Convergence of Kähler-Ricci flow

Tian, Gang; Zhu, Xiaohua

doi:10.1090/S0894-0347-06-00552-2

Convergence of Kähler-Ricci flow
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by Gang Tian and Xiaohua Zhu

J. Amer. Math. Soc. 20 (2007), 675-699

DOI: https://doi.org/10.1090/S0894-0347-06-00552-2

Published electronically: November 17, 2006

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Abstract:

In this paper, we prove a theorem on convergence of Kähler-Ricci flow on a compact Kähler manifold which admits a Kähler-Ricci soliton.

References

Herbert J. Alexander and B. A. Taylor, Comparison of two capacities in $\textbf {C}^{n}$, Math. Z. 186 (1984), no. 3, 407–417. MR 744831, DOI 10.1007/BF01174894
Eric Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1-2, 1–40. MR 674165, DOI 10.1007/BF02392348
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 G. Tian, Ricci flow on Kähler-Einstein surfaces, Invent. Math. 147 (2002), no. 3, 487–544. MR 1893004, DOI 10.1007/s002220100181
X. X. Chen and G. Tian, Ricci flow on Kähler-Einstein manifolds, Duke Math. J. 131 (2006), no. 1, 17–73. MR 2219236, DOI 10.1215/S0012-7094-05-13112-X
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
Akito Futaki, Kähler-Einstein metrics and integral invariants, Lecture Notes in Mathematics, vol. 1314, Springer-Verlag, Berlin, 1988. MR 947341, DOI 10.1007/BFb0078084
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
Sławomir Kołodziej, The complex Monge-Ampère equation, Acta Math. 180 (1998), no. 1, 69–117. MR 1618325, DOI 10.1007/BF02392879
Toshiki Mabuchi, $K$-energy maps integrating Futaki invariants, Tohoku Math. J. (2) 38 (1986), no. 4, 575–593. MR 867064, DOI 10.2748/tmj/1178228410
Gang Tian, Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), no. 1, 1–37. MR 1471884, DOI 10.1007/s002220050176
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
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
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