A Concentration‐Collapse Decomposition for L 2 Flow Singularities

We exhibit a concentration‐collapse decomposition of singularities of fourth‐order curvature flows, including the L 2 curvature flow and Calabi flow, in dimensions n  ≤ 4. The proof requires the development of several new a priori estimates. First, we develop a smoothing result for initial metrics with small energy and a volume growth lower bound, in the vein of Perelman's pseudo locality result. Next, we generalize our technique from prior work to exhibit local smoothing estimates for the L 2 flow in the presence of a curvature‐related bound. A final key ingredient is a new local ϵ‐regularity result for L 2 critical metrics with possibly nonconstant scalar curvature. Applications of these results include new compactness and diffeomorphism‐finiteness theorems for smooth compact 4‐manifolds satisfying the necessary and effectively minimal hypotheses of L 2 curvature pinching and a volume‐noncollapsing condition. © 2015 Wiley Periodicals, Inc.