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We establish the full global non-linear stability of the Kerr-de Sitterfamily of black holes, as solutions of the initial value problem for theEinstein vacuum equations with positive cosmological constant, for smallangular momenta, and without any symmetry assumptions on the initial data. Weachieve this by extending the linear and non-linear analysis on black holespacetimes described in a sequence of earlier papers by the authors: We developa general framework which enables us to deal systematically with thediffeomorphism invariance of Einstein's equations. In particular, the iterationscheme used to solve Einstein's equations automatically finds the parameters ofthe Kerr-de Sitter black hole that the solution is asymptotic to, theexponentially decaying tail of the solution, and the gauge in which we are ableto find the solution; the gauge here is a wave map/DeTurck type gauge, modifiedby source terms which are treated as unknowns, lying in a suitablefinite-dimensional space.

The global non-linear stability of the Kerr-de Sitter family of black holes