A Concentration‐Collapse Decomposition for L2 Flow Singularities

Abstract

We exhibit a concentration‐collapse decomposition of singularities of fourth‐order curvature flows, including the L2 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 L2 flow in the presence of a curvature‐related bound. A final key ingredient is a new local ϵ‐regularity result for L2 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 L2 curvature pinching and a volume‐noncollapsing condition. © 2015 Wiley Periodicals, Inc.