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

Abstract

A Kerr–de Sitter black hole is a solution $(M,g_{\Lambda ,\mathfrak{m},\mathfrak{a}})$ of the Einstein vacuum equations with cosmological constant $\Lambda >0$. It describes a black hole with mass $\mathfrak{m}>0$ and specific angular momentum $\mathfrak{a}\in \mathbb{R}$. We show that for any $\epsilon >0$ there exists $\delta >0$ so that mode stability holds for the linear scalar wave equation ${\square }_{g_{\Lambda ,\mathfrak{m},\mathfrak{a}}}\varphi =0$ when $|\mathfrak{a}/\mathfrak{m}|\in [0,1-\epsilon ]$ and $\Lambda {\mathfrak{m}}^2<\delta$. In fact, we show that all quasinormal modes $\sigma$ in any fixed half-space $\mathrm{Im}\sigma >-C\sqrt{\Lambda }$ are equal to $0$ or $-i\sqrt{\Lambda /3}(n+o(1))$, $n\in \mathbb{N}$, as $\Lambda {\mathfrak{m}}^2\searrow 0$. We give an analogous description of quasinormal modes for the Klein–Gordon equation. We regard a Kerr–de Sitter black hole with small $\Lambda {\mathfrak{m}}^2$ 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 $\Lambda {\mathfrak{m}}^2\searrow 0$.