Are Kerr-AdS black holes nonlinearly stable with reflecting boundary conditions?
In plain words
In anti-de Sitter space, a model universe with a reflecting boundary, waves cannot escape and may concentrate into ever smaller scales. Even empty anti-de Sitter space is unstable, and the fate of its black holes is unknown.
Precise statement
For Kerr-AdS with reflecting boundary conditions at conformal infinity, linear waves decay only logarithmically in time (Holzegel and Smulevici), and superradiant instabilities occur when $\Omega_{H}\ell > 1$, with $\Omega_{H}$ the horizon angular velocity and l the AdS radius. Determine whether small perturbations of Kerr-AdS with $\Omega_{H}\ell < 1$ remain small for all time or develop turbulent cascades, and identify the end state when $\Omega_{H}\ell > 1$. An answer is a theorem or convergent numerical evidence.
What would settle it
A nonlinear stability or instability theorem for Kerr-AdS, or long-time convergent simulations resolving the cascade.
Status in the literature
Unverified note
Pure AdS is nonlinearly unstable (Bizon and Rostworowski 2011 numerics; proved for Einstein-null dust by Moschidis 2017-2018); Kerr-AdS is open as of 2026.