Mode stability and shallow quasinormal modes of Kerr–de Sitter black holes away from extremality

A Kerr–de Sitter black hole is a solution (M,gΛ,m,a) of the Einstein vacuum equations with cosmological constant Λ>0 . It describes a black hole with mass m>0 and specific angular momentum a . We show that for any ε>0 there exists δ>0 so that mode stability holds for the linear scalar wave equation _{gΛ,m,a}φ=0 when |a/m|∈[0,1-ε] and ²<δ . In fact, we show that all quasinormal modes σ in any fixed half-space Imσ>-C√Λ are equal to 0 or -i√Λ/3(n+o(1)) , n , as ² 0 . We give an analogous description of quasinormal modes for the Klein–Gordon equation. We regard a Kerr–de Sitter black hole with small ² as a singular perturbation either of a Kerr black hole with the same angular momentum-to-mass ratio, or of de Sitter spacetime without any black hole present. We use the mode stability of subextremal Kerr black holes, proved by Whiting and Shlapentokh-Rothman, as a black box; the quasinormal modes described by our main result are perturbations of those of de Sitter space. Our proof is based on careful uniform a priori estimates, in a variety of asymptotic regimes, for the spectral family and its de Sitter and Kerr model problems in the singular limit ² 0 .