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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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Existence, lifespan, and transfer rate of Ricci flows on manifolds with small Ricci curvature
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by Fei He PDF
Trans. Amer. Math. Soc. 371 (2019), 4059-4095 Request permission

Abstract:

We show that in dimension 4 and above, the lifespan of Ricci flows depends on the relative smallness of the Ricci curvature compared to the Riemann curvature on the initial manifold. We can generalize this lifespan estimate to the local Ricci flow, using what we prove as the short-time existence of Ricci flow solutions on complete noncompact Riemannian manifolds with at most quadratic curvature growth, where the Ricci curvature and its first two derivatives are sufficiently small in regions where the Riemann curvature is large. Those Ricci flow solutions may have unbounded curvature. Moreover, our method implies that, under some appropriate assumptions, the spatial transfer rate (the rate at which high curvature regions affect low curvature regions) of the Ricci flow resembles that of the heat equation.
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Additional Information
  • Fei He
  • Affiliation: School of Mathematical Science, Xiamen University, 422 Siming South Road, Xiamen, Fujian 361005, China
  • MR Author ID: 1047999
  • Email: hefei@xmu.edu.cn
  • Received by editor(s): August 29, 2016
  • Received by editor(s) in revised form: November 12, 2017, and November 20, 2017
  • Published electronically: August 14, 2018
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 371 (2019), 4059-4095
  • MSC (2010): Primary 53C44
  • DOI: https://doi.org/10.1090/tran/7466
  • MathSciNet review: 3917217