We prove the spacetime positive mass theorem in dimensions less than eight. This theorem asserts that for any asymptotically flat initial data set that satisfies the dominant energy condition, the inequality E ≥ |P | holds, where (E, P ) is the ADM energy-momentum vector. Previously, this theorem was only known for spin manifolds [38]. Our approach is a modification of the minimal hypersurface technique that was used by the last named author and S.-T. Yau to establish the time-symmetric case of this theorem [30,27]. Instead of minimal hypersurfaces, we use marginally outer trapped hypersurfaces (MOTS) whose existence is guaranteed by earlier work of the first named author [14]. An important part of our proof is to introduce an appropriate substitute for the area functional that is used in the time-symmetric case to single out certain minimal hypersurfaces. We also establish a density theorem of independent interest and use it to reduce the general case of the spacetime positive mass theorem to the special case of initial data that has harmonic asymptotics and satisfies the strict dominant energy condition.

The rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We study the stability of this statement for spaces that can be realized as graphical hypersurfaces in E n+1 . We prove (under certain technical hypotheses) that if a sequence of complete asymptotically flat graphs of nonnegative scalar curvature has mass approaching zero, then the sequence must converge to Euclidean space in the pointed intrinsic flat sense. The appendix includes a new Gromov-Hausdorff and intrinsic flat compactness theorem for sequences of metric spaces with uniform Lipschitz bounds on their metrics.

Abstract. We prove the existence and uniqueness of constant mean curvature foliations for initial data sets which are asymptotically flat satisfying the Regge-Teitelboim condition near infinity. It is known that the (Hamiltonian) center of mass is well-defined for manifolds satisfying this condition. We also show that the foliation is asymptotically concentric, and its geometric center is the center of mass. The construction of the foliation generalizes the results of Huisken-Yau, Ye, and Metzger, where strongly asymptotically flat manifolds and their small perturbations were studied.

Abstract. The rigidity of the positive mass theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We prove a corresponding stability theorem for spaces that can be realized as graphical hypersurfaces in R n+1 . Specifically, for an asymptotically flat graphical hypersurface M n ⊂ R n+1 of nonnegative scalar curvature (satisfying certain technical conditions), there is a horizontal hyperplane Π ⊂ R n+1 such that the flat distance between M and Π in any ball of radius ρ can be bounded purely in terms of n, ρ, and the mass of M . In particular, this means that if the masses of a sequence of such graphs approach zero, then the sequence weakly converges (in the sense of currents, after a suitable vertical normalization) to a flat plane in R n+1 . This result generalizes some of the earlier findings of the second author and C. Sormani [14] and provides some evidence for a conjecture stated there.