for $\nu\not=0$. In particular, the causality condition is violated, and there exist closed time-like curves in the region satisfying$f(r)>0$, corresponding to the outside of a black hole. Second, each $t=\const$ section of this $S^3$ is actually a cylinder with two boundaries for $f(r)>0$, although they shrink to points at the bifurcation surface of the horizons. Further, this section becomes time-like near the boundary. Due to these features, taking the limit $\nu\tend 0$ is a singular procedure.

In the present paper, we only consider $L^2$-normalizable perturbations. Therefore, solutions with peculiarities like this Taub-NUT solution are excluded from the argument of perturbative uniqueness.

Asymptotically Flat Case
========================

The analysis of the perturbative uniqueness of asymptotically flat black hole solutions for $K=1$ and $\Lambda=0$ is rather straightforward, because the perturbation equations are exactly soluble. We mainly consider the case of a spherically symmetric black hole, i.e., $\K^n=S^n$, and only briefly mention generic Einstein cases.

Tensor perturbations
--------------------

In terms of the variable the master equation can be written This equation is of the Fuchs type, and its general solution can be expressed in terms of hypergeometric functions as where $l$ is a solution of the equation and For $\K^n=S^n$, $l$ takes the discrete 