# Classification of static vacuum black holes

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):

1. $$\Sigma$$ has only one end,
2. every horizon is weakly outermost, and
3. 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.