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, g, ₌, ₀) of the Einstein vacuum equations with cosmological constant >0. It describes a black hole with mass m>0 and specific angular momentum a. We show that for any >0 there exists >0 so that mode stability holds for the linear scalar wave equation ₆_, ₌, ₀=0 when |a/m|0, 1- and ^2-C are equal to 0 or -i/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.