horizon is relevant.

Tensor perturbations
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For a tensor perturbation, by multiplying by $r^{n/2}H_T$, integrating it over $r$, and using the regularity condition , we obtain the identity When $\K^n=S^n$, $U_T$ is always positive in the range $r_h<r<r_c$, because $k_T^2\ge n-2$. Hence, from this integral identity, it follows that there exists no regular stationary tensor perturbation.

By contrast, in the case in which $\K^n$ is not a constant curvature space, $k_T^2$ may become negative, and no general lower bound on it is known. Therefore, there may exist a regular stationary perturbation.

Vector perturbations {#vector-perturbations-2}
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The argument concerning vector perturbations is almost the same as that for tensor perturbations. We multiply by $F_t/r$ and integrate it over $r$. Then, using the regularity condition , we obtain As in the case $\Lambda=0$, we can restrict our consideration to the case in which $m_V=k_V^2-n+1>0$. Then, only the trivial solution $F^t=0$ satisfies this integral identity. Unlike the case of tensor perturbations, this result holds even if $\K^n$ is a generic Einstein space with $K=1$.

Scalar perturbations {#scalar-perturbations-2}
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The argument for a scalar perturbation is also similar to that given above. Now, we utilise the master equation for $X$, , which leads to 