The spacetime positive mass theorem in dimensions less than eight
The spacetime positive mass theorem in dimensions less than eight
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DOI：
10.4171/jems/584
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发表时间：
2016-01-01
影响因子：
2.6
通讯作者：
Schoen, Richard
中科院分区：
文献类型：
--
作者：
Eichmair, Michael;Huang, Lan-Hsuan;Schoen, Richard
我们证明了在小于8维的时空中的正质量定理。这个定理断言，对于任何满足主导能量条件的渐近平坦的初始数据集，不等式E >= vertical bar P vertical bar成立，其中（E，P）是ADM能量-动量向量。在此之前，这个定理只在自旋流形中才被发现[38]。我们的方法是最后一位作者和S. T. Yau建立了该定理的时间对称情况[30，27]。代替极小超曲面，我们使用边缘外捕获超曲面（MOTS），其存在性由第一命名作者的早期工作保证[14]。我们的证明的一个重要部分是引入一个适当的替代品的面积功能，用于在时间对称的情况下，挑选出某些最小的超曲面。我们还建立了一个独立的兴趣密度定理，并使用它来减少时空正质量定理的一般情况下的特殊情况下的初始数据，具有调和渐近，并满足严格的主导能量条件。
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 >= vertical bar P vertical bar 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.