  A necessary condition for a globally hyperbolic spacetime ${\mathbb R}\times
\Sigma$ to admit a maximal slice is that the Cauchy slice $\Sigma$ admit a
metric with nonnegative scalar curvature, $R\ge 0$. In this paper, the two
cases considered are the closed spatial manifold and the asymptotically flat
spatial manifold. Although most results here will apply in four or more
spacetime dimensions, this work will mainly consider 4-dimensional spacetimes.
For $\Sigma$ closed or asymptotically flat, all topologies are allowed by the
field equations. Since all $\Sigma$ occur as Cauchy slices of solutions to the
Einstein equations and most $\Sigma$ do not admit metrics with $R\ge 0$, it
follows that most globally hyperbolic spacetimes never admit a maximal slice,
i.e. a slice with zero mean extrinsic curvature. In particular, asymptotically
flat globally hyperbolic spacetimes which admit maximal slices are the
exception rather than the rule. The reason for this is due to topological
obstructions to constructing such slices. In the asymptotically flat case, this
will be shown by smooth compactification of the manifold in order to use the
results for spatially closed manifolds.
