A classification theorem for static vacuum black holes. Part II: the study of the asymptotic

This is the second article of a series or two, proving a generalisation of the uniqueness theorem of the Schwarzschild solution. The theorem to be shown classifies all (metrically complete) solutions of the static vacuum Einstein equations with compact but non-necessarily connected horizon without any further assumption on the topology or the asymptotic. Specifically, it is shown that any such solution is either: (i) a Boost, (ii) a Schwarzschild black hole, or (iii) is of Myers/Korotkin-Nicolai type, that is, it has the same topology and Kasner asymptotic as the Myers/Korotkin-Nicolai black holes. In this Part II we show that the only end of a static black hole data set is either asymptotically flat or asymptotically Kasner. This proves the third and last step required in the proof of the classification theorem, as was explained in Part I. The analysis requires a thorough study of static data sets with a free $S^{1}$-symmetry and a delicate investigation of the geometry of static black hole ends with sub-cubic volume growth. Many of the conclusions of this article are hitherto unknown and have their own interest.