Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-2013-06027-8
Published electronically: August 15, 2013
MathSciNet review: 3105766
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [Ca] Huai Dong Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81 (1985), no. 2, 359-372. MR 799272 (87d:58051), https://doi.org/10.1007/BF01389058
  • [CS] X.X. Chen and C. Sun, Calabi flow, geodesic rays, and uniqueness of constant scalar curvature Kähler metrics, arXiv:1004.2012v1, 2010.
  • [CTZ] Huai-Dong Cao, Gang Tian, and Xiaohua Zhu, Kähler-Ricci solitons on compact complex manifolds with $ C_1(M)>0$, Geom. Funct. Anal. 15 (2005), no. 3, 697-719. MR 2221147 (2007h:32031), https://doi.org/10.1007/s00039-005-0522-y
  • [Fu] A. Futaki, An obstruction to the existence of Einstein Kähler metrics, Invent. Math. 73 (1983), no. 3, 437-443. MR 718940 (84j:53072), https://doi.org/10.1007/BF01388438
  • [Ma] Toshiki Mabuchi, Some symplectic geometry on compact Kähler manifolds. I, Osaka J. Math. 24 (1987), no. 2, 227-252. MR 909015 (88m:53126)
  • [Pe] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arxiv:math.DG/0211159.
  • [PSSW] D.H. Phong, J. Song, J. Sturm and B. Weinkove, The modified Kähler-Ricci flow and solitons, arxiv:0809.0941v1.
  • [Ro] O. S. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators, J. Funct. Anal. 42 (1981), no. 1, 110-120. MR 620582 (83f:58080b), https://doi.org/10.1016/0022-1236(81)90050-1
  • [ST] Natasa Sesum and Gang Tian, Bounding scalar curvature and diameter along the Kähler Ricci flow (after Perelman), J. Inst. Math. Jussieu 7 (2008), no. 3, 575-587. MR 2427424 (2009c:53092), https://doi.org/10.1017/S1474748008000133
  • [TZha] G. Tian and Z.L. Zhang, Degeneration of Kähler-Ricci Solitons, Int. Math. Res. Notices, (2011) doi:10.1093/imrn/rnr036
  • [TZhu1] Gang Tian and Xiaohua Zhu, Uniqueness of Kähler-Ricci solitons, Acta Math. 184 (2000), no. 2, 271-305. MR 1768112 (2001h:32040), https://doi.org/10.1007/BF02392630
  • [TZhu2] Gang Tian and Xiaohua Zhu, A new holomorphic invariant and uniqueness of Kähler-Ricci solitons, Comment. Math. Helv. 77 (2002), no. 2, 297-325. MR 1915043 (2003i:32042), https://doi.org/10.1007/s00014-002-8341-3
  • [TZhu3] Gang Tian and Xiaohua Zhu, Convergence of Kähler-Ricci flow, J. Amer. Math. Soc. 20 (2007), no. 3, 675-699. MR 2291916 (2007k:53107), https://doi.org/10.1090/S0894-0347-06-00552-2
  • [TZhu4] G. Tian and X.H. Zhu, Perelman's W-functional and stability of Kähler-Ricci flow, arxiv:0801.3504v1.
  • [TZhu5] G. Tian and X.H. Zhu, Convergence of Kähler-Ricci flow on Fano manifolds,, J. Reine Angew. Math. 678 (2013), 223-245. DOI 10.1515.
  • [Ya] Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339-411. MR 480350 (81d:53045), https://doi.org/10.1002/cpa.3160310304
  • [Zha] Qi S. Zhang, A uniform Sobolev inequality under Ricci flow, Int. Math. Res. Not. IMRN 17 (2007), Art. ID rnm056, 17. MR 2354801 (2008g:53083), https://doi.org/10.1093/imrn/rnm056
  • [Zhu1] Xiaohua Zhu, Kähler-Ricci soliton typed equations on compact complex manifolds with $ C_1(M)>0$, J. Geom. Anal. 10 (2000), no. 4, 759-774. MR 1817785 (2002c:32042), https://doi.org/10.1007/BF02921996
  • [Zhu2] X.H. Zhu, Stability of Kähler-Ricci flow on a Fano manifold, Math. Ann. 356 (2013), 1425-1454. DOI 10.1007. MR 3072807
  • [ZZ] Bin Zhou and Xiaohua Zhu, Relative $ K$-stability and modified $ K$-energy on toric manifolds, Adv. Math. 219 (2008), no. 4, 1327-1362. MR 2450612 (2010b:32038), https://doi.org/10.1016/j.aim.2008.06.016

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C25, 53C55, 58J05

Retrieve articles in all journals with MSC (2010): 53C25, 53C55, 58J05


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: https://doi.org/10.1090/S0002-9947-2013-06027-8
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.

American Mathematical Society