2022/07/16 by Fang, Allen Juntao
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2207.07902
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 Λ>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 Hk 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. 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.