In a series of two papers ( arXiv:1806.00818 and arXiv:1806.00819 ) Martin Reiris Ithurralde classifies all metrically complete solutions of the static vacuum Einstein equations with compact (but not necessarily connected) horizon.

**Main Theorem**

Any static vacuum black hole is either

- a Schwarzschild black hole,
- a Boost, or
- of Myers/Korotkin-Nicolai type.

**Basic definition**

A static vacuum black hole is a pair \(\big((\Sigma,g),N\big)\) consisting of

- an orientable, complete Riemannian \(3\)-manifold \((\Sigma,g)\), possibly with boundary,
- a function \(N\) on \(\Sigma\) such that
- \(N\) is strictly positive on \(\Sigma \setminus \partial\Sigma\) and
- \(N\) satisfies the vacuum static Einstein equation \[N \cdot \mathrm{Ric} = \nabla \nabla N, \quad \text{and} \quad \Delta N = 0,\]

- and the boundary \(\partial\Sigma\) is compact and given by \(\partial\Sigma = \{N = 0\}\).

**Rough outline of the proof**

The following three facts are proved in the two articles (the first two points in the first article, and the third in the second one):

- \(\Sigma\) has only one end,
- every horizon is weakly outermost, and
- the end is either asymptotically flat or asymptotically Kasner.

A horizon \(H\) is a connected component of \(\partial \Sigma\). It is called weakly outermost, if there are no embedded surfaces \(S\) in \(\Sigma\) which are homologous to \(H\) and have negative outwards mean curvature.

Having proved the above three points, one can go on as follows: in the case the end is asymptotically flat one can invoke a well-known uniqueness theorem that it must be Schwarzschild. If the end is asymptotically Kasner, then it follows by previous results that it either is a Boost or every horizon is a totally geodesic sphere. In the latter case one can then go on and conclude that the static vacuum black hole is of Myers/Korotkin-Nicolai type.