We prove that solutions to linear wave equations in a subextremal Kerr-de Sitter spacetime have asymptotic expansions in quasinormal modes up to a decay order given by the normally hyperbolic trapping, extending the result of the second named author (2013). The main novelties are a different way of obtaining a Fredholm setup that defines the quasinormal modes and a new analysis of the trapping of lightlike geodesics in the Kerr-de Sitter spacetime, both of which apply in the full subextremal range. In particular, this reduces the question of decay for solutions to wave equations to the question of mode stability.

We prove that quasinormal modes (or resonant states) for linear wave equations in the subextremal Kerr and Kerr–de Sitter spacetimes are real analytic. The main novelty of this paper is the observation that the bicharacteristic flow associated to the linear wave equations for quasinormal modes with respect to a suitable Killing vector field has a stable radial point source/sink structure rather than merely a generalized normal source/sink structure. The analyticity then follows by a recent result in the microlocal analysis of radial points by Galkowski and Zworski. The results can then be recast with respect to the standard Killing vector field.

We prove that any smooth vacuum spacetime containing a compact Cauchy horizon with surface gravity that can be normalised to a nonzero constant admits a Killing vector field. This proves a conjecture by Moncrief and Isenberg from 1983 under the assumption on the surface gravity and generalises previous results due to Moncrief–Isenberg and Friedrich–Rácz–Wald, where the generators of the Cauchy horizon were closed or densely filled a 2-torus. Consequently, the maximal globally hyperbolic vacuum development of generic initial data cannot be extended across a compact Cauchy horizon with surface gravity that can be normalised to a nonzero constant. Our result supports, thereby, the validity of the strong cosmic censorship conjecture in the considered special case. The proof consists of two main steps. First, we show that the Killing equation can be solved up to infinite order at the Cauchy horizon. Second, by applying a recent result of the first author on wave equations with initial data on a compact Cauchy horizon, we show that this Killing vector field extends to the globally hyperbolic region.

Hawking’s singularity theorem says that cosmological solutions satisfying the strong energy condition and corresponding to initial data with positive mean curvature have a past singularity; any past timelike curve emanating from the initial hypersurface has length at most equal to the inverse of the mean curvature. However, the nature of the singularity remains unclear. We therefore ask the following question: If the initial hypersurface has sufficiently large mean curvature, does the curvature necessarily blow up towards the singularity?

In case the eigenvalues of the expansion-normalized Weingarten map are everywhere distinct and satisfy a certain algebraic condition (which in 3+1 dimensions is equivalent to them being positive), we prove that this is indeed the case in the CMC Einstein-non-linear scalar field setting. More specifically, we associate a set of geometric expansion-normalized quantities to any initial data set with positive mean curvature. These quantities are expected to converge, in the quiescent setting, in the direction of crushing big bang singularities; i.e. as the mean curvature diverges. Our main result says that if the mean curvature is large enough, relative to an appropriate Sobolev norm of these geometric quantities, and if the algebraic condition on the eigenvalues is satisfied, then a quiescent (as opposed to oscillatory) big bang singularity with curvature blow-up necessarily forms. This provides a stable regime of big bang formation without requiring proximity to any particular class of background solutions.

An important recent result by Fournodavlos, Rodnianski and Speck demonstrates stable big bang formation for all the spatially flat and spatially homogeneous solutions to the Einstein-scalar field equations satisfying the algebraic condition. As an application of our analysis, we obtain analogous stability results for any solution with induced data at a quiescent big bang singularity, in the sense introduced by the third author. In particular, we conclude stable big bang formation of large classes of spatially locally homogeneous solutions, of which the result by Fournodavlos, Rodnianski and Speck is a special case. Finally, since we here consider the Einstein-non-linear scalar field setting, we are also, combining the results of this article withan analysis of Bianchi class A solutions, able to prove both future and past global non-linear stability of a large class of spatially locally homogeneous solutions.