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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Perelman's entropy and Kähler-Ricci flow on a Fano manifold


Authors: Gang Tian, Shijin Zhang, Zhenlei Zhang and Xiaohua Zhu
Journal: Trans. Amer. Math. Soc. 365 (2013), 6669-6695
MSC (2010): Primary 53C25; Secondary 53C55, 58J05
Published electronically: August 15, 2013
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Abstract: In this paper, we extend the method in a recent paper of Tian and Zhu to study the energy level $ L(\cdot )$ of Perelman's entropy $ \lambda (\cdot )$ for the Kähler-Ricci flow on a Fano manifold $ M$. We prove that $ L(\cdot )$ is independent of the initial metric of the Kähler-Ricci flow under an assumption that the modified Mabuchi's K-energy is bounded from below on $ M$. As an application of the above result, we give an alternative proof to the main theorem about the convergence of Kähler-Ricci flow found in a 2007 paper by Tian and Zhu.


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Additional Information

Gang Tian
Affiliation: School of Mathematical Sciences and BICMR, Peking University, Beijing, 100871, People’s Republic of China – and – Department of Mathematics, Princeton University, Princeton, New Jersey 02139
Email: tian@math.mit.edu

Shijin Zhang
Affiliation: Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, People’s Republic of China
Address at time of publication: School of Mathematics and Systems Science, Beijing University of Aeronautics & Astronautics, Beijing, 100191, People’s Republic of China
Email: zhangshj.1982@yahoo.com.cn

Zhenlei Zhang
Affiliation: Department of Mathematics, Beijing Capital Normal University, Beijing, People’s Republic of China
Email: zhleigo@aliyun.com

Xiaohua Zhu
Affiliation: School of Mathematical Sciences and BICMR, Peking University, Beijing, 100871, People’s Republic of China
Email: xhzhu@math.pku.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-06027-8
PII: S 0002-9947(2013)06027-8
Keywords: K\"ahler-Ricci flow, K\"ahler-Ricci solitons, Perelman's entropy
Received by editor(s): January 30, 2012
Received by editor(s) in revised form: June 22, 2012, and August 29, 2012
Published electronically: August 15, 2013
Additional Notes: The third author was supported in part by a grant of BMCE 11224010007 in China.
The fourth author was supported in part by NSFC Grants 10990013 and 11271022.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.