A complete classification of S1-symmetric static vacuum black holes

Abstract

In a seminal paper of 1917, Weyl presented a remarkable reduction of the static axisymmetric vacuum Einstein equations, serving as a relatively straightforward technique to generate and explore new solutions. Weyl’s reduction was used by Myers in 1987, and independently by Korotkin–Nicolai in 1994, to construct a new family of static and axisymmetric solutions with compact non-empty horizon, however with non-trivial topology and asymptotically Kasner. This family, together with the Schwarzschild and the Boost families, remained until now as the only known -symmetric static black hole solutions, namely, (metrically complete) -symmetric static vacuum solutions with compact and non-empty horizon. In this article we prove that, indeed, these three families exhaust all the examples of -symmetric static vacuum black holes.