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This note is a continuation of the author’s paper (Li, Adv. Math. 223(6):1924–1957, 2010). We prove that if the metric g of a compact 4-manifold has bounded Ricci curvature and its curvature has no local concentration everywhere, then it can be smoothed to a metric with bounded sectional curvature. Here we don’t assume the bound for local Sobolev constant of g and hence this smoothing result can be applied to the collapsing case.

Smoothing Riemannian metrics with bounded Ricci curvatures in dimension four, II