We establish the full global non-linear stability of the Kerr-de Sitter family of black holes, as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta, and without any symmetry assumptions on the initial data. We achieve this by extending the linear and non-linear analysis on black hole spacetimes described in a sequence of earlier papers by the authors: We develop a general framework which enables us to deal systematically with the diffeomorphism invariance of Einstein's equations. In particular, the iteration scheme used to solve Einstein's equations automatically finds the parameters of the Kerr-de Sitter black hole that the solution is asymptotic to, the exponentially decaying tail of the solution, and the gauge in which we are able to find the solution; the gauge here is a wave map/DeTurck type gauge, modified by source terms which are treated as unknowns, lying in a suitable finite-dimensional space.

中文翻译：

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克尔-德西特黑洞家族的全局非线性稳定性

我们建立了 Kerr-de Sitter 黑洞的完整全局非线性稳定性，作为具有正宇宙常数的爱因斯坦真空方程的初值问题的解，对于小角动量，并且对初始值没有任何对称性假设数据。我们通过扩展作者之前的一系列论文中描述的对黑洞时空的线性和非线性分析来实现这一点：我们开发了一个通用框架，使我们能够系统地处理爱因斯坦方程的微分同胚不变性。特别是，用于求解爱因斯坦方程的迭代方案会自动找到解渐近的 Kerr-de Sitter 黑洞的参数、解的指数衰减尾部以及我们能够找到解的规范;