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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Perelman’s entropy and Kähler-Ricci flow on a Fano manifold
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by Gang Tian, Shijin Zhang, Zhenlei Zhang and Xiaohua Zhu PDF
Trans. Amer. Math. Soc. 365 (2013), 6669-6695 Request permission

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.
References
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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
  • MR Author ID: 220655
  • 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
  • MR Author ID: 887805
  • Email: zhangshj.1982@yahoo.com.cn
  • Zhenlei Zhang
  • Affiliation: Department of Mathematics, Beijing Capital Normal University, Beijing, People’s Republic of China
  • MR Author ID: 794099
  • 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
  • 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