# A complete classification of S1-symmetric static vacuum black holes

## TL;DR

This paper proves that the known families of S1-symmetric static vacuum black holes, including Schwarzschild, Boost, and a family with non-trivial topology, are the only such solutions.

## Contribution

It provides a complete classification of all S1-symmetric static vacuum black hole solutions, confirming no other solutions exist beyond the known families.

## Key findings

- Schwarzschild, Boost, and non-trivial topology solutions are exhaustive.
- No additional S1-symmetric static vacuum black holes exist.
- The classification confirms the uniqueness of these solutions.

## Abstract

In a seminal paper of 1917, H. 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 S1-symmetric static black hole solutions, namely, (metrically complete) S1-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 S1-symmetric static vacuum black holes.