The classification of static vacuum space-times containing an   asymptotically flat spacelike hypersurface with compact interior
  We prove non-existence of static, vacuum, appropriately regular, asymptotically flat black hole space-times with degenerate (not necessarily connected) components of the event horizon. This finishes the classification of static, vacuum, asymptotically flat domains of outer communication in an appropriate class of space-times, showing that the domains of outer communication of the Schwarzschild black holes exhaust the space of appropriately regular black hole exteriors. 
Piotr T. Chru\'sciel