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

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.

Keywords

Einstein’s equation, black hole stability, constraint damping, global iteration, gauge modification, Nash–Moser iteration, microlocal analysis

2010 Mathematics Subject Classification

Primary 83C57. Secondary 35B40, 58J47, 83C05, 83C35.

Full Text (PDF format)

Received 1 July 2016

Accepted 30 December 2017

Published 1 June 2018