## 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

- 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**, DOI 10.1007/BF01389058 - X.X. Chen and C. Sun,
*Calabi flow, geodesic rays, and uniqueness of constant scalar curvature Kähler metrics*, arXiv:1004.2012v1, 2010. - 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**, DOI 10.1007/s00039-005-0522-y - A. Futaki,
*An obstruction to the existence of Einstein Kähler metrics*, Invent. Math.**73**(1983), no. 3, 437–443. MR**718940**, DOI 10.1007/BF01388438 - Toshiki Mabuchi,
*Some symplectic geometry on compact Kähler manifolds. I*, Osaka J. Math.**24**(1987), no. 2, 227–252. MR**909015** - G. Perelman,
*The entropy formula for the Ricci flow and its geometric applications*, arxiv:math.DG/0211159. - D.H. Phong, J. Song, J. Sturm and B. Weinkove,
*The modified Kähler-Ricci flow and solitons*, arxiv:0809.0941v1. - O. S. Rothaus,
*Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators*, J. Functional Analysis**42**(1981), no. 1, 110–120. MR**620582**, DOI 10.1016/0022-1236(81)90050-1 - 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**, DOI 10.1017/S1474748008000133 - G. Tian and Z.L. Zhang,
*Degeneration of Kähler-Ricci Solitons*, Int. Math. Res. Notices, (2011) doi:10.1093/imrn/rnr036 - Gang Tian and Xiaohua Zhu,
*Uniqueness of Kähler-Ricci solitons*, Acta Math.**184**(2000), no. 2, 271–305. MR**1768112**, DOI 10.1007/BF02392630 - 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**, DOI 10.1007/s00014-002-8341-3 - Gang Tian and Xiaohua Zhu,
*Convergence of Kähler-Ricci flow*, J. Amer. Math. Soc.**20**(2007), no. 3, 675–699. MR**2291916**, DOI 10.1090/S0894-0347-06-00552-2 - G. Tian and X.H. Zhu,
*Perelman’s W-functional and stability of Kähler-Ricci flow*, arxiv:0801.3504v1. - 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. - 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**, DOI 10.1002/cpa.3160310304 - Qi S. Zhang,
*A uniform Sobolev inequality under Ricci flow*, Int. Math. Res. Not. IMRN**17**(2007), Art. ID rnm056, 17. MR**2354801**, DOI 10.1093/imrn/rnm056 - 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**, DOI 10.1007/BF02921996 - Xiaohua Zhu,
*Stability of Kähler-Ricci flow on a Fano manifold*, Math. Ann.**356**(2013), no. 4, 1425–1454. MR**3072807**, DOI 10.1007/s00208-012-0889-7 - 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**, DOI 10.1016/j.aim.2008.06.016

## 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