Linear stability of the slowly-rotating Kerr-de Sitter family

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In this paper, we prove that the slowly-rotating Kerr-de Sitter family of black holes are 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 and Vasy 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.
• In particular, we avoid the construction of a meromorphic continuation of the resolvent.