Non-constant mean curvature trumpet solutions for the Einstein   constraint equations
  We prove the existence of a large class of initial data for the vacuum Einstein equations which possess a finite number of asymptotically Euclidean and asymptotically conformally cylindrical or periodic ends. Aside from being asymptotically constant, only mild conditions on the mean curvature of these initial data sets are imposed. 
Jeremy Leach
gr-qc math.DG