When Do Spacetimes Have Constant Mean Curvature Slices?

TL;DR

This paper reviews known results and open questions regarding the existence of constant mean curvature slices in spacetimes, which are crucial for many problems in mathematical relativity.

Contribution

It provides an overview of existing theorems, discusses evidence for unknown cases, and presents conjectures about the existence and generality of CMC slices.

Findings

Some spacetimes lack CMC slices.

Existence of CMC slices is not guaranteed in all spacetimes.

Conjectures suggest conditions under which CMC slices may exist.

Abstract

Many results in mathematical relativity, including results for both the initial data problem and for the evolution problem, rely on the existence of a constant mean curvature (CMC) Cauchy surface in the underlying spacetime. However, it is known that some spacetimes have no CMC Cauchy surfaces (slices). This is an obstacle for many results and constructions with these types of spacetimes, and is particularly worrisome since it is not known whether spacetimes that do have CMC slices are in any sense generic. In this expository paper, we will discuss the known results about the existence (and non-existence) of CMC slices, examine the evidence for cases which are unknown, and make several conjectures concerning the existence of CMC slices and their generality.