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) over cap -genus. Another condition is that the fundamental group is infinite or, more generally. of sufficiently large cardinality.