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Transactions of the American Mathematical Society

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Ambient obstruction flow


Author: Christopher Lopez
Journal: Trans. Amer. Math. Soc. 370 (2018), 4111-4145
MSC (2010): Primary 53C44
DOI: https://doi.org/10.1090/tran/7106
Published electronically: January 18, 2018
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Abstract: We establish fundamental results for a parabolic flow of Riemannian metrics introduced by Bahuaud-Helliwell which is based on the Fefferman-Graham ambient obstruction tensor. First, we obtain local L2 smoothing estimates for the curvature tensor and use them to prove pointwise smoothing estimates for the curvature tensor. We use the pointwise smoothing estimates to show that the curvature must blow up for a finite time singular solution. We also use the pointwise smoothing estimates to prove a compactness theorem for a sequence of solutions with bounded C0 curvature norm and injectivity radius bounded from below at one point. Finally, we use the compactness theorem to obtain a singularity model from a finite time singular solution and to characterize the behavior at infinity of a nonsingular solution.


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Additional Information

Christopher Lopez
Affiliation: Department of Mathematics 419 Rowland Hall University of California, Irvine Irvine, California 92697-3875
Address at time of publication: Department of Mathematics, South Hall, Room 6502, University of California, Santa Barbara, Santa Barbara, California 93106-3080
Email: clopez00@math.ucsb.edu

DOI: https://doi.org/10.1090/tran/7106
Received by editor(s): October 26, 2015
Received by editor(s) in revised form: September 11, 2016, and October 18, 2016
Published electronically: January 18, 2018
Additional Notes: The author was supported by an NSF AGEP Supplement under DMS-1301864.
Article copyright: © Copyright 2018 American Mathematical Society

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