Non-trivial, static, geodesically complete, vacuum space-times with a   negative cosmological constant
  We construct a large class of new singularity-free static Lorentzian four-dimensional solutions of the vacuum Einstein equations with a negative cosmological constant. The new families of metrics contain space-times with, or without, black hole regions. Two uniqueness results are also established. 
M. Anderson, P.T. Chrusciel, E. Delay