Mode stability results for the Teukolsky equations on Kerr-anti-de Sitter spacetimes

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 root Hawking-Reall bound and |a root-lambda |<root 3/20 . We conclude with the statement of 3 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.