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In a seminal paper of 1917, H. Weyl presented a remarkable reduction of thestatic axisymmetric vacuum Einstein equations, serving as a relativelystraightforward technique to generate and explore new solutions. Weyl'sreduction was used by Myers in 1987, and independently by Korotkin-Nicolai in1994, to construct a new family of static and axisymmetric solutions withcompact non-empty horizon, however with non-trivial topology and asymptoticallyKasner. This family, together with the Schwarzschild and the Boost families,remained until now as the only known S1-symmetric static black hole solutions,namely, (metrically complete) S1-symmetric static vacuum solutions with compactand non-empty horizon. In this article we prove that, indeed, these threefamilies exhaust all the examples of S1-symmetric static vacuum black holes.

A complete classification of S1-symmetric static vacuum black holes