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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Ambient obstruction flow
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by Christopher Lopez PDF
Trans. Amer. Math. Soc. 370 (2018), 4111-4145 Request permission

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
  • MR Author ID: 951587
  • Email: clopez00@math.ucsb.edu
  • 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.
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 4111-4145
  • MSC (2010): Primary 53C44
  • DOI: https://doi.org/10.1090/tran/7106
  • MathSciNet review: 3811522