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
√
−Λ

<
√
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.