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Some curvature estimates of Kähler-Ricci flow


Authors: Man-Chun Lee and Luen-Fai Tam
Journal: Proc. Amer. Math. Soc. 147 (2019), 2641-2654
MSC (2010): Primary 32Q15; Secondary 53C44
DOI: https://doi.org/10.1090/proc/14436
Published electronically: March 1, 2019
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Abstract: In this work, first we will obtain some local curvature estimates for Kähler-Ricci flow on Kähler manifolds with initial metrics of nonnegative bisectional curvature. As a corollary, we prove that if $ g(t)$ is a complete solution of the Kähler Ricci flow which satisfies $ \vert$$ \text {Rm}(g(t))\vert\leq at^{-\theta }$ for some $ 0<\theta <2$, $ a>0$ and $ g(0)$ has nonnegative bisectional curvature, then $ g(t)$ also has nonnegative bisectional curvature. This generalizes results in [Amer. J. Math. 140 (2018), pp. 189-220] and [J. Differential Geom. 45 (1997), pp. 94-220]. Using the local curvature estimate, we prove that for a complete solution $ g(t)$ of the Kähler-Ricci flow with $ g(0)$ to have nonnegative bisectional curvature, to be noncollapsing, and $ \sup _{M\times [\tau ,T]} \vert$$ \text {Rm}(g(t))\vert<+\infty $ for all $ \tau >0$, then the curvature of $ g(t)$ is in fact bounded by $ at^{-1}$ for some $ a>0$. In particular, $ g(t)$ has nonnegative bisectional curvature for $ t>0$. This result is similar to a result by Simon and Topping in the Kähler category.


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Additional Information

Man-Chun Lee
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, People’s Republic of China
Address at time of publication: Department of Mathematics, University of British Columbia, 121-1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada
Email: mclee@math.ubc.ca

Luen-Fai Tam
Affiliation: The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, People’s Republic of China
Email: lftam@math.cuhk.edu.hk

DOI: https://doi.org/10.1090/proc/14436
Keywords: K\"ahler-Ricci flow, K\"ahler manifold, holomorphic bisectional curvature, uniformization
Received by editor(s): April 9, 2018
Received by editor(s) in revised form: October 4, 2018
Published electronically: March 1, 2019
Additional Notes: Research of the second author was partially supported by Hong Kong RGC General Research Fund #CUHK 14301517
Communicated by: Jia-Ping Wang
Article copyright: © Copyright 2019 American Mathematical Society