Linear Stability of the Slowly-Rotating Kerr-de Sitter Family

In this paper, we prove that the slowly-rotating Kerr-de Sitter family of black holes is linearly stable as a family of solutions to the Einstein vacuum equations with $\Lambda >0$ in harmonic (wave) gauge. This article is part of a series that provides a novel proof of the full nonlinear stability of the slowly-rotating Kerr-de Sitter family. This paper and its follow-up offer a self-contained alternative approach to nonlinear stability of the Kerr-de Sitter family from the original work of Hintz, Vasy Hintz and Vasy (Acta Math. 220(1), 1-206 (2018a). 10.4310/ACTA.2018.v220.n1.a1) by interpreting quasinormal modes as $H^k$ eigenvalues of an operator on a Hilbert space, and using integrated local energy decay estimates to prove the existence of a spectral gap. We also do not compactify the spacetime, thus avoiding the use of b-calculus and instead only use standard pseudo-differential arguments in a neighborhood of the trapped set; and avoid constraint damping altogether. The methods in the current paper offer an explicit example of how to use the vectorfield method to achieve resolvent estimates on a trapping background.