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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- 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
- MR Author ID: 220655
- Email: email@example.com
- 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
- MR Author ID: 887805
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- Zhenlei Zhang
- Affiliation: Department of Mathematics, Beijing Capital Normal University, Beijing, People’s Republic of China
- MR Author ID: 794099
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- Xiaohua Zhu
- Affiliation: School of Mathematical Sciences and BICMR, Peking University, Beijing, 100871, People’s Republic of China
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- 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.
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- Journal: Trans. Amer. Math. Soc. 365 (2013), 6669-6695
- MSC (2010): Primary 53C25; Secondary 53C55, 58J05
- DOI: https://doi.org/10.1090/S0002-9947-2013-06027-8
- MathSciNet review: 3105766