A note on Wu-Zheng's splitting conjecture
Author:
Chengjie Yu
Journal:
Proc. Amer. Math. Soc. 141 (2013), 1791-1793
MSC (2010):
Primary 53C44; Secondary 53C55
DOI:
https://doi.org/10.1090/S0002-9939-2012-11570-9
Published electronically:
October 25, 2012
MathSciNet review:
3020864
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Cao's splitting theorem says that for any complete Kähler-Ricci flow with
,
simply connected and nonnegative bounded holomorphic bisectional curvature,
is holomorphically isometric to
, where
is a Kähler-Ricci flow with positive Ricci curvature for
. In this article, we show that
, where
is the Ricci rank of the initial metric. As a corollary, we also confirm a splitting conjecture of Wu and Zheng when curvature is assumed to be bounded.
- 1. Cao, H.-D. On dimension reduction in the Kähler-Ricci flow. Comm. Anal. Geom. 12 (2004), no. 1-2, 305-320. MR 2074880 (2005d:53104)
- 2. Ferus, D. On the completeness of nullity foliations. Michigan Math. J. 18 (1971) 61-64. MR 0279733 (43:5454)
- 3. Howard, A.; Smyth, B.; Wu, H. On compact Kähler manifolds of nonnegative bisectional curvature. I. Acta Math. 147 (1981), no. 1-2, 51-56. MR 631087 (83e:53064a)
- 4. Ni, L.; Tam, L.-F. Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature. J. Differential Geom. 64 (2003), no. 3, 457-524. MR 2032112 (2005a:32023)
- 5. Shi, Wan-Xiong. Deforming the metric on complete Riemannian manifolds. J. Differential Geom. 30 (1989), no. 1, 223-301. MR 1001277 (90i:58202)
- 6. Wu, H.; Zheng, F. Kähler manifolds with slightly positive bisectional curvature. Explorations in complex and Riemannian geometry, 305-325, Contemp. Math., 332, Amer. Math. Soc., Providence, RI, 2003. MR 2018347 (2005d:32043)
- 7.
Wu, H.; Zheng, F. Compact Kähler manifolds with nonpositive bisectional curvature.
J. Differential Geom. 61 (2002), no. 2, 263-287. MR 1972147 (2004b:53128)
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C44, 53C55
Retrieve articles in all journals with MSC (2010): 53C44, 53C55
Additional Information
Chengjie Yu
Affiliation:
Department of Mathematics, Shantou University, Shantou, Guangdong, People’s Republic of China
Email:
cjyu@stu.edu.cn
DOI:
https://doi.org/10.1090/S0002-9939-2012-11570-9
Keywords:
Kähler manifolds,
bisectional curvature,
splitting theorem
Received by editor(s):
September 3, 2011
Published electronically:
October 25, 2012
Additional Notes:
The author’s research was partially supported by the National Natural Science Foundation of China (11001161) and (10901072).
Communicated by:
Lei Ni
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.