Maximal Hypersurfaces in Spacetimes with a Nonvanishing Spacelike Killing Field

Abstract

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.