  We prove that if a closed, smooth, simply-connected 4-manifold with a circle
action admits an almost non-negatively curved sequence of invariant Riemannian
metrics, then it also admits a non-negatively curved Riemannian metric
invariant with respect to the same action. The same is shown for torus actions
of higher rank, giving a classification of closed, smooth, simply-connected
4-manifolds of almost non-negative curvature under the assumption of torus
symmetry.
