Mode stability results for the Teukolsky equations on Kerr–anti-de Sitter spacetimes
Résumé
Abstract We prove that there are no non-stationary (with respect to the Hawking vectorfield), real mode solutions to the Teukolsky equations on all ( 3 + 1 ) -dimensional subextremal Kerr–anti-de Sitter spacetimes. We further prove that stationary solutions do not exist if the black hole parameters satisfy the Hawking–Reall bound and $|a \sqrt{-\Lambda}| < \sqrt{3}/20 . We conclude with the statement of mode stability which preludes boundedness and decay estimates for general solutions which will be proven in a separate paper. Our boundary conditions are the standard ones which follow from fixing the conformal class of the metric at infinity and lead to a coupling of the two Teukolsky equations. The proof relies on combining the Teukolsky–Starobinsky identities with the coupled boundary conditions. In the stationary case the proof exploits elliptic estimates which fail if the Hawking–Reall bound is violated. This is consistent with the superradiant instabilities expected in that regime.