Semyon Dyatlov

Quasi-Normal Modes and Exponential Energy Decay for the Kerr-de Sitter Black Hole

We provide a rigorous definition of quasi-normal modes for a rotating black hole. They are given by the poles of a certain meromorphic family of operators and agree with the heuristic definition in the physics literature. If the black hole rotates slowly enough, we show that these poles form a discrete subset of

Pollicott–Ruelle Resonances for Open Systems

{\mathbb C}

r

. As an application we prove that the local energy of linear waves in that background decays exponentially once orthogonality to the zero resonance is imposed.

-normally hyperbolic trapped sets

We describe asymptotic behavior of linear waves on Kerr(–de Sitter) black holes and more general Lorentzian manifolds, providing a quantitative analysis of the ringdown phenomenon. In particular we prove that if the initial data is localized at frequencies

Spectral gaps, additive energy, and a fractal uncertainty principle

{\sim \lambda \gg 1}

Stochastic stability of Pollicott–Ruelle resonances

∼λ≫1, then the energy norm of the solution is bounded by