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.

We propose a definition of center of mass for asymptotically flat manifolds satisfying Regge-Teitelboim condition at infinity. This definition has a coordinate-free expression and natural properties. Furthermore, we prove that our definition is consistent both with the one proposed by Corvino and Schoen and another by Huisken and Yau. The main tool is a new density theorem for data satisfying the Regge-Teitelboim condition.

We prove the equality case of the Penrose inequality in all dimensions for asymptotically flat hypersurfaces. It was recently proven by G. Lam [19] that the Penrose inequality holds for asymptotically flat graphical hypersurfaces in Euclidean space with non-negative scalar curvature and with a minimal boundary. Our main theorem states that if the equality holds, then the hypersurface is a Schwarzschild solution. As part of our proof, we show that asymptotically flat graphical hypersurfaces with a minimal boundary and non-negative scalar curvature must be mean convex, using the argument that we developed in [15]. This enables us to obtain the ellipticity for the linearized scalar curvature operator and to establish the strong maximum principles for the scalar curvature equation . 1 THE EQUALITY CASE OF THE PENROSE INEQUALITY 2 where ω n−1 is the volume of the unit (n − 1)-sphere in Euclidean space. Moreover, the equality holds if and only if (M, g) is isometric to the region of a Schwarzschild metric outside its minimal hypersurface.G. Huisken and T. Ilmanen proved the conjecture in dimension three for a connected minimal boundary [17]. H. Bray used a different approach and proved the conjecture in dimension three for any number of components of the minimal boundary [2]. In dimensions less than 8, the inequality was proved by H. Bray and D. Lee, with the extra spin assumption for the equality case [5]. In the case that (M, g) is conformally flat, H. Bray and K. Iga derived new properties of superharmonic functions in R n and proved the Penrose inequality with a suboptimal constant for n = 3 [4], F. Schwartz obtained a lower bound of the ADM mass in terms of the Euclidean volume of the region enclosed by the minimal boundary [25], and J. Jauregui proved a Penrose-like inequality [18]. For the Penrose inequality (with the sharp constant) in dimensions higher than 8, the only result that we know, other than the spherically symmetric case, is the result of G. Lam [19] (cf. [8]), where he proved that the Penrose inequality for graphical asymptotically flat hypersurfaces. (Note some related work regarding the Penrose inequality for asymptotically hyperbolic graphs in [7,9].)

Abstract. We show that it is possible to perturb arbitrary vacuum asymptotically flat spacetimes to new ones having exactly the same energy and linear momentum, but with center of mass and angular momentum equal to any preassigned values measured with respect to a fixed affine frame at infinity. This is in contrast to the axisymmetric situation where a bound on the angular momentum by the mass has been shown to hold for black hole solutions. Our construction involves changing the solution at the linear level in a shell near infinity, and perturbing to impose the vacuum constraint equations. The procedure involves the perturbation correction of an approximate solution which is given explicitly.

We affirm the rigidity conjecture of the spacetime positive mass theorem in dimensions less than eight. Namely, if an asymptotically flat initial data set satisfies the dominant energy condition and has E = |P |, then E = |P | = 0, where (E, P ) is the ADM energy-momentum vector. The dimensional restriction can be removed if we assume the positive mass inequality holds. Previously the result was only known for spin manifolds [5,6].