimes over large future intervals, and extends results to cases without these bounds.

## Contribution

It provides new insights into the asymptotic structure of expanding vacuum spacetimes under different geometric conditions.

## Key findings

- Spacetimes with type-III bounds approximate flat or Kasner models over large times.
- Results extend to certain type-II expanding vacuum spacetimes.
- Demonstrates the role of geometric bounds in spacetime evolution.

## Abstract

We consider expanding vacuum spacetimes with a CMC foliation by compact spacelike hypersurfaces. Under scale invariant a priori geometric bounds (type-III), we show that there are arbitrarily large future time intervals that are modelled by a flat spacetime or a Kasner spacetime. We give related results for a class of expanding vacuum spacetimes that do not satisfy the a priori bounds (type-II).