This paper proves a version of the spacetime positive mass theorem for initial data that have a “corner” along a hypersurface Σ. The authors show that if the dominant energy condition holds on each side of Σ and a precise matching condition for the boundary data (called Bartnik data) holds across Σ, then the ADM energy E of the exterior end is at least the length of the ADM linear momentum P: E ≥ |P|. The result is valid in every space dimension n ≥ 3.

The work is about asymptotically flat initial data for general relativity. An initial dataset is a Riemannian manifold with a symmetric tensor (g, k) that encodes the initial geometry and extrinsic curvature. Local energy and current densities µ and J are built from g and k. The dominant energy condition (DEC) is the local inequality µ ≥ |J| that one expects from physically reasonable matter. The global quantities E and P are defined by surface integrals far out along the asymptotically flat end and measure total energy and linear momentum of the spacetime.

A key input is the Bartnik boundary data on the hypersurface Σ. This is the quadruple (γ, H, ω, τ) made of the induced metric γ, the mean curvature H, a connection one-form ω, and the tangential trace τ of k. The corner condition used in the paper requires the tangential traces to match and a jump inequality for the mean curvatures and connection forms: τ− = τ+ and H− − H+ ≥ |ω− − ω+|γ on Σ. In the time-symmetric case k = 0 this reduces to the known corner condition H− ≥ H+.

The proof proceeds by careful deformations and smoothing. First the interior region is deformed so the DEC becomes strict there. This uses a conformal change and solving a semilinear elliptic partial differential equation to produce a positive conformal factor u with u = 1 on Σ. The exterior region is then deformed to strict DEC while keeping control of the Bartnik data, by applying a proposition of Hirsch and Huang that uses a modified constraint operator introduced by Corvino and Huang. Next the authors apply a mollification (a smoothing) inspired by Miao to smooth across Σ. Mollification can temporarily break the DEC in a small neighborhood of Σ, so they correct that region by solving a linear PDE and applying another conformal change. After these steps they obtain smooth data arbitrarily close in ADM energy–momentum to the original exterior data, and then invoke known spacetime positive mass theorems to conclude E ≥ |P|. For dimensions 3 ≤ n ≤ 7 they cite Eichmair–Huang–Lee–Schoen. For n ≥ 8 they reduce to a setting covered by Brendle–Wang provided extra asymptotic regularity is assumed.