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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Bergman metric on a Stein manifold with a bounded plurisubharmonic function


Authors: Bo-Yong Chen and Jin-Hao Zhang
Journal: Trans. Amer. Math. Soc. 354 (2002), 2997-3009
MSC (2000): Primary 32H10
Published electronically: March 29, 2002
MathSciNet review: 1897387
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Abstract: In this article, we use the pluricomplex Green function to give a sufficient condition for the existence and the completeness of the Bergman metric. As a consequence, we proved that a simply connected complete Kähler manifold possesses a complete Bergman metric provided that the Riemann sectional curvature $\le -A/\rho^2$, which implies a conjecture of Greene and Wu. Moreover, we obtain a sharp estimate for the Bergman distance on such manifolds.


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Additional Information

Bo-Yong Chen
Affiliation: Department of Applied Mathematics, Tongji University, Shanghai 200092, China
Email: chenboy@online.sh.cn

Jin-Hao Zhang
Affiliation: Department of Mathematics, Fudan University, Shanghai 200433, China
Email: zhangjhk@online.sh.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-02-02989-6
PII: S 0002-9947(02)02989-6
Keywords: Bergman metric, pluricomplex Green function, sectional curvature, K\"ahler manifold
Received by editor(s): August 1, 2001
Published electronically: March 29, 2002
Additional Notes: The first author was supported by an NSF grant TY10126005 and a grant from Tongji Univ. No.\ 1390104014
The second author was supported by project G1998030600
Article copyright: © Copyright 2002 American Mathematical Society