Ames Ellery, Beyer Florian, Isenberg James, Oliynyk Todd A
Department of Mathematics, Humboldt State University, 1 Harpst St. Arcata, CA 95521, USA.
Department of Mathematics and Statistics, 730 Cumberland St, University of Otago, Dunedin 9016, New Zealand.
Philos Trans A Math Phys Eng Sci. 2022 May 2;380(2222):20210173. doi: 10.1098/rsta.2021.0173. Epub 2022 Mar 14.
We prove the nonlinear stability of the asymptotic behaviour of perturbations of subfamilies of Kasner solutions in the contracting time direction within the class of polarized [Formula: see text]-symmetric solutions of the vacuum Einstein equations with arbitrary cosmological constant [Formula: see text]. This stability result generalizes the results proven in Ames E (2022 Stability of AVTD Behavior within the Polarized [Formula: see text]-symmetric vacuum spacetimes. . (doi:10.1007/s00023-021-01142-0)), which focus on the [Formula: see text] case, and as in that article, the proof relies on an areal time foliation and Fuchsian techniques. Even for [Formula: see text], the results established here apply to a wider class of perturbations of Kasner solutions within the family of polarized [Formula: see text]-symmetric vacuum solutions than those considered in Ames E (2022 Stability of AVTD Behavior within the Polarized [Formula: see text]-symmetric vacuum spacetimes. . (doi:10.1007/s00023-021-01142-0)) and Fournodavlos G (2020 . Preprint. (http://arxiv.org/abs/2012.05888)). Our results establish that the areal time coordinate takes all values in [Formula: see text] for some [Formula: see text], for certain families of polarized [Formula: see text]-symmetric solutions with cosmological constant. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'.
我们证明了在具有任意宇宙学常数(\Lambda)的真空爱因斯坦方程的极化(U(1))-对称解类中，Kasner解子族在收缩时间方向上的扰动渐近行为的非线性稳定性。这个稳定性结果推广了Ames E（2022年，极化(U(1))-对称真空时空中AVTD行为的稳定性。。（doi:10.1007/s00023-021-01142-0））中证明的结果，该结果专注于(\Lambda = 0)的情况，并且与那篇文章一样，证明依赖于面积时间叶状结构和富克斯技术。即使对于(\Lambda = 0)，这里建立的结果也适用于极化(U(1))-对称真空解族中比Ames E（2022年，极化(U(1))-对称真空时空中AVTD行为的稳定性。。（doi:10.1007/s00023-021-01142-0））和Fournodavlos G（2020年。预印本。（http://arxiv.org/abs/2012.05888））中考虑的更广泛的Kasner解扰动类。我们的结果表明，对于某些具有宇宙学常数的极化\(U(1)\)-对称解族，面积时间坐标在\([t_0, +\infty))内取所有值，其中(t_0)为某个值。本文是主题特刊“数学宇宙学的未来，第1卷”的一部分。
