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

TL;DR

This paper proves the full nonlinear stability of Kerr-de Sitter black holes with small angular momentum, using a new framework that handles Einstein's equations' diffeomorphism invariance without symmetry assumptions.

Contribution

The authors develop a systematic approach to prove nonlinear stability of Kerr-de Sitter black holes, extending previous linear and nonlinear analyses and handling gauge issues effectively.

Findings

Established global nonlinear stability of Kerr-de Sitter black holes.

Developed a framework for dealing with Einstein's equations' diffeomorphism invariance.

Automatically determines black hole parameters and gauge conditions in the solution process.

Abstract

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.