of the value of the cosmological constant $\Lambda$, where $[(d-1)/2]$ is the rank of $\SO(d-1)$. A similar result holds for a regular static topological black hole spacetime for which a spatial section of the horizon is an $n$-dimensional compact space with non-positive constant curvature $K$. The only difference between the two cases is the number of degrees of freedom of the exceptional vector perturbation, which is $d-2$ for $K=0$ and zero for $K<0$.*

For the asymptotically constant-curvature vacuum case, this theorem implies that the Myers-Perry solution for $\Lambda=0$ and the Gibbons-Lü-Page-Pope solution for $\Lambda\not=0$ are the only regular black hole solutions near the static and spherically symmetric limit. Further, it shows that a similar perturbative uniqueness holds for topological black holes as well. Because no uniqueness theorem has been proved for asymptotically de Sitter black holes in four dimensions, this theorem also has a non-trivial implication for the four-dimensional case.

The present paper is organised as follows. In the next section, we give a brief summary of a gauge-invariant formulation for stationary perturbations of generalised static vacuum black holes. The basic equations given there are basically specialisations to the case of stationary perturbations of the equations for generic perturbations derived 