  We consider four-dimensional vacuum spacetimes which admit a nonvanishing
spacelike Killing field. The quotient with respect to the Killing action is a
three-dimensional quotient spacetime $(M,g)$. We establish several results
regarding maximal hypersurfaces (spacelike hypersurfaces of zero mean
curvature) in such quotient spacetimes. First, we show that a complete
noncompact maximal hypersurface must either be a cylinder $S^1 \times
\mathbb{R}$ with flat metric or else conformal to the Euclidean plane
$\mathbb{R}^2$. Second, we establish positivity of mass for certain maximal
hypersurfaces, referring to a analogue of ADM mass adapted for the quotient
setting. Finally, while lapse functions corresponding to the maximal
hypersurface gauge are necessarily bounded in the four-dimensional
asymptotically Euclidean setting, we show that nontrivial quotient spacetimes
admit the maximal hypersurface gauge only with unbounded lapse.
