标题： 利用时空泊松方程将初始数据集进行共形变形以满足严格主导能量条件 显示英文标题

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

摘要： 我们给出[16]中一个结果的另一种证明，表明满足主导能量条件的爱因斯坦方程带边界的初始数据集$(M, g, k)$可以通过共形变形为严格主导能量条件，同时保持边界性质（极小、未来捕获或过去捕获），并且可以任意小地改变边界的面积和初始数据集的ADM能量。 该证明依赖于求解一个类似于[7]中研究的时空调和函数方程，但带有诺伊曼边界条件和非零右边，我们称之为时空泊松方程。 这种方法的一个优点是共形变形被显式构造为一个偏微分方程的解，而不是像[16]中那样仅通过隐函数定理的应用知道解的存在性。 我们只考虑物理相关的$3$-流形$M$的情况，尽管该证明可以推广到更高维度。 显示英文摘要

摘要： We give an alternate proof of one of the results given in [16] showing that initial data sets with boundary for the Einstein equations $(M, g, k)$ satisfying the dominant energy condition can be conformally deformed to the strict dominant energy condition, while preserving the character of the boundary (minimal, future trapped, or past trapped) while changing the area of the boundary and ADM energy 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 [7], 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 [16]. We restrict ourselves to the physically relevant case of a $3$-manifold $M$, though the proof can be generalized to higher dimensions.