Existence and non-existence results for global constant mean curvature foliations

TL;DR

This paper proves the existence of spatially compact solutions to Einstein-dust equations with limited CMC time, highlighting differences from other matter models and providing bounds on the lapse function.

Contribution

It establishes that Einstein-dust solutions can have arbitrarily small CMC time and derives a lower bound for the lapse function in inhomogeneous spacetimes.

Findings

Existence of solutions with minimal CMC time.

Lower bound for the lapse function in Einstein-dust spacetimes.

Contrast with positive results for other matter models.

Abstract

The main result of this paper is a proof that there are examples of spatially compact solutions of the Einstein-dust equations which only exist for an arbitrarily small amount of CMC time. While this fact is plausible, it is not trivial to prove. It is necessary to obtain a lower bound for the lapse function of a CMC foliation in a suitable class of inhomogeneous spacetimes. This bound, which shows that in these spacetimes the lapse cannot collapse in finite CMC time, may be of independent interest. This fact is contrasted with the positive results previously obtained for other matter models, e.g. collisionless matter or wave maps.