Schwarzschild-Tangherlini black hole, there exists no stationary perturbation that is regular and bounded outside the horizon, except for the exceptional vector perturbations.*

Taking account of the uniqueness theorem for asymptotically flat static vacuum regular solutions in higher dimensions, this result implies that the Myers-Perry solution is the only regular stationary solution for an asymptotically flat vacuum system near the static limit in a spacetime with any number of dimensions. However, for a generic system for which $\K^n$ is a non-spherically symmetric Einstein space with $K=1$, there may exist a regular static tensor perturbation that falls off at infinity.

Asymptotically de Sitter Case
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In the case with a non-vanishing cosmological constant $\Lambda$, we cannot express general solutions to the perturbation equations in terms of known functions explicitly. Nevertheless, we can show the non-existence of stationary perturbations satisfying the boundary conditions with the help of integral identities derived from the master equations.

We first consider the case $\Lambda>0$, for which $K$ must be unity. In this case, we can demonstrate uniqueness only by studying the behaviour of perturbations in the region bounded by the black hole horizon $r=r_h$ and the cosmological horizon $r=r_c$, for which only the regularity condition at the 