On noncollapsed almost Ricci-flat 4-manifolds
  We give topological conditions to ensure that a noncollapsed almost Ricci-flat 4-manifold admits a Ricci-flat metric. One sufficient condition is that the manifold is spin and has a nonzero A-hat genus. Another condition is that the fundamental group is infinite or, more generally, of sufficiently large cardinality. 
Vitali Kapovitch, John Lott
math.DG