Toroidal black holes

Recently Thorne presented a class of toroidal solutions which are cylindrically symmetric near the ring singularity. Although the class is a three-parameter family, there is a one-parameter constraint in order to eliminate an unwanted singularity. We show that for parameters satisfying a different one-parameter constraint, the ring singularity is replaced by a nonsingular toroidal horizon. We also show that such a constraint is inconsistent with Thorne's constraint, by finding the latter constraint explicitly in the limit opposite to that considered by Thorne. We exhibit the two-parameter class of solutions with nonsingular toroidal horizons which do not violate the black hole uniqueness theorems, since there is a singularity outside the horizon