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

A Kerr-de Sitter black hole is a solution $(M,g_{\Lambda,\mathfrak{m},\mathfrak{a}})$ 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 $\Box_{g_{\Lambda,\mathfrak{m},\mathfrak{a}}}\phi=0$ 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 $\Im\sigma>-C\sqrt\Lambda$ 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 $\Lambda\mathfrak{m}^2\searrow 0$.