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Transactions of the American Mathematical Society

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Existence, lifespan, and transfer rate of Ricci flows on manifolds with small Ricci curvature


Author: Fei He
Journal: Trans. Amer. Math. Soc. 371 (2019), 4059-4095
MSC (2010): Primary 53C44
DOI: https://doi.org/10.1090/tran/7466
Published electronically: August 14, 2018
MathSciNet review: 3917217
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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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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

Keywords: Ricci flow, local Ricci flow, existence, noncompact manifolds
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
Article copyright: © Copyright 2018 American Mathematical Society