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

TL;DR

The paper proves the global nonlinear stability of slowly rotating Kerr–de Sitter black holes as solutions to the Einstein vacuum equations with a positive cosmological constant, under small angular momentum perturbations and without symmetry restrictions. It develops a robust framework combining a dynamically adjusted DeTurck-type gauge with Nash–Moser iteration, and identifies three core stability ingredients (ungauged Einstein mode stability, stable constraint propagation, and an essential spectral gap) to control linear and nonlinear behavior. A linear stability warm-up, detailed microlocal analysis of resonances, and explicit geometric operator computations underpin the argument, enabling exponential decay toward a nearby Kerr–de Sitter metric. The results imply uniqueness of nearby stationary solutions, potential progress toward Penrose’s cosmic censorship in cosmological settings, and open paths for deeper understanding of ringdown and stability for broader black hole spacetimes.

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.

Figures (24)

• Figure 1.1: Setup for the initial value problem for perturbations of a Schwarzschild--de Sitter spacetime $(M^\circ,g_{b_0})$, showing the Cauchy surface $\Sigma_0$ of $\Omega$ and a few translates $\Sigma_{t_*}$; here $\epsilon_M>0$ is small. Left: Product-type picture, illustrating the stationary nature of $g_{b_0}$. Right: Penrose diagram of the same setup. The event horizon is $\mathcal{H}^+=\{r=r_-\}$, the cosmological horizon is $\overline\mathcal{H}{}^+=\{r=r_+\}$, and the (idealized) future timelike infinity is $i^+$.
• Figure 3.1: Penrose diagram of Schwarzschild--de Sitter space. The form \ref{['EqSdSMetric']} of the metric is valid in the shaded region. Here, $\mathcal{H}^\pm$ denotes the future/past event horizon, $\overline\mathcal{H}{}^\pm$ the future/past cosmological horizon, and $i^\pm$ future/past timelike infinity. Also indicated are two level sets of the static time coordinate $t$, and a level set of $r$ (dashed).
• Figure 3.2: The smooth manifold $M^\circ$ (shaded) within Schwarz-schild--de Sitter space, and two exemplary level sets of the timelike function $t_*$. The form \ref{['EqSdSExt']} of the metric in fact extends beyond the dashed boundary of $M^\circ$, $r=r_\pm\pm 3\epsilon_M$, all the way up to (but excluding) the black hole singularity $r=0$ and the conformal boundary $r=\infty$ of the cosmological region.
• Figure 3.3: The b-conormal bundles $\mathcal{L}_\pm^+$ of the horizons as well as their boundaries $\mathcal{R}_\pm^+$ in ${}^{{\mathrm{b}}}T^*_X M\setminus o$. The arrows indicate the Hamilton vector field $H_{G_b}$. In $\Gamma^-$, the subscripts are replaced by '$-$', and the directions of the arrows are reversed.
• Figure 3.4: The domain $\Omega$ (shaded), with its boundary $Y=\Omega\cap X$ at future infinity, as a smooth domain with corners within $M$. The Cauchy surface is $\Sigma_0$, extending a bit beyond the horizons of slowly rotating Kerr--de Sitter spacetimes.

Theorems & Definitions (139)

• Theorem 1.1: Stability of the Kerr--de Sitter family for small $a$; informal version
• Theorem 1.2
• Remark 1.3
• Theorem 1.4: Stability of the Kerr--de Sitter family for small $a$; precise version
• Remark 1.5
• Remark 2.1
• Lemma 3.1
• proof
• Remark 3.2
• Lemma 3.3