Conformal deformations of initial data sets to the strict dominant energy condition using a spacetime Poisson equation

We give an alternate proof of one of the results given in [D. A. Lee, M. Lesourd and R. Unger, Density and positive mass theorems for initial data sets with boundary, Commun. Math. Phys. 395 (2022) 643–677] showing that initial data sets with boundary for the Einstein equations [Formula: see text] satisfying the dominant energy condition can be conformally deformed to the strict dominant energy condition, while preserving the sign of the null expansions at the boundary, while changing the area of the boundary, the ADM energy, and the ADM momentum of the initial data set by an arbitrarily small amount. The proof relies on solving an equation that looks like the equation for spacetime harmonic functions studied in [S. Hirsch, D. Kazaras and M. Khuri, Spacetime harmonic functions and the mass of 3-dimensional asymptotically flat initial data for the Einstein equations, J. Differ. Geom. 122(2) (2021) 223–258], but with a Neumann boundary condition and non-zero right-hand side, which we refer to as a spacetime Poisson equation. One advantage of this method of proof is that the conformal deformation is explicitly constructed as a solution to a PDE, as opposed to only knowing the solution exists via an application of the implicit function theorem as in [D. A. Lee, M. Lesourd and R. Unger, Density and positive mass theorems for initial data sets with boundary, Commun. Math. Phys. 395 (2022) 643–677]. We prove the theorem in arbitrary dimension [Formula: see text]