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A Concentration‐Collapse Decomposition for L 2 Flow Singularities
Author(s) -
Streets Jeffrey
Publication year - 2016
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21557
Subject(s) - mathematics , curvature , diffeomorphism , gravitational singularity , smoothing , compact space , scalar curvature , flow (mathematics) , mean curvature flow , upper and lower bounds , geometric flow , mathematical analysis , pure mathematics , geometry , statistics
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.