Some curvature estimates of Kähler-Ricci flow
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- by Man-Chun Lee and Luen-Fai Tam PDF
- Proc. Amer. Math. Soc. 147 (2019), 2641-2654 Request permission
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 $|\text {Rm}(g(t))|\leq at^{-\theta }$ 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 $\sup _{M\times [\tau ,T]} |\text {Rm}(g(t))|<+\infty$ 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.References
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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
- MR Author ID: 1322380
- 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
- MR Author ID: 170445
- Email: lftam@math.cuhk.edu.hk
- 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
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 2641-2654
- MSC (2010): Primary 32Q15; Secondary 53C44
- DOI: https://doi.org/10.1090/proc/14436
- MathSciNet review: 3951439