Acta Mathematica

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

Peter Hintz and András Vasy

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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.

Article information

Acta Math., Volume 220, Number 1 (2018), 1-206.

Received: 1 July 2016
First available in Project Euclid: 19 June 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 83C57: Black holes
Secondary: 35B40: Asymptotic behavior of solutions 58J47: Propagation of singularities; initial value problems 83C05: Einstein's equations (general structure, canonical formalism, Cauchy problems) 83C35: Gravitational waves

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


Hintz, Peter; Vasy, András. The global non-linear stability of the Kerr–de Sitter family of black holes. Acta Math. 220 (2018), no. 1, 1--206. doi:10.4310/ACTA.2018.v220.n1.a1.

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